Integrand size = 36, antiderivative size = 819 \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {a^2 e f x}{2 b^3 d}-\frac {3 e f x}{16 b d}+\frac {a^2 f^2 x^2}{4 b^3 d}-\frac {3 f^2 x^2}{32 b d}-\frac {a^2 \left (a^2+b^2\right ) (e+f x)^3}{3 b^5 f}+\frac {2 a^3 f (e+f x) \cosh (c+d x)}{b^4 d^2}+\frac {4 a f (e+f x) \cosh (c+d x)}{3 b^2 d^2}+\frac {3 f^2 \cosh ^2(c+d x)}{32 b d^3}+\frac {2 a f (e+f x) \cosh ^3(c+d x)}{9 b^2 d^2}+\frac {f^2 \cosh ^4(c+d x)}{32 b d^3}+\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 b d}+\frac {a^2 \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d}+\frac {a^2 \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d}+\frac {2 a^2 \left (a^2+b^2\right ) f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d^2}+\frac {2 a^2 \left (a^2+b^2\right ) f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d^2}-\frac {2 a^2 \left (a^2+b^2\right ) f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d^3}-\frac {2 a^2 \left (a^2+b^2\right ) f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d^3}-\frac {2 a^3 f^2 \sinh (c+d x)}{b^4 d^3}-\frac {14 a f^2 \sinh (c+d x)}{9 b^2 d^3}-\frac {a^3 (e+f x)^2 \sinh (c+d x)}{b^4 d}-\frac {2 a (e+f x)^2 \sinh (c+d x)}{3 b^2 d}-\frac {a^2 f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^3 d^2}-\frac {3 f (e+f x) \cosh (c+d x) \sinh (c+d x)}{16 b d^2}-\frac {a (e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^2 d}-\frac {f (e+f x) \cosh ^3(c+d x) \sinh (c+d x)}{8 b d^2}+\frac {a^2 f^2 \sinh ^2(c+d x)}{4 b^3 d^3}+\frac {a^2 (e+f x)^2 \sinh ^2(c+d x)}{2 b^3 d}-\frac {2 a f^2 \sinh ^3(c+d x)}{27 b^2 d^3} \]
-2/3*a*(f*x+e)^2*sinh(d*x+c)/b^2/d-3/16*e*f*x/b/d-14/9*a*f^2*sinh(d*x+c)/b ^2/d^3+4/3*a*f*(f*x+e)*cosh(d*x+c)/b^2/d^2-3/16*f*(f*x+e)*cosh(d*x+c)*sinh (d*x+c)/b/d^2-1/2*a^2*f*(f*x+e)*cosh(d*x+c)*sinh(d*x+c)/b^3/d^2+2*a^2*(a^2 +b^2)*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^5/d^2+2*a^2 *(a^2+b^2)*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^5/d^2+ 3/32*f^2*cosh(d*x+c)^2/b/d^3+1/32*f^2*cosh(d*x+c)^4/b/d^3+1/4*(f*x+e)^2*co sh(d*x+c)^4/b/d+1/4*a^2*f^2*x^2/b^3/d-1/3*a^2*(a^2+b^2)*(f*x+e)^3/b^5/f-2* a^3*f^2*sinh(d*x+c)/b^4/d^3+1/4*a^2*f^2*sinh(d*x+c)^2/b^3/d^3+1/2*a^2*(f*x +e)^2*sinh(d*x+c)^2/b^3/d-2/27*a*f^2*sinh(d*x+c)^3/b^2/d^3+1/2*a^2*e*f*x/b ^3/d+2*a^3*f*(f*x+e)*cosh(d*x+c)/b^4/d^2+2/9*a*f*(f*x+e)*cosh(d*x+c)^3/b^2 /d^2-1/3*a*(f*x+e)^2*cosh(d*x+c)^2*sinh(d*x+c)/b^2/d-1/8*f*(f*x+e)*cosh(d* x+c)^3*sinh(d*x+c)/b/d^2-2*a^2*(a^2+b^2)*f^2*polylog(3,-b*exp(d*x+c)/(a-(a ^2+b^2)^(1/2)))/b^5/d^3-2*a^2*(a^2+b^2)*f^2*polylog(3,-b*exp(d*x+c)/(a+(a^ 2+b^2)^(1/2)))/b^5/d^3+a^2*(a^2+b^2)*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a-(a^2+b ^2)^(1/2)))/b^5/d+a^2*(a^2+b^2)*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^( 1/2)))/b^5/d-a^3*(f*x+e)^2*sinh(d*x+c)/b^4/d-3/32*f^2*x^2/b/d
Leaf count is larger than twice the leaf count of optimal. \(5113\) vs. \(2(819)=1638\).
