Integrand size = 34, antiderivative size = 792 \[ \int \frac {(e+f x)^3 \cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {3 a f^3 x}{8 b^2 d^3}-\frac {a (e+f x)^3}{4 b^2 d}+\frac {a^3 (e+f x)^4}{4 b^4 f}-\frac {6 a^2 f^3 \cosh (c+d x)}{b^3 d^4}+\frac {14 f^3 \cosh (c+d x)}{9 b d^4}-\frac {3 a^2 f (e+f x)^2 \cosh (c+d x)}{b^3 d^2}+\frac {2 f (e+f x)^2 \cosh (c+d x)}{3 b d^2}-\frac {2 f^3 \cosh ^3(c+d x)}{27 b d^4}-\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {3 a^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {3 a^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^2}+\frac {6 a^3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^3}+\frac {6 a^3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^3}-\frac {6 a^3 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^4}-\frac {6 a^3 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^4}+\frac {6 a^2 f^2 (e+f x) \sinh (c+d x)}{b^3 d^3}-\frac {4 f^2 (e+f x) \sinh (c+d x)}{3 b d^3}+\frac {a^2 (e+f x)^3 \sinh (c+d x)}{b^3 d}+\frac {3 a f^3 \cosh (c+d x) \sinh (c+d x)}{8 b^2 d^4}+\frac {3 a f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 b^2 d^2}-\frac {3 a f^2 (e+f x) \sinh ^2(c+d x)}{4 b^2 d^3}-\frac {a (e+f x)^3 \sinh ^2(c+d x)}{2 b^2 d}-\frac {f (e+f x)^2 \cosh (c+d x) \sinh ^2(c+d x)}{3 b d^2}+\frac {2 f^2 (e+f x) \sinh ^3(c+d x)}{9 b d^3}+\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 b d} \]
-1/3*f*(f*x+e)^2*cosh(d*x+c)*sinh(d*x+c)^2/b/d^2-3*a^3*f*(f*x+e)^2*polylog (2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^4/d^2-3*a^3*f*(f*x+e)^2*polylog(2, -b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^4/d^2+6*a^3*f^2*(f*x+e)*polylog(3,-b* exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^4/d^3+6*a^3*f^2*(f*x+e)*polylog(3,-b*exp (d*x+c)/(a+(a^2+b^2)^(1/2)))/b^4/d^3-3*a^2*f*(f*x+e)^2*cosh(d*x+c)/b^3/d^2 +2/3*f*(f*x+e)^2*cosh(d*x+c)/b/d^2-4/3*f^2*(f*x+e)*sinh(d*x+c)/b/d^3-3/8*a *f^3*x/b^2/d^3-6*a^2*f^3*cosh(d*x+c)/b^3/d^4-1/2*a*(f*x+e)^3*sinh(d*x+c)^2 /b^2/d+1/3*(f*x+e)^3*sinh(d*x+c)^3/b/d+a^2*(f*x+e)^3*sinh(d*x+c)/b^3/d-a^3 *(f*x+e)^3*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^4/d-a^3*(f*x+e)^3*ln(1 +b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^4/d+1/4*a^3*(f*x+e)^4/b^4/f+2/9*f^2*( f*x+e)*sinh(d*x+c)^3/b/d^3-6*a^3*f^3*polylog(4,-b*exp(d*x+c)/(a-(a^2+b^2)^ (1/2)))/b^4/d^4-6*a^3*f^3*polylog(4,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^4 /d^4-1/4*a*(f*x+e)^3/b^2/d-2/27*f^3*cosh(d*x+c)^3/b/d^4+3/4*a*f*(f*x+e)^2* cosh(d*x+c)*sinh(d*x+c)/b^2/d^2+14/9*f^3*cosh(d*x+c)/b/d^4+6*a^2*f^2*(f*x+ e)*sinh(d*x+c)/b^3/d^3+3/8*a*f^3*cosh(d*x+c)*sinh(d*x+c)/b^2/d^4-3/4*a*f^2 *(f*x+e)*sinh(d*x+c)^2/b^2/d^3
Leaf count is larger than twice the leaf count of optimal. \(5656\) vs. \(2(792)=1584\).
Time = 28.49 (sec) , antiderivative size = 5656, normalized size of antiderivative = 7.14 \[ \int \frac {(e+f x)^3 \cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show} \]
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.
