Integrand size = 34, antiderivative size = 641 \[ \int \frac {(e+f x) \cosh ^3(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a^3 f x}{4 b^4 d}+\frac {3 a f x}{32 b^2 d}+\frac {a^3 \left (a^2+b^2\right ) (e+f x)^2}{2 b^6 f}-\frac {a^4 f \cosh (c+d x)}{b^5 d^2}-\frac {2 a^2 f \cosh (c+d x)}{3 b^3 d^2}+\frac {f \cosh (c+d x)}{8 b d^2}-\frac {a^2 f \cosh ^3(c+d x)}{9 b^3 d^2}-\frac {a (e+f x) \cosh ^4(c+d x)}{4 b^2 d}-\frac {f \cosh (3 c+3 d x)}{144 b d^2}-\frac {f \cosh (5 c+5 d x)}{400 b d^2}-\frac {a^3 \left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^6 d}-\frac {a^3 \left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^6 d}-\frac {a^3 \left (a^2+b^2\right ) f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^6 d^2}-\frac {a^3 \left (a^2+b^2\right ) f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^6 d^2}+\frac {a^4 (e+f x) \sinh (c+d x)}{b^5 d}+\frac {2 a^2 (e+f x) \sinh (c+d x)}{3 b^3 d}-\frac {(e+f x) \sinh (c+d x)}{8 b d}+\frac {a^3 f \cosh (c+d x) \sinh (c+d x)}{4 b^4 d^2}+\frac {3 a f \cosh (c+d x) \sinh (c+d x)}{32 b^2 d^2}+\frac {a^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{3 b^3 d}+\frac {a f \cosh ^3(c+d x) \sinh (c+d x)}{16 b^2 d^2}-\frac {a^3 (e+f x) \sinh ^2(c+d x)}{2 b^4 d}+\frac {(e+f x) \sinh (3 c+3 d x)}{48 b d}+\frac {(e+f x) \sinh (5 c+5 d x)}{80 b d} \] Output:
-1/4*a^3*f*x/b^4/d+3/32*a*f*x/b^2/d+1/2*a^3*(a^2+b^2)*(f*x+e)^2/b^6/f-a^4* f*cosh(d*x+c)/b^5/d^2-2/3*a^2*f*cosh(d*x+c)/b^3/d^2+1/8*f*cosh(d*x+c)/b/d^ 2-1/9*a^2*f*cosh(d*x+c)^3/b^3/d^2-1/4*a*(f*x+e)*cosh(d*x+c)^4/b^2/d-1/144* f*cosh(3*d*x+3*c)/b/d^2-1/400*f*cosh(5*d*x+5*c)/b/d^2-a^3*(a^2+b^2)*(f*x+e )*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^6/d-a^3*(a^2+b^2)*(f*x+e)*ln(1+ b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^6/d-a^3*(a^2+b^2)*f*polylog(2,-b*exp(d *x+c)/(a-(a^2+b^2)^(1/2)))/b^6/d^2-a^3*(a^2+b^2)*f*polylog(2,-b*exp(d*x+c) /(a+(a^2+b^2)^(1/2)))/b^6/d^2+a^4*(f*x+e)*sinh(d*x+c)/b^5/d+2/3*a^2*(f*x+e )*sinh(d*x+c)/b^3/d-1/8*(f*x+e)*sinh(d*x+c)/b/d+1/4*a^3*f*cosh(d*x+c)*sinh (d*x+c)/b^4/d^2+3/32*a*f*cosh(d*x+c)*sinh(d*x+c)/b^2/d^2+1/3*a^2*(f*x+e)*c osh(d*x+c)^2*sinh(d*x+c)/b^3/d+1/16*a*f*cosh(d*x+c)^3*sinh(d*x+c)/b^2/d^2- 1/2*a^3*(f*x+e)*sinh(d*x+c)^2/b^4/d+1/48*(f*x+e)*sinh(3*d*x+3*c)/b/d+1/80* (f*x+e)*sinh(5*d*x+5*c)/b/d
Leaf count is larger than twice the leaf count of optimal. \(2704\) vs. \(2(641)=1282\).