Time = 15.71 (sec) , antiderivative size = 5113, normalized size of antiderivative = 6.24 \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show} \]
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.
| \(\displaystyle \int \frac {(e+f x)^2 \sinh ^2(c+d x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6113 |
| \(\displaystyle \frac {\int (e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 5970 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \int (e+f x) \cosh ^4(c+d x)dx}{2 d}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )^4dx}{2 d}}{b}\) |
| \(\Big \downarrow \) 3791 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \int (e+f x) \cosh ^2(c+d x)dx-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )^2dx-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}\) |
| \(\Big \downarrow \) 3791 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (\frac {1}{2} \int (e+f x)dx-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 6113 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (\frac {\int (e+f x)^2 \cosh ^3(c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (\frac {\frac {2 f^2 \int \cosh ^3(c+d x)dx}{9 d^2}+\frac {2}{3} \int (e+f x)^2 \cosh (c+d x)dx-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2 f^2 \int \sin \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{9 d^2}+\frac {2}{3} \int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2 i f^2 \int \left (\sinh ^2(c+d x)+1\right )d(-i \sinh (c+d x))}{9 d^3}+\frac {2}{3} \int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2}{3} \int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 i f \int -i (e+f x) \sinh (c+d x)dx}{d}\right )+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int (e+f x) \sinh (c+d x)dx}{d}\right )+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int -i (e+f x) \sin (i c+i d x)dx}{d}\right )+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \int (e+f x) \sin (i c+i d x)dx}{d}\right )+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \cosh (c+d x)dx}{d}\right )}{d}\right )+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{d}\right )+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 6099 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \int (e+f x)^2 \cosh (c+d x)dx}{b^2}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{b}+\frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (\frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (\frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 i f \int -i (e+f x) \sinh (c+d x)dx}{d}\right )}{b^2}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int (e+f x) \sinh (c+d x)dx}{d}\right )}{b^2}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{b}+\frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (\frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int -i (e+f x) \sin (i c+i d x)dx}{d}\right )}{b^2}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (\frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \int (e+f x) \sin (i c+i d x)dx}{d}\right )}{b^2}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (\frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \cosh (c+d x)dx}{d}\right )}{d}\right )}{b^2}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (\frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{d}\right )}{b^2}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (\frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx}{b}-\frac {a \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{b^2}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 5969 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (\frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}+\frac {\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 d}-\frac {f \int (e+f x) \sinh ^2(c+d x)dx}{d}}{b}-\frac {a \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{b^2}\right )}{b}\right )}{b}\) |
3.4.73.3.1 Defintions of rubi rules used
Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 )/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
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[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
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[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)* (x_)]^(n_.), x_Symbol] :> Simp[(c + d*x)^m*(Sinh[a + b*x]^(n + 1)/(b*(n + 1 ))), x] - Simp[d*(m/(b*(n + 1))) Int[(c + d*x)^(m - 1)*Sinh[a + b*x]^(n + 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
Int[Cosh[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[(c + d*x)^m*(Cosh[a + b*x]^(n + 1)/(b*(n + 1 ))), x] - Simp[d*(m/(b*(n + 1))) Int[(c + d*x)^(m - 1)*Cosh[a + b*x]^(n + 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -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]
\[\int \frac {\left (f x +e \right )^{2} \cosh \left (d x +c \right )^{3} \sinh \left (d x +c \right )^{2}}{a +b \sinh \left (d x +c \right )}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 7645 vs. \(2 (761) = 1522\).
