Integrand size = 26, antiderivative size = 420 \[ \int \frac {(e+f x) \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {(e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a^3 d}+\frac {b (e+f x) \coth (c+d x)}{a^2 d}-\frac {f \text {csch}(c+d x)}{2 a d^2}-\frac {(e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {b^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2} d}+\frac {b^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2} d}-\frac {b f \log (\sinh (c+d x))}{a^2 d^2}+\frac {f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 a d^2}-\frac {b^2 f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^3 d^2}-\frac {f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 a d^2}+\frac {b^2 f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^3 d^2}-\frac {b^3 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2} d^2}+\frac {b^3 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2} d^2} \] 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^3*(f*x+e)*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^3/(a^ 2+b^2)^(1/2)/d+b^3*(f*x+e)*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^3/(a^2 +b^2)^(1/2)/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^3*f*polylog(2,-b*exp(d*x+c)/(a-(a^2+ b^2)^(1/2)))/a^3/(a^2+b^2)^(1/2)/d^2+b^3*f*polylog(2,-b*exp(d*x+c)/(a+(a^2 +b^2)^(1/2)))/a^3/(a^2+b^2)^(1/2)/d^2
Time = 8.22 (sec) , antiderivative size = 617, normalized size of antiderivative = 1.47 \[ \int \frac {(e+f x) \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\left (2 b d e \cosh \left (\frac {1}{2} (c+d x)\right )-a f \cosh \left (\frac {1}{2} (c+d x)\right )-2 b c f \cosh \left (\frac {1}{2} (c+d x)\right )+2 b f (c+d x) \cosh \left (\frac {1}{2} (c+d x)\right )\right ) \text {csch}\left (\frac {1}{2} (c+d x)\right )}{4 a^2 d^2}+\frac {(-d e+c f-f (c+d x)) \text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{8 a d^2}-\frac {2 a b f (c+d x)+\left (2 a b f+a^2 (d e+d f x)-2 b^2 (d e+d f x)\right ) \log \left (1-e^{-c-d x}\right )+\left (2 a b f-a^2 (d e+d f x)+2 b^2 (d e+d f x)\right ) \log \left (1+e^{-c-d x}\right )+\left (a^2-2 b^2\right ) f \operatorname {PolyLog}\left (2,-e^{-c-d x}\right )-\left (a^2-2 b^2\right ) f \operatorname {PolyLog}\left (2,e^{-c-d x}\right )}{2 a^3 d^2}-\frac {b^3 \left (-2 d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+2 c f \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+f \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{a^3 \sqrt {a^2+b^2} d^2}+\frac {(-d e+c f-f (c+d x)) \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{8 a d^2}+\frac {\text {sech}\left (\frac {1}{2} (c+d x)\right ) \left (2 b d e \sinh \left (\frac {1}{2} (c+d x)\right )+a f \sinh \left (\frac {1}{2} (c+d x)\right )-2 b c f \sinh \left (\frac {1}{2} (c+d x)\right )+2 b f (c+d x) \sinh \left (\frac {1}{2} (c+d x)\right )\right )}{4 a^2 d^2} \] Input:
Integrate[((e + f*x)*Csch[c + d*x]^3)/(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^3*(-2*d*e*ArcTanh[(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*PolyLog[2, (b*E^( c + d*x))/(-a + Sqrt[a^2 + b^2])] - f*PolyLog[2, -((b*E^(c + d*x))/(a + Sq rt[a^2 + b^2]))]))/(a^3*Sqrt[a^2 + b^2]*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.
