Integrand size = 28, antiderivative size = 864 \[ \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=-\frac {3 b f^3 x}{8 a^2 d^3}-\frac {b (e+f x)^3}{4 a^2 d}+\frac {b \left (a^2+b^2\right ) (e+f x)^4}{4 a^4 f}-\frac {40 f^3 \cosh (c+d x)}{9 a d^4}-\frac {6 b^2 f^3 \cosh (c+d x)}{a^3 d^4}-\frac {2 f (e+f x)^2 \cosh (c+d x)}{a d^2}-\frac {3 b^2 f (e+f x)^2 \cosh (c+d x)}{a^3 d^2}-\frac {2 f^3 \cosh ^3(c+d x)}{27 a d^4}-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 a d^2}-\frac {b \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {b \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {3 b \left (a^2+b^2\right ) f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^2}-\frac {3 b \left (a^2+b^2\right ) f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^2}+\frac {6 b \left (a^2+b^2\right ) f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^3}+\frac {6 b \left (a^2+b^2\right ) f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^3}-\frac {6 b \left (a^2+b^2\right ) f^3 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^4}-\frac {6 b \left (a^2+b^2\right ) f^3 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^4}+\frac {40 f^2 (e+f x) \sinh (c+d x)}{9 a d^3}+\frac {6 b^2 f^2 (e+f x) \sinh (c+d x)}{a^3 d^3}+\frac {2 (e+f x)^3 \sinh (c+d x)}{3 a d}+\frac {b^2 (e+f x)^3 \sinh (c+d x)}{a^3 d}+\frac {3 b f^3 \cosh (c+d x) \sinh (c+d x)}{8 a^2 d^4}+\frac {3 b f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 a^2 d^2}+\frac {2 f^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 a d^3}+\frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 a d}-\frac {3 b f^2 (e+f x) \sinh ^2(c+d x)}{4 a^2 d^3}-\frac {b (e+f x)^3 \sinh ^2(c+d x)}{2 a^2 d} \]
-b*(a^2+b^2)*(f*x+e)^3*ln(1+a*exp(d*x+c)/(b+(a^2+b^2)^(1/2)))/a^4/d-3*b^2* f*(f*x+e)^2*cosh(d*x+c)/a^3/d^2+6*b^2*f^2*(f*x+e)*sinh(d*x+c)/a^3/d^3+3/8* b*f^3*cosh(d*x+c)*sinh(d*x+c)/a^2/d^4+2/9*f^2*(f*x+e)*cosh(d*x+c)^2*sinh(d *x+c)/a/d^3-3/4*b*f^2*(f*x+e)*sinh(d*x+c)^2/a^2/d^3-6*b*(a^2+b^2)*f^3*poly log(4,-a*exp(d*x+c)/(b-(a^2+b^2)^(1/2)))/a^4/d^4-6*b*(a^2+b^2)*f^3*polylog (4,-a*exp(d*x+c)/(b+(a^2+b^2)^(1/2)))/a^4/d^4-2*f*(f*x+e)^2*cosh(d*x+c)/a/ d^2+40/9*f^2*(f*x+e)*sinh(d*x+c)/a/d^3+1/3*(f*x+e)^3*cosh(d*x+c)^2*sinh(d* x+c)/a/d-1/2*b*(f*x+e)^3*sinh(d*x+c)^2/a^2/d-1/3*f*(f*x+e)^2*cosh(d*x+c)^3 /a/d^2-6*b^2*f^3*cosh(d*x+c)/a^3/d^4+1/4*b*(a^2+b^2)*(f*x+e)^4/a^4/f-3/8*b *f^3*x/a^2/d^3+2/3*(f*x+e)^3*sinh(d*x+c)/a/d-3*b*(a^2+b^2)*f*(f*x+e)^2*pol ylog(2,-a*exp(d*x+c)/(b-(a^2+b^2)^(1/2)))/a^4/d^2-3*b*(a^2+b^2)*f*(f*x+e)^ 2*polylog(2,-a*exp(d*x+c)/(b+(a^2+b^2)^(1/2)))/a^4/d^2+6*b*(a^2+b^2)*f^2*( f*x+e)*polylog(3,-a*exp(d*x+c)/(b-(a^2+b^2)^(1/2)))/a^4/d^3+6*b*(a^2+b^2)* f^2*(f*x+e)*polylog(3,-a*exp(d*x+c)/(b+(a^2+b^2)^(1/2)))/a^4/d^3+3/4*b*f*( f*x+e)^2*cosh(d*x+c)*sinh(d*x+c)/a^2/d^2+b^2*(f*x+e)^3*sinh(d*x+c)/a^3/d-b *(a^2+b^2)*(f*x+e)^3*ln(1+a*exp(d*x+c)/(b-(a^2+b^2)^(1/2)))/a^4/d-40/9*f^3 *cosh(d*x+c)/a/d^4-1/4*b*(f*x+e)^3/a^2/d-2/27*f^3*cosh(d*x+c)^3/a/d^4
Leaf count is larger than twice the leaf count of optimal. \(7404\) vs. \(2(864)=1728\).
