Integrand size = 36, antiderivative size = 1443 \[ \int \frac {(e+f x)^3 \cosh ^3(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {3 a^3 f^3 x}{8 b^4 d^3}+\frac {45 a f^3 x}{256 b^2 d^3}-\frac {a^3 (e+f x)^3}{4 b^4 d}+\frac {3 a (e+f x)^3}{32 b^2 d}+\frac {a^3 \left (a^2+b^2\right ) (e+f x)^4}{4 b^6 f}-\frac {6 a^4 f^3 \cosh (c+d x)}{b^5 d^4}-\frac {40 a^2 f^3 \cosh (c+d x)}{9 b^3 d^4}+\frac {3 f^3 \cosh (c+d x)}{4 b d^4}-\frac {3 a^4 f (e+f x)^2 \cosh (c+d x)}{b^5 d^2}-\frac {2 a^2 f (e+f x)^2 \cosh (c+d x)}{b^3 d^2}+\frac {3 f (e+f x)^2 \cosh (c+d x)}{8 b d^2}-\frac {9 a f^2 (e+f x) \cosh ^2(c+d x)}{32 b^2 d^3}-\frac {2 a^2 f^3 \cosh ^3(c+d x)}{27 b^3 d^4}-\frac {a^2 f (e+f x)^2 \cosh ^3(c+d x)}{3 b^3 d^2}-\frac {3 a f^2 (e+f x) \cosh ^4(c+d x)}{32 b^2 d^3}-\frac {a (e+f x)^3 \cosh ^4(c+d x)}{4 b^2 d}-\frac {f^3 \cosh (3 c+3 d x)}{216 b d^4}-\frac {f (e+f x)^2 \cosh (3 c+3 d x)}{48 b d^2}-\frac {3 f^3 \cosh (5 c+5 d x)}{5000 b d^4}-\frac {3 f (e+f x)^2 \cosh (5 c+5 d x)}{400 b d^2}-\frac {a^3 \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^6 d}-\frac {a^3 \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^6 d}-\frac {3 a^3 \left (a^2+b^2\right ) f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^6 d^2}-\frac {3 a^3 \left (a^2+b^2\right ) f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^6 d^2}+\frac {6 a^3 \left (a^2+b^2\right ) f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^6 d^3}+\frac {6 a^3 \left (a^2+b^2\right ) f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^6 d^3}-\frac {6 a^3 \left (a^2+b^2\right ) f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^6 d^4}-\frac {6 a^3 \left (a^2+b^2\right ) f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^6 d^4}+\frac {6 a^4 f^2 (e+f x) \sinh (c+d x)}{b^5 d^3}+\frac {40 a^2 f^2 (e+f x) \sinh (c+d x)}{9 b^3 d^3}-\frac {3 f^2 (e+f x) \sinh (c+d x)}{4 b d^3}+\frac {a^4 (e+f x)^3 \sinh (c+d x)}{b^5 d}+\frac {2 a^2 (e+f x)^3 \sinh (c+d x)}{3 b^3 d}-\frac {(e+f x)^3 \sinh (c+d x)}{8 b d}+\frac {3 a^3 f^3 \cosh (c+d x) \sinh (c+d x)}{8 b^4 d^4}+\frac {45 a f^3 \cosh (c+d x) \sinh (c+d x)}{256 b^2 d^4}+\frac {3 a^3 f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 b^4 d^2}+\frac {9 a f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{32 b^2 d^2}+\frac {2 a^2 f^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 b^3 d^3}+\frac {a^2 (e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^3 d}+\frac {3 a f^3 \cosh ^3(c+d x) \sinh (c+d x)}{128 b^2 d^4}+\frac {3 a f (e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{16 b^2 d^2}-\frac {3 a^3 f^2 (e+f x) \sinh ^2(c+d x)}{4 b^4 d^3}-\frac {a^3 (e+f x)^3 \sinh ^2(c+d x)}{2 b^4 d}+\frac {f^2 (e+f x) \sinh (3 c+3 d x)}{72 b d^3}+\frac {(e+f x)^3 \sinh (3 c+3 d x)}{48 b d}+\frac {3 f^2 (e+f x) \sinh (5 c+5 d x)}{1000 b d^3}+\frac {(e+f x)^3 \sinh (5 c+5 d x)}{80 b d} \]
-2*a^2*f*(f*x+e)^2*cosh(d*x+c)/b^3/d^2+3/8*f*(f*x+e)^2*cosh(d*x+c)/b/d^2-3 /4*f^2*(f*x+e)*sinh(d*x+c)/b/d^3+45/256*a*f^3*x/b^2/d^3-40/9*a^2*f^3*cosh( d*x+c)/b^3/d^4-1/8*(f*x+e)^3*sinh(d*x+c)/b/d+2/3*a^2*(f*x+e)^3*sinh(d*x+c) /b^3/d-3*a^4*f*(f*x+e)^2*cosh(d*x+c)/b^5/d^2-9/32*a*f^2*(f*x+e)*cosh(d*x+c )^2/b^2/d^3-1/3*a^2*f*(f*x+e)^2*cosh(d*x+c)^3/b^3/d^2-3/32*a*f^2*(f*x+e)*c osh(d*x+c)^4/b^2/d^3+6*a^4*f^2*(f*x+e)*sinh(d*x+c)/b^5/d^3+3/8*a^3*f^3*cos h(d*x+c)*sinh(d*x+c)/b^4/d^4+1/3*a^2*(f*x+e)^3*cosh(d*x+c)^2*sinh(d*x+c)/b ^3/d+3/128*a*f^3*cosh(d*x+c)^3*sinh(d*x+c)/b^2/d^4-3/4*a^3*f^2*(f*x+e)*sin h(d*x+c)^2/b^4/d^3-6*a^3*(a^2+b^2)*f^3*polylog(4,-b*exp(d*x+c)/(a-(a^2+b^2 )^(1/2)))/b^6/d^4-6*a^3*(a^2+b^2)*f^3*polylog(4,-b*exp(d*x+c)/(a+(a^2+b^2) ^(1/2)))/b^6/d^4+1/48*(f*x+e)^3*sinh(3*d*x+3*c)/b/d+1/80*(f*x+e)^3*sinh(5* d*x+5*c)/b/d-1/4*a^3*(f*x+e)^3/b^4/d-1/216*f^3*cosh(3*d*x+3*c)/b/d^4-3/500 0*f^3*cosh(5*d*x+5*c)/b/d^4+3/4*a^3*f*(f*x+e)^2*cosh(d*x+c)*sinh(d*x+c)/b^ 4/d^2+2/9*a^2*f^2*(f*x+e)*cosh(d*x+c)^2*sinh(d*x+c)/b^3/d^3+3/16*a*f*(f*x+ e)^2*cosh(d*x+c)^3*sinh(d*x+c)/b^2/d^2-3*a^3*(a^2+b^2)*f*(f*x+e)^2*polylog (2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^6/d^2-3*a^3*(a^2+b^2)*f*(f*x+e)^2* polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^6/d^2+6*a^3*(a^2+b^2)*f^2*( f*x+e)*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^6/d^3+6*a^3*(a^2+b^2 )*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^6/d^3+3/32*a* (f*x+e)^3/b^2/d-a^3*(a^2+b^2)*(f*x+e)^3*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^...
Leaf count is larger than twice the leaf count of optimal. \(5147\) vs. \(2(1443)=2886\).
Time = 11.07 (sec) , antiderivative size = 5147, normalized size of antiderivative = 3.57 \[ \int \frac {(e+f x)^3 \cosh ^3(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show} \]
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.