| \(\displaystyle \int \frac {(e+f x) \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6109 |
| \(\displaystyle \frac {\int (e+f x) \text {csch}^3(c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int -i (e+f x) \csc (i c+i d x)^3dx}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {i \int (e+f x) \csc (i c+i d x)^3dx}{a}\) |
| \(\Big \downarrow \) 4673 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {i \left (\frac {1}{2} \int -i (e+f x) \text {csch}(c+d x)dx-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {i \left (-\frac {1}{2} i \int (e+f x) \text {csch}(c+d x)dx-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {i \left (-\frac {1}{2} i \int i (e+f x) \csc (i c+i d x)dx-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {i \left (\frac {1}{2} \int (e+f x) \csc (i c+i d x)dx-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {i \left (\frac {1}{2} \left (\frac {i f \int \log \left (1-e^{c+d x}\right )dx}{d}-\frac {i f \int \log \left (1+e^{c+d x}\right )dx}{d}+\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {i \left (\frac {1}{2} \left (\frac {i f \int e^{-c-d x} \log \left (1-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {i f \int e^{-c-d x} \log \left (1+e^{c+d x}\right )de^{c+d x}}{d^2}+\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 6109 |
| \(\displaystyle -\frac {b \left (\frac {\int (e+f x) \text {csch}^2(c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}-\frac {i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \left (-\frac {b \int \frac {(e+f x) \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int -\left ((e+f x) \csc (i c+i d x)^2\right )dx}{a}\right )}{a}-\frac {i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {b \left (-\frac {b \int \frac {(e+f x) \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\int (e+f x) \csc (i c+i d x)^2dx}{a}\right )}{a}-\frac {i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 4672 |
| \(\displaystyle -\frac {b \left (-\frac {b \int \frac {(e+f x) \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\frac {(e+f x) \coth (c+d x)}{d}-\frac {i f \int -i \coth (c+d x)dx}{d}}{a}\right )}{a}-\frac {i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \left (-\frac {b \int \frac {(e+f x) \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\frac {(e+f x) \coth (c+d x)}{d}-\frac {f \int \coth (c+d x)dx}{d}}{a}\right )}{a}-\frac {i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \left (-\frac {b \int \frac {(e+f x) \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\frac {(e+f x) \coth (c+d x)}{d}-\frac {f \int -i \tan \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}}{a}\right )}{a}-\frac {i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \left (-\frac {b \int \frac {(e+f x) \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\frac {(e+f x) \coth (c+d x)}{d}+\frac {i f \int \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )dx}{d}}{a}\right )}{a}-\frac {i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle -\frac {b \left (-\frac {b \int \frac {(e+f x) \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\frac {(e+f x) \coth (c+d x)}{d}-\frac {f \log (-i \sinh (c+d x))}{d^2}}{a}\right )}{a}-\frac {i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 6109 |
| \(\displaystyle -\frac {b \left (-\frac {b \left (\frac {\int (e+f x) \text {csch}(c+d x)dx}{a}-\frac {b \int \frac {e+f x}{a+b \sinh (c+d x)}dx}{a}\right )}{a}-\frac {\frac {(e+f x) \coth (c+d x)}{d}-\frac {f \log (-i \sinh (c+d x))}{d^2}}{a}\right )}{a}-\frac {i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \left (-\frac {b \left (\frac {\int i (e+f x) \csc (i c+i d x)dx}{a}-\frac {b \int \frac {e+f x}{a-i b \sin (i c+i d x)}dx}{a}\right )}{a}-\frac {\frac {(e+f x) \coth (c+d x)}{d}-\frac {f \log (-i \sinh (c+d x))}{d^2}}{a}\right )}{a}-\frac {i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \left (-\frac {b \left (\frac {i \int (e+f x) \csc (i c+i d x)dx}{a}-\frac {b \int \frac {e+f x}{a-i b \sin (i c+i d x)}dx}{a}\right )}{a}-\frac {\frac {(e+f x) \coth (c+d x)}{d}-\frac {f \log (-i \sinh (c+d x))}{d^2}}{a}\right )}{a}-\frac {i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 3803 |