\(\displaystyle \int \frac {(e+f x)^3 \sinh ^3(c+d x) \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx\) |
\(\Big \downarrow \) 6113 |
\(\displaystyle \frac {\int (e+f x)^3 \cosh (c+d x) \sinh ^2(c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^3 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
\(\Big \downarrow \) 5969 |
\(\displaystyle \frac {\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 d}-\frac {f \int (e+f x)^2 \sinh ^3(c+d x)dx}{d}}{b}-\frac {a \int \frac {(e+f x)^3 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 d}-\frac {f \int i (e+f x)^2 \sin (i c+i d x)^3dx}{d}}{b}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 d}-\frac {i f \int (e+f x)^2 \sin (i c+i d x)^3dx}{d}}{b}\) |
\(\Big \downarrow \) 3792 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 d}-\frac {i f \left (\frac {2 f^2 \int -i \sinh ^3(c+d x)dx}{9 d^2}+\frac {2}{3} \int i (e+f x)^2 \sinh (c+d x)dx+\frac {2 i f (e+f x) \sinh ^3(c+d x)}{9 d^2}-\frac {i (e+f x)^2 \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 d}-\frac {i f \left (-\frac {2 i f^2 \int \sinh ^3(c+d x)dx}{9 d^2}+\frac {2}{3} i \int (e+f x)^2 \sinh (c+d x)dx+\frac {2 i f (e+f x) \sinh ^3(c+d x)}{9 d^2}-\frac {i (e+f x)^2 \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 d}-\frac {i f \left (-\frac {2 i f^2 \int i \sin (i c+i d x)^3dx}{9 d^2}+\frac {2}{3} i \int -i (e+f x)^2 \sin (i c+i d x)dx+\frac {2 i f (e+f x) \sinh ^3(c+d x)}{9 d^2}-\frac {i (e+f x)^2 \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 d}-\frac {i f \left (\frac {2 f^2 \int \sin (i c+i d x)^3dx}{9 d^2}+\frac {2}{3} \int (e+f x)^2 \sin (i c+i d x)dx+\frac {2 i f (e+f x) \sinh ^3(c+d x)}{9 d^2}-\frac {i (e+f x)^2 \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 3113 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 d}-\frac {i f \left (\frac {2 i f^2 \int \left (1-\cosh ^2(c+d x)\right )d\cosh (c+d x)}{9 d^3}+\frac {2}{3} \int (e+f x)^2 \sin (i c+i d x)dx+\frac {2 i f (e+f x) \sinh ^3(c+d x)}{9 d^2}-\frac {i (e+f x)^2 \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 d}-\frac {i f \left (\frac {2}{3} \int (e+f x)^2 \sin (i c+i d x)dx+\frac {2 i f^2 \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{9 d^3}+\frac {2 i f (e+f x) \sinh ^3(c+d x)}{9 d^2}-\frac {i (e+f x)^2 \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 d}-\frac {i f \left (\frac {2}{3} \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \int (e+f x) \cosh (c+d x)dx}{d}\right )+\frac {2 i f^2 \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{9 d^3}+\frac {2 i f (e+f x) \sinh ^3(c+d x)}{9 d^2}-\frac {i (e+f x)^2 \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 d}-\frac {i f \left (\frac {2}{3} \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )+\frac {2 i f^2 \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{9 d^3}+\frac {2 i f (e+f x) \sinh ^3(c+d x)}{9 d^2}-\frac {i (e+f x)^2 \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 d}-\frac {i f \left (\frac {2}{3} \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {i f \int -i \sinh (c+d x)dx}{d}\right )}{d}\right )+\frac {2 i f^2 \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{9 d^3}+\frac {2 i f (e+f x) \sinh ^3(c+d x)}{9 d^2}-\frac {i (e+f x)^2 \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 d}-\frac {i f \left (\frac {2}{3} \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int \sinh (c+d x)dx}{d}\right )}{d}\right )+\frac {2 i f^2 \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{9 d^3}+\frac {2 i f (e+f x) \sinh ^3(c+d x)}{9 d^2}-\frac {i (e+f x)^2 \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 d}-\frac {i f \left (\frac {2}{3} \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int -i \sin (i c+i d x)dx}{d}\right )}{d}\right )+\frac {2 i f^2 \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{9 d^3}+\frac {2 i f (e+f x) \sinh ^3(c+d x)}{9 d^2}-\frac {i (e+f x)^2 \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 d}-\frac {i f \left (\frac {2}{3} \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}+\frac {i f \int \sin (i c+i d x)dx}{d}\right )}{d}\right )+\frac {2 i f^2 \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{9 d^3}+\frac {2 i f (e+f x) \sinh ^3(c+d x)}{9 d^2}-\frac {i (e+f x)^2 \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 3118 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 d}-\frac {i f \left (\frac {2 i f^2 \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{9 d^3}+\frac {2 i f (e+f x) \sinh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )-\frac {i (e+f x)^2 \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 6113 |
\(\displaystyle -\frac {a \left (\frac {\int (e+f x)^3 \cosh (c+d x) \sinh (c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}+\frac {\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 d}-\frac {i f \left (\frac {2 i f^2 \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{9 d^3}+\frac {2 i f (e+f x) \sinh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )-\frac {i (e+f x)^2 \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 5969 |
\(\displaystyle -\frac {a \left (\frac {\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 d}-\frac {3 f \int (e+f x)^2 \sinh ^2(c+d x)dx}{2 d}}{b}-\frac {a \int \frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}+\frac {\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 d}-\frac {i f \left (\frac {2 i f^2 \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{9 d^3}+\frac {2 i f (e+f x) \sinh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )-\frac {i (e+f x)^2 \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 d}-\frac {i f \left (\frac {2 i f^2 \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{9 d^3}+\frac {2 i f (e+f x) \sinh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )-\frac {i (e+f x)^2 \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 d}-\frac {3 f \int -(e+f x)^2 \sin (i c+i d x)^2dx}{2 d}}{b}\right )}{b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 d}-\frac {i f \left (\frac {2 i f^2 \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{9 d^3}+\frac {2 i f (e+f x) \sinh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )-\frac {i (e+f x)^2 \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 d}+\frac {3 f \int (e+f x)^2 \sin (i c+i d x)^2dx}{2 d}}{b}\right )}{b}\) |