| \(\displaystyle \int \frac {(e+f x)^3 \sinh ^3(c+d x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6113 |
| \(\displaystyle \frac {\int (e+f x)^3 \cosh ^3(c+d x) \sinh ^2(c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^3 \cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle \frac {\int \left (-\frac {1}{8} \cosh (c+d x) (e+f x)^3+\frac {1}{16} \cosh (3 c+3 d x) (e+f x)^3+\frac {1}{16} \cosh (5 c+5 d x) (e+f x)^3\right )dx}{b}-\frac {a \int \frac {(e+f x)^3 \cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {3 f^3 \cosh (c+d x)}{4 d^4}-\frac {f^3 \cosh (3 c+3 d x)}{216 d^4}-\frac {3 f^3 \cosh (5 c+5 d x)}{5000 d^4}-\frac {3 f^2 (e+f x) \sinh (c+d x)}{4 d^3}+\frac {f^2 (e+f x) \sinh (3 c+3 d x)}{72 d^3}+\frac {3 f^2 (e+f x) \sinh (5 c+5 d x)}{1000 d^3}+\frac {3 f (e+f x)^2 \cosh (c+d x)}{8 d^2}-\frac {f (e+f x)^2 \cosh (3 c+3 d x)}{48 d^2}-\frac {3 f (e+f x)^2 \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x)^3 \sinh (c+d x)}{8 d}+\frac {(e+f x)^3 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^3 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \int \frac {(e+f x)^3 \cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 6113 |
| \(\displaystyle \frac {\frac {3 f^3 \cosh (c+d x)}{4 d^4}-\frac {f^3 \cosh (3 c+3 d x)}{216 d^4}-\frac {3 f^3 \cosh (5 c+5 d x)}{5000 d^4}-\frac {3 f^2 (e+f x) \sinh (c+d x)}{4 d^3}+\frac {f^2 (e+f x) \sinh (3 c+3 d x)}{72 d^3}+\frac {3 f^2 (e+f x) \sinh (5 c+5 d x)}{1000 d^3}+\frac {3 f (e+f x)^2 \cosh (c+d x)}{8 d^2}-\frac {f (e+f x)^2 \cosh (3 c+3 d x)}{48 d^2}-\frac {3 f (e+f x)^2 \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x)^3 \sinh (c+d x)}{8 d}+\frac {(e+f x)^3 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^3 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\int (e+f x)^3 \cosh ^3(c+d x) \sinh (c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^3 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 5970 |
| \(\displaystyle \frac {\frac {3 f^3 \cosh (c+d x)}{4 d^4}-\frac {f^3 \cosh (3 c+3 d x)}{216 d^4}-\frac {3 f^3 \cosh (5 c+5 d x)}{5000 d^4}-\frac {3 f^2 (e+f x) \sinh (c+d x)}{4 d^3}+\frac {f^2 (e+f x) \sinh (3 c+3 d x)}{72 d^3}+\frac {3 f^2 (e+f x) \sinh (5 c+5 d x)}{1000 d^3}+\frac {3 f (e+f x)^2 \cosh (c+d x)}{8 d^2}-\frac {f (e+f x)^2 \cosh (3 c+3 d x)}{48 d^2}-\frac {3 f (e+f x)^2 \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x)^3 \sinh (c+d x)}{8 d}+\frac {(e+f x)^3 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^3 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^3 \cosh ^4(c+d x)}{4 d}-\frac {3 f \int (e+f x)^2 \cosh ^4(c+d x)dx}{4 d}}{b}-\frac {a \int \frac {(e+f x)^3 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 f^3 \cosh (c+d x)}{4 d^4}-\frac {f^3 \cosh (3 c+3 d x)}{216 d^4}-\frac {3 f^3 \cosh (5 c+5 d x)}{5000 d^4}-\frac {3 f^2 (e+f x) \sinh (c+d x)}{4 d^3}+\frac {f^2 (e+f x) \sinh (3 c+3 d x)}{72 d^3}+\frac {3 f^2 (e+f x) \sinh (5 c+5 d x)}{1000 d^3}+\frac {3 f (e+f x)^2 \cosh (c+d x)}{8 d^2}-\frac {f (e+f x)^2 \cosh (3 c+3 d x)}{48 d^2}-\frac {3 f (e+f x)^2 \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x)^3 \sinh (c+d x)}{8 d}+\frac {(e+f x)^3 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^3 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \cosh ^4(c+d x)}{4 d}-\frac {3 f \int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )^4dx}{4 d}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle \frac {\frac {3 f^3 \cosh (c+d x)}{4 d^4}-\frac {f^3 \cosh (3 c+3 d x)}{216 d^4}-\frac {3 f^3 \cosh (5 c+5 d x)}{5000 d^4}-\frac {3 f^2 (e+f x) \sinh (c+d x)}{4 d^3}+\frac {f^2 (e+f x) \sinh (3 c+3 d x)}{72 d^3}+\frac {3 f^2 (e+f x) \sinh (5 c+5 d x)}{1000 d^3}+\frac {3 f (e+f x)^2 \cosh (c+d x)}{8 d^2}-\frac {f (e+f x)^2 \cosh (3 c+3 d x)}{48 d^2}-\frac {3 f (e+f x)^2 \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x)^3 \sinh (c+d x)}{8 d}+\frac {(e+f x)^3 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^3 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^3 \cosh ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {f^2 \int \cosh ^4(c+d x)dx}{8 d^2}+\frac {3}{4} \int (e+f x)^2 \cosh ^2(c+d x)dx-\frac {f (e+f x) \cosh ^4(c+d x)}{8 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{4 d}}{b}-\frac {a \int \frac {(e+f x)^3 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 f^3 \cosh (c+d x)}{4 d^4}-\frac {f^3 \cosh (3 c+3 d x)}{216 d^4}-\frac {3 f^3 \cosh (5 c+5 d x)}{5000 d^4}-\frac {3 f^2 (e+f x) \sinh (c+d x)}{4 d^3}+\frac {f^2 (e+f x) \sinh (3 c+3 d x)}{72 d^3}+\frac {3 f^2 (e+f x) \sinh (5 c+5 d x)}{1000 d^3}+\frac {3 f (e+f x)^2 \cosh (c+d x)}{8 d^2}-\frac {f (e+f x)^2 \cosh (3 c+3 d x)}{48 d^2}-\frac {3 f (e+f x)^2 \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x)^3 \sinh (c+d x)}{8 d}+\frac {(e+f x)^3 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^3 