Integrand size = 34, antiderivative size = 499 \[ \int \frac {(e+f x) \cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {a^2 f x}{4 b^3 d}-\frac {3 f x}{32 b d}-\frac {a^2 \left (a^2+b^2\right ) (e+f x)^2}{2 b^5 f}+\frac {a^3 f \cosh (c+d x)}{b^4 d^2}+\frac {2 a f \cosh (c+d x)}{3 b^2 d^2}+\frac {a f \cosh ^3(c+d x)}{9 b^2 d^2}+\frac {(e+f x) \cosh ^4(c+d x)}{4 b d}+\frac {a^2 \left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d}+\frac {a^2 \left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d}+\frac {a^2 \left (a^2+b^2\right ) f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d^2}+\frac {a^2 \left (a^2+b^2\right ) f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d^2}-\frac {a^3 (e+f x) \sinh (c+d x)}{b^4 d}-\frac {2 a (e+f x) \sinh (c+d x)}{3 b^2 d}-\frac {a^2 f \cosh (c+d x) \sinh (c+d x)}{4 b^3 d^2}-\frac {3 f \cosh (c+d x) \sinh (c+d x)}{32 b d^2}-\frac {a (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{3 b^2 d}-\frac {f \cosh ^3(c+d x) \sinh (c+d x)}{16 b d^2}+\frac {a^2 (e+f x) \sinh ^2(c+d x)}{2 b^3 d} \] Output:
1/4*a^2*f*x/b^3/d-3/32*f*x/b/d-1/2*a^2*(a^2+b^2)*(f*x+e)^2/b^5/f+a^3*f*cos h(d*x+c)/b^4/d^2+2/3*a*f*cosh(d*x+c)/b^2/d^2+1/9*a*f*cosh(d*x+c)^3/b^2/d^2 +1/4*(f*x+e)*cosh(d*x+c)^4/b/d+a^2*(a^2+b^2)*(f*x+e)*ln(1+b*exp(d*x+c)/(a- (a^2+b^2)^(1/2)))/b^5/d+a^2*(a^2+b^2)*(f*x+e)*ln(1+b*exp(d*x+c)/(a+(a^2+b^ 2)^(1/2)))/b^5/d+a^2*(a^2+b^2)*f*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2 )))/b^5/d^2+a^2*(a^2+b^2)*f*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b ^5/d^2-a^3*(f*x+e)*sinh(d*x+c)/b^4/d-2/3*a*(f*x+e)*sinh(d*x+c)/b^2/d-1/4*a ^2*f*cosh(d*x+c)*sinh(d*x+c)/b^3/d^2-3/32*f*cosh(d*x+c)*sinh(d*x+c)/b/d^2- 1/3*a*(f*x+e)*cosh(d*x+c)^2*sinh(d*x+c)/b^2/d-1/16*f*cosh(d*x+c)^3*sinh(d* x+c)/b/d^2+1/2*a^2*(f*x+e)*sinh(d*x+c)^2/b^3/d
Time = 1.60 (sec) , antiderivative size = 904, normalized size of antiderivative = 1.81 \[ \int \frac {(e+f x) \cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {-144 b^4 d e \log (a+b \sinh (c+d x))+72 b^4 f \left (d x \left (d x-2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )+72 b^2 d e \left (\left (4 a^2+b^2\right ) \log (a+b \sinh (c+d x))-4 a b \sinh (c+d x)+2 b^2 \sinh ^2(c+d x)\right )+24 d e \left (3 \left (16 a^4+12 a^2 b^2+b^4\right ) \log (a+b \sinh (c+d x))-12 a b \left (4 a^2+3 b^2\right ) \sinh (c+d x)+6 b^2 \left (4 a^2+3 b^2\right ) \sinh ^2(c+d x)-16 a b^3 \sinh ^3(c+d x)+12 b^4 \sinh ^4(c+d x)\right )+36 b^2 f \left (8 a b \cosh (c+d x)+2 b^2 d x \cosh (2 (c+d x))+\left (4 a^2+b^2\right ) \left (2 c (c+d x)-(c+d x)^2+2 (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+2 (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-2 c \log \left (b-2 a e^{c+d x}-b e^{2 (c+d x)}\right )+2 \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )-8 a b d x \sinh (c+d x)-b^2 \sinh (2 (c+d x))\right )+f \left (576 a b \left (2 a^2+b^2\right ) \cosh (c+d x)+72 b^2 \left (4 a^2+b^2\right ) d x \cosh (2 (c+d x))+32 a b^3 \cosh (3 (c+d x))+36 b^4 d x \cosh (4 (c+d x))+36 \left (16 a^4+12 a^2 b^2+b^4\right ) \left (2 c (c+d x)-(c+d x)^2+2 (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+2 (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-2 c \log \left (b-2 a e^{c+d x}-b e^{2 (c+d x)}\right )+2 \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )-576 a b \left (2 a^2+b^2\right ) d x \sinh (c+d x)-36 b^2 \left (4 a^2+b^2\right ) \sinh (2 (c+d x))-96 a b^3 d x \sinh (3 (c+d x))-9 b^4 \sinh (4 (c+d x))\right )}{1152 b^5 d^2} \] Input:
Integrate[((e + f*x)*Cosh[c + d*x]^3*Sinh[c + d*x]^2)/(a + b*Sinh[c + d*x] ),x]
Output:
(-144*b^4*d*e*Log[a + b*Sinh[c + d*x]] + 72*b^4*f*(d*x*(d*x - 2*Log[1 + (b *E^(c + d*x))/(a - Sqrt[a^2 + b^2])] - 2*Log[1 + (b*E^(c + d*x))/(a + Sqrt [a^2 + b^2])]) - 2*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - 2* PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]) + 72*b^2*d*e*((4*a^2 + b^2)*Log[a + b*Sinh[c + d*x]] - 4*a*b*Sinh[c + d*x] + 2*b^2*Sinh[c + d* x]^2) + 24*d*e*(3*(16*a^4 + 12*a^2*b^2 + b^4)*Log[a + b*Sinh[c + d*x]] - 1 2*a*b*(4*a^2 + 3*b^2)*Sinh[c + d*x] + 6*b^2*(4*a^2 + 3*b^2)*Sinh[c + d*x]^ 2 - 16*a*b^3*Sinh[c + d*x]^3 + 12*b^4*Sinh[c + d*x]^4) + 36*b^2*f*(8*a*b*C osh[c + d*x] + 2*b^2*d*x*Cosh[2*(c + d*x)] + (4*a^2 + b^2)*(2*c*(c + d*x) - (c + d*x)^2 + 2*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + 2*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] - 2*c*Log[b - 2*a*E^(c + d*x) - b*E^(2*(c + d*x))] + 2*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + 2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]) )]) - 8*a*b*d*x*Sinh[c + d*x] - b^2*Sinh[2*(c + d*x)]) + f*(576*a*b*(2*a^2 + b^2)*Cosh[c + d*x] + 72*b^2*(4*a^2 + b^2)*d*x*Cosh[2*(c + d*x)] + 32*a* b^3*Cosh[3*(c + d*x)] + 36*b^4*d*x*Cosh[4*(c + d*x)] + 36*(16*a^4 + 12*a^2 *b^2 + b^4)*(2*c*(c + d*x) - (c + d*x)^2 + 2*(c + d*x)*Log[1 + (b*E^(c + d *x))/(a - Sqrt[a^2 + b^2])] + 2*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a + Sqr t[a^2 + b^2])] - 2*c*Log[b - 2*a*E^(c + d*x) - b*E^(2*(c + d*x))] + 2*Poly Log[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + 2*PolyLog[2, -((b*E^(c...
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.