\(\Big \downarrow \) 3792 |
\(\displaystyle -\frac {a \left (\frac {\frac {3 f \left (\frac {f^2 \int -\sinh ^2(c+d x)dx}{2 d^2}+\frac {1}{2} \int (e+f x)^2dx+\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}+\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 d}}{b}-\frac {a \int \frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}+\frac {\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 d}-\frac {i f \left (\frac {2 i f^2 \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{9 d^3}+\frac {2 i f (e+f x) \sinh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )-\frac {i (e+f x)^2 \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 17 |
\(\displaystyle -\frac {a \left (\frac {\frac {3 f \left (\frac {f^2 \int -\sinh ^2(c+d x)dx}{2 d^2}+\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{2 d}+\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 d}}{b}-\frac {a \int \frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}+\frac {\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 d}-\frac {i f \left (\frac {2 i f^2 \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{9 d^3}+\frac {2 i f (e+f x) \sinh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )-\frac {i (e+f x)^2 \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {a \left (\frac {\frac {3 f \left (-\frac {f^2 \int \sinh ^2(c+d x)dx}{2 d^2}+\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{2 d}+\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 d}}{b}-\frac {a \int \frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}+\frac {\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 d}-\frac {i f \left (\frac {2 i f^2 \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{9 d^3}+\frac {2 i f (e+f x) \sinh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )-\frac {i (e+f x)^2 \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 d}-\frac {i f \left (\frac {2 i f^2 \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{9 d^3}+\frac {2 i f (e+f x) \sinh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )-\frac {i (e+f x)^2 \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 d}+\frac {3 f \left (-\frac {f^2 \int -\sin (i c+i d x)^2dx}{2 d^2}+\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{2 d}}{b}\right )}{b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 d}-\frac {i f \left (\frac {2 i f^2 \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{9 d^3}+\frac {2 i f (e+f x) \sinh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )-\frac {i (e+f x)^2 \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 d}+\frac {3 f \left (\frac {f^2 \int \sin (i c+i d x)^2dx}{2 d^2}+\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{2 d}}{b}\right )}{b}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle -\frac {a \left (\frac {\frac {3 f \left (\frac {f^2 \left (\frac {\int 1dx}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}+\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{2 d}+\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 d}}{b}-\frac {a \int \frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}+\frac {\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 d}-\frac {i f \left (\frac {2 i f^2 \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{9 d^3}+\frac {2 i f (e+f x) \sinh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )-\frac {i (e+f x)^2 \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle -\frac {a \left (\frac {\frac {3 f \left (\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{2 d}+\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 d}}{b}-\frac {a \int \frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}+\frac {\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 d}-\frac {i f \left (\frac {2 i f^2 \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{9 d^3}+\frac {2 i f (e+f x) \sinh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )-\frac {i (e+f x)^2 \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 6113 |
\(\displaystyle -\frac {a \left (\frac {\frac {3 f \left (\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{2 d}+\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 d}}{b}-\frac {a \left (\frac {\int (e+f x)^3 \cosh (c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\right )}{b}+\frac {\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 d}-\frac {i f \left (\frac {2 i f^2 \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{9 d^3}+\frac {2 i f (e+f x) \sinh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )-\frac {i (e+f x)^2 \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 d}-\frac {i f \left (\frac {2 i f^2 \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{9 d^3}+\frac {2 i f (e+f x) \sinh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )-\frac {i (e+f x)^2 \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{d}}{b}-\frac {a \left (\frac {\frac {3 f \left (\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{2 d}+\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int (e+f x)^3 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{b}\right )}{b}\right )}{b}\) |
3.4.91.3.1 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[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
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_))^(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[(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]
\[\int \frac {\left (f x +e \right )^{3} \cosh \left (d x +c \right ) \sinh \left (d x +c \right )^{3}}{a +b \sinh \left (d x +c \right )}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 7020 vs. \(2 (738) = 1476\).