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \cosh ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {f^2 \int \sin \left (i c+i d x+\frac {\pi }{2}\right )^4dx}{8 d^2}+\frac {3}{4} \int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )^2dx-\frac {f (e+f x) \cosh ^4(c+d x)}{8 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{4 d}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {3 f^3 \cosh (c+d x)}{4 d^4}-\frac {f^3 \cosh (3 c+3 d x)}{216 d^4}-\frac {3 f^3 \cosh (5 c+5 d x)}{5000 d^4}-\frac {3 f^2 (e+f x) \sinh (c+d x)}{4 d^3}+\frac {f^2 (e+f x) \sinh (3 c+3 d x)}{72 d^3}+\frac {3 f^2 (e+f x) \sinh (5 c+5 d x)}{1000 d^3}+\frac {3 f (e+f x)^2 \cosh (c+d x)}{8 d^2}-\frac {f (e+f x)^2 \cosh (3 c+3 d x)}{48 d^2}-\frac {3 f (e+f x)^2 \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x)^3 \sinh (c+d x)}{8 d}+\frac {(e+f x)^3 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^3 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \cosh ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {f^2 \left (\frac {3}{4} \int \cosh ^2(c+d x)dx+\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{8 d^2}+\frac {3}{4} \int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )^2dx-\frac {f (e+f x) \cosh ^4(c+d x)}{8 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{4 d}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 f^3 \cosh (c+d x)}{4 d^4}-\frac {f^3 \cosh (3 c+3 d x)}{216 d^4}-\frac {3 f^3 \cosh (5 c+5 d x)}{5000 d^4}-\frac {3 f^2 (e+f x) \sinh (c+d x)}{4 d^3}+\frac {f^2 (e+f x) \sinh (3 c+3 d x)}{72 d^3}+\frac {3 f^2 (e+f x) \sinh (5 c+5 d x)}{1000 d^3}+\frac {3 f (e+f x)^2 \cosh (c+d x)}{8 d^2}-\frac {f (e+f x)^2 \cosh (3 c+3 d x)}{48 d^2}-\frac {3 f (e+f x)^2 \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x)^3 \sinh (c+d x)}{8 d}+\frac {(e+f x)^3 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^3 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \cosh ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {f^2 \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \int \sin \left (i c+i d x+\frac {\pi }{2}\right )^2dx\right )}{8 d^2}+\frac {3}{4} \int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )^2dx-\frac {f (e+f x) \cosh ^4(c+d x)}{8 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{4 d}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {3 f^3 \cosh (c+d x)}{4 d^4}-\frac {f^3 \cosh (3 c+3 d x)}{216 d^4}-\frac {3 f^3 \cosh (5 c+5 d x)}{5000 d^4}-\frac {3 f^2 (e+f x) \sinh (c+d x)}{4 d^3}+\frac {f^2 (e+f x) \sinh (3 c+3 d x)}{72 d^3}+\frac {3 f^2 (e+f x) \sinh (5 c+5 d x)}{1000 d^3}+\frac {3 f (e+f x)^2 \cosh (c+d x)}{8 d^2}-\frac {f (e+f x)^2 \cosh (3 c+3 d x)}{48 d^2}-\frac {3 f (e+f x)^2 \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x)^3 \sinh (c+d x)}{8 d}+\frac {(e+f x)^3 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^3 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \cosh ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {f^2 \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )+\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{8 d^2}+\frac {3}{4} \int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )^2dx-\frac {f (e+f x) \cosh ^4(c+d x)}{8 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{4 d}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {3 f^3 \cosh (c+d x)}{4 d^4}-\frac {f^3 \cosh (3 c+3 d x)}{216 d^4}-\frac {3 f^3 \cosh (5 c+5 d x)}{5000 d^4}-\frac {3 f^2 (e+f x) \sinh (c+d x)}{4 d^3}+\frac {f^2 (e+f x) \sinh (3 c+3 d x)}{72 d^3}+\frac {3 f^2 (e+f x) \sinh (5 c+5 d x)}{1000 d^3}+\frac {3 f (e+f x)^2 \cosh (c+d x)}{8 d^2}-\frac {f (e+f x)^2 \cosh (3 c+3 d x)}{48 d^2}-\frac {3 f (e+f x)^2 \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x)^3 \sinh (c+d x)}{8 d}+\frac {(e+f x)^3 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^3 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \cosh ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {3}{4} \int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )^2dx-\frac {f (e+f x) \cosh ^4(c+d x)}{8 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{8 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{4 d}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle \frac {\frac {3 f^3 \cosh (c+d x)}{4 d^4}-\frac {f^3 \cosh (3 c+3 d x)}{216 d^4}-\frac {3 f^3 \cosh (5 c+5 d x)}{5000 d^4}-\frac {3 f^2 (e+f x) \sinh (c+d x)}{4 d^3}+\frac {f^2 (e+f x) \sinh (3 c+3 d x)}{72 d^3}+\frac {3 f^2 (e+f x) \sinh (5 c+5 d x)}{1000 d^3}+\frac {3 f (e+f x)^2 \cosh (c+d x)}{8 d^2}-\frac {f (e+f x)^2 \cosh (3 c+3 d x)}{48 d^2}-\frac {3 f (e+f x)^2 \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x)^3 \sinh (c+d x)}{8 d}+\frac {(e+f x)^3 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^3 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^3 \cosh ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {3}{4} \left (\frac {f^2 \int \cosh ^2(c+d x)dx}{2 d^2}+\frac {1}{2} \int (e+f x)^2dx-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}\right )-\frac {f (e+f x) \cosh ^4(c+d x)}{8 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{8 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{4 d}}{b}-\frac {a \int \frac {(e+f x)^3 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle \frac {\frac {3 f^3 \cosh (c+d x)}{4 d^4}-\frac {f^3 \cosh (3 c+3 d x)}{216 d^4}-\frac {3 f^3 \cosh (5 c+5 d x)}{5000 d^4}-\frac {3 f^2 (e+f x) \sinh (c+d x)}{4 d^3}+\frac {f^2 (e+f x) \sinh (3 c+3 d x)}{72 d^3}+\frac {3 f^2 (e+f x) \sinh (5 c+5 d x)}{1000 d^3}+\frac {3 f (e+f x)^2 \cosh (c+d x)}{8 d^2}-\frac {f (e+f x)^2 \cosh (3 c+3 d x)}{48 d^2}-\frac {3 f (e+f x)^2 \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x)^3 \sinh (c+d x)}{8 d}+\frac {(e+f x)^3 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^3 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^3 \cosh ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {3}{4} \left (\frac {f^2 \int \cosh ^2(c+d x)dx}{2 d^2}-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )-\frac {f (e+f x) \cosh ^4(c+d x)}{8 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{8 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{4 d}}{b}-\frac {a \int \frac {(e+f x)^3 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 f^3 \cosh (c+d x)}{4 d^4}-\frac {f^3 \cosh (3 c+3 d x)}{216 d^4}-\frac {3 f^3 \cosh (5 c+5 d x)}{5000 d^4}-\frac {3 f^2 (e+f x) \sinh (c+d x)}{4 d^3}+\frac {f^2 (e+f x) \sinh (3 c+3 d x)}{72 d^3}+\frac {3 f^2 (e+f x) \sinh (5 c+5 d x)}{1000 d^3}+\frac {3 f (e+f x)^2 \cosh (c+d x)}{8 d^2}-\frac {f (e+f x)^2 \cosh (3 c+3 d x)}{48 d^2}-\frac {3 f (e+f x)^2 \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x)^3 \sinh (c+d x)}{8 d}+\frac {(e+f x)^3 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^3 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \cosh ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {3}{4} \left (\frac {f^2 \int \sin \left (i c+i d x+\frac {\pi }{2}\right )^2dx}{2 d^2}-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )-\frac {f (e+f x) \cosh ^4(c+d x)}{8 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{8 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{4 d}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {3 f^3 \cosh (c+d x)}{4 d^4}-\frac {f^3 \cosh (3 c+3 d x)}{216 d^4}-\frac {3 f^3 \cosh (5 c+5 d x)}{5000 d^4}-\frac {3 f^2 (e+f x) \sinh (c+d x)}{4 d^3}+\frac {f^2 (e+f x) \sinh (3 c+3 d x)}{72 d^3}+\frac {3 f^2 (e+f x) \sinh (5 c+5 d x)}{1000 d^3}+\frac {3 f (e+f x)^2 \cosh (c+d x)}{8 d^2}-\frac {f (e+f x)^2 \cosh (3 c+3 d x)}{48 d^2}-\frac {3 f (e+f x)^2 \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x)^3 \sinh (c+d x)}{8 d}+\frac {(e+f x)^3 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^3 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^3 \cosh ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {3}{4} \left (\frac {f^2 \left (\frac {\int 1dx}{2}+\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )-\frac {f (e+f x) \cosh ^4(c+d x)}{8 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{8 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{4 d}}{b}-\frac {a \int \frac {(e+f x)^3 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {3 f^3 \cosh (c+d x)}{4 d^4}-\frac {f^3 \cosh (3 c+3 d x)}{216 d^4}-\frac {3 f^3 \cosh (5 c+5 d x)}{5000 d^4}-\frac {3 f^2 (e+f x) \sinh (c+d x)}{4 d^3}+\frac {f^2 (e+f x) \sinh (3 c+3 d x)}{72 d^3}+\frac {3 f^2 (e+f x) \sinh (5 c+5 d x)}{1000 d^3}+\frac {3 f (e+f x)^2 \cosh (c+d x)}{8 d^2}-\frac {f (e+f x)^2 \cosh (3 c+3 d x)}{48 d^2}-\frac {3 f (e+f x)^2 \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x)^3 \sinh (c+d x)}{8 d}+\frac {(e+f x)^3 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^3 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^3 \cosh ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {3}{4} \left (-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )-\frac {f (e+f x) \cosh ^4(c+d x)}{8 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{8 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{4 d}}{b}-\frac {a \int \frac {(e+f x)^3 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 6113 |
