Integrand size = 17, antiderivative size = 1147 \[ \int \frac {x \cosh (c+d x)}{\left (a+b x^3\right )^3} \, dx =\text {Too large to display} \]
-2/27*Chi(a^(1/3)*d/b^(1/3)+d*x)*cosh(c-a^(1/3)*d/b^(1/3))/a^(7/3)/b^(2/3) -1/54*d^2*Chi(a^(1/3)*d/b^(1/3)+d*x)*cosh(c-a^(1/3)*d/b^(1/3))/a^(5/3)/b^( 4/3)-2/27*(-1)^(2/3)*Chi((-1)^(1/3)*a^(1/3)*d/b^(1/3)-d*x)*cosh(c+(-1)^(1/ 3)*a^(1/3)*d/b^(1/3))/a^(7/3)/b^(2/3)+1/54*(-1)^(1/3)*d^2*Chi((-1)^(1/3)*a ^(1/3)*d/b^(1/3)-d*x)*cosh(c+(-1)^(1/3)*a^(1/3)*d/b^(1/3))/a^(5/3)/b^(4/3) +2/27*(-1)^(1/3)*Chi(-(-1)^(2/3)*a^(1/3)*d/b^(1/3)-d*x)*cosh(c-(-1)^(2/3)* a^(1/3)*d/b^(1/3))/a^(7/3)/b^(2/3)-1/54*(-1)^(2/3)*d^2*Chi(-(-1)^(2/3)*a^( 1/3)*d/b^(1/3)-d*x)*cosh(c-(-1)^(2/3)*a^(1/3)*d/b^(1/3))/a^(5/3)/b^(4/3)-1 /18*cosh(d*x+c)/a/b^2/x^4+2/9*cosh(d*x+c)/a^2/b/x-1/6*cosh(d*x+c)/b/x/(b*x ^3+a)^2+1/18*cosh(d*x+c)/b^2/x^4/(b*x^3+a)-2/27*d*cosh(c+(-1)^(1/3)*a^(1/3 )*d/b^(1/3))*Shi(-(-1)^(1/3)*a^(1/3)*d/b^(1/3)+d*x)/a^2/b-2/27*d*cosh(c-a^ (1/3)*d/b^(1/3))*Shi(a^(1/3)*d/b^(1/3)+d*x)/a^2/b-2/27*d*cosh(c-(-1)^(2/3) *a^(1/3)*d/b^(1/3))*Shi((-1)^(2/3)*a^(1/3)*d/b^(1/3)+d*x)/a^2/b-2/27*d*Chi (a^(1/3)*d/b^(1/3)+d*x)*sinh(c-a^(1/3)*d/b^(1/3))/a^2/b-2/27*Shi(a^(1/3)*d /b^(1/3)+d*x)*sinh(c-a^(1/3)*d/b^(1/3))/a^(7/3)/b^(2/3)-1/54*d^2*Shi(a^(1/ 3)*d/b^(1/3)+d*x)*sinh(c-a^(1/3)*d/b^(1/3))/a^(5/3)/b^(4/3)-2/27*d*Chi((-1 )^(1/3)*a^(1/3)*d/b^(1/3)-d*x)*sinh(c+(-1)^(1/3)*a^(1/3)*d/b^(1/3))/a^2/b- 2/27*(-1)^(2/3)*Shi(-(-1)^(1/3)*a^(1/3)*d/b^(1/3)+d*x)*sinh(c+(-1)^(1/3)*a ^(1/3)*d/b^(1/3))/a^(7/3)/b^(2/3)+1/54*(-1)^(1/3)*d^2*Shi(-(-1)^(1/3)*a^(1 /3)*d/b^(1/3)+d*x)*sinh(c+(-1)^(1/3)*a^(1/3)*d/b^(1/3))/a^(5/3)/b^(4/3)...
Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
Time = 0.35 (sec) , antiderivative size = 669, normalized size of antiderivative = 0.58 \[ \int \frac {x \cosh (c+d x)}{\left (a+b x^3\right )^3} \, dx=\frac {\text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {-a d^2 \cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1}))+a d^2 \text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1})+a d^2 \cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))-a d^2 \sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))+4 b \cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1})) \text {$\#$1}-4 b \text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1}) \text {$\#$1}-4 b \cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1})) \text {$\#$1}+4 b \sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1})) \text {$\#$1}+4 b d \cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1})) \text {$\#$1}^2-4 b d \text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1}) \text {$\#$1}^2-4 b d \cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1})) \text {$\#$1}^2+4 b d \sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1})) \text {$\#$1}^2}{\text {$\#$1}^2}\&\right ]-\text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {a d^2 \cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1}))+a d^2 \text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1})+a d^2 \cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))+a d^2 \sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))-4 b \cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1})) \text {$\#$1}-4 b \text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1}) \text {$\#$1}-4 b \cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1})) \text {$\#$1}-4 b \sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1})) \text {$\#$1}+4 b d \cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1})) \text {$\#$1}^2+4 b d \text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1}) \text {$\#$1}^2+4 b d \cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1})) \text {$\#$1}^2+4 b d \sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1})) \text {$\#$1}^2}{\text {$\#$1}^2}\&\right ]+\frac {6 b \cosh (d x) \left (b x^2 \left (7 a+4 b x^3\right ) \cosh (c)+a d \left (a+b x^3\right ) \sinh (c)\right )}{\left (a+b x^3\right )^2}+\frac {6 b \left (a d \left (a+b x^3\right ) \cosh (c)+b x^2 \left (7 a+4 b x^3\right ) \sinh (c)\right ) \sinh (d x)}{\left (a+b x^3\right )^2}}{108 a^2 b^2} \]
(RootSum[a + b*#1^3 & , (-(a*d^2*Cosh[c + d*#1]*CoshIntegral[d*(x - #1)]) + a*d^2*CoshIntegral[d*(x - #1)]*Sinh[c + d*#1] + a*d^2*Cosh[c + d*#1]*Sin hIntegral[d*(x - #1)] - a*d^2*Sinh[c + d*#1]*SinhIntegral[d*(x - #1)] + 4* b*Cosh[c + d*#1]*CoshIntegral[d*(x - #1)]*#1 - 4*b*CoshIntegral[d*(x - #1) ]*Sinh[c + d*#1]*#1 - 4*b*Cosh[c + d*#1]*SinhIntegral[d*(x - #1)]*#1 + 4*b *Sinh[c + d*#1]*SinhIntegral[d*(x - #1)]*#1 + 4*b*d*Cosh[c + d*#1]*CoshInt egral[d*(x - #1)]*#1^2 - 4*b*d*CoshIntegral[d*(x - #1)]*Sinh[c + d*#1]*#1^ 2 - 4*b*d*Cosh[c + d*#1]*SinhIntegral[d*(x - #1)]*#1^2 + 4*b*d*Sinh[c + d* #1]*SinhIntegral[d*(x - #1)]*#1^2)/#1^2 & ] - RootSum[a + b*#1^3 & , (a*d^ 2*Cosh[c + d*#1]*CoshIntegral[d*(x - #1)] + a*d^2*CoshIntegral[d*(x - #1)] *Sinh[c + d*#1] + a*d^2*Cosh[c + d*#1]*SinhIntegral[d*(x - #1)] + a*d^2*Si nh[c + d*#1]*SinhIntegral[d*(x - #1)] - 4*b*Cosh[c + d*#1]*CoshIntegral[d* (x - #1)]*#1 - 4*b*CoshIntegral[d*(x - #1)]*Sinh[c + d*#1]*#1 - 4*b*Cosh[c + d*#1]*SinhIntegral[d*(x - #1)]*#1 - 4*b*Sinh[c + d*#1]*SinhIntegral[d*( x - #1)]*#1 + 4*b*d*Cosh[c + d*#1]*CoshIntegral[d*(x - #1)]*#1^2 + 4*b*d*C oshIntegral[d*(x - #1)]*Sinh[c + d*#1]*#1^2 + 4*b*d*Cosh[c + d*#1]*SinhInt egral[d*(x - #1)]*#1^2 + 4*b*d*Sinh[c + d*#1]*SinhIntegral[d*(x - #1)]*#1^ 2)/#1^2 & ] + (6*b*Cosh[d*x]*(b*x^2*(7*a + 4*b*x^3)*Cosh[c] + a*d*(a + b*x ^3)*Sinh[c]))/(a + b*x^3)^2 + (6*b*(a*d*(a + b*x^3)*Cosh[c] + b*x^2*(7*a + 4*b*x^3)*Sinh[c])*Sinh[d*x])/(a + b*x^3)^2)/(108*a^2*b^2)
Time = 4.55 (sec) , antiderivative size = 1813, normalized size of antiderivative = 1.58, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {5814, 5813, 5814, 5815, 2009, 5816, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x \cosh (c+d x)}{\left (a+b x^3\right )^3} \, dx\) |
| \(\Big \downarrow \) 5814 |
| \(\displaystyle \frac {d \int \frac {\sinh (c+d x)}{x \left (b x^3+a\right )^2}dx}{6 b}-\frac {\int \frac {\cosh (c+d x)}{x^2 \left (b x^3+a\right )^2}dx}{6 b}-\frac {\cosh (c+d x)}{6 b x \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 5813 |
| \(\displaystyle \frac {d \left (\frac {d \int \frac {\cosh (c+d x)}{x^3 \left (b x^3+a\right )}dx}{3 b}-\frac {\int \frac {\sinh (c+d x)}{x^4 \left (b x^3+a\right )}dx}{b}-\frac {\sinh (c+d x)}{3 b x^3 \left (a+b x^3\right )}\right )}{6 b}-\frac {\int \frac {\cosh (c+d x)}{x^2 \left (b x^3+a\right )^2}dx}{6 b}-\frac {\cosh (c+d x)}{6 b x \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 5814 |
| \(\displaystyle \frac {d \left (\frac {d \int \frac {\cosh (c+d x)}{x^3 \left (b x^3+a\right )}dx}{3 b}-\frac {\int \frac {\sinh (c+d x)}{x^4 \left (b x^3+a\right )}dx}{b}-\frac {\sinh (c+d x)}{3 b x^3 \left (a+b x^3\right )}\right )}{6 b}-\frac {-\frac {4 \int \frac {\cosh (c+d x)}{x^5 \left (b x^3+a\right )}dx}{3 b}+\frac {d \int \frac {\sinh (c+d x)}{x^4 \left (b x^3+a\right )}dx}{3 b}-\frac {\cosh (c+d x)}{3 b x^4 \left (a+b x^3\right )}}{6 b}-\frac {\cosh (c+d x)}{6 b x \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 5815 |
| \(\displaystyle \frac {d \left (-\frac {\int \left (\frac {b^2 \sinh (c+d x) x^2}{a^2 \left (b x^3+a\right )}-\frac {b \sinh (c+d x)}{a^2 x}+\frac {\sinh (c+d x)}{a x^4}\right )dx}{b}+\frac {d \int \frac {\cosh (c+d x)}{x^3 \left (b x^3+a\right )}dx}{3 b}-\frac {\sinh (c+d x)}{3 b x^3 \left (a+b x^3\right )}\right )}{6 b}-\frac {\frac {d \int \left (\frac {b^2 \sinh (c+d x) x^2}{a^2 \left (b x^3+a\right )}-\frac {b \sinh (c+d x)}{a^2 x}+\frac {\sinh (c+d x)}{a x^4}\right )dx}{3 b}-\frac {4 \int \frac {\cosh (c+d x)}{x^5 \left (b x^3+a\right )}dx}{3 b}-\frac {\cosh (c+d x)}{3 b x^4 \left (a+b x^3\right )}}{6 b}-\frac {\cosh (c+d x)}{6 b x \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\cosh (c+d x)}{6 b x \left (b x^3+a\right )^2}-\frac {-\frac {\cosh (c+d x)}{3 b x^4 \left (b x^3+a\right )}+\frac {d \left (\frac {\cosh (c) \text {Chi}(d x) d^3}{6 a}+\frac {\sinh (c) \text {Shi}(d x) d^3}{6 a}-\frac {\sinh (c+d x) d^2}{6 a x}-\frac {\cosh (c+d x) d}{6 a x^2}-\frac {b \text {Chi}(d x) \sinh (c)}{a^2}+\frac {b \text {Chi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}+\frac {b \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sinh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}+\frac {b \text {Chi}\left (-x d-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}-\frac {\sinh (c+d x)}{3 a x^3}-\frac {b \cosh (c) \text {Shi}(d x)}{a^2}-\frac {b \cosh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^2}+\frac {b \cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}+\frac {b \cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}\right )}{3 b}-\frac {4 \int \frac {\cosh (c+d x)}{x^5 \left (b x^3+a\right )}dx}{3 b}}{6 b}+\frac {d \left (-\frac {\sinh (c+d x)}{3 b x^3 \left (b x^3+a\right )}-\frac {\frac {\cosh (c) \text {Chi}(d x) d^3}{6 a}+\frac {\sinh (c) \text {Shi}(d x) d^3}{6 a}-\frac {\sinh (c+d x) d^2}{6 a x}-\frac {\cosh (c+d x) d}{6 a x^2}-\frac {b \text {Chi}(d x) \sinh (c)}{a^2}+\frac {b \text {Chi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}+\frac {b \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sinh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}+\frac {b \text {Chi}\left (-x d-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}-\frac {\sinh (c+d x)}{3 a x^3}-\frac {b \cosh (c) \text {Shi}(d x)}{a^2}-\frac {b \cosh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^2}+\frac {b \cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}+\frac {b \cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}}{b}+\frac {d \int \frac {\cosh (c+d x)}{x^3 \left (b x^3+a\right )}dx}{3 b}\right )}{6 b}\) |
| \(\Big \downarrow \) 5816 |
| \(\displaystyle -\frac {\cosh (c+d x)}{6 b x \left (b x^3+a\right )^2}+\frac {d \left (-\frac {\sinh (c+d x)}{3 b x^3 \left (b x^3+a\right )}-\frac {\frac {\cosh (c) \text {Chi}(d x) d^3}{6 a}+\frac {\sinh (c) \text {Shi}(d x) d^3}{6 a}-\frac {\sinh (c+d x) d^2}{6 a x}-\frac {\cosh (c+d x) d}{6 a x^2}-\frac {b \text {Chi}(d x) \sinh (c)}{a^2}+\frac {b \text {Chi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}+\frac {b \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sinh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}+\frac {b \text {Chi}\left (-x d-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}-\frac {\sinh (c+d x)}{3 a x^3}-\frac {b \cosh (c) \text {Shi}(d x)}{a^2}-\frac {b \cosh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^2}+\frac {b \cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}+\frac {b \cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}}{b}+\frac {d \int \left (\frac {\cosh (c+d x)}{a x^3}-\frac {b \cosh (c+d x)}{a \left (b x^3+a\right )}\right )dx}{3 b}\right )}{6 b}-\frac {-\frac {\cosh (c+d x)}{3 b x^4 \left (b x^3+a\right )}+\frac {d \left (\frac {\cosh (c) \text {Chi}(d x) d^3}{6 a}+\frac {\sinh (c) \text {Shi}(d x) d^3}{6 a}-\frac {\sinh (c+d x) d^2}{6 a x}-\frac {\cosh (c+d x) d}{6 a x^2}-\frac {b \text {Chi}(d x) \sinh (c)}{a^2}+\frac {b \text {Chi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}+\frac {b \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sinh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}+\frac {b \text {Chi}\left (-x d-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}-\frac {\sinh (c+d x)}{3 a x^3}-\frac {b \cosh (c) \text {Shi}(d x)}{a^2}-\frac {b \cosh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^2}+\frac {b \cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}+\frac {b \cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}\right )}{3 b}-\frac {4 \int \left (\frac {x \cosh (c+d x) b^2}{a^2 \left (b x^3+a\right )}-\frac {\cosh (c+d x) b}{a^2 x^2}+\frac {\cosh (c+d x)}{a x^5}\right )dx}{3 b}}{6 b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\cosh (c+d x)}{6 b x \left (b x^3+a\right )^2}+\frac {d \left (-\frac {\sinh (c+d x)}{3 b x^3 \left (b x^3+a\right )}-\frac {\frac {\cosh (c) \text {Chi}(d x) d^3}{6 a}+\frac {\sinh (c) \text {Shi}(d x) d^3}{6 a}-\frac {\sinh (c+d x) d^2}{6 a x}-\frac {\cosh (c+d x) d}{6 a x^2}-\frac {b \text {Chi}(d x) \sinh (c)}{a^2}+\frac {b \text {Chi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}+\frac {b \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sinh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}+\frac {b \text {Chi}\left (-x d-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}-\frac {\sinh (c+d x)}{3 a x^3}-\frac {b \cosh (c) \text {Shi}(d x)}{a^2}-\frac {b \cosh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^2}+\frac {b \cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}+\frac {b \cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}}{b}+\frac {d \left (\frac {\cosh (c) \text {Chi}(d x) d^2}{2 a}+\frac {\sinh (c) \text {Shi}(d x) d^2}{2 a}-\frac {\sinh (c+d x) d}{2 a x}-\frac {\cosh (c+d x)}{2 a x^2}+\frac {\sqrt [3]{-1} b^{2/3} \cosh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{5/3}}-\frac {(-1)^{2/3} b^{2/3} \cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (-x d-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{5/3}}-\frac {b^{2/3} \cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{5/3}}-\frac {\sqrt [3]{-1} b^{2/3} \sinh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{5/3}}-\frac {b^{2/3} \sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{5/3}}-\frac {(-1)^{2/3} b^{2/3} \sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{5/3}}\right )}{3 b}\right )}{6 b}-\frac {-\frac {\cosh (c+d x)}{3 b x^4 \left (b x^3+a\right )}+\frac {d \left (\frac {\cosh (c) \text {Chi}(d x) d^3}{6 a}+\frac {\sinh (c) \text {Shi}(d x) d^3}{6 a}-\frac {\sinh (c+d x) d^2}{6 a x}-\frac {\cosh (c+d x) d}{6 a x^2}-\frac {b \text {Chi}(d x) \sinh (c)}{a^2}+\frac {b \text {Chi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}+\frac {b \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sinh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}+\frac {b \text {Chi}\left (-x d-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}-\frac {\sinh (c+d x)}{3 a x^3}-\frac {b \cosh (c) \text {Shi}(d x)}{a^2}-\frac {b \cosh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^2}+\frac {b \cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}+\frac {b \cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}\right )}{3 b}-\frac {4 \left (\frac {\cosh (c) \text {Chi}(d x) d^4}{24 a}+\frac {\sinh (c) \text {Shi}(d x) d^4}{24 a}-\frac {\sinh (c+d x) d^3}{24 a x}-\frac {\cosh (c+d x) d^2}{24 a x^2}-\frac {b \text {Chi}(d x) \sinh (c) d}{a^2}-\frac {\sinh (c+d x) d}{12 a x^3}-\frac {b \cosh (c) \text {Shi}(d x) d}{a^2}+\frac {b \cosh (c+d x)}{a^2 x}-\frac {\cosh (c+d x)}{4 a x^4}-\frac {(-1)^{2/3} b^{4/3} \cosh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{7/3}}+\frac {\sqrt [3]{-1} b^{4/3} \cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (-x d-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{7/3}}-\frac {b^{4/3} \cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{7/3}}+\frac {(-1)^{2/3} b^{4/3} \sinh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{7/3}}-\frac {b^{4/3} \sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{7/3}}+\frac {\sqrt [3]{-1} b^{4/3} \sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{7/3}}\right )}{3 b}}{6 b}\) |
-1/6*Cosh[c + d*x]/(b*x*(a + b*x^3)^2) + (d*(-1/3*Sinh[c + d*x]/(b*x^3*(a + b*x^3)) - (-1/6*(d*Cosh[c + d*x])/(a*x^2) + (d^3*Cosh[c]*CoshIntegral[d* x])/(6*a) - (b*CoshIntegral[d*x]*Sinh[c])/a^2 + (b*CoshIntegral[(a^(1/3)*d )/b^(1/3) + d*x]*Sinh[c - (a^(1/3)*d)/b^(1/3)])/(3*a^2) + (b*CoshIntegral[ ((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*Sinh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1 /3)])/(3*a^2) + (b*CoshIntegral[-(((-1)^(2/3)*a^(1/3)*d)/b^(1/3)) - d*x]*S inh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)])/(3*a^2) - Sinh[c + d*x]/(3*a*x^3) - (d^2*Sinh[c + d*x])/(6*a*x) - (b*Cosh[c]*SinhIntegral[d*x])/a^2 + (d^3* Sinh[c]*SinhIntegral[d*x])/(6*a) - (b*Cosh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1 /3)]*SinhIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(3*a^2) + (b*Cosh [c - (a^(1/3)*d)/b^(1/3)]*SinhIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*a^2) + (b*Cosh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(2/3)*a^ (1/3)*d)/b^(1/3) + d*x])/(3*a^2))/b + (d*(-1/2*Cosh[c + d*x]/(a*x^2) + (d^ 2*Cosh[c]*CoshIntegral[d*x])/(2*a) + ((-1)^(1/3)*b^(2/3)*Cosh[c + ((-1)^(1 /3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x] )/(3*a^(5/3)) - ((-1)^(2/3)*b^(2/3)*Cosh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3 )]*CoshIntegral[-(((-1)^(2/3)*a^(1/3)*d)/b^(1/3)) - d*x])/(3*a^(5/3)) - (b ^(2/3)*Cosh[c - (a^(1/3)*d)/b^(1/3)]*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d* x])/(3*a^(5/3)) - (d*Sinh[c + d*x])/(2*a*x) + (d^2*Sinh[c]*SinhIntegral[d* x])/(2*a) - ((-1)^(1/3)*b^(2/3)*Sinh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)...
3.2.11.3.1 Defintions of rubi rules used