Time = 8.32 (sec) , antiderivative size = 2704, normalized size of antiderivative = 4.22 \[ \int \frac {(e+f x) \cosh ^3(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show} \] Input:
Integrate[((e + f*x)*Cosh[c + d*x]^3*Sinh[c + d*x]^3)/(a + b*Sinh[c + d*x] ),x]
Output:
((-4*a^3*(a^2 + b^2)*(-2*d*e*(c + d*x) + 2*c*f*(c + d*x) - f*(c + d*x)^2 + (4*a*Sqrt[a^2 + b^2]*d*e*ArcTan[(a + b*E^(c + d*x))/Sqrt[-a^2 - b^2]])/Sq rt[-(a^2 + b^2)^2] - (4*a*Sqrt[-(a^2 + b^2)^2]*d*e*ArcTanh[(a + b*E^(c + d *x))/Sqrt[a^2 + b^2]])/(-a^2 - b^2)^(3/2) + 2*f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + 2*f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] - 2*c*f*Log[b - 2*a*E^(c + d*x) - b*E^(2*(c + d*x))] + 2*d*e*Log[2*a*E^(c + d*x) + b*(-1 + E^(2*(c + d*x)))] + 2*f*PolyLog[2, (b *E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + 2*f*PolyLog[2, -((b*E^(c + d*x))/( a + Sqrt[a^2 + b^2]))]))/(b^6*d^2) + ((Cosh[5*(c + d*x)]/(7200*b^5*d) - Si nh[5*(c + d*x)]/(7200*b^5*d))*(-360*b^4*d*e - 72*b^4*f + 360*b^4*c*f - 360 *b^4*f*(c + d*x) - 900*a*b^3*d*e*Cosh[c + d*x] - 225*a*b^3*f*Cosh[c + d*x] + 900*a*b^3*c*f*Cosh[c + d*x] - 900*a*b^3*f*(c + d*x)*Cosh[c + d*x] - 240 0*a^2*b^2*d*e*Cosh[2*(c + d*x)] - 600*b^4*d*e*Cosh[2*(c + d*x)] - 800*a^2* b^2*f*Cosh[2*(c + d*x)] - 200*b^4*f*Cosh[2*(c + d*x)] + 2400*a^2*b^2*c*f*C osh[2*(c + d*x)] + 600*b^4*c*f*Cosh[2*(c + d*x)] - 2400*a^2*b^2*f*(c + d*x )*Cosh[2*(c + d*x)] - 600*b^4*f*(c + d*x)*Cosh[2*(c + d*x)] - 7200*a^3*b*d *e*Cosh[3*(c + d*x)] - 3600*a*b^3*d*e*Cosh[3*(c + d*x)] - 3600*a^3*b*f*Cos h[3*(c + d*x)] - 1800*a*b^3*f*Cosh[3*(c + d*x)] + 7200*a^3*b*c*f*Cosh[3*(c + d*x)] + 3600*a*b^3*c*f*Cosh[3*(c + d*x)] - 7200*a^3*b*f*(c + d*x)*Cosh[ 3*(c + d*x)] - 3600*a*b^3*f*(c + d*x)*Cosh[3*(c + d*x)] - 28800*a^4*d*e...
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.
