Integrand size = 32, antiderivative size = 413 \[ \int \frac {(e+f x) \coth ^2(c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {(e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a^3 d}+\frac {b (e+f x) \coth (c+d x)}{a^2 d}-\frac {f \text {csch}(c+d x)}{2 a d^2}-\frac {(e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {b \sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d}+\frac {b \sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d}-\frac {b f \log (\sinh (c+d x))}{a^2 d^2}-\frac {f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 a d^2}-\frac {b^2 f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^3 d^2}+\frac {f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 a d^2}+\frac {b^2 f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^3 d^2}-\frac {b \sqrt {a^2+b^2} f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {b \sqrt {a^2+b^2} f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^2} \] Output:
-(f*x+e)*arctanh(exp(d*x+c))/a/d-2*b^2*(f*x+e)*arctanh(exp(d*x+c))/a^3/d+b *(f*x+e)*coth(d*x+c)/a^2/d-1/2*f*csch(d*x+c)/a/d^2-1/2*(f*x+e)*coth(d*x+c) *csch(d*x+c)/a/d-b*(a^2+b^2)^(1/2)*(f*x+e)*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^ (1/2)))/a^3/d+b*(a^2+b^2)^(1/2)*(f*x+e)*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/ 2)))/a^3/d-b*f*ln(sinh(d*x+c))/a^2/d^2-1/2*f*polylog(2,-exp(d*x+c))/a/d^2- b^2*f*polylog(2,-exp(d*x+c))/a^3/d^2+1/2*f*polylog(2,exp(d*x+c))/a/d^2+b^2 *f*polylog(2,exp(d*x+c))/a^3/d^2-b*(a^2+b^2)^(1/2)*f*polylog(2,-b*exp(d*x+ c)/(a-(a^2+b^2)^(1/2)))/a^3/d^2+b*(a^2+b^2)^(1/2)*f*polylog(2,-b*exp(d*x+c )/(a+(a^2+b^2)^(1/2)))/a^3/d^2
Time = 8.64 (sec) , antiderivative size = 615, normalized size of antiderivative = 1.49 \[ \int \frac {(e+f x) \coth ^2(c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\left (2 b d e \cosh \left (\frac {1}{2} (c+d x)\right )-a f \cosh \left (\frac {1}{2} (c+d x)\right )-2 b c f \cosh \left (\frac {1}{2} (c+d x)\right )+2 b f (c+d x) \cosh \left (\frac {1}{2} (c+d x)\right )\right ) \text {csch}\left (\frac {1}{2} (c+d x)\right )}{4 a^2 d^2}+\frac {(-d e+c f-f (c+d x)) \text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{8 a d^2}+\frac {-2 a b f (c+d x)+\left (-2 a b f+a^2 (d e+d f x)+2 b^2 (d e+d f x)\right ) \log \left (1-e^{-c-d x}\right )-\left (2 a b f+a^2 (d e+d f x)+2 b^2 (d e+d f x)\right ) \log \left (1+e^{-c-d x}\right )+\left (a^2+2 b^2\right ) f \operatorname {PolyLog}\left (2,-e^{-c-d x}\right )-\left (a^2+2 b^2\right ) f \operatorname {PolyLog}\left (2,e^{-c-d x}\right )}{2 a^3 d^2}-\frac {b \sqrt {a^2+b^2} \left (-2 d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+2 c f \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+f \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{a^3 d^2}+\frac {(-d e+c f-f (c+d x)) \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{8 a d^2}+\frac {\text {sech}\left (\frac {1}{2} (c+d x)\right ) \left (2 b d e \sinh \left (\frac {1}{2} (c+d x)\right )+a f \sinh \left (\frac {1}{2} (c+d x)\right )-2 b c f \sinh \left (\frac {1}{2} (c+d x)\right )+2 b f (c+d x) \sinh \left (\frac {1}{2} (c+d x)\right )\right )}{4 a^2 d^2} \] Input:
Integrate[((e + f*x)*Coth[c + d*x]^2*Csch[c + d*x])/(a + b*Sinh[c + d*x]), x]
Output:
((2*b*d*e*Cosh[(c + d*x)/2] - a*f*Cosh[(c + d*x)/2] - 2*b*c*f*Cosh[(c + d* x)/2] + 2*b*f*(c + d*x)*Cosh[(c + d*x)/2])*Csch[(c + d*x)/2])/(4*a^2*d^2) + ((-(d*e) + c*f - f*(c + d*x))*Csch[(c + d*x)/2]^2)/(8*a*d^2) + (-2*a*b*f *(c + d*x) + (-2*a*b*f + a^2*(d*e + d*f*x) + 2*b^2*(d*e + d*f*x))*Log[1 - E^(-c - d*x)] - (2*a*b*f + a^2*(d*e + d*f*x) + 2*b^2*(d*e + d*f*x))*Log[1 + E^(-c - d*x)] + (a^2 + 2*b^2)*f*PolyLog[2, -E^(-c - d*x)] - (a^2 + 2*b^2 )*f*PolyLog[2, E^(-c - d*x)])/(2*a^3*d^2) - (b*Sqrt[a^2 + b^2]*(-2*d*e*Arc Tanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] + 2*c*f*ArcTanh[(a + b*E^(c + d* x))/Sqrt[a^2 + b^2]] + f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] - f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + f*P olyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]))/(a^3*d^2) + ((-(d*e) + c*f - f*(c + d*x ))*Sech[(c + d*x)/2]^2)/(8*a*d^2) + (Sech[(c + d*x)/2]*(2*b*d*e*Sinh[(c + d*x)/2] + a*f*Sinh[(c + d*x)/2] - 2*b*c*f*Sinh[(c + d*x)/2] + 2*b*f*(c + d *x)*Sinh[(c + d*x)/2]))/(4*a^2*d^2)
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.
| \(\displaystyle \int \frac {(e+f x) \coth ^2(c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6121 |
| \(\displaystyle \frac {\int (e+f x) \coth ^2(c+d x) \text {csch}(c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 5980 |
| \(\displaystyle \frac {\int (e+f x) \text {csch}^3(c+d x)dx+\int (e+f x) \text {csch}(c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int i (e+f x) \csc (i c+i d x)dx+\int -i (e+f x) \csc (i c+i d x)^3dx}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x) \csc (i c+i d x)dx-i \int (e+f x) \csc (i c+i d x)^3dx}{a}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (\frac {i f \int \log \left (1-e^{c+d x}\right )dx}{d}-\frac {i f \int \log \left (1+e^{c+d x}\right )dx}{d}+\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \int (e+f x) \csc (i c+i d x)^3dx}{a}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (\frac {i f \int e^{-c-d x} \log \left (1-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {i f \int e^{-c-d x} \log \left (1+e^{c+d x}\right )de^{c+d x}}{d^2}+\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \int (e+f x) \csc (i c+i d x)^3dx}{a}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-i \int (e+f x) \csc (i c+i d x)^3dx}{a}\) |
| \(\Big \downarrow \) 4673 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-i \left (\frac {1}{2} \int -i (e+f x) \text {csch}(c+d x)dx-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-i \left (-\frac {1}{2} i \int (e+f x) \text {csch}(c+d x)dx-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-i \left (-\frac {1}{2} i \int i (e+f x) \csc (i c+i d x)dx-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-i \left (\frac {1}{2} \int (e+f x) \csc (i c+i d x)dx-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-i \left (\frac {1}{2} \left (\frac {i f \int \log \left (1-e^{c+d x}\right )dx}{d}-\frac {i f \int \log \left (1+e^{c+d x}\right )dx}{d}+\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-i \left (\frac {1}{2} \left (\frac {i f \int e^{-c-d x} \log \left (1-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {i f \int e^{-c-d x} \log \left (1+e^{c+d x}\right )de^{c+d x}}{d^2}+\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 6103 |
| \(\displaystyle -\frac {b \left (\frac {\int (e+f x) \coth ^2(c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}+\frac {i \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {i \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int -\left ((e+f x) \tan \left (i c+i d x+\frac {\pi }{2}\right )^2\right )dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {i \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\int (e+f x) \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )^2dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 4203 |
