Integrand size = 28, antiderivative size = 721 \[ \int \frac {(e+f x)^3 \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {(e+f x)^3}{a d}+\frac {2 b (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a^2 d}-\frac {(e+f x)^3 \coth (c+d x)}{a d}+\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^2}-\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^2}+\frac {3 \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^2}-\frac {3 \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^2}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a^2 d^3}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a^2 d^3}-\frac {6 \sqrt {a^2+b^2} f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^3}+\frac {6 \sqrt {a^2+b^2} f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^3}-\frac {3 f^3 \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a d^4}+\frac {6 b f^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )}{a^2 d^4}-\frac {6 b f^3 \operatorname {PolyLog}\left (4,e^{c+d x}\right )}{a^2 d^4}+\frac {6 \sqrt {a^2+b^2} f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^4}-\frac {6 \sqrt {a^2+b^2} f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^4} \] Output:
-(f*x+e)^3/a/d+2*b*(f*x+e)^3*arctanh(exp(d*x+c))/a^2/d-(f*x+e)^3*coth(d*x+ c)/a/d+(a^2+b^2)^(1/2)*(f*x+e)^3*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^ 2/d-(a^2+b^2)^(1/2)*(f*x+e)^3*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^2/d +3*f*(f*x+e)^2*ln(1-exp(2*d*x+2*c))/a/d^2+3*b*f*(f*x+e)^2*polylog(2,-exp(d *x+c))/a^2/d^2-3*b*f*(f*x+e)^2*polylog(2,exp(d*x+c))/a^2/d^2+3*(a^2+b^2)^( 1/2)*f*(f*x+e)^2*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^2/d^2-3*(a ^2+b^2)^(1/2)*f*(f*x+e)^2*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^2 /d^2+3*f^2*(f*x+e)*polylog(2,exp(2*d*x+2*c))/a/d^3-6*b*f^2*(f*x+e)*polylog (3,-exp(d*x+c))/a^2/d^3+6*b*f^2*(f*x+e)*polylog(3,exp(d*x+c))/a^2/d^3-6*(a ^2+b^2)^(1/2)*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^2 /d^3+6*(a^2+b^2)^(1/2)*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1 /2)))/a^2/d^3-3/2*f^3*polylog(3,exp(2*d*x+2*c))/a/d^4+6*b*f^3*polylog(4,-e xp(d*x+c))/a^2/d^4-6*b*f^3*polylog(4,exp(d*x+c))/a^2/d^4+6*(a^2+b^2)^(1/2) *f^3*polylog(4,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^2/d^4-6*(a^2+b^2)^(1/2 )*f^3*polylog(4,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^2/d^4
Leaf count is larger than twice the leaf count of optimal. \(1490\) vs. \(2(721)=1442\).
Time = 7.03 (sec) , antiderivative size = 1490, normalized size of antiderivative = 2.07 \[ \int \frac {(e+f x)^3 \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Input:
Integrate[((e + f*x)^3*Coth[c + d*x]^2)/(a + b*Sinh[c + d*x]),x]
Output:
-((-(d^3*e^2*(-1 + E^(2*c))*(b*d*e - 3*a*f)*x) + d^3*e^2*(-1 + E^(2*c))*(b *d*e + 3*a*f)*x + 2*a*d^3*(e + f*x)^3 + 3*d^2*e*(-1 + E^(2*c))*f*(b*d*e - 2*a*f)*x*Log[1 - E^(-c - d*x)] + 3*d^2*(-1 + E^(2*c))*f^2*(b*d*e - a*f)*x^ 2*Log[1 - E^(-c - d*x)] + b*d^3*(-1 + E^(2*c))*f^3*x^3*Log[1 - E^(-c - d*x )] - 3*d^2*e*(-1 + E^(2*c))*f*(b*d*e + 2*a*f)*x*Log[1 + E^(-c - d*x)] - 3* d^2*(-1 + E^(2*c))*f^2*(b*d*e + a*f)*x^2*Log[1 + E^(-c - d*x)] - b*d^3*(-1 + E^(2*c))*f^3*x^3*Log[1 + E^(-c - d*x)] + d^2*e^2*(-1 + E^(2*c))*(b*d*e - 3*a*f)*Log[1 - E^(c + d*x)] - d^2*e^2*(-1 + E^(2*c))*(b*d*e + 3*a*f)*Log [1 + E^(c + d*x)] + 3*d*e*(-1 + E^(2*c))*f*(b*d*e + 2*a*f)*PolyLog[2, -E^( -c - d*x)] + 6*d*(-1 + E^(2*c))*f^2*(b*d*e + a*f)*x*PolyLog[2, -E^(-c - d* x)] + 3*b*d^2*(-1 + E^(2*c))*f^3*x^2*PolyLog[2, -E^(-c - d*x)] - 3*d*e*(-1 + E^(2*c))*f*(b*d*e - 2*a*f)*PolyLog[2, E^(-c - d*x)] - 6*d*(-1 + E^(2*c) )*f^2*(b*d*e - a*f)*x*PolyLog[2, E^(-c - d*x)] - 3*b*d^2*(-1 + E^(2*c))*f^ 3*x^2*PolyLog[2, E^(-c - d*x)] + 6*(-1 + E^(2*c))*f^2*(b*d*e + a*f)*PolyLo g[3, -E^(-c - d*x)] + 6*b*d*(-1 + E^(2*c))*f^3*x*PolyLog[3, -E^(-c - d*x)] + 6*(-1 + E^(2*c))*f^2*(-(b*d*e) + a*f)*PolyLog[3, E^(-c - d*x)] - 6*b*d* (-1 + E^(2*c))*f^3*x*PolyLog[3, E^(-c - d*x)] + 6*b*(-1 + E^(2*c))*f^3*Pol yLog[4, -E^(-c - d*x)] - 6*b*(-1 + E^(2*c))*f^3*PolyLog[4, E^(-c - d*x)])/ (a^2*d^4*(-1 + E^(2*c)))) + (Sqrt[a^2 + b^2]*(-2*d^3*e^3*ArcTanh[(a + b*E^ (c + d*x))/Sqrt[a^2 + b^2]] + 3*d^3*e^2*f*x*Log[1 + (b*E^(c + d*x))/(a ...
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.
| \(\displaystyle \int \frac {(e+f x)^3 \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6103 |
| \(\displaystyle \frac {\int (e+f x)^3 \coth ^2(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int -(e+f x)^3 \tan \left (i c+i d x+\frac {\pi }{2}\right )^2dx}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\int (e+f x)^3 \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )^2dx}{a}\) |
| \(\Big \downarrow \) 4203 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\frac {3 i f \int i (e+f x)^2 \coth (c+d x)dx}{d}-\int (e+f x)^3dx+\frac {(e+f x)^3 \coth (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\frac {3 i f \int i (e+f x)^2 \coth (c+d x)dx}{d}+\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {-\frac {3 f \int (e+f x)^2 \coth (c+d x)dx}{d}+\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {-\frac {3 f \int -i (e+f x)^2 \tan \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}+\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\frac {3 i f \int (e+f x)^2 \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )dx}{d}+\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{a}\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\frac {3 i f \left (2 i \int \frac {e^{2 c+2 d x-i \pi } (e+f x)^2}{1+e^{2 c+2 d x-i \pi }}dx-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \int (e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )dx}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )dx}{2 d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{a}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{a}\) |
| \(\Big \downarrow \) 6119 |
| \(\displaystyle -\frac {b \left (\frac {\int (e+f x)^3 \cosh (c+d x) \coth (c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{a}\) |
| \(\Big \downarrow \) 5973 |
| \(\displaystyle -\frac {b \left (\frac {\int (e+f x)^3 \sinh (c+d x)dx+\int (e+f x)^3 \text {csch}(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int -i (e+f x)^3 \sin (i c+i d x)dx+\int i (e+f x)^3 \csc (i c+i d x)dx}{a}\right )}{a}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x)^3 \csc (i c+i d x)dx-i \int (e+f x)^3 \sin (i c+i d x)dx}{a}\right )}{a}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x)^3 \csc (i c+i d x)dx-i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \int (e+f x)^2 \cosh (c+d x)dx}{d}\right )}{a}\right )}{a}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x)^3 \csc (i c+i d x)dx-i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{a}\right )}{a}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x)^3 \csc (i c+i d x)dx-i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 i f \int -i (e+f x) \sinh (c+d x)dx}{d}\right )}{d}\right )}{a}\right )}{a}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x)^3 \csc (i c+i d x)dx-i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int (e+f x) \sinh (c+d x)dx}{d}\right )}{d}\right )}{a}\right )}{a}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x)^3 \csc (i c+i d