| \(\displaystyle -\frac {b \left (-\frac {b \left (-\frac {2 b \int -\frac {e^{c+d x} (e+f x)}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{a}+\frac {i \int (e+f x) \csc (i c+i d x)dx}{a}\right )}{a}-\frac {\frac {(e+f x) \coth (c+d x)}{d}-\frac {f \log (-i \sinh (c+d x))}{d^2}}{a}\right )}{a}-\frac {i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {b \left (-\frac {b \left (\frac {2 b \int \frac {e^{c+d x} (e+f x)}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{a}+\frac {i \int (e+f x) \csc (i c+i d x)dx}{a}\right )}{a}-\frac {\frac {(e+f x) \coth (c+d x)}{d}-\frac {f \log (-i \sinh (c+d x))}{d^2}}{a}\right )}{a}-\frac {i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 2694 |
| \(\displaystyle -\frac {b \left (-\frac {b \left (\frac {2 b \left (\frac {b \int -\frac {e^{c+d x} (e+f x)}{2 \left (a+b e^{c+d x}-\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}-\frac {b \int -\frac {e^{c+d x} (e+f x)}{2 \left (a+b e^{c+d x}+\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}\right )}{a}+\frac {i \int (e+f x) \csc (i c+i d x)dx}{a}\right )}{a}-\frac {\frac {(e+f x) \coth (c+d x)}{d}-\frac {f \log (-i \sinh (c+d x))}{d^2}}{a}\right )}{a}-\frac {i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b \left (-\frac {b \left (\frac {2 b \left (\frac {b \int \frac {e^{c+d x} (e+f x)}{a+b e^{c+d x}+\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}-\frac {b \int \frac {e^{c+d x} (e+f x)}{a+b e^{c+d x}-\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}\right )}{a}+\frac {i \int (e+f x) \csc (i c+i d x)dx}{a}\right )}{a}-\frac {\frac {(e+f x) \coth (c+d x)}{d}-\frac {f \log (-i \sinh (c+d x))}{d^2}}{a}\right )}{a}-\frac {i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {b \left (-\frac {b \left (\frac {2 b \left (\frac {b \left (\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {f \int \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {f \int \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a}+\frac {i \int (e+f x) \csc (i c+i d x)dx}{a}\right )}{a}-\frac {\frac {(e+f x) \coth (c+d x)}{d}-\frac {f \log (-i \sinh (c+d x))}{d^2}}{a}\right )}{a}-\frac {i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {b \left (-\frac {b \left (\frac {2 b \left (\frac {b \left (\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2}\right )}{2 \sqrt {a^2+b^2}}\right )}{a}+\frac {i \int (e+f x) \csc (i c+i d x)dx}{a}\right )}{a}-\frac {\frac {(e+f x) \coth (c+d x)}{d}-\frac {f \log (-i \sinh (c+d x))}{d^2}}{a}\right )}{a}-\frac {i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {b \left (-\frac {b \left (\frac {2 b \left (\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a}+\frac {i \int (e+f x) \csc (i c+i d x)dx}{a}\right )}{a}-\frac {\frac {(e+f x) \coth (c+d x)}{d}-\frac {f \log (-i \sinh (c+d x))}{d^2}}{a}\right )}{a}-\frac {i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle -\frac {b \left (-\frac {b \left (\frac {2 b \left (\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a}+\frac {i \left (\frac {i f \int \log \left (1-e^{c+d x}\right )dx}{d}-\frac {i f \int \log \left (1+e^{c+d x}\right )dx}{d}+\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{a}\right )}{a}-\frac {\frac {(e+f x) \coth (c+d x)}{d}-\frac {f \log (-i \sinh (c+d x))}{d^2}}{a}\right )}{a}-\frac {i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {b \left (-\frac {b \left (\frac {2 b \left (\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a}+\frac {i \left (\frac {i f \int e^{-c-d x} \log \left (1-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {i f \int e^{-c-d x} \log \left (1+e^{c+d x}\right )de^{c+d x}}{d^2}+\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{a}\right )}{a}-\frac {\frac {(e+f x) \coth (c+d x)}{d}-\frac {f \log (-i \sinh (c+d x))}{d^2}}{a}\right )}{a}-\frac {i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
Input:
Int[((e + f*x)*Csch[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(860\) vs. \(2(386)=772\).