Time = 0.36 (sec) , antiderivative size = 7020, normalized size of antiderivative = 8.86 \[ \int \frac {(e+f x)^3 \cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {(e+f x)^3 \cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {(e+f x)^3 \cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a} \,d x } \]
-1/24*e^3*(24*(d*x + c)*a^3/(b^4*d) + 24*a^3*log(-2*a*e^(-d*x - c) + b*e^( -2*d*x - 2*c) - b)/(b^4*d) + (3*a*b*e^(-d*x - c) - b^2 - 3*(4*a^2 - b^2)*e ^(-2*d*x - 2*c))*e^(3*d*x + 3*c)/(b^3*d) + (3*a*b*e^(-2*d*x - 2*c) + b^2*e ^(-3*d*x - 3*c) + 3*(4*a^2 - b^2)*e^(-d*x - c))/(b^3*d)) - 1/864*(216*a^3* d^4*f^3*x^4*e^(3*c) + 864*a^3*d^4*e*f^2*x^3*e^(3*c) + 1296*a^3*d^4*e^2*f*x ^2*e^(3*c) - 4*(9*b^3*d^3*f^3*x^3*e^(6*c) + 9*(3*d^3*e*f^2 - d^2*f^3)*b^3* x^2*e^(6*c) + 3*(9*d^3*e^2*f - 6*d^2*e*f^2 + 2*d*f^3)*b^3*x*e^(6*c) - (9*d ^2*e^2*f - 6*d*e*f^2 + 2*f^3)*b^3*e^(6*c))*e^(3*d*x) + 27*(4*a*b^2*d^3*f^3 *x^3*e^(5*c) + 6*(2*d^3*e*f^2 - d^2*f^3)*a*b^2*x^2*e^(5*c) + 6*(2*d^3*e^2* f - 2*d^2*e*f^2 + d*f^3)*a*b^2*x*e^(5*c) - 3*(2*d^2*e^2*f - 2*d*e*f^2 + f^ 3)*a*b^2*e^(5*c))*e^(2*d*x) + 108*(12*(d^2*e^2*f - 2*d*e*f^2 + 2*f^3)*a^2* b*e^(4*c) - 3*(d^2*e^2*f - 2*d*e*f^2 + 2*f^3)*b^3*e^(4*c) - (4*a^2*b*d^3*f ^3*e^(4*c) - b^3*d^3*f^3*e^(4*c))*x^3 - 3*(4*(d^3*e*f^2 - d^2*f^3)*a^2*b*e ^(4*c) - (d^3*e*f^2 - d^2*f^3)*b^3*e^(4*c))*x^2 - 3*(4*(d^3*e^2*f - 2*d^2* e*f^2 + 2*d*f^3)*a^2*b*e^(4*c) - (d^3*e^2*f - 2*d^2*e*f^2 + 2*d*f^3)*b^3*e ^(4*c))*x)*e^(d*x) + 108*(12*(d^2*e^2*f + 2*d*e*f^2 + 2*f^3)*a^2*b*e^(2*c) - 3*(d^2*e^2*f + 2*d*e*f^2 + 2*f^3)*b^3*e^(2*c) + (4*a^2*b*d^3*f^3*e^(2*c ) - b^3*d^3*f^3*e^(2*c))*x^3 + 3*(4*(d^3*e*f^2 + d^2*f^3)*a^2*b*e^(2*c) - (d^3*e*f^2 + d^2*f^3)*b^3*e^(2*c))*x^2 + 3*(4*(d^3*e^2*f + 2*d^2*e*f^2 + 2 *d*f^3)*a^2*b*e^(2*c) - (d^3*e^2*f + 2*d^2*e*f^2 + 2*d*f^3)*b^3*e^(2*c)...
\[ \int \frac {(e+f x)^3 \cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int \frac {(e+f x)^3 \cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )\,{\mathrm {sinh}\left (c+d\,x\right )}^3\,{\left (e+f\,x\right )}^3}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]