Time = 32.30 (sec) , antiderivative size = 7404, normalized size of antiderivative = 8.57 \[ \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\text {Result too large to show} \]
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.
| \(\displaystyle \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 6128 |
| \(\displaystyle \int \frac {(e+f x)^3 \sinh (c+d x) \cosh ^3(c+d x)}{a \sinh (c+d x)+b}dx\) |
| \(\Big \downarrow \) 6113 |
| \(\displaystyle \frac {\int (e+f x)^3 \cosh ^3(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{b+a \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\int (e+f x)^3 \sin \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{a}\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle \frac {\frac {2 f^2 \int (e+f x) \cosh ^3(c+d x)dx}{3 d^2}+\frac {2}{3} \int (e+f x)^3 \cosh (c+d x)dx-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{b+a \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\frac {2 f^2 \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{3 d^2}+\frac {2}{3} \int (e+f x)^3 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\frac {2 f^2 \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{3 d^2}+\frac {2}{3} \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}-\frac {3 i f \int -i (e+f x)^2 \sinh (c+d x)dx}{d}\right )-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\frac {2 f^2 \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{3 d^2}+\frac {2}{3} \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}-\frac {3 f \int (e+f x)^2 \sinh (c+d x)dx}{d}\right )-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\frac {2 f^2 \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{3 d^2}+\frac {2}{3} \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}-\frac {3 f \int -i (e+f x)^2 \sin (i c+i d x)dx}{d}\right )-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\frac {2 f^2 \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{3 d^2}+\frac {2}{3} \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \int (e+f x)^2 \sin (i c+i d x)dx}{d}\right )-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\frac {2 f^2 \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{3 d^2}+\frac {2}{3} \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \int (e+f x) \cosh (c+d x)dx}{d}\right )}{d}\right )-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\frac {2 f^2 \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{3 d^2}+\frac {2}{3} \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{d}\right )-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\frac {2 f^2 \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{3 d^2}+\frac {2}{3} \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {i f \int -i \sinh (c+d x)dx}{d}\right )}{d}\right )}{d}\right )-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\frac {2 f^2 \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{3 d^2}+\frac {2}{3} \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int \sinh (c+d x)dx}{d}\right )}{d}\right )}{d}\right )-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\frac {2 f^2 \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{3 d^2}+\frac {2}{3} \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int -i \sin (i c+i d x)dx}{d}\right )}{d}\right )}{d}\right )-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\frac {2 f^2 \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{3 d^2}+\frac {2}{3} \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}+\frac {i f \int \sin (i c+i d x)dx}{d}\right )}{d}\right )}{d}\right )-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\frac {2 f^2 \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{3 d^2}-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 d^2}+\frac {2}{3} \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )+\frac {(e+f x)^3 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 3791 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\frac {2 f^2 \left (\frac {2}{3} \int (e+f x) \cosh (c+d x)dx-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d^2}-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 d^2}+\frac {2}{3} \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )+\frac {(e+f x)^3 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\frac {2 f^2 \left (\frac {2}{3} \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )dx-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d^2}-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 d^2}+\frac {2}{3} \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )+\frac {(e+f x)^3 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\frac {2 f^2 \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {i f \int -i \sinh (c+d x)dx}{d}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d^2}-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 d^2}+\frac {2}{3} \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )+\frac {(e+f x)^3 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\frac {2 f^2 \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int \sinh (c+d x)dx}{d}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d^2}-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 d^2}+\frac {2}{3} \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )+\frac {(e+f x)^3 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\frac {2 f^2 \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int -i \sin (i c+i d x)dx}{d}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d^2}-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 d^2}+\frac {2}{3} \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )+\frac {(e+f x)^3 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\frac {2 f^2 \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}+\frac {i f \int \sin (i c+i d x)dx}{d}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d^2}-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 d^2}+\frac {2}{3} \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )+\frac {(e+f x)^3 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\frac {2 f^2 \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d^2}-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 d^2}+\frac {2}{3} \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )+\frac {(e+f x)^3 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 6099 |
