Integrand size = 34, antiderivative size = 518 \[ \int \frac {(e+f x)^2 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b (e+f x)^3}{3 a^2 f}-\frac {\left (a^2+b^2\right ) (e+f x)^3}{3 a^2 b f}-\frac {4 f (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d^2}-\frac {(e+f x)^2 \text {csch}(c+d x)}{a d}+\frac {\left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 b d}+\frac {\left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 b d}-\frac {b (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}-\frac {2 f^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^3}+\frac {2 f^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^3}+\frac {2 \left (a^2+b^2\right ) f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 b d^2}+\frac {2 \left (a^2+b^2\right ) f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 b d^2}-\frac {b f (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a^2 d^2}-\frac {2 \left (a^2+b^2\right ) f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 b d^3}-\frac {2 \left (a^2+b^2\right ) f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 b d^3}+\frac {b f^2 \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a^2 d^3} \] Output:
1/3*b*(f*x+e)^3/a^2/f-1/3*(a^2+b^2)*(f*x+e)^3/a^2/b/f-4*f*(f*x+e)*arctanh( exp(d*x+c))/a/d^2-(f*x+e)^2*csch(d*x+c)/a/d+(a^2+b^2)*(f*x+e)^2*ln(1+b*exp (d*x+c)/(a-(a^2+b^2)^(1/2)))/a^2/b/d+(a^2+b^2)*(f*x+e)^2*ln(1+b*exp(d*x+c) /(a+(a^2+b^2)^(1/2)))/a^2/b/d-b*(f*x+e)^2*ln(1-exp(2*d*x+2*c))/a^2/d-2*f^2 *polylog(2,-exp(d*x+c))/a/d^3+2*f^2*polylog(2,exp(d*x+c))/a/d^3+2*(a^2+b^2 )*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^2/b/d^2+2*(a^2+ b^2)*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^2/b/d^2-b*f* (f*x+e)*polylog(2,exp(2*d*x+2*c))/a^2/d^2-2*(a^2+b^2)*f^2*polylog(3,-b*exp (d*x+c)/(a-(a^2+b^2)^(1/2)))/a^2/b/d^3-2*(a^2+b^2)*f^2*polylog(3,-b*exp(d* x+c)/(a+(a^2+b^2)^(1/2)))/a^2/b/d^3+1/2*b*f^2*polylog(3,exp(2*d*x+2*c))/a^ 2/d^3
Leaf count is larger than twice the leaf count of optimal. \(1806\) vs. \(2(518)=1036\).
Time = 9.80 (sec) , antiderivative size = 1806, normalized size of antiderivative = 3.49 \[ \int \frac {(e+f x)^2 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Input:
Integrate[((e + f*x)^2*Cosh[c + d*x]*Coth[c + d*x]^2)/(a + b*Sinh[c + d*x] ),x]
Output:
(3*d^2*e*(-1 + E^(2*c))*f*(b*d*e - 2*a*f)*x + 3*d^2*e*(-1 + E^(2*c))*f*(b* d*e + 2*a*f)*x + 2*b*d^3*(e + f*x)^3 - 6*d*(-1 + E^(2*c))*f^2*(b*d*e - a*f )*x*Log[1 - E^(-c - d*x)] - 3*b*d^2*(-1 + E^(2*c))*f^3*x^2*Log[1 - E^(-c - d*x)] - 6*d*(-1 + E^(2*c))*f^2*(b*d*e + a*f)*x*Log[1 + E^(-c - d*x)] - 3* b*d^2*(-1 + E^(2*c))*f^3*x^2*Log[1 + E^(-c - d*x)] - 3*d*e*(-1 + E^(2*c))* f*(b*d*e - 2*a*f)*Log[1 - E^(c + d*x)] - 3*d*e*(-1 + E^(2*c))*f*(b*d*e + 2 *a*f)*Log[1 + E^(c + d*x)] + 6*(-1 + E^(2*c))*f^2*(b*d*e + a*f)*PolyLog[2, -E^(-c - d*x)] + 6*b*d*(-1 + E^(2*c))*f^3*x*PolyLog[2, -E^(-c - d*x)] - 6 *(-1 + E^(2*c))*f^2*(-(b*d*e) + a*f)*PolyLog[2, E^(-c - d*x)] + 6*b*d*(-1 + E^(2*c))*f^3*x*PolyLog[2, E^(-c - d*x)] + 6*b*(-1 + E^(2*c))*f^3*PolyLog [3, -E^(-c - d*x)] + 6*b*(-1 + E^(2*c))*f^3*PolyLog[3, E^(-c - d*x)])/(3*a ^2*d^3*(-1 + E^(2*c))*f) - ((a^2 + b^2)*(6*e^2*E^(2*c)*x + 6*e*E^(2*c)*f*x ^2 + 2*E^(2*c)*f^2*x^3 + (6*a*Sqrt[a^2 + b^2]*e^2*ArcTan[(a + b*E^(c + d*x ))/Sqrt[-a^2 - b^2]])/(Sqrt[-(a^2 + b^2)^2]*d) + (6*a*Sqrt[-(a^2 + b^2)^2] *e^2*E^(2*c)*ArcTan[(a + b*E^(c + d*x))/Sqrt[-a^2 - b^2]])/((a^2 + b^2)^(3 /2)*d) - (6*a*Sqrt[-(a^2 + b^2)^2]*e^2*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^ 2 + b^2]])/((-a^2 - b^2)^(3/2)*d) + (6*a*Sqrt[-(a^2 + b^2)^2]*e^2*E^(2*c)* ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]])/((-a^2 - b^2)^(3/2)*d) + (3* e^2*Log[2*a*E^(c + d*x) + b*(-1 + E^(2*(c + d*x)))])/d - (3*e^2*E^(2*c)*Lo g[2*a*E^(c + d*x) + b*(-1 + E^(2*(c + d*x)))])/d + (6*e*f*x*Log[1 + (b*...