| \(\displaystyle \int \frac {(e+f x) \sinh ^2(c+d x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6113 |
| \(\displaystyle \frac {\int (e+f x) \cosh ^3(c+d x) \sinh (c+d x)dx}{b}-\frac {a \int \frac {(e+f x) \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 5970 |
| \(\displaystyle \frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \int \cosh ^4(c+d x)dx}{4 d}}{b}-\frac {a \int \frac {(e+f x) \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \int \frac {(e+f x) \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \int \sin \left (i c+i d x+\frac {\pi }{2}\right )^4dx}{4 d}}{b}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \int \cosh ^2(c+d x)dx+\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{4 d}}{b}-\frac {a \int \frac {(e+f x) \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \int \frac {(e+f x) \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \int \sin \left (i c+i d x+\frac {\pi }{2}\right )^2dx\right )}{4 d}}{b}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )+\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{4 d}}{b}-\frac {a \int \frac {(e+f x) \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{b}-\frac {a \int \frac {(e+f x) \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 6113 |
| \(\displaystyle \frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{b}-\frac {a \left (\frac {\int (e+f x) \cosh ^3(c+d x)dx}{b}-\frac {a \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3791 |
| \(\displaystyle \frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{b}-\frac {a \left (\frac {\frac {2}{3} \int (e+f x) \cosh (c+d x)dx-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2}{3} \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )dx-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {i f \int -i \sinh (c+d x)dx}{d}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{b}-\frac {a \left (\frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int \sinh (c+d x)dx}{d}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int -i \sin (i c+i d x)dx}{d}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}+\frac {i f \int \sin (i c+i d x)dx}{d}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle \frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{b}-\frac {a \left (\frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 6099 |
| \(\displaystyle \frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{b}-\frac {a \left (\frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \int (e+f x) \cosh (c+d x)dx}{b^2}+\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{b}-\frac {a \left (\frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{b}-\frac {a \left (\frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {i f \int -i \sinh (c+d x)dx}{d}\right )}{b^2}+\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{b}-\frac {a \left (\frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int \sinh (c+d x)dx}{d}\right )}{b^2}+\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{b}-\frac {a \left (\frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int -i \sin (i c+i d x)dx}{d}\right )}{b^2}+\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{b}-\frac {a \left (\frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \left (\frac {(e+f x) \sinh (c+d x)}{d}+\frac {i f \int \sin (i c+i d x)dx}{d}\right )}{b^2}+\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle \frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{b}-\frac {a \left (\frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}+\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx}{b}-\frac {a \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{b^2}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 5969 |
| \(\displaystyle \frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{b}-\frac {a \left (\frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}+\frac {\frac {(e+f x) \sinh ^2(c+d x)}{2 d}-\frac {f \int \sinh ^2(c+d x)dx}{2 d}}{b}-\frac {a \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{b^2}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{b}-\frac {a \left (\frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}+\frac {\frac {(e+f x) \sinh ^2(c+d x)}{2 d}-\frac {f \int -\sin (i c+i d x)^2dx}{2 d}}{b}-\frac {a \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{b^2}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{b}-\frac {a \left (\frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}+\frac {\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \int \sin (i c+i d x)^2dx}{2 d}}{b}-\frac {a \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{b^2}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{b}-\frac {a \left (\frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}+\frac {\frac {f \left (\frac {\int 1dx}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}+\frac {(e+f x) \sinh ^2(c+d x)}{2 d}}{b}-\frac {a \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{b^2}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{b}-\frac {a \left (\frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{b^2}+\frac {\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 6095 |
| \(\displaystyle \frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{b}-\frac {a \left (\frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \left (\int \frac {e^{c+d x} (e+f x)}{a+b e^{c+d x}-\sqrt {a^2+b^2}}dx+\int \frac {e^{c+d x} (e+f x)}{a+b e^{c+d x}+\sqrt {a^2+b^2}}dx-\frac {(e+f x)^2}{2 b f}\right )}{b^2}-\frac {a \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{b^2}+\frac {\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {\frac {(e+f x) \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{b}-\frac {a \left (\frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \left (-\frac {f \int \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )dx}{b d}-\frac {f \int \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )dx}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^2}{2 b f}\right )}{b^2}-\frac {a \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{b^2}+\frac {\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}}{b}\right )}{b}\right )}{b}\) |
Input:
Int[((e + f*x)*Cosh[c + d*x]^3*Sinh[c + d*x]^2)/(a + b*Sinh[c + d*x]),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(1216\) vs. \(2(463)=926\).