Time = 0.38 (sec) , antiderivative size = 7645, normalized size of antiderivative = 9.33 \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a} \,d x } \]
-1/192*e^2*((8*a*b^2*e^(-d*x - c) - 3*b^3 - 12*(2*a^2*b + b^3)*e^(-2*d*x - 2*c) + 24*(4*a^3 + 3*a*b^2)*e^(-3*d*x - 3*c))*e^(4*d*x + 4*c)/(b^4*d) - 1 92*(a^4 + a^2*b^2)*(d*x + c)/(b^5*d) - (8*a*b^2*e^(-3*d*x - 3*c) + 3*b^3*e ^(-4*d*x - 4*c) + 24*(4*a^3 + 3*a*b^2)*e^(-d*x - c) + 12*(2*a^2*b + b^3)*e ^(-2*d*x - 2*c))/(b^4*d) - 192*(a^4 + a^2*b^2)*log(-2*a*e^(-d*x - c) + b*e ^(-2*d*x - 2*c) - b)/(b^5*d)) + 1/13824*(4608*(a^4*d^3*f^2*e^(4*c) + a^2*b ^2*d^3*f^2*e^(4*c))*x^3 + 13824*(a^4*d^3*e*f*e^(4*c) + a^2*b^2*d^3*e*f*e^( 4*c))*x^2 + 27*(8*b^4*d^2*f^2*x^2*e^(8*c) + 4*(4*d^2*e*f - d*f^2)*b^4*x*e^ (8*c) - (4*d*e*f - f^2)*b^4*e^(8*c))*e^(4*d*x) - 64*(9*a*b^3*d^2*f^2*x^2*e ^(7*c) + 6*(3*d^2*e*f - d*f^2)*a*b^3*x*e^(7*c) - 2*(3*d*e*f - f^2)*a*b^3*e ^(7*c))*e^(3*d*x) - 432*(2*(2*d*e*f - f^2)*a^2*b^2*e^(6*c) + (2*d*e*f - f^ 2)*b^4*e^(6*c) - 2*(2*a^2*b^2*d^2*f^2*e^(6*c) + b^4*d^2*f^2*e^(6*c))*x^2 - 2*(2*(2*d^2*e*f - d*f^2)*a^2*b^2*e^(6*c) + (2*d^2*e*f - d*f^2)*b^4*e^(6*c ))*x)*e^(2*d*x) + 1728*(8*(d*e*f - f^2)*a^3*b*e^(5*c) + 6*(d*e*f - f^2)*a* b^3*e^(5*c) - (4*a^3*b*d^2*f^2*e^(5*c) + 3*a*b^3*d^2*f^2*e^(5*c))*x^2 - 2* (4*(d^2*e*f - d*f^2)*a^3*b*e^(5*c) + 3*(d^2*e*f - d*f^2)*a*b^3*e^(5*c))*x) *e^(d*x) + 1728*(8*(d*e*f + f^2)*a^3*b*e^(3*c) + 6*(d*e*f + f^2)*a*b^3*e^( 3*c) + (4*a^3*b*d^2*f^2*e^(3*c) + 3*a*b^3*d^2*f^2*e^(3*c))*x^2 + 2*(4*(d^2 *e*f + d*f^2)*a^3*b*e^(3*c) + 3*(d^2*e*f + d*f^2)*a*b^3*e^(3*c))*x)*e^(-d* x) + 432*(2*(2*d*e*f + f^2)*a^2*b^2*e^(2*c) + (2*d*e*f + f^2)*b^4*e^(2*...
\[ \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^3\,{\mathrm {sinh}\left (c+d\,x\right )}^2\,{\left (e+f\,x\right )}^2}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]