| \(\displaystyle \int \frac {(e+f x) \sinh ^3(c+d x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6113 |
| \(\displaystyle \frac {\int (e+f x) \cosh ^3(c+d x) \sinh ^2(c+d x)dx}{b}-\frac {a \int \frac {(e+f x) \cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle \frac {\int \left (-\frac {1}{8} (e+f x) \cosh (c+d x)+\frac {1}{16} (e+f x) \cosh (3 c+3 d x)+\frac {1}{16} (e+f x) \cosh (5 c+5 d x)\right )dx}{b}-\frac {a \int \frac {(e+f x) \cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {f \cosh (c+d x)}{8 d^2}-\frac {f \cosh (3 c+3 d x)}{144 d^2}-\frac {f \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x) \sinh (c+d x)}{8 d}+\frac {(e+f x) \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x) \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \int \frac {(e+f x) \cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 6113 |
| \(\displaystyle \frac {\frac {f \cosh (c+d x)}{8 d^2}-\frac {f \cosh (3 c+3 d x)}{144 d^2}-\frac {f \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x) \sinh (c+d x)}{8 d}+\frac {(e+f x) \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x) \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\int (e+f x) \cosh ^3(c+d x) \sinh (c+d x)dx}{b}-\frac {a \int \frac {(e+f x) \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 5970 |
| \(\displaystyle \frac {\frac {f \cosh (c+d x)}{8 d^2}-\frac {f \cosh (3 c+3 d x)}{144 d^2}-\frac {f \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x) \sinh (c+d x)}{8 d}+\frac {(e+f x) \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x) \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \int \cosh ^4(c+d x)dx}{4 d}}{b}-\frac {a \int \frac {(e+f x) \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {f \cosh (c+d x)}{8 d^2}-\frac {f \cosh (3 c+3 d x)}{144 d^2}-\frac {f \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x) \sinh (c+d x)}{8 d}+\frac {(e+f x) \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x) \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \int \sin \left (i c+i d x+\frac {\pi }{2}\right )^4dx}{4 d}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {f \cosh (c+d x)}{8 d^2}-\frac {f \cosh (3 c+3 d x)}{144 d^2}-\frac {f \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x) \sinh (c+d x)}{8 d}+\frac {(e+f x) \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x) \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \int \cosh ^2(c+d x)dx+\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{4 d}}{b}-\frac {a \int \frac {(e+f x) \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {f \cosh (c+d x)}{8 d^2}-\frac {f \cosh (3 c+3 d x)}{144 d^2}-\frac {f \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x) \sinh (c+d x)}{8 d}+\frac {(e+f x) \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x) \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \int \sin \left (i c+i d x+\frac {\pi }{2}\right )^2dx\right )}{4 d}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {f \cosh (c+d x)}{8 d^2}-\frac {f \cosh (3 c+3 d x)}{144 d^2}-\frac {f \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x) \sinh (c+d x)}{8 d}+\frac {(e+f x) \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x) \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )+\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{4 d}}{b}-\frac {a \int \frac {(e+f x) \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {f \cosh (c+d x)}{8 d^2}-\frac {f \cosh (3 c+3 d x)}{144 d^2}-\frac {f \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x) \sinh (c+d x)}{8 d}+\frac {(e+f x) \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x) \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{b}-\frac {a \int \frac {(e+f x) \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 6113 |
| \(\displaystyle \frac {\frac {f \cosh (c+d x)}{8 d^2}-\frac {f \cosh (3 c+3 d x)}{144 d^2}-\frac {f \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x) \sinh (c+d x)}{8 d}+\frac {(e+f x) \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x) \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{b}-\frac {a \left (\frac {\int (e+f x) \cosh ^3(c+d x)dx}{b}-\frac {a \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {f \cosh (c+d x)}{8 d^2}-\frac {f \cosh (3 c+3 d x)}{144 d^2}-\frac {f \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x) \sinh (c+d x)}{8 d}+\frac {(e+f x) \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x) \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3791 |