| \(\displaystyle -\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3 \cosh (c+d x)}{b+a \sinh (c+d x)}dx}{a^2}-\frac {b \int (e+f x)^3 \cosh (c+d x)dx}{a^2}+\frac {\int (e+f x)^3 \cosh (c+d x) \sinh (c+d x)dx}{a}\right )}{a}+\frac {\frac {2 f^2 \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d^2}-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 d^2}+\frac {2}{3} \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )+\frac {(e+f x)^3 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 f^2 \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d^2}-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 d^2}+\frac {2}{3} \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )+\frac {(e+f x)^3 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3 \cosh (c+d x)}{b+a \sinh (c+d x)}dx}{a^2}-\frac {b \int (e+f x)^3 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{a^2}+\frac {\int (e+f x)^3 \cosh (c+d x) \sinh (c+d x)dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {\frac {2 f^2 \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d^2}-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 d^2}+\frac {2}{3} \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )+\frac {(e+f x)^3 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3 \cosh (c+d x)}{b+a \sinh (c+d x)}dx}{a^2}-\frac {b \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}-\frac {3 i f \int -i (e+f x)^2 \sinh (c+d x)dx}{d}\right )}{a^2}+\frac {\int (e+f x)^3 \cosh (c+d x) \sinh (c+d x)dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3 \cosh (c+d x)}{b+a \sinh (c+d x)}dx}{a^2}-\frac {b \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}-\frac {3 f \int (e+f x)^2 \sinh (c+d x)dx}{d}\right )}{a^2}+\frac {\int (e+f x)^3 \cosh (c+d x) \sinh (c+d x)dx}{a}\right )}{a}+\frac {\frac {2 f^2 \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d^2}-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 d^2}+\frac {2}{3} \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )+\frac {(e+f x)^3 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 f^2 \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d^2}-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 d^2}+\frac {2}{3} \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )+\frac {(e+f x)^3 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3 \cosh (c+d x)}{b+a \sinh (c+d x)}dx}{a^2}-\frac {b \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}-\frac {3 f \int -i (e+f x)^2 \sin (i c+i d x)dx}{d}\right )}{a^2}+\frac {\int (e+f x)^3 \cosh (c+d x) \sinh (c+d x)dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {2 f^2 \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d^2}-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 d^2}+\frac {2}{3} \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )+\frac {(e+f x)^3 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3 \cosh (c+d x)}{b+a \sinh (c+d x)}dx}{a^2}-\frac {b \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \int (e+f x)^2 \sin (i c+i d x)dx}{d}\right )}{a^2}+\frac {\int (e+f x)^3 \cosh (c+d x) \sinh (c+d x)dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {\frac {2 f^2 \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d^2}-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 d^2}+\frac {2}{3} \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )+\frac {(e+f x)^3 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3 \cosh (c+d x)}{b+a \sinh (c+d x)}dx}{a^2}-\frac {b \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \int (e+f x) \cosh (c+d x)dx}{d}\right )}{d}\right )}{a^2}+\frac {\int (e+f x)^3 \cosh (c+d x) \sinh (c+d x)dx}{a}\right )}{a}\) |
3.1.26.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( -(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*C os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Simp[b*(c + d*x)*Cos[e + f*x ]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^2*((n - 1)/n) Int[(c + d* x)*(b*Sin[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]
Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 2*((n - 1)/n) Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 *m*((m - 1)/(f^2*n^2)) Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
Int[(Cosh[(c_.) + (d_.)*(x_)]^(n_)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_. )*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[-a/b^2 Int[(e + f*x)^m*Cos h[c + d*x]^(n - 2), x], x] + (Simp[1/b Int[(e + f*x)^m*Cosh[c + d*x]^(n - 2)*Sinh[c + d*x], x], x] + Simp[(a^2 + b^2)/b^2 Int[(e + f*x)^m*(Cosh[c + d*x]^(n - 2)/(a + b*Sinh[c + d*x])), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 1] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0]
Int[(Cosh[(c_.) + (d_.)*(x_)]^(p_.)*((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> S imp[1/b Int[(e + f*x)^m*Cosh[c + d*x]^p*Sinh[c + d*x]^(n - 1), x], x] - S imp[a/b Int[(e + f*x)^m*Cosh[c + d*x]^p*(Sinh[c + d*x]^(n - 1)/(a + b*Sin h[c + d*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[ n, 0] && IGtQ[p, 0]
Int[(((e_.) + (f_.)*(x_))^(m_.)*(F_)[(c_.) + (d_.)*(x_)]^(n_.))/(Csch[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Int[(e + f*x)^m*Sinh[c + d*x]*(F [c + d*x]^n/(b + a*Sinh[c + d*x])), x] /; FreeQ[{a, b, c, d, e, f}, x] && H yperbolicQ[F] && IntegersQ[m, n]
\[\int \frac {\left (f x +e \right )^{3} \cosh \left (d x +c \right )^{3}}{a +b \,\operatorname {csch}\left (d x +c \right )}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 7980 vs. \(2 (810) = 1620\).