| \(\displaystyle \frac {\frac {3 f^3 \cosh (c+d x)}{4 d^4}-\frac {f^3 \cosh (3 c+3 d x)}{216 d^4}-\frac {3 f^3 \cosh (5 c+5 d x)}{5000 d^4}-\frac {3 f^2 (e+f x) \sinh (c+d x)}{4 d^3}+\frac {f^2 (e+f x) \sinh (3 c+3 d x)}{72 d^3}+\frac {3 f^2 (e+f x) \sinh (5 c+5 d x)}{1000 d^3}+\frac {3 f (e+f x)^2 \cosh (c+d x)}{8 d^2}-\frac {f (e+f x)^2 \cosh (3 c+3 d x)}{48 d^2}-\frac {3 f (e+f x)^2 \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x)^3 \sinh (c+d x)}{8 d}+\frac {(e+f x)^3 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^3 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^3 \cosh ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {3}{4} \left (-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )-\frac {f (e+f x) \cosh ^4(c+d x)}{8 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{8 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{4 d}}{b}-\frac {a \left (\frac {\int (e+f x)^3 \cosh ^3(c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 f^3 \cosh (c+d x)}{4 d^4}-\frac {f^3 \cosh (3 c+3 d x)}{216 d^4}-\frac {3 f^3 \cosh (5 c+5 d x)}{5000 d^4}-\frac {3 f^2 (e+f x) \sinh (c+d x)}{4 d^3}+\frac {f^2 (e+f x) \sinh (3 c+3 d x)}{72 d^3}+\frac {3 f^2 (e+f x) \sinh (5 c+5 d x)}{1000 d^3}+\frac {3 f (e+f x)^2 \cosh (c+d x)}{8 d^2}-\frac {f (e+f x)^2 \cosh (3 c+3 d x)}{48 d^2}-\frac {3 f (e+f x)^2 \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x)^3 \sinh (c+d x)}{8 d}+\frac {(e+f x)^3 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^3 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^3 \cosh ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {3}{4} \left (-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )-\frac {f (e+f x) \cosh ^4(c+d x)}{8 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{8 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{4 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int (e+f x)^3 \sin \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle \frac {\frac {3 f^3 \cosh (c+d x)}{4 d^4}-\frac {f^3 \cosh (3 c+3 d x)}{216 d^4}-\frac {3 f^3 \cosh (5 c+5 d x)}{5000 d^4}-\frac {3 f^2 (e+f x) \sinh (c+d x)}{4 d^3}+\frac {f^2 (e+f x) \sinh (3 c+3 d x)}{72 d^3}+\frac {3 f^2 (e+f x) \sinh (5 c+5 d x)}{1000 d^3}+\frac {3 f (e+f x)^2 \cosh (c+d x)}{8 d^2}-\frac {f (e+f x)^2 \cosh (3 c+3 d x)}{48 d^2}-\frac {3 f (e+f x)^2 \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x)^3 \sinh (c+d x)}{8 d}+\frac {(e+f x)^3 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^3 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^3 \cosh ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {3}{4} \left (-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )-\frac {f (e+f x) \cosh ^4(c+d x)}{8 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{8 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{4 d}}{b}-\frac {a \left (\frac {\frac {2 f^2 \int (e+f x) \cosh ^3(c+d x)dx}{3 d^2}+\frac {2}{3} \int (e+f x)^3 \cosh (c+d x)dx-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 f^3 \cosh (c+d x)}{4 d^4}-\frac {f^3 \cosh (3 c+3 d x)}{216 d^4}-\frac {3 f^3 \cosh (5 c+5 d x)}{5000 d^4}-\frac {3 f^2 (e+f x) \sinh (c+d x)}{4 d^3}+\frac {f^2 (e+f x) \sinh (3 c+3 d x)}{72 d^3}+\frac {3 f^2 (e+f x) \sinh (5 c+5 d x)}{1000 d^3}+\frac {3 f (e+f x)^2 \cosh (c+d x)}{8 d^2}-\frac {f (e+f x)^2 \cosh (3 c+3 d x)}{48 d^2}-\frac {3 f (e+f x)^2 \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x)^3 \sinh (c+d x)}{8 d}+\frac {(e+f x)^3 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^3 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^3 \cosh ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {3}{4} \left (-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )-\frac {f (e+f x) \cosh ^4(c+d x)}{8 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{8 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{4 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2 f^2 \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{3 d^2}+\frac {2}{3} \int (e+f x)^3 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {\frac {3 f^3 \cosh (c+d x)}{4 d^4}-\frac {f^3 \cosh (3 c+3 d x)}{216 d^4}-\frac {3 f^3 \cosh (5 c+5 d x)}{5000 d^4}-\frac {3 f^2 (e+f x) \sinh (c+d x)}{4 d^3}+\frac {f^2 (e+f x) \sinh (3 c+3 d x)}{72 d^3}+\frac {3 f^2 (e+f x) \sinh (5 c+5 d x)}{1000 d^3}+\frac {3 f (e+f x)^2 \cosh (c+d x)}{8 d^2}-\frac {f (e+f x)^2 \cosh (3 c+3 d x)}{48 d^2}-\frac {3 f (e+f x)^2 \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x)^3 \sinh (c+d x)}{8 d}+\frac {(e+f x)^3 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^3 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^3 \cosh ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {3}{4} \left (-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )-\frac {f (e+f x) \cosh ^4(c+d x)}{8 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{8 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{4 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2 f^2 \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{3 d^2}+\frac {2}{3} \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}-\frac {3 i f \int -i (e+f x)^2 \sinh (c+d x)dx}{d}\right )-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {3 f^3 \cosh (c+d x)}{4 d^4}-\frac {f^3 \cosh (3 c+3 d x)}{216 d^4}-\frac {3 f^3 \cosh (5 c+5 d x)}{5000 d^4}-\frac {3 f^2 (e+f x) \sinh (c+d x)}{4 d^3}+\frac {f^2 (e+f x) \sinh (3 c+3 d x)}{72 d^3}+\frac {3 f^2 (e+f x) \sinh (5 c+5 d x)}{1000 d^3}+\frac {3 f (e+f x)^2 \cosh (c+d x)}{8 d^2}-\frac {f (e+f x)^2 \cosh (3 c+3 d x)}{48 d^2}-\frac {3 f (e+f x)^2 \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x)^3 \sinh (c+d x)}{8 d}+\frac {(e+f x)^3 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^3 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^3 \cosh ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {3}{4} \left (-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )-\frac {f (e+f x) \cosh ^4(c+d x)}{8 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{8 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{4 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2 f^2 \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{3 d^2}+\frac {2}{3} \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}-\frac {3 f \int (e+f x)^2 \sinh (c+d x)dx}{d}\right )-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 f^3 \cosh (c+d x)}{4 d^4}-\frac {f^3 \cosh (3 c+3 d x)}{216 d^4}-\frac {3 f^3 \cosh (5 c+5 d x)}{5000 d^4}-\frac {3 f^2 (e+f x) \sinh (c+d x)}{4 d^3}+\frac {f^2 (e+f x) \sinh (3 c+3 d x)}{72 d^3}+\frac {3 f^2 (e+f x) \sinh (5 c+5 d x)}{1000 d^3}+\frac {3 f (e+f x)^2 \cosh (c+d x)}{8 d^2}-\frac {f (e+f x)^2 \cosh (3 c+3 d x)}{48 d^2}-\frac {3 f (e+f x)^2 \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x)^3 \sinh (c+d x)}{8 d}+\frac {(e+f x)^3 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^3 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^3 \cosh ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {3}{4} \left (-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )-\frac {f (e+f x) \cosh ^4(c+d x)}{8 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{8 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{4 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2 f^2 \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{3 d^2}+\frac {2}{3} \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}-\frac {3 f \int -i (e+f x)^2 \sin (i c+i d x)dx}{d}\right )-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {3 f^3 \cosh (c+d x)}{4 d^4}-\frac {f^3 \cosh (3 c+3 d x)}{216 d^4}-\frac {3 f^3 \cosh (5 c+5 d x)}{5000 d^4}-\frac {3 f^2 (e+f x) \sinh (c+d x)}{4 d^3}+\frac {f^2 (e+f x) \sinh (3 c+3 d x)}{72 d^3}+\frac {3 f^2 (e+f x) \sinh (5 c+5 d x)}{1000 d^3}+\frac {3 f (e+f x)^2 \cosh (c+d x)}{8 d^2}-\frac {f (e+f x)^2 \cosh (3 c+3 d x)}{48 d^2}-\frac {3 f (e+f x)^2 \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x)^3 \sinh (c+d x)}{8 d}+\frac {(e+f x)^3 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^3 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^3 \cosh ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {3}{4} \left (-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )-\frac {f (e+f x) \cosh ^4(c+d x)}{8 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{8 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{4 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2 f^2 \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{3 d^2}+\frac {2}{3} \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \int (e+f x)^2 \sin (i c+i d x)dx}{d}\right )-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {\frac {3 f^3 \cosh (c+d x)}{4 d^4}-\frac {f^3 \cosh (3 c+3 d x)}{216 d^4}-\frac {3 f^3 \cosh (5 c+5 d x)}{5000 d^4}-\frac {3 f^2 (e+f x) \sinh (c+d x)}{4 d^3}+\frac {f^2 (e+f x) \sinh (3 c+3 d x)}{72 d^3}+\frac {3 f^2 (e+f x) \sinh (5 c+5 d x)}{1000 d^3}+\frac {3 f (e+f x)^2 \cosh (c+d x)}{8 d^2}-\frac {f (e+f x)^2 \cosh (3 c+3 d x)}{48 d^2}-\frac {3 f (e+f x)^2 \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x)^3 \sinh (c+d x)}{8 d}+\frac {(e+f x)^3 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^3 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^3 \cosh ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {3}{4} \left (-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )-\frac {f (e+f x) \cosh ^4(c+d x)}{8 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{8 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{4 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2 f^2 \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{3 d^2}+\frac {2}{3} \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \int (e+f x) \cosh (c+d x)dx}{d}\right )}{d}\right )-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 f^3 \cosh (c+d x)}{4 d^4}-\frac {f^3 \cosh (3 c+3 d x)}{216 d^4}-\frac {3 f^3 \cosh (5 c+5 d x)}{5000 d^4}-\frac {3 f^2 (e+f x) \sinh (c+d x)}{4 d^3}+\frac {f^2 (e+f x) \sinh (3 c+3 d x)}{72 d^3}+\frac {3 f^2 (e+f x) \sinh (5 c+5 d x)}{1000 d^3}+\frac {3 f (e+f x)^2 \cosh (c+d x)}{8 d^2}-\frac {f (e+f x)^2 \cosh (3 c+3 d x)}{48 d^2}-\frac {3 f (e+f x)^2 \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x)^3 \sinh (c+d x)}{8 d}+\frac {(e+f x)^3 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^3 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^3 \cosh ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {3}{4} \left (-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )-\frac {f (e+f x) \cosh ^4(c+d x)}{8 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{8 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{4 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2 f^2 \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{3 d^2}+\frac {2}{3} \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{d}\right )-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {\frac {3 f^3 \cosh (c+d x)}{4 d^4}-\frac {f^3 \cosh (3 c+3 d x)}{216 d^4}-\frac {3 f^3 \cosh (5 c+5 d x)}{5000 d^4}-\frac {3 f^2 (e+f x) \sinh (c+d x)}{4 d^3}+\frac {f^2 (e+f x) \sinh (3 c+3 d x)}{72 d^3}+\frac {3 f^2 (e+f x) \sinh (5 c+5 d x)}{1000 d^3}+\frac {3 f (e+f x)^2 \cosh (c+d x)}{8 d^2}-\frac {f (e+f x)^2 \cosh (3 c+3 d x)}{48 d^2}-\frac {3 f (e+f x)^2 \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x)^3 \sinh (c+d x)}{8 d}+\frac {(e+f x)^3 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^3 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^3 \cosh ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {3}{4} \left (-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )-\frac {f (e+f x) \cosh ^4(c+d x)}{8 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{8 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{4 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2 f^2 \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{3 d^2}+\frac {2}{3} \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {i f \int -i \sinh (c+d x)dx}{d}\right )}{d}\right )}{d}\right )-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {3 f^3 \cosh (c+d x)}{4 d^4}-\frac {f^3 \cosh (3 c+3 d x)}{216 d^4}-\frac {3 f^3 \cosh (5 c+5 d x)}{5000 d^4}-\frac {3 f^2 (e+f x) \sinh (c+d x)}{4 d^3}+\frac {f^2 (e+f x) \sinh (3 c+3 d x)}{72 d^3}+\frac {3 f^2 (e+f x) \sinh (5 c+5 d x)}{1000 d^3}+\frac {3 f (e+f x)^2 \cosh (c+d x)}{8 d^2}-\frac {f (e+f x)^2 \cosh (3 c+3 d x)}{48 d^2}-\frac {3 f (e+f x)^2 \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x)^3 \sinh (c+d x)}{8 d}+\frac {(e+f x)^3 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^3 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^3 \cosh ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {3}{4} \left (-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )-\frac {f (e+f x) \cosh ^4(c+d x)}{8 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{8 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{4 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2 f^2 \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{3 d^2}+\frac {2}{3} \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int \sinh (c+d x)dx}{d}\right )}{d}\right )}{d}\right )-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 f^3 \cosh (c+d x)}{4 d^4}-\frac {f^3 \cosh (3 c+3 d x)}{216 d^4}-\frac {3 f^3 \cosh (5 c+5 d x)}{5000 d^4}-\frac {3 f^2 (e+f x) \sinh (c+d x)}{4 d^3}+\frac {f^2 (e+f x) \sinh (3 c+3 d x)}{72 d^3}+\frac {3 f^2 (e+f x) \sinh (5 c+5 d x)}{1000 d^3}+\frac {3 f (e+f x)^2 \cosh (c+d x)}{8 d^2}-\frac {f (e+f x)^2 \cosh (3 c+3 d x)}{48 d^2}-\frac {3 f (e+f x)^2 \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x)^3 \sinh (c+d x)}{8 d}+\frac {(e+f x)^3 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^3 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^3 \cosh ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {3}{4} \left (-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )-\frac {f (e+f x) \cosh ^4(c+d x)}{8 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{8 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{4 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2 f^2 \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{3 d^2}+\frac {2}{3} \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int -i \sin (i c+i d x)dx}{d}\right )}{d}\right )}{d}\right )-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\right )}{b}\) |
3.5.1.3.1 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[((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_)]^(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[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & & IGtQ[p, 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]
\[\int \frac {\left (f x +e \right )^{3} \cosh \left (d x +c \right )^{3} \sinh \left (d x +c \right )^{3}}{a +b \sinh \left (d x +c \right )}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 18801 vs. \(2 (1347) = 2694\).