| \(\displaystyle \frac {\frac {f \cosh (c+d x)}{8 d^2}-\frac {f \cosh (3 c+3 d x)}{144 d^2}-\frac {f \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x) \sinh (c+d x)}{8 d}+\frac {(e+f x) \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x) \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{b}-\frac {a \left (\frac {\frac {2}{3} \int (e+f x) \cosh (c+d x)dx-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {f \cosh (c+d x)}{8 d^2}-\frac {f \cosh (3 c+3 d x)}{144 d^2}-\frac {f \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x) \sinh (c+d x)}{8 d}+\frac {(e+f x) \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x) \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2}{3} \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )dx-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {\frac {f \cosh (c+d x)}{8 d^2}-\frac {f \cosh (3 c+3 d x)}{144 d^2}-\frac {f \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x) \sinh (c+d x)}{8 d}+\frac {(e+f x) \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x) \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {i f \int -i \sinh (c+d x)dx}{d}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {f \cosh (c+d x)}{8 d^2}-\frac {f \cosh (3 c+3 d x)}{144 d^2}-\frac {f \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x) \sinh (c+d x)}{8 d}+\frac {(e+f x) \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x) \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{b}-\frac {a \left (\frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int \sinh (c+d x)dx}{d}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {f \cosh (c+d x)}{8 d^2}-\frac {f \cosh (3 c+3 d x)}{144 d^2}-\frac {f \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x) \sinh (c+d x)}{8 d}+\frac {(e+f x) \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x) \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int -i \sin (i c+i d x)dx}{d}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {f \cosh (c+d x)}{8 d^2}-\frac {f \cosh (3 c+3 d x)}{144 d^2}-\frac {f \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x) \sinh (c+d x)}{8 d}+\frac {(e+f x) \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x) \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}+\frac {i f \int \sin (i c+i d x)dx}{d}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle \frac {\frac {f \cosh (c+d x)}{8 d^2}-\frac {f \cosh (3 c+3 d x)}{144 d^2}-\frac {f \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x) \sinh (c+d x)}{8 d}+\frac {(e+f x) \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x) \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{b}-\frac {a \left (\frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 6099 |
| \(\displaystyle \frac {\frac {f \cosh (c+d x)}{8 d^2}-\frac {f \cosh (3 c+3 d x)}{144 d^2}-\frac {f \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x) \sinh (c+d x)}{8 d}+\frac {(e+f x) \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x) \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{b}-\frac {a \left (\frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \int (e+f x) \cosh (c+d x)dx}{b^2}+\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {f \cosh (c+d x)}{8 d^2}-\frac {f \cosh (3 c+3 d x)}{144 d^2}-\frac {f \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x) \sinh (c+d x)}{8 d}+\frac {(e+f x) \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x) \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{b}-\frac {a \left (\frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {\frac {f \cosh (c+d x)}{8 d^2}-\frac {f \cosh (3 c+3 d x)}{144 d^2}-\frac {f \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x) \sinh (c+d x)}{8 d}+\frac {(e+f x) \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x) \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{b}-\frac {a \left (\frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {i f \int -i \sinh (c+d x)dx}{d}\right )}{b^2}+\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {f \cosh (c+d x)}{8 d^2}-\frac {f \cosh (3 c+3 d x)}{144 d^2}-\frac {f \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x) \sinh (c+d x)}{8 d}+\frac {(e+f x) \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x) \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{b}-\frac {a \left (\frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int \sinh (c+d x)dx}{d}\right )}{b^2}+\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {f \cosh (c+d x)}{8 d^2}-\frac {f \cosh (3 c+3 d x)}{144 d^2}-\frac {f \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x) \sinh (c+d x)}{8 d}+\frac {(e+f x) \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x) \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{b}-\frac {a \left (\frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int -i \sin (i c+i d x)dx}{d}\right )}{b^2}+\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {f \cosh (c+d x)}{8 d^2}-\frac {f \cosh (3 c+3 d x)}{144 d^2}-\frac {f \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x) \sinh (c+d x)}{8 d}+\frac {(e+f x) \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x) \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{b}-\frac {a \left (\frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \left (\frac {(e+f x) \sinh (c+d x)}{d}+\frac {i f \int \sin (i c+i d x)dx}{d}\right )}{b^2}+\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle \frac {\frac {f \cosh (c+d x)}{8 d^2}-\frac {f \cosh (3 c+3 d x)}{144 d^2}-\frac {f \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x) \sinh (c+d x)}{8 d}+\frac {(e+f x) \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x) \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{b}-\frac {a \left (\frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}+\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx}{b}-\frac {a \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{b^2}\right )}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 5969 |