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*Sinh[(c_.) + (d_.)*(x_)], x_Sy mbol] :> Simp[x^(m - n + 1)*(a + b*x^n)^(p + 1)*(Sinh[c + d*x]/(b*n*(p + 1) )), x] + (-Simp[(m - n + 1)/(b*n*(p + 1)) Int[x^(m - n)*(a + b*x^n)^(p + 1)*Sinh[c + d*x], x], x] - Simp[d/(b*n*(p + 1)) Int[x^(m - n + 1)*(a + b* x^n)^(p + 1)*Cosh[c + d*x], x], x]) /; FreeQ[{a, b, c, d}, x] && ILtQ[p, -1 ] && IGtQ[n, 0] && RationalQ[m] && (GtQ[m - n + 1, 0] || GtQ[n, 2])
Int[Cosh[(c_.) + (d_.)*(x_)]*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Sy mbol] :> Simp[x^(m - n + 1)*(a + b*x^n)^(p + 1)*(Cosh[c + d*x]/(b*n*(p + 1) )), x] + (-Simp[(m - n + 1)/(b*n*(p + 1)) Int[x^(m - n)*(a + b*x^n)^(p + 1)*Cosh[c + d*x], x], x] - Simp[d/(b*n*(p + 1)) Int[x^(m - n + 1)*(a + b* x^n)^(p + 1)*Sinh[c + d*x], x], x]) /; FreeQ[{a, b, c, d}, x] && ILtQ[p, -1 ] && IGtQ[n, 0] && RationalQ[m] && (GtQ[m - n + 1, 0] || GtQ[n, 2])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*Sinh[(c_.) + (d_.)*(x_)], x_Sy mbol] :> Int[ExpandIntegrand[Sinh[c + d*x], x^m*(a + b*x^n)^p, x], x] /; Fr eeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IntegerQ[m] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -1])
Int[Cosh[(c_.) + (d_.)*(x_)]*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Sy mbol] :> Int[ExpandIntegrand[Cosh[c + d*x], x^m*(a + b*x^n)^p, x], x] /; Fr eeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IntegerQ[m] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -1])
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.46 (sec) , antiderivative size = 1416, normalized size of antiderivative = 1.23
-1/108*(-sum((_R1^2-2*_R1*c+c^2-6*_R1+6*c+10)/(_R1^2-2*_R1*c+c^2)*exp(_R1) *Ei(1,-d*x+_R1-c),_R1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))*b^ 3*c*d*x^6-sum((_R1^2-2*_R1*c+c^2+6*_R1-6*c+10)/(_R1^2-2*_R1*c+c^2)*exp(-_R 1)*Ei(1,d*x-_R1+c),_R1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))*b ^3*c*d*x^6+sum((_R1^2*b*c-2*_R1*b*c^2-a*d^3+b*c^3-4*_R1^2*b+2*_R1*b*c+2*b* c^2+4*_R1*b+6*b*c)/(_R1^2-2*_R1*c+c^2)*exp(_R1)*Ei(1,-d*x+_R1-c),_R1=RootO f(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))*b^2*d*x^6+sum((_R1^2*b*c-2*_R 1*b*c^2-a*d^3+b*c^3+4*_R1^2*b-2*_R1*b*c-2*b*c^2+4*_R1*b+6*b*c)/(_R1^2-2*_R 1*c+c^2)*exp(-_R1)*Ei(1,d*x-_R1+c),_R1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2 +a*d^3-b*c^3))*b^2*d*x^6-2*sum((_R1^2-2*_R1*c+c^2-6*_R1+6*c+10)/(_R1^2-2*_ R1*c+c^2)*exp(_R1)*Ei(1,-d*x+_R1-c),_R1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^ 2+a*d^3-b*c^3))*a*b^2*c*d*x^3-2*sum((_R1^2-2*_R1*c+c^2+6*_R1-6*c+10)/(_R1^ 2-2*_R1*c+c^2)*exp(-_R1)*Ei(1,d*x-_R1+c),_R1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z *b*c^2+a*d^3-b*c^3))*a*b^2*c*d*x^3-12*exp(-d*x-c)*b^3*x^5-12*exp(d*x+c)*b^ 3*x^5+3*exp(-d*x-c)*a*b^2*d*x^3-3*exp(d*x+c)*a*b^2*d*x^3+2*sum((_R1^2*b*c- 2*_R1*b*c^2-a*d^3+b*c^3-4*_R1^2*b+2*_R1*b*c+2*b*c^2+4*_R1*b+6*b*c)/(_R1^2- 2*_R1*c+c^2)*exp(_R1)*Ei(1,-d*x+_R1-c),_R1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b *c^2+a*d^3-b*c^3))*a*b*d*x^3+2*sum((_R1^2*b*c-2*_R1*b*c^2-a*d^3+b*c^3+4*_R 1^2*b-2*_R1*b*c-2*b*c^2+4*_R1*b+6*b*c)/(_R1^2-2*_R1*c+c^2)*exp(-_R1)*Ei(1, d*x-_R1+c),_R1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))*a*b*d*...
Leaf count of result is larger than twice the leaf count of optimal. 4691 vs. \(2 (843) = 1686\).