Time = 0.38 (sec) , antiderivative size = 7980, normalized size of antiderivative = 9.24 \[ \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \cosh \left (d x + c\right )^{3}}{b \operatorname {csch}\left (d x + c\right ) + a} \,d x } \]
-1/24*e^3*((3*a*b*e^(-d*x - c) - a^2 - 3*(3*a^2 + 4*b^2)*e^(-2*d*x - 2*c)) *e^(3*d*x + 3*c)/(a^3*d) + 24*(a^2*b + b^3)*(d*x + c)/(a^4*d) + (3*a*b*e^( -2*d*x - 2*c) + a^2*e^(-3*d*x - 3*c) + 3*(3*a^2 + 4*b^2)*e^(-d*x - c))/(a^ 3*d) + 24*(a^2*b + b^3)*log(-2*b*e^(-d*x - c) + a*e^(-2*d*x - 2*c) - a)/(a ^4*d)) - 1/864*(216*(a^2*b*d^4*f^3*e^(3*c) + b^3*d^4*f^3*e^(3*c))*x^4 + 86 4*(a^2*b*d^4*e*f^2*e^(3*c) + b^3*d^4*e*f^2*e^(3*c))*x^3 + 1296*(a^2*b*d^4* e^2*f*e^(3*c) + b^3*d^4*e^2*f*e^(3*c))*x^2 - 4*(9*a^3*d^3*f^3*x^3*e^(6*c) + 9*(3*d^3*e*f^2 - d^2*f^3)*a^3*x^2*e^(6*c) + 3*(9*d^3*e^2*f - 6*d^2*e*f^2 + 2*d*f^3)*a^3*x*e^(6*c) - (9*d^2*e^2*f - 6*d*e*f^2 + 2*f^3)*a^3*e^(6*c)) *e^(3*d*x) + 27*(4*a^2*b*d^3*f^3*x^3*e^(5*c) + 6*(2*d^3*e*f^2 - d^2*f^3)*a ^2*b*x^2*e^(5*c) + 6*(2*d^3*e^2*f - 2*d^2*e*f^2 + d*f^3)*a^2*b*x*e^(5*c) - 3*(2*d^2*e^2*f - 2*d*e*f^2 + f^3)*a^2*b*e^(5*c))*e^(2*d*x) + 108*(9*(d^2* e^2*f - 2*d*e*f^2 + 2*f^3)*a^3*e^(4*c) + 12*(d^2*e^2*f - 2*d*e*f^2 + 2*f^3 )*a*b^2*e^(4*c) - (3*a^3*d^3*f^3*e^(4*c) + 4*a*b^2*d^3*f^3*e^(4*c))*x^3 - 3*(3*(d^3*e*f^2 - d^2*f^3)*a^3*e^(4*c) + 4*(d^3*e*f^2 - d^2*f^3)*a*b^2*e^( 4*c))*x^2 - 3*(3*(d^3*e^2*f - 2*d^2*e*f^2 + 2*d*f^3)*a^3*e^(4*c) + 4*(d^3* e^2*f - 2*d^2*e*f^2 + 2*d*f^3)*a*b^2*e^(4*c))*x)*e^(d*x) + 108*(9*(d^2*e^2 *f + 2*d*e*f^2 + 2*f^3)*a^3*e^(2*c) + 12*(d^2*e^2*f + 2*d*e*f^2 + 2*f^3)*a *b^2*e^(2*c) + (3*a^3*d^3*f^3*e^(2*c) + 4*a*b^2*d^3*f^3*e^(2*c))*x^3 + 3*( 3*(d^3*e*f^2 + d^2*f^3)*a^3*e^(2*c) + 4*(d^3*e*f^2 + d^2*f^3)*a*b^2*e^(...
\[ \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \cosh \left (d x + c\right )^{3}}{b \operatorname {csch}\left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^3\,{\left (e+f\,x\right )}^3}{a+\frac {b}{\mathrm {sinh}\left (c+d\,x\right )}} \,d x \]