| \(\displaystyle \frac {i \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\frac {i f \int i \coth (c+d x)dx}{d}-\int (e+f x)dx+\frac {(e+f x) \coth (c+d x)}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle \frac {i \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\frac {i f \int i \coth (c+d x)dx}{d}+\frac {(e+f x) \coth (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \left (-\frac {b \int \frac {(e+f x) \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {-\frac {f \int \coth (c+d x)dx}{d}+\frac {(e+f x) \coth (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{a}\right )}{a}+\frac {i \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {i \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {-\frac {f \int -i \tan \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}+\frac {(e+f x) \coth (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {i \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\frac {i f \int \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )dx}{d}+\frac {(e+f x) \coth (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle \frac {i \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {-\frac {f \log (-i \sinh (c+d x))}{d^2}+\frac {(e+f x) \coth (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 6119 |
| \(\displaystyle \frac {i \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \left (\frac {\int (e+f x) \cosh (c+d x) \coth (c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}-\frac {-\frac {f \log (-i \sinh (c+d x))}{d^2}+\frac {(e+f x) \coth (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 5973 |
| \(\displaystyle \frac {i \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \left (\frac {\int (e+f x) \sinh (c+d x)dx+\int (e+f x) \text {csch}(c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}-\frac {-\frac {f \log (-i \sinh (c+d x))}{d^2}+\frac {(e+f x) \coth (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {i \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \left (-\frac {b \int \frac {(e+f x) \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int -i (e+f x) \sin (i c+i d x)dx+\int i (e+f x) \csc (i c+i d x)dx}{a}\right )}{a}-\frac {-\frac {f \log (-i \sinh (c+d x))}{d^2}+\frac {(e+f x) \coth (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {i \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \left (-\frac {b \int \frac {(e+f x) \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x) \csc (i c+i d x)dx-i \int (e+f x) \sin (i c+i d x)dx}{a}\right )}{a}-\frac {-\frac {f \log (-i \sinh (c+d x))}{d^2}+\frac {(e+f x) \coth (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {i \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \left (-\frac {b \int \frac {(e+f x) \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x) \csc (i c+i d x)dx-i \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \cosh (c+d x)dx}{d}\right )}{a}\right )}{a}-\frac {-\frac {f \log (-i \sinh (c+d x))}{d^2}+\frac {(e+f x) \coth (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {i \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \left (-\frac {b \int \frac {(e+f x) \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x) \csc (i c+i d x)dx-i \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{a}\right )}{a}-\frac {-\frac {f \log (-i \sinh (c+d x))}{d^2}+\frac {(e+f x) \coth (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle \frac {i \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \left (-\frac {b \int \frac {(e+f x) \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x) \csc (i c+i d x)dx-i \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{a}\right )}{a}-\frac {-\frac {f \log (-i \sinh (c+d x))}{d^2}+\frac {(e+f x) \coth (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{a}\right )}{a}\) |
Input:
Int[((e + f*x)*Coth[c + d*x]^2*Csch[c + d*x])/(a + b*Sinh[c + d*x]),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(1283\) vs. \(2(379)=758\).