Time = 56.19 (sec) , antiderivative size = 1217, normalized size of antiderivative = 2.44
Input:
int((f*x+e)*cosh(d*x+c)^3*sinh(d*x+c)^2/(a+b*sinh(d*x+c)),x,method=_RETURN VERBOSE)
Output:
1/d^2/b^3*a^2*f*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c +2/d^2/b^3*c*a^2*f*ln(exp(d*x+c))+2/d^2*a^4/b^5*c*f*ln(exp(d*x+c))-1/d^2*a ^4/b^5*c*f*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)+1/d*a^4/b^5*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^4/b^5*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^4/b^5*f*ln((-b*ex p(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c+1/d^2*a^4/b^5*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/b^3*a^2*f*c^2-2 /d/b^3*a^2*e*ln(exp(d*x+c))+1/d/b^3*a^2*e*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+ c)-b)+1/d^2/b^3*a^2*f*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2 )^(1/2)))+1/d^2/b^3*a^2*f*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b ^2)^(1/2)))-2/d/b^3*a^2*f*c*x+1/d/b^3*a^2*f*ln((-b*exp(d*x+c)+(a^2+b^2)^(1 /2)-a)/(-a+(a^2+b^2)^(1/2)))*x+1/d/b^3*a^2*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/b^3*a^2*f*ln((-b*exp(d*x+c)+(a^2+b^2)^ (1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c-1/8*a*(4*a^2*d*f*x+3*b^2*d*f*x+4*a^2*d*e+ 3*b^2*d*e-4*a^2*f-3*b^2*f)/b^4/d^2*exp(d*x+c)-1/d^2/b^3*c*a^2*f*ln(b*exp(2 *d*x+2*c)+2*a*exp(d*x+c)-b)+1/8*a*(4*a^2+3*b^2)*(d*f*x+d*e+f)/b^4/d^2*exp( -d*x-c)+1/256*(4*d*f*x+4*d*e-f)/b/d^2*exp(4*d*x+4*c)+1/32*(4*a^2*d*f*x+2*b ^2*d*f*x+4*a^2*d*e+2*b^2*d*e-2*a^2*f-b^2*f)/b^3/d^2*exp(2*d*x+2*c)+1/256*( 4*d*f*x+4*d*e+f)/b/d^2*exp(-4*d*x-4*c)-1/72*a*(3*d*f*x+3*d*e-f)/b^2/d^2*ex p(3*d*x+3*c)+1/32*(2*a^2+b^2)*(2*d*f*x+2*d*e+f)/b^3/d^2*exp(-2*d*x-2*c)...
Leaf count of result is larger than twice the leaf count of optimal. 3795 vs. \(2 (461) = 922\).
Time = 0.15 (sec) , antiderivative size = 3795, normalized size of antiderivative = 7.61 \[ \int \frac {(e+f x) \cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)*cosh(d*x+c)^3*sinh(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorit hm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {(e+f x) \cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)*cosh(d*x+c)**3*sinh(d*x+c)**2/(a+b*sinh(d*x+c)),x)
Output:
Timed out
\[ \int \frac {(e+f x) \cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)*cosh(d*x+c)^3*sinh(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorit hm="maxima")
Output:
-1/192*e*((8*a*b^2*e^(-d*x - c) - 3*b^3 - 12*(2*a^2*b + b^3)*e^(-2*d*x - 2 *c) + 24*(4*a^3 + 3*a*b^2)*e^(-3*d*x - 3*c))*e^(4*d*x + 4*c)/(b^4*d) - 192 *(a^4 + a^2*b^2)*(d*x + c)/(b^5*d) - (8*a*b^2*e^(-3*d*x - 3*c) + 3*b^3*e^( -4*d*x - 4*c) + 24*(4*a^3 + 3*a*b^2)*e^(-d*x - c) + 12*(2*a^2*b + b^3)*e^( -2*d*x - 2*c))/(b^4*d) - 192*(a^4 + a^2*b^2)*log(-2*a*e^(-d*x - c) + b*e^( -2*d*x - 2*c) - b)/(b^5*d)) + 1/2304*f*((1152*(a^4*d^2*e^(4*c) + a^2*b^2*d ^2*e^(4*c))*x^2 + 9*(4*b^4*d*x*e^(8*c) - b^4*e^(8*c))*e^(4*d*x) - 32*(3*a* b^3*d*x*e^(7*c) - a*b^3*e^(7*c))*e^(3*d*x) - 