| \(\displaystyle \frac {\frac {f \cosh (c+d x)}{8 d^2}-\frac {f \cosh (3 c+3 d x)}{144 d^2}-\frac {f \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x) \sinh (c+d x)}{8 d}+\frac {(e+f x) \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x) \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{b}-\frac {a \left (\frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}+\frac {\frac {(e+f x) \sinh ^2(c+d x)}{2 d}-\frac {f \int \sinh ^2(c+d x)dx}{2 d}}{b}-\frac {a \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{b^2}\right )}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {f \cosh (c+d x)}{8 d^2}-\frac {f \cosh (3 c+3 d x)}{144 d^2}-\frac {f \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x) \sinh (c+d x)}{8 d}+\frac {(e+f x) \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x) \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{b}-\frac {a \left (\frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}+\frac {\frac {(e+f x) \sinh ^2(c+d x)}{2 d}-\frac {f \int -\sin (i c+i d x)^2dx}{2 d}}{b}-\frac {a \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{b^2}\right )}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {f \cosh (c+d x)}{8 d^2}-\frac {f \cosh (3 c+3 d x)}{144 d^2}-\frac {f \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x) \sinh (c+d x)}{8 d}+\frac {(e+f x) \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x) \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{b}-\frac {a \left (\frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}+\frac {\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \int \sin (i c+i d x)^2dx}{2 d}}{b}-\frac {a \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{b^2}\right )}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {f \cosh (c+d x)}{8 d^2}-\frac {f \cosh (3 c+3 d x)}{144 d^2}-\frac {f \cosh (5 c+5 d x)}{400 d^2}-\frac {(e+f x) \sinh (c+d x)}{8 d}+\frac {(e+f x) \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x) \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{b}-\frac {a \left (\frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}+\frac {\frac {f \left (\frac {\int 1dx}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}+\frac {(e+f x) \sinh ^2(c+d x)}{2 d}}{b}-\frac {a \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{b^2}\right )}{b}\right )}{b}\right )}{b}\) |
Input:
Int[((e + f*x)*Cosh[c + d*x]^3*Sinh[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(1362\) vs. \(2(593)=1186\).
Time = 284.30 (sec) , antiderivative size = 1363, normalized size of antiderivative = 2.13
Input:
int((f*x+e)*cosh(d*x+c)^3*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x,method=_RETURN VERBOSE)
Output:
2/d*a^5/b^6*f*c*x-1/32*a*(2*a^2+b^2)*(2*d*f*x+2*d*e+f)/b^4/d^2*exp(-2*d*x- 2*c)-1/16*(8*a^4+6*a^2*b^2-b^4)*(d*f*x+d*e+f)/b^5/d^2*exp(-d*x-c)-1/288*(4 *a^2+b^2)*(3*d*f*x+3*d*e+f)/b^3/d^2*exp(-3*d*x-3*c)-1/256*a*(4*d*f*x+4*d*e +f)/b^2/d^2*exp(-4*d*x-4*c)+1/d^2*a^5/b^6*f*c^2-1/d^2*a^5/b^6*f*dilog((-b* exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))-1/d^2*a^5/b^6*f*dilog( (b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))+2/d*a^5/b^6*e*ln(exp (d*x+c))-1/d*a^5/b^6*e*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)+1/d^2*a^3/b^4 *f*c^2+2/d*a^3/b^4*e*ln(exp(d*x+c))-1/d^2*a^3/b^4*f*dilog((-b*exp(d*x+c)+( a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))-1/d^2*a^3/b^4*f*dilog((b*exp(d*x+c )+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))-1/d*a^3/b^4*e*ln(b*exp(2*d*x+2*c )+2*a*exp(d*x+c)-b)+1/2*a^3/b^4*f*x^2-a^3/b^4*e*x+1/2*a^5/b^6*f*x^2-a^5/b^ 6*e*x+2/d*a^3/b^4*f*c*x-1/d*a^3/b^4*f*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a) /(-a+(a^2+b^2)^(1/2)))*x-1/d*a^3/b^4*f*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a) /(a+(a^2+b^2)^(1/2)))*x-1/d^2*a^3/b^4*f*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)- a)/(-a+(a^2+b^2)^(1/2)))*c-1/d^2*a^3/b^4*f*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2 )+a)/(a+(a^2+b^2)^(1/2)))*c+1/d^2*a^3/b^4*c*f*ln(b*exp(2*d*x+2*c)+2*a*exp( d*x+c)-b)-2/d^2*a^3/b^4*c*f*ln(exp(d*x+c))+1/288*(12*a^2*d*f*x+3*b^2*d*f*x +12*a^2*d*e+3*b^2*d*e-4*a^2*f-b^2*f)/b^3/d^2*exp(3*d*x+3*c)-1/800*(5*d*f*x +5*d*e+f)/b/d^2*exp(-5*d*x-5*c)+1/800*(5*d*f*x+5*d*e-f)/b/d^2*exp(5*d*x+5* c)+1/16*(8*a^4*d*f*x+6*a^2*b^2*d*f*x-b^4*d*f*x+8*a^4*d*e+6*a^2*b^2*d*e-...
