These theorems reduce integrals over a real interval $(y,x)$ of certain integrands containing the square root of a quartic or cubic polynomial to symmetric integrals over $(0,\mathrm{\infty })$ containing the square root of a cubic polynomial (compare §19.16(i)). Let

and assume that the line segment with endpoints $a_{\alpha }+b_{\alpha }x$ and $a_{\alpha }+b_{\alpha }y$ lies in $ℂ\setminus (-\mathrm{\infty },0)$ for $1\le \alpha \le 4$. If

and $\alpha ,\beta ,\gamma ,\delta$ is any permutation of the numbers $1,2,3,4$, then

where

There are only three distinct $U$’s with subscripts $\le 4$, and at most one of them can be 0 because the $d$’s are nonzero. Then

where

The Cauchy principal value is taken when $U_{\alpha 5}^2$ or $Q_{\alpha 5}^2$ is real and negative. Cubic cases of these formulas are obtained by setting one of the factors in (19.29.3) equal to 1.

The advantages of symmetric integrals for tables of integrals and symbolic integration are illustrated by (19.29.4) and its cubic case, which replace the $8+8+12=28$ formulas in Gradshteyn and Ryzhik (2015, 3.147, 3.131, 3.152) after taking $x^2$ as the variable of integration in 3.152. Moreover, the requirement that one limit of integration be a branch point of the integrand is eliminated without doubling the number of standard integrals in the result. (19.29.7) subsumes all 72 formulas in Gradshteyn and Ryzhik (2015, 3.168), and its cubic cases similarly replace the $18+36+18=72$ formulas in Gradshteyn and Ryzhik (2015, 3.133, 3.142, and 3.141(1-18)). For example, 3.142(2) is included as

where the arguments of the $R_D$ function are, in order, $(a-b)(u-c)$, $(b-c)(a-u)$, $(a-b)(b-c)$.

(19.2.3) can be written

where $x>y$, $h=3$ or 4, $n\ge h$, and $m_j$ is an integer. Define

where ${𝐞}_j$ is an $n$-tuple with 1 in the $j$th position and 0’s elsewhere. Define also $\mathrm{𝟎}=(0,\mathrm{\dots },0)$ and retain the notation and conditions associated with (19.29.1) and (19.29.2). The integrals in (19.29.4), (19.29.7), and (19.29.8) are $I(\mathrm{𝟎})$, $I({𝐞}_{\alpha }-{𝐞}_{\delta })$, and $I({𝐞}_{\alpha }-{𝐞}_5)$, respectively.

The only cases of $I(𝐦)$ that are integrals of the first kind are the two ($h=3$ or 4) with $𝐦=\mathrm{𝟎}$. The only cases that are integrals of the third kind are those in which at least one $m_j$ with $j>h$ is a negative integer and those in which $h=4$ and ${\sum }_{j=1}^n{m}_j$ is a positive integer. All other cases are integrals of the second kind.

$I(𝐦)$ can be reduced to a linear combination of basic integrals and algebraic functions. In the cubic case ($h=3$) the basic integrals are

In the quartic case ($h=4$) the basic integrals are

Basic integrals of type $I(-{𝐞}_j)$, $1\le j\le h$, are not linearly independent, nor are those of type $I({𝐞}_j)$, $1\le j\le 4$.

The reduction of $I(𝐦)$ is carried out by a relation derived from partial fractions and by use of two recurrence relations. These are given in Carlson (1999, (2.19), (3.5), (3.11)) and simplified in Carlson (2002, (1.10), (1.7), (1.8)) by means of modified definitions. Partial fractions provide a reduction to integrals in which $𝐦$ has at most one nonzero component, and these are then reduced to basic integrals by the recurrence relations. A special case of Carlson (1999, (2.19)) is given by

which shows how to express the basic integral $I(-{𝐞}_j)$ in terms of symmetric integrals by using (19.29.4) and either (19.29.7) or (19.29.8). The first choice gives a formula that includes the 18+9+18 = 45 formulas in Gradshteyn and Ryzhik (2015, 3.133, 3.156, 3.158), and the second choice includes the 8+8+8+12 = 36 formulas in Gradshteyn and Ryzhik (2015, 3.151, 3.149, 3.137, 3.157) (after setting $x^2=t$ in some cases).

If $h=3$, then the recurrence relation (Carlson (1999, (3.5))) has the special case

where $\alpha ,\beta ,\gamma$ is any permutation of the numbers $1,2,3$, and

(This shows why $I({𝐞}_{\alpha })$ is not needed as a basic integral in the cubic case.) In the quartic case this recurrence relation has an extra term in $I(2{𝐞}_{\alpha })$, and hence $I({𝐞}_{\alpha })$, $1\le \alpha \le 4$, is a basic integral. It can be expressed in terms of symmetric integrals by setting $a_5=1$ and $b_5=0$ in (19.29.8).

The other recurrence relation is

see Carlson (1999, (3.11)). An example that uses (19.29.15)–(19.29.18) is given in §19.34.

For an implementation by James FitzSimons of the method for reducing $I(𝐦)$ to basic integrals and extensive tables of such reductions, see Carlson (1999) and Carlson and FitzSimons (2000).

Another method of reduction is given in Gray (2002). It depends primarily on multivariate recurrence relations that replace one integral by two or more.

The first formula replaces (19.14.4)–(19.14.10). Define $Q_j(t)=a_j+b_j{t}^2$, $j=1,2$, and assume both $Q$’s are positive for . Then

and

where

If both square roots in (19.29.22) are 0, then the indeterminacy in the two preceding equations can be removed by using (19.27.8) to evaluate the integral as $R_G(a_1{b}_2,a_2{b}_1,0)$ multiplied either by $-2/(b_1{b}_2)$ or by $-2/(a_1{a}_2)$ in the cases of (19.29.20) or (19.29.21), respectively. If $x=\mathrm{\infty }$, then $U$ is found by taking the limit. For example,

Next, for $j=1,2$, define $Q_j(t)=f_j+g_{j}t+h_j{t}^2$, and assume both $Q$’s are positive for . If each has real zeros, then (19.29.4) may be simpler than

where

(The variables of $R_F$ are real and nonnegative unless both $Q$’s have real zeros and those of $Q_1$ interlace those of $Q_2$.) If $Q_1(t)=(a_1+b_{1}t)(a_2+b_{2}t)$, where both linear factors are positive for , and $Q_2(t)=f_2+g_{2}t+h_2{t}^2$, then (19.29.25) is modified so that

with other quantities remaining as in (19.29.25). In the cubic case, in which $a_2=1$, $b_2=0$, (19.29.26) reduces further to

For example, because $t^3-a^3=(t-a)(t^2+at+a^2)$, we find that when

where

Lastly, define $Q(t^2)=f+g{t}^2+h{t}^4$ and assume $Q(t^2)$ is positive and monotonic for . Then

where

For example, if $0\le y\le x$ and $a^4\ge 0$, then

where