72*(2*a^2*b^2*e^(6*c) + b^4*e ^(6*c) - 2*(2*a^2*b^2*d*e^(6*c) + b^4*d*e^(6*c))*x)*e^(2*d*x) + 288*(4*a^3 *b*e^(5*c) + 3*a*b^3*e^(5*c) - (4*a^3*b*d*e^(5*c) + 3*a*b^3*d*e^(5*c))*x)* e^(d*x) + 288*(4*a^3*b*e^(3*c) + 3*a*b^3*e^(3*c) + (4*a^3*b*d*e^(3*c) + 3* a*b^3*d*e^(3*c))*x)*e^(-d*x) + 72*(2*a^2*b^2*e^(2*c) + b^4*e^(2*c) + 2*(2* a^2*b^2*d*e^(2*c) + b^4*d*e^(2*c))*x)*e^(-2*d*x) + 32*(3*a*b^3*d*x*e^c + a *b^3*e^c)*e^(-3*d*x) + 9*(4*b^4*d*x + b^4)*e^(-4*d*x))*e^(-4*c)/(b^5*d^2) - 72*integrate(64*((a^5*e^c + a^3*b^2*e^c)*x*e^(d*x) - (a^4*b + a^2*b^3)*x )/(b^6*e^(2*d*x + 2*c) + 2*a*b^5*e^(d*x + c) - b^6), x))
\[ \int \frac {(e+f x) \cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)*cosh(d*x+c)^3*sinh(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorit hm="giac")
Output:
integrate((f*x + e)*cosh(d*x + c)^3*sinh(d*x + c)^2/(b*sinh(d*x + c) + a), x)
Timed out. \[ \int \frac {(e+f x) \cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^3\,{\mathrm {sinh}\left (c+d\,x\right )}^2\,\left (e+f\,x\right )}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \] Input:
int((cosh(c + d*x)^3*sinh(c + d*x)^2*(e + f*x))/(a + b*sinh(c + d*x)),x)
Output:
int((cosh(c + d*x)^3*sinh(c + d*x)^2*(e + f*x))/(a + b*sinh(c + d*x)), x)
\[ \int \frac {(e+f x) \cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {too large to display} \] Input:
int((f*x+e)*cosh(d*x+c)^3*sinh(d*x+c)^2/(a+b*sinh(d*x+c)),x)
Output:
(36*e**(8*c + 8*d*x)*b**8*d*e + 36*e**(8*c + 8*d*x)*b**8*d*f*x - 9*e**(8*c + 8*d*x)*b**8*f - 96*e**(7*c + 7*d*x)*a*b**7*d*e - 96*e**(7*c + 7*d*x)*a* b**7*d*f*x + 32*e**(7*c + 7*d*x)*a*b**7*f + 288*e**(6*c + 6*d*x)*a**2*b**6 *d*e + 288*e**(6*c + 6*d*x)*a**2*b**6*d*f*x - 144*e**(6*c + 6*d*x)*a**2*b* *6*f + 144*e**(6*c + 6*d*x)*b**8*d*e + 144*e**(6*c + 6*d*x)*b**8*d*f*x - 7 2*e**(6*c + 6*d*x)*b**8*f - 1152*e**(5*c + 5*d*x)*a**3*b**5*d*e - 1152*e** (5*c + 5*d*x)*a**3*b**5*d*f*x + 1152*e**(5*c + 5*d*x)*a**3*b**5*f - 864*e* *(5*c + 5*d*x)*a*b**7*d*e - 864*e**(5*c + 5*d*x)*a*b**7*d*f*x + 864*e**(5* c + 5*d*x)*a*b**7*f + 36864*e**(4*c + 4*d*x)*int(x/(e**(6*c + 6*d*x)*b + 2 *e**(5*c + 5*d*x)*a - e**(4*c + 4*d*x)*b),x)*a**8*b*d**2*f + 73728*e**(4*c + 4*d*x)*int(x/(e**(6*c + 6*d*x)*b + 2*e**(5*c + 5*d*x)*a - e**(4*c + 4*d *x)*b),x)*a**6*b**3*d**2*f + 41472*e**(4*c + 4*d*x)*int(x/(e**(6*c + 6*d*x )*b + 2*e**(5*c + 5*d*x)*a - e**(4*c + 4*d*x)*b),x)*a**4*b**5*d**2*f + 460 8*e**(4*c + 4*d*x)*int(x/(e**(6*c + 6*d*x)*b + 2*e**(5*c + 5*d*x)*a - e**( 4*c + 4*d*x)*b),x)*a**2*b**7*d**2*f + 2304*e**(4*c + 4*d*x)*log(e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b)*a**4*b**4*d*e + 2304*e**(4*c + 4*d*x)*log (e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b)*a**2*b**6*d*e - 2304*e**(4*c + 4*d*x)*a**4*b**4*d**2*e*x + 1152*e**(4*c + 4*d*x)*a**4*b**4*d**2*f*x**2 - 2304*e**(4*c + 4*d*x)*a**2*b**6*d**2*e*x + 1152*e**(4*c + 4*d*x)*a**2*b** 6*d**2*f*x**2 - 73728*e**(c + 4*d*x)*int(x/(e**(2*c + 5*d*x)*b + 2*e**(...