Time = 0.53 (sec) , antiderivative size = 18801, normalized size of antiderivative = 13.03 \[ \int \frac {(e+f x)^3 \cosh ^3(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {(e+f x)^3 \cosh ^3(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {(e+f x)^3 \cosh ^3(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a} \,d x } \]
-1/960*e^3*((15*a*b^3*e^(-d*x - c) - 6*b^4 - 10*(4*a^2*b^2 + b^4)*e^(-2*d* x - 2*c) + 60*(2*a^3*b + a*b^3)*e^(-3*d*x - 3*c) - 60*(8*a^4 + 6*a^2*b^2 - b^4)*e^(-4*d*x - 4*c))*e^(5*d*x + 5*c)/(b^5*d) + 960*(a^5 + a^3*b^2)*(d*x + c)/(b^6*d) + (15*a*b^3*e^(-4*d*x - 4*c) + 6*b^4*e^(-5*d*x - 5*c) + 60*( 8*a^4 + 6*a^2*b^2 - b^4)*e^(-d*x - c) + 60*(2*a^3*b + a*b^3)*e^(-2*d*x - 2 *c) + 10*(4*a^2*b^2 + b^4)*e^(-3*d*x - 3*c))/(b^5*d) + 960*(a^5 + a^3*b^2) *log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(b^6*d)) - 1/34560000*(86 40000*(a^5*d^4*f^3*e^(5*c) + a^3*b^2*d^4*f^3*e^(5*c))*x^4 + 34560000*(a^5* d^4*e*f^2*e^(5*c) + a^3*b^2*d^4*e*f^2*e^(5*c))*x^3 + 51840000*(a^5*d^4*e^2 *f*e^(5*c) + a^3*b^2*d^4*e^2*f*e^(5*c))*x^2 - 1728*(125*b^5*d^3*f^3*x^3*e^ (10*c) + 75*(5*d^3*e*f^2 - d^2*f^3)*b^5*x^2*e^(10*c) + 15*(25*d^3*e^2*f - 10*d^2*e*f^2 + 2*d*f^3)*b^5*x*e^(10*c) - 3*(25*d^2*e^2*f - 10*d*e*f^2 + 2* f^3)*b^5*e^(10*c))*e^(5*d*x) + 16875*(32*a*b^4*d^3*f^3*x^3*e^(9*c) + 24*(4 *d^3*e*f^2 - d^2*f^3)*a*b^4*x^2*e^(9*c) + 12*(8*d^3*e^2*f - 4*d^2*e*f^2 + d*f^3)*a*b^4*x*e^(9*c) - 3*(8*d^2*e^2*f - 4*d*e*f^2 + f^3)*a*b^4*e^(9*c))* e^(4*d*x) + 40000*(4*(9*d^2*e^2*f - 6*d*e*f^2 + 2*f^3)*a^2*b^3*e^(8*c) + ( 9*d^2*e^2*f - 6*d*e*f^2 + 2*f^3)*b^5*e^(8*c) - 9*(4*a^2*b^3*d^3*f^3*e^(8*c ) + b^5*d^3*f^3*e^(8*c))*x^3 - 9*(4*(3*d^3*e*f^2 - d^2*f^3)*a^2*b^3*e^(8*c ) + (3*d^3*e*f^2 - d^2*f^3)*b^5*e^(8*c))*x^2 - 3*(4*(9*d^3*e^2*f - 6*d^2*e *f^2 + 2*d*f^3)*a^2*b^3*e^(8*c) + (9*d^3*e^2*f - 6*d^2*e*f^2 + 2*d*f^3)...
\[ \int \frac {(e+f x)^3 \cosh ^3(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int \frac {(e+f x)^3 \cosh ^3(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^3\,{\mathrm {sinh}\left (c+d\,x\right )}^3\,{\left (e+f\,x\right )}^3}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]