Leaf count of result is larger than twice the leaf count of optimal. 5548 vs. \(2 (591) = 1182\).
Time = 0.17 (sec) , antiderivative size = 5548, normalized size of antiderivative = 8.66 \[ \int \frac {(e+f x) \cosh ^3(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)*cosh(d*x+c)^3*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorit hm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {(e+f x) \cosh ^3(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)*cosh(d*x+c)**3*sinh(d*x+c)**3/(a+b*sinh(d*x+c)),x)
Output:
Timed out
\[ \int \frac {(e+f x) \cosh ^3(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)*cosh(d*x+c)^3*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorit hm="maxima")
Output:
-1/960*e*((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)*l og(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(b^6*d)) - 1/57600*f*((2880 0*(a^5*d^2*e^(5*c) + a^3*b^2*d^2*e^(5*c))*x^2 - 72*(5*b^5*d*x*e^(10*c) - b ^5*e^(10*c))*e^(5*d*x) + 225*(4*a*b^4*d*x*e^(9*c) - a*b^4*e^(9*c))*e^(4*d* x) + 200*(4*a^2*b^3*e^(8*c) + b^5*e^(8*c) - 3*(4*a^2*b^3*d*e^(8*c) + b^5*d *e^(8*c))*x)*e^(3*d*x) - 1800*(2*a^3*b^2*e^(7*c) + a*b^4*e^(7*c) - 2*(2*a^ 3*b^2*d*e^(7*c) + a*b^4*d*e^(7*c))*x)*e^(2*d*x) + 3600*(8*a^4*b*e^(6*c) + 6*a^2*b^3*e^(6*c) - b^5*e^(6*c) - (8*a^4*b*d*e^(6*c) + 6*a^2*b^3*d*e^(6*c) - b^5*d*e^(6*c))*x)*e^(d*x) + 3600*(8*a^4*b*e^(4*c) + 6*a^2*b^3*e^(4*c) - b^5*e^(4*c) + (8*a^4*b*d*e^(4*c) + 6*a^2*b^3*d*e^(4*c) - b^5*d*e^(4*c))*x )*e^(-d*x) + 1800*(2*a^3*b^2*e^(3*c) + a*b^4*e^(3*c) + 2*(2*a^3*b^2*d*e^(3 *c) + a*b^4*d*e^(3*c))*x)*e^(-2*d*x) + 200*(4*a^2*b^3*e^(2*c) + b^5*e^(2*c ) + 3*(4*a^2*b^3*d*e^(2*c) + b^5*d*e^(2*c))*x)*e^(-3*d*x) + 225*(4*a*b^4*d *x*e^c + a*b^4*e^c)*e^(-4*d*x) + 72*(5*b^5*d*x + b^5)*e^(-5*d*x))*e^(-5*c) /(b^6*d^2) - 900*integrate(128*((a^6*e^c + a^4*b^2*e^c)*x*e^(d*x) - (a^...