Time = 0.91 (sec) , antiderivative size = 1284, normalized size of antiderivative = 3.11
Input:
int((f*x+e)*coth(d*x+c)^2*csch(d*x+c)/(a+b*sinh(d*x+c)),x,method=_RETURNVE RBOSE)
Output:
1/d^2/a*b*f/(a^2+b^2)^(1/2)*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2 +b^2)^(1/2)))+2/d/a^3*b^3*e/(a^2+b^2)^(1/2)*arctanh(1/2*(2*b*exp(d*x+c)+2* a)/(a^2+b^2)^(1/2))-1/d^2*b^2/a^3*f*dilog(exp(d*x+c)+1)+1/d*b^2/a^3*e*ln(e xp(d*x+c)-1)-1/d*b^2/a^3*e*ln(exp(d*x+c)+1)-1/d^2*b/a^2*f*ln(exp(d*x+c)-1) -1/d^2*b/a^2*f*ln(exp(d*x+c)+1)-1/d^2*b^2/a^3*f*dilog(exp(d*x+c))+1/d^2/a* b*f/(a^2+b^2)^(1/2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2) ))*c-2/d^2/a^3*c*b^3*f/(a^2+b^2)^(1/2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a ^2+b^2)^(1/2))-1/d^2/a^3*b^3*f/(a^2+b^2)^(1/2)*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^3*f/(a^2+b^2)^(1/2)*ln((b*ex p(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c-1/d/a*b*f/(a^2+b^2)^(1/ 2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x+1/d/a*b*f/ (a^2+b^2)^(1/2)*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^3*f/(a^2+b^2)^(1/2)*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^3*f/(a^2+b^2)^(1/2)*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*b*f/(a^2+b^2)^(1/2)*ln((-b*exp(d* x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c-2/a/d^2*c*f*b/(a^2+b^2)^(1 /2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-1/2/d^2*c*f/a*ln(exp (d*x+c)-1)-1/2/d^2/a*f*dilog(exp(d*x+c))-1/2/d/a*e*ln(exp(d*x+c)+1)+1/2/d/ a*e*ln(exp(d*x+c)-1)-1/d^2*b^2/a^3*c*f*ln(exp(d*x+c)-1)-1/d*b^2/a^3*f*ln(e xp(d*x+c)+1)*x+2/d^2/a^2*b*f*ln(exp(d*x+c))-1/2/d/a*f*ln(exp(d*x+c)+1)*...
Leaf count of result is larger than twice the leaf count of optimal. 3585 vs. \(2 (373) = 746\).
Time = 0.22 (sec) , antiderivative size = 3585, normalized size of antiderivative = 8.68 \[ \int \frac {(e+f x) \coth ^2(c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)*coth(d*x+c)^2*csch(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm ="fricas")
Output:
Too large to include
\[ \int \frac {(e+f x) \coth ^2(c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (e + f x\right ) \coth ^{2}{\left (c + d x \right )} \operatorname {csch}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \] Input:
integrate((f*x+e)*coth(d*x+c)**2*csch(d*x+c)/(a+b*sinh(d*x+c)),x)
Output:
Integral((e + f*x)*coth(c + d*x)**2*csch(c + d*x)/(a + b*sinh(c + d*x)), x )
\[ \int \frac {(e+f x) \coth ^2(c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \coth \left (d x + c\right )^{2} \operatorname {csch}\left (d x + c\right )}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)*coth(d*x+c)^2*csch(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm ="maxima")
Output:
(2*a^2*d*integrate(1/4*x/(a^3*d*e^(d*x + c) + a^3*d), x) + 4*b^2*d*integra te(1/4*x/(a^3*d*e^(d*x + c) + a^3*d), x) + 2*a^2*d*integrate(1/4*x/(a^3*d* e^(d*x + c) - a^3*d), x) + 4*b^2*d*integrate(1/4*x/(a^3*d*e^(d*x + c) - a^ 3*d), x) + a*b*((d*x + c)/(a^3*d^2) - log(e^(d*x + c) + 1)/(a^3*d^2)) + a* b*((d*x + c)/(a^3*d^2) - log(e^(d*x + c) - 1)/(a^3*d^2)) - 2*(a^2*b*e^c + b^3*e^c)*integrate(x*e^(d*x)/(a^3*b*e^(2*d*x + 2*c) + 2*a^4*e^(d*x + c) - a^3*b), x) + (2*b*d*x*e^(2*d*x + 2*c) - 2*b*d*x - (a*d*x*e^(3*c) + a*e^(3* c))*e^(3*d*x) - (a*d*x*e^c - a*e^c)*e^(d*x))/(a^2*d^2*e^(4*d*x + 4*c) - 2* a^2*d^2*e^(2*d*x + 2*c) + a^2*d^2))*f + 1/2*e*(2*(a*e^(-d*x - c) + 2*b*e^( -2*d*x - 2*c) + a*e^(-3*d*x - 3*c) - 2*b)/((2*a^2*e^(-2*d*x - 2*c) - a^2*e ^(-4*d*x - 4*c) - a^2)*d) - (a^2 + 2*b^2)*log(e^(-d*x - c) + 1)/(a^3*d) + (a^2 + 2*b^2)*log(e^(-d*x - c) - 1)/(a^3*d) - 2*(a^2*b + b^3)*log((b*e^(-d *x - c) - a - sqrt(a^2 + b^2))/(b*e^(-d*x - c) - a + sqrt(a^2 + b^2)))/(sq rt(a^2 + b^2)*a^3*d))
Timed out. \[ \int \frac {(e+f x) \coth ^2(c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)*coth(d*x+c)^2*csch(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm ="giac")
Output:
Timed out
Timed out. \[ \int \frac {(e+f x) \coth ^2(c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\mathrm {coth}\left (c+d\,x\right )}^2\,\left (e+f\,x\right )}{\mathrm {sinh}\left (c+d\,x\right )\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \] Input:
int((coth(c + d*x)^2*(e + f*x))/(sinh(c + d*x)*(a + b*sinh(c + d*x))),x)
Output:
int((coth(c + d*x)^2*(e + f*x))/(sinh(c + d*x)*(a + b*sinh(c + d*x))), x)
\[ \int \frac {(e+f x) \coth ^2(c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {too large to display} \] Input:
int((f*x+e)*coth(d*x+c)^2*csch(d*x+c)/(a+b*sinh(d*x+c)),x)
Output:
( - 12*e**(4*c + 4*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sq rt(a**2 + b**2))*b**4*d*e*i + 24*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**4*d*e*i - 12*sqrt(a**2 + b** 2)*atan((e**(c + d*x)*b*i + a*i)/sqrt(a**2 + b**2))*b**4*d*e*i - 192*e**(7 *c + 4*d*x)*int((e**(3*d*x)*x)/(e**(8*c + 8*d*x)*b + 2*e**(7*c + 7*d*x)*a - 4*e**(6*c + 6*d*x)*b - 6*e**(5*c + 5*d*x)*a + 6*e**(4*c + 4*d*x)*b + 6*e **(3*c + 3*d*x)*a - 4*e**(2*c + 2*d*x)*b - 2*e**(c + d*x)*a + b),x)*a**6*d **2*f - 176*e**(7*c + 4*d*x)*int((e**(3*d*x)*x)/(e**(8*c + 8*d*x)*b + 2*e* *(7*c + 7*d*x)*a - 4*e**(6*c + 6*d*x)*b - 6*e**(5*c + 5*d*x)*a + 6*e**(4*c + 4*d*x)*b + 6*e**(3*c + 3*d*x)*a - 4*e**(2*c + 2*d*x)*b - 2*e**(c + d*x) *a + b),x)*a**4*b**2*d**2*f + 128*e**(6*c + 4*d*x)*int((e**(2*d*x)*x)/(e** (8*c + 8*d*x)*b + 2*e**(7*c + 7*d*x)*a - 4*e**(6*c + 6*d*x)*b - 6*e**(5*c + 5*d*x)*a + 6*e**(4*c + 4*d*x)*b + 6*e**(3*c + 3*d*x)*a - 4*e**(2*c + 2*d *x)*b - 2*e**(c + d*x)*a + b),x)*a**5*b*d**2*f + 96*e**(6*c + 4*d*x)*int(( e**(2*d*x)*x)/(e**(8*c + 8*d*x)*b + 2*e**(7*c + 7*d*x)*a - 4*e**(6*c + 6*d *x)*b - 6*e**(5*c + 5*d*x)*a + 6*e**(4*c + 4*d*x)*b + 6*e**(3*c + 3*d*x)*a - 4*e**(2*c + 2*d*x)*b - 2*e**(c + d*x)*a + b),x)*a**3*b**3*d**2*f - 16*e **(5*c + 4*d*x)*int((e**(d*x)*x)/(e**(8*c + 8*d*x)*b + 2*e**(7*c + 7*d*x)* a - 4*e**(6*c + 6*d*x)*b - 6*e**(5*c + 5*d*x)*a + 6*e**(4*c + 4*d*x)*b + 6 *e**(3*c + 3*d*x)*a - 4*e**(2*c + 2*d*x)*b - 2*e**(c + d*x)*a + b),x)*a...