x)dx-i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int -i (e+f x) \sin (i c+i d x)dx}{d}\right )}{d}\right )}{a}\right )}{a}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x)^3 \csc (i c+i d x)dx-i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \int (e+f x) \sin (i c+i d x)dx}{d}\right )}{d}\right )}{a}\right )}{a}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x)^3 \csc (i c+i d x)dx-i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \cosh (c+d x)dx}{d}\right )}{d}\right )}{d}\right )}{a}\right )}{a}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x)^3 \csc (i c+i d x)dx-i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{d}\right )}{d}\right )}{a}\right )}{a}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{a}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle -\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x)^3 \csc (i c+i d x)dx-i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{a}\right )}{a}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{a}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle -\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (\frac {3 i f \int (e+f x)^2 \log \left (1-e^{c+d x}\right )dx}{d}-\frac {3 i f \int (e+f x)^2 \log \left (1+e^{c+d x}\right )dx}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{a}\right )}{a}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{a}\right )}{a}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{a}\) |
| \(\Big \downarrow \) 6099 |
| \(\displaystyle -\frac {b \left (-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \int (e+f x)^3dx}{b^2}+\frac {\int (e+f x)^3 \sinh (c+d x)dx}{b}\right )}{a}+\frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{a}\right )}{a}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{a}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle -\frac {b \left (-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a+b \sinh (c+d x)}dx}{b^2}+\frac {\int (e+f x)^3 \sinh (c+d x)dx}{b}-\frac {a (e+f x)^4}{4 b^2 f}\right )}{a}+\frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{a}\right )}{a}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \left (\frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a-i b \sin (i c+i d x)}dx}{b^2}+\frac {\int -i (e+f x)^3 \sin (i c+i d x)dx}{b}-\frac {a (e+f x)^4}{4 b^2 f}\right )}{a}\right )}{a}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{a}\) |
Input:
Int[((e + f*x)^3*Coth[c + d*x]^2)/(a + b*Sinh[c + d*x]),x]
Output:
$Aborted
\[\int \frac {\left (f x +e \right )^{3} \coth \left (d x +c \right )^{2}}{a +b \sinh \left (d x +c \right )}d x\]
Input:
int((f*x+e)^3*coth(d*x+c)^2/(a+b*sinh(d*x+c)),x)
Output:
int((f*x+e)^3*coth(d*x+c)^2/(a+b*sinh(d*x+c)),x)
Leaf count of result is larger than twice the leaf count of optimal. 4612 vs. \(2 (667) = 1334\).
Time = 0.23 (sec) , antiderivative size = 4612, normalized size of antiderivative = 6.40 \[ \int \frac {(e+f x)^3 \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^3*coth(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {(e+f x)^3 \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (e + f x\right )^{3} \coth ^{2}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \] Input:
integrate((f*x+e)**3*coth(d*x+c)**2/(a+b*sinh(d*x+c)),x)
Output:
Integral((e + f*x)**3*coth(c + d*x)**2/(a + b*sinh(c + d*x)), x)
\[ \int \frac {(e+f x)^3 \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \coth \left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)^3*coth(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="maxima")
Output:
e^3*(b*log(e^(-d*x - c) + 1)/(a^2*d) - b*log(e^(-d*x - c) - 1)/(a^2*d) + s qrt(a^2 + b^2)*log((b*e^(-d*x - c) - a - sqrt(a^2 + b^2))/(b*e^(-d*x - c) - a + sqrt(a^2 + b^2)))/(a^2*d) + 2/((a*e^(-2*d*x - 2*c) - a)*d)) - 6*e^2* f*x/(a*d) + 3*e^2*f*log(e^(d*x + c) + 1)/(a*d^2) + 3*e^2*f*log(e^(d*x + c) - 1)/(a*d^2) - 2*(f^3*x^3 + 3*e*f^2*x^2 + 3*e^2*f*x)/(a*d*e^(2*d*x + 2*c) - a*d) + (d^3*x^3*log(e^(d*x + c) + 1) + 3*d^2*x^2*dilog(-e^(d*x + c)) - 6*d*x*polylog(3, -e^(d*x + c)) + 6*polylog(4, -e^(d*x + c)))*b*f^3/(a^2*d^ 4) - (d^3*x^3*log(-e^(d*x + c) + 1) + 3*d^2*x^2*dilog(e^(d*x + c)) - 6*d*x *polylog(3, e^(d*x + c)) + 6*polylog(4, e^(d*x + c)))*b*f^3/(a^2*d^4) + 3* (b*d*e^2*f + 2*a*e*f^2)*(d*x*log(e^(d*x + c) + 1) + dilog(-e^(d*x + c)))/( a^2*d^3) - 3*(b*d*e^2*f - 2*a*e*f^2)*(d*x*log(-e^(d*x + c) + 1) + dilog(e^ (d*x + c)))/(a^2*d^3) + 3*(b*d*e*f^2 + a*f^3)*(d^2*x^2*log(e^(d*x + c) + 1 ) + 2*d*x*dilog(-e^(d*x + c)) - 2*polylog(3, -e^(d*x + c)))/(a^2*d^4) - 3* (b*d*e*f^2 - a*f^3)*(d^2*x^2*log(-e^(d*x + c) + 1) + 2*d*x*dilog(e^(d*x + c)) - 2*polylog(3, e^(d*x + c)))/(a^2*d^4) - 1/4*(b*d^4*f^3*x^4 + 4*(b*d*e *f^2 + a*f^3)*d^3*x^3 + 6*(b*d^2*e^2*f + 2*a*d*e*f^2)*d^2*x^2)/(a^2*d^4) + 1/4*(b*d^4*f^3*x^4 + 4*(b*d*e*f^2 - a*f^3)*d^3*x^3 + 6*(b*d^2*e^2*f - 2*a *d*e*f^2)*d^2*x^2)/(a^2*d^4) + integrate(2*((a^2*f^3*e^c + b^2*f^3*e^c)*x^ 3 + 3*(a^2*e*f^2*e^c + b^2*e*f^2*e^c)*x^2 + 3*(a^2*e^2*f*e^c + b^2*e^2*f*e ^c)*x)*e^(d*x)/(a^2*b*e^(2*d*x + 2*c) + 2*a^3*e^(d*x + c) - a^2*b), x)
Timed out. \[ \int \frac {(e+f x)^3 \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)^3*coth(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {(e+f x)^3 \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\mathrm {coth}\left (c+d\,x\right )}^2\,{\left (e+f\,x\right )}^3}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \] Input:
int((coth(c + d*x)^2*(e + f*x)^3)/(a + b*sinh(c + d*x)),x)
Output:
int((coth(c + d*x)^2*(e + f*x)^3)/(a + b*sinh(c + d*x)), x)
\[ \int \frac {(e+f x)^3 \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {too large to display} \] Input:
int((f*x+e)^3*coth(d*x+c)^2/(a+b*sinh(d*x+c)),x)
Output:
(4*e**(2*c + 2*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqrt(a **2 + b**2))*b**2*d**3*e**3*i - 4*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqrt(a**2 + b**2))*b**2*d**3*e**3*i + 16*e**(5*c + 2*d*x)*int((e** (3*d*x)*x**3)/(e**(6*c + 6*d*x)*b + 2*e**(5*c + 5*d*x)*a - 3*e**(4*c + 4*d *x)*b - 4*e**(3*c + 3*d*x)*a + 3*e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b ),x)*a**4*d**4*f**3 + 8*e**(5*c + 2*d*x)*int((e**(3*d*x)*x**3)/(e**(6*c + 6*d*x)*b + 2*e**(5*c + 5*d*x)*a - 3*e**(4*c + 4*d*x)*b - 4*e**(3*c + 3*d*x )*a + 3*e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b),x)*a**2*b**2*d**4*f**3 + 48*e**(5*c + 2*d*x)*int((e**(3*d*x)*x**2)/(e**(6*c + 6*d*x)*b + 2*e**(5* c + 5*d*x)*a - 3*e**(4*c + 4*d*x)*b - 4*e**(3*c + 3*d*x)*a + 3*e**(2*c + 2 *d*x)*b + 2*e**(c + d*x)*a - b),x)*a**4*d**4*e*f**2 + 24*e**(5*c + 2*d*x)* int((e**(3*d*x)*x**2)/(e**(6*c + 6*d*x)*b + 2*e**(5*c + 5*d*x)*a - 3*e**(4 *c + 4*d*x)*b - 4*e**(3*c + 3*d*x)*a + 3*e**(2*c + 2*d*x)*b + 2*e**(c + d* x)*a - b),x)*a**2*b**2*d**4*e*f**2 - 24*e**(5*c + 2*d*x)*int((e**(3*d*x)*x **2)/(e**(6*c + 6*d*x)*b + 2*e**(5*c + 5*d*x)*a - 3*e**(4*c + 4*d*x)*b - 4 *e**(3*c + 3*d*x)*a + 3*e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b),x)*a**2 *b**2*d**3*f**3 + 48*e**(5*c + 2*d*x)*int((e**(3*d*x)*x)/(e**(6*c + 6*d*x) *b + 2*e**(5*c + 5*d*x)*a - 3*e**(4*c + 4*d*x)*b - 4*e**(3*c + 3*d*x)*a + 3*e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b),x)*a**4*d**4*e**2*f + 24*e**( 5*c + 2*d*x)*int((e**(3*d*x)*x)/(e**(6*c + 6*d*x)*b + 2*e**(5*c + 5*d*x...