\[ \int \frac {(e+f x) \cosh ^3(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)*cosh(d*x+c)^3*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorit hm="giac")
Output:
integrate((f*x + e)*cosh(d*x + c)^3*sinh(d*x + c)^3/(b*sinh(d*x + c) + a), x)
Timed out. \[ \int \frac {(e+f x) \cosh ^3(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^3\,{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (e+f\,x\right )}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \] Input:
int((cosh(c + d*x)^3*sinh(c + d*x)^3*(e + f*x))/(a + b*sinh(c + d*x)),x)
Output:
int((cosh(c + d*x)^3*sinh(c + d*x)^3*(e + f*x))/(a + b*sinh(c + d*x)), x)
\[ \int \frac {(e+f x) \cosh ^3(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {too large to display} \] Input:
int((f*x+e)*cosh(d*x+c)^3*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x)
Output:
(360*e**(10*c + 10*d*x)*b**10*d*e + 360*e**(10*c + 10*d*x)*b**10*d*f*x - 7 2*e**(10*c + 10*d*x)*b**10*f - 900*e**(9*c + 9*d*x)*a*b**9*d*e - 900*e**(9 *c + 9*d*x)*a*b**9*d*f*x + 225*e**(9*c + 9*d*x)*a*b**9*f + 2400*e**(8*c + 8*d*x)*a**2*b**8*d*e + 2400*e**(8*c + 8*d*x)*a**2*b**8*d*f*x - 800*e**(8*c + 8*d*x)*a**2*b**8*f + 600*e**(8*c + 8*d*x)*b**10*d*e + 600*e**(8*c + 8*d *x)*b**10*d*f*x - 200*e**(8*c + 8*d*x)*b**10*f - 7200*e**(7*c + 7*d*x)*a** 3*b**7*d*e - 7200*e**(7*c + 7*d*x)*a**3*b**7*d*f*x + 3600*e**(7*c + 7*d*x) *a**3*b**7*f - 3600*e**(7*c + 7*d*x)*a*b**9*d*e - 3600*e**(7*c + 7*d*x)*a* b**9*d*f*x + 1800*e**(7*c + 7*d*x)*a*b**9*f + 28800*e**(6*c + 6*d*x)*a**4* b**6*d*e + 28800*e**(6*c + 6*d*x)*a**4*b**6*d*f*x - 28800*e**(6*c + 6*d*x) *a**4*b**6*f + 21600*e**(6*c + 6*d*x)*a**2*b**8*d*e + 21600*e**(6*c + 6*d* x)*a**2*b**8*d*f*x - 21600*e**(6*c + 6*d*x)*a**2*b**8*f - 3600*e**(6*c + 6 *d*x)*b**10*d*e - 3600*e**(6*c + 6*d*x)*b**10*d*f*x + 3600*e**(6*c + 6*d*x )*b**10*f + 1843200*e**(5*c + 5*d*x)*int(x/(e**(7*c + 7*d*x)*b + 2*e**(6*c + 6*d*x)*a - e**(5*c + 5*d*x)*b),x)*a**10*b*d**2*f + 4147200*e**(5*c + 5* d*x)*int(x/(e**(7*c + 7*d*x)*b + 2*e**(6*c + 6*d*x)*a - e**(5*c + 5*d*x)*b ),x)*a**8*b**3*d**2*f + 2880000*e**(5*c + 5*d*x)*int(x/(e**(7*c + 7*d*x)*b + 2*e**(6*c + 6*d*x)*a - e**(5*c + 5*d*x)*b),x)*a**6*b**5*d**2*f + 576000 *e**(5*c + 5*d*x)*int(x/(e**(7*c + 7*d*x)*b + 2*e**(6*c + 6*d*x)*a - e**(5 *c + 5*d*x)*b),x)*a**4*b**7*d**2*f - 57600*e**(5*c + 5*d*x)*log(e**(2*c...