Time = 0.59 (sec) , antiderivative size = 861, normalized size of antiderivative = 2.05
| method | result | size |
| risch | \(-\frac {a d f x \,{\mathrm e}^{3 d x +3 c}+a d e \,{\mathrm e}^{3 d x +3 c}-2 b d f x \,{\mathrm e}^{2 d x +2 c}+a d f x \,{\mathrm e}^{d x +c}+a f \,{\mathrm e}^{3 d x +3 c}-2 b d e \,{\mathrm e}^{2 d x +2 c}+a d e \,{\mathrm e}^{d x +c}+2 b d f x -a f \,{\mathrm e}^{d x +c}+2 b d e}{d^{2} a^{2} \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}+\frac {2 b^{3} e \,\operatorname {arctanh}\left (\frac {2 b \,{\mathrm e}^{d x +c}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{d \,a^{3} \sqrt {a^{2}+b^{2}}}-\frac {2 c \,b^{3} f \,\operatorname {arctanh}\left (\frac {2 b \,{\mathrm e}^{d x +c}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{d^{2} a^{3} \sqrt {a^{2}+b^{2}}}-\frac {b^{3} f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} a^{3} \sqrt {a^{2}+b^{2}}}+\frac {e \ln \left ({\mathrm e}^{d x +c}+1\right )}{2 a d}-\frac {e \ln \left ({\mathrm e}^{d x +c}-1\right )}{2 a d}+\frac {b^{3} f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} a^{3} \sqrt {a^{2}+b^{2}}}-\frac {b^{2} f \ln \left ({\mathrm e}^{d x +c}+1\right ) x}{d \,a^{3}}-\frac {b f \ln \left ({\mathrm e}^{d x +c}-1\right )}{d^{2} a^{2}}-\frac {b f \ln \left ({\mathrm e}^{d x +c}+1\right )}{d^{2} a^{2}}-\frac {b^{2} f \operatorname {dilog}\left ({\mathrm e}^{d x +c}+1\right )}{d^{2} a^{3}}-\frac {b^{2} f \operatorname {dilog}\left ({\mathrm e}^{d x +c}\right )}{d^{2} a^{3}}+\frac {2 b f \ln \left ({\mathrm e}^{d x +c}\right )}{d^{2} a^{2}}+\frac {c f \ln \left ({\mathrm e}^{d x +c}-1\right )}{2 a \,d^{2}}+\frac {f \ln \left ({\mathrm e}^{d x +c}+1\right ) x}{2 a d}-\frac {b^{2} e \ln \left ({\mathrm e}^{d x +c}+1\right )}{d \,a^{3}}+\frac {b^{2} e \ln \left ({\mathrm e}^{d x +c}-1\right )}{d \,a^{3}}-\frac {c \,b^{2} f \ln \left ({\mathrm e}^{d x +c}-1\right )}{d^{2} a^{3}}-\frac {b^{3} f \operatorname {dilog}\left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} a^{3} \sqrt {a^{2}+b^{2}}}+\frac {b^{3} f \operatorname {dilog}\left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} a^{3} \sqrt {a^{2}+b^{2}}}+\frac {f \operatorname {dilog}\left ({\mathrm e}^{d x +c}+1\right )}{2 d^{2} a}+\frac {f \operatorname {dilog}\left ({\mathrm e}^{d x +c}\right )}{2 d^{2} a}-\frac {b^{3} f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,a^{3} \sqrt {a^{2}+b^{2}}}+\frac {b^{3} f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,a^{3} \sqrt {a^{2}+b^{2}}}\) | \(861\) |
Input:
int((f*x+e)*csch(d*x+c)^3/(a+b*sinh(d*x+c)),x,method=_RETURNVERBOSE)
Output:
-(a*d*f*x*exp(3*d*x+3*c)+a*d*e*exp(3*d*x+3*c)-2*b*d*f*x*exp(2*d*x+2*c)+a*d *f*x*exp(d*x+c)+a*f*exp(3*d*x+3*c)-2*b*d*e*exp(2*d*x+2*c)+a*d*e*exp(d*x+c) +2*b*d*f*x-a*f*exp(d*x+c)+2*b*d*e)/d^2/a^2/(exp(2*d*x+2*c)-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))-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+1/2/a/d*e*ln(exp(d*x+c)+1)-1/2/a/d*e*ln(exp(d*x+c )-1)+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/a^3*b^2*f*ln(exp(d*x+c)+1)*x-1/d^2/a^2*b*f*ln(ex p(d*x+c)-1)-1/d^2/a^2*b*f*ln(exp(d*x+c)+1)-1/d^2/a^3*b^2*f*dilog(exp(d*x+c )+1)-1/d^2/a^3*b^2*f*dilog(exp(d*x+c))+2/d^2/a^2*b*f*ln(exp(d*x+c))+1/2/a/ d^2*c*f*ln(exp(d*x+c)-1)+1/2/a/d*f*ln(exp(d*x+c)+1)*x-1/d/a^3*b^2*e*ln(exp (d*x+c)+1)+1/d/a^3*b^2*e*ln(exp(d*x+c)-1)-1/d^2/a^3*c*b^2*f*ln(exp(d*x+c)- 1)-1/d^2/a^3*b^3*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)))+1/d^2/a^3*b^3*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)))+1/2/d^2*f/a*dilog(exp(d*x+c)+1)+1 /2/d^2*f*dilog(exp(d*x+c))/a-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
Leaf count of result is larger than twice the leaf count of optimal. 4720 vs. \(2 (380) = 760\).
Time = 0.20 (sec) , antiderivative size = 4720, normalized size of antiderivative = 11.24 \[ \int \frac {(e+f x) \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)*csch(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {(e+f x) \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (e + f x\right ) \operatorname {csch}^{3}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \] Input:
integrate((f*x+e)*csch(d*x+c)**3/(a+b*sinh(d*x+c)),x)
Output:
Integral((e + f*x)*csch(c + d*x)**3/(a + b*sinh(c + d*x)), x)
\[ \int \frac {(e+f x) \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \operatorname {csch}\left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)*csch(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="maxima")
Output:
-(8*b^3*integrate(1/4*x*e^(d*x + c)/(a^3*b*e^(2*d*x + 2*c) + 2*a^4*e^(d*x + c) - a^3*b), x) + 8*a^2*d*integrate(1/16*x/(a^3*d*e^(d*x + c) + a^3*d), x) - 16*b^2*d*integrate(1/16*x/(a^3*d*e^(d*x + c) + a^3*d), x) + 8*a^2*d*i ntegrate(1/16*x/(a^3*d*e^(d*x + c) - a^3*d), x) - 16*b^2*d*integrate(1/16* 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*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*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)))/(sqrt(a^2 + b^2 )*a^3*d) - 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))
Timed out. \[ \int \frac {(e+f x) \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)*csch(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {(e+f x) \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {e+f\,x}{{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \] Input:
int((e + f*x)/(sinh(c + d*x)^3*(a + b*sinh(c + d*x))),x)
Output:
int((e + f*x)/(sinh(c + d*x)^3*(a + b*sinh(c + d*x))), x)
\[ \int \frac {(e+f x) \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {too large to display} \] Input:
int((f*x+e)*csch(d*x+c)^3/(a+b*sinh(d*x+c)),x)
Output:
( - 4*e**(4*c + 4*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqr t(a**2 + b**2))*b**4*d*e*i + 8*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 - 4*sqrt(a**2 + b**2)* atan((e**(c + d*x)*b*i + a*i)/sqrt(a**2 + b**2))*b**4*d*e*i - 64*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 - 64*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 + 32*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 + 32*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 - 4*e**(4*c + 4*d*x)*log(e**(c + d*x) - 1)*a**5*f - e**(4*c + 4*d*x)*log(e**(c + d*x) - 1)*a**4*b*d*e - 4*e**(4*c + 4*d*x)*log(e**(c + d*x) - 1)*a**3*b**2*f + e **(4*c + 4*d*x)*log(e**(c + d*x) - 1)*a**2*b**3*d*e + 2*e**(4*c + 4*d*x...