Time = 0.33 (sec) , antiderivative size = 4691, normalized size of antiderivative = 4.09 \[ \int \frac {x \cosh (c+d x)}{\left (a+b x^3\right )^3} \, dx=\text {Too large to display} \]
-1/216*((8*(a*b^2*d^3*x^6 + 2*a^2*b*d^3*x^3 + a^3*d^3)*cosh(d*x + c)^2 - 8 *(a*b^2*d^3*x^6 + 2*a^2*b*d^3*x^3 + a^3*d^3)*sinh(d*x + c)^2 - 4*(a*d^3/b) ^(2/3)*((b^3*x^6 + 2*a*b^2*x^3 + a^2*b - sqrt(-3)*(b^3*x^6 + 2*a*b^2*x^3 + a^2*b))*cosh(d*x + c)^2 - (b^3*x^6 + 2*a*b^2*x^3 + a^2*b - sqrt(-3)*(b^3* x^6 + 2*a*b^2*x^3 + a^2*b))*sinh(d*x + c)^2) - (a*d^3/b)^(1/3)*((a*b^2*d^3 *x^6 + 2*a^2*b*d^3*x^3 + a^3*d^3 + sqrt(-3)*(a*b^2*d^3*x^6 + 2*a^2*b*d^3*x ^3 + a^3*d^3))*cosh(d*x + c)^2 - (a*b^2*d^3*x^6 + 2*a^2*b*d^3*x^3 + a^3*d^ 3 + sqrt(-3)*(a*b^2*d^3*x^6 + 2*a^2*b*d^3*x^3 + a^3*d^3))*sinh(d*x + c)^2) )*Ei(d*x - 1/2*(a*d^3/b)^(1/3)*(sqrt(-3) + 1))*cosh(1/2*(a*d^3/b)^(1/3)*(s qrt(-3) + 1) + c) - (8*(a*b^2*d^3*x^6 + 2*a^2*b*d^3*x^3 + a^3*d^3)*cosh(d* x + c)^2 - 8*(a*b^2*d^3*x^6 + 2*a^2*b*d^3*x^3 + a^3*d^3)*sinh(d*x + c)^2 + 4*(-a*d^3/b)^(2/3)*((b^3*x^6 + 2*a*b^2*x^3 + a^2*b - sqrt(-3)*(b^3*x^6 + 2*a*b^2*x^3 + a^2*b))*cosh(d*x + c)^2 - (b^3*x^6 + 2*a*b^2*x^3 + a^2*b - s qrt(-3)*(b^3*x^6 + 2*a*b^2*x^3 + a^2*b))*sinh(d*x + c)^2) - (-a*d^3/b)^(1/ 3)*((a*b^2*d^3*x^6 + 2*a^2*b*d^3*x^3 + a^3*d^3 + sqrt(-3)*(a*b^2*d^3*x^6 + 2*a^2*b*d^3*x^3 + a^3*d^3))*cosh(d*x + c)^2 - (a*b^2*d^3*x^6 + 2*a^2*b*d^ 3*x^3 + a^3*d^3 + sqrt(-3)*(a*b^2*d^3*x^6 + 2*a^2*b*d^3*x^3 + a^3*d^3))*si nh(d*x + c)^2))*Ei(-d*x - 1/2*(-a*d^3/b)^(1/3)*(sqrt(-3) + 1))*cosh(1/2*(- a*d^3/b)^(1/3)*(sqrt(-3) + 1) - c) + (8*(a*b^2*d^3*x^6 + 2*a^2*b*d^3*x^3 + a^3*d^3)*cosh(d*x + c)^2 - 8*(a*b^2*d^3*x^6 + 2*a^2*b*d^3*x^3 + a^3*d^...
Timed out. \[ \int \frac {x \cosh (c+d x)}{\left (a+b x^3\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {x \cosh (c+d x)}{\left (a+b x^3\right )^3} \, dx=\int { \frac {x \cosh \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{3}} \,d x } \]
1/2*(x*e^(d*x + 2*c) - x*e^(-d*x))/(b^3*d*x^9*e^c + 3*a*b^2*d*x^6*e^c + 3* a^2*b*d*x^3*e^c + a^3*d*e^c) + 1/2*integrate((8*b*x^3*e^c - a*e^c)*e^(d*x) /(b^4*d*x^12 + 4*a*b^3*d*x^9 + 6*a^2*b^2*d*x^6 + 4*a^3*b*d*x^3 + a^4*d), x ) - 1/2*integrate((8*b*x^3 - a)*e^(-d*x)/(b^4*d*x^12*e^c + 4*a*b^3*d*x^9*e ^c + 6*a^2*b^2*d*x^6*e^c + 4*a^3*b*d*x^3*e^c + a^4*d*e^c), x)
\[ \int \frac {x \cosh (c+d x)}{\left (a+b x^3\right )^3} \, dx=\int { \frac {x \cosh \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {x \cosh (c+d x)}{\left (a+b x^3\right )^3} \, dx=\int \frac {x\,\mathrm {cosh}\left (c+d\,x\right )}{{\left (b\,x^3+a\right )}^3} \,d x \]