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.
| \(\displaystyle \int \frac {(e+f x)^2 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6119 |
| \(\displaystyle \frac {\int (e+f x)^2 \cosh (c+d x) \coth ^2(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 5973 |
| \(\displaystyle \frac {\int (e+f x)^2 \cosh (c+d x)dx+\int (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)dx+\int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-\frac {2 i f \int -i (e+f x) \sinh (c+d x)dx}{d}+\int (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)dx+\frac {(e+f x)^2 \sinh (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {-\frac {2 f \int (e+f x) \sinh (c+d x)dx}{d}+\int (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)dx+\frac {(e+f x)^2 \sinh (c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-\frac {2 f \int -i (e+f x) \sin (i c+i d x)dx}{d}+\int (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)dx+\frac {(e+f x)^2 \sinh (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\frac {2 i f \int (e+f x) \sin (i c+i d x)dx}{d}+\int (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)dx+\frac {(e+f x)^2 \sinh (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \cosh (c+d x)dx}{d}\right )}{d}+\int (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)dx+\frac {(e+f x)^2 \sinh (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{d}+\int (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)dx+\frac {(e+f x)^2 \sinh (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)dx+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 5975 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\frac {2 f \int (e+f x) \text {csch}(c+d x)dx}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {(e+f x)^2 \text {csch}(c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\frac {2 f \int i (e+f x) \csc (i c+i d x)dx}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {(e+f x)^2 \text {csch}(c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\frac {2 i f \int (e+f x) \csc (i c+i d x)dx}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {(e+f x)^2 \text {csch}(c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\frac {2 i f \left (\frac {i f \int \log \left (1-e^{c+d x}\right )dx}{d}-\frac {i f \int \log \left (1+e^{c+d x}\right )dx}{d}+\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {(e+f x)^2 \text {csch}(c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\frac {2 i f \left (\frac {i f \int e^{-c-d x} \log \left (1-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {i f \int e^{-c-d x} \log \left (1+e^{c+d x}\right )de^{c+d x}}{d^2}+\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {(e+f x)^2 \text {csch}(c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\frac {2 i f \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {(e+f x)^2 \text {csch}(c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 6119 |
| \(\displaystyle -\frac {b \left (\frac {\int (e+f x)^2 \cosh ^2(c+d x) \coth (c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}+\frac {\frac {2 i f \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {(e+f x)^2 \text {csch}(c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 5973 |
| \(\displaystyle -\frac {b \left (\frac {\int (e+f x)^2 \coth (c+d x)dx+\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}+\frac {\frac {2 i f \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {(e+f x)^2 \text {csch}(c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 i f \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {(e+f x)^2 \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx+\int -i (e+f x)^2 \tan \left (i c+i d x+\frac {\pi }{2}\right )dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {2 i f \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {(e+f x)^2 \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx-i \int (e+f x)^2 \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle \frac {\frac {2 i f \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {(e+f x)^2 \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx-i \left (2 i \int \frac {e^{2 c+2 d x-i \pi } (e+f x)^2}{1+e^{2 c+2 d x-i \pi }}dx-\frac {i (e+f x)^3}{3 f}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {\frac {2 i f \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {(e+f x)^2 \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx-i \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \int (e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )dx}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {\frac {2 i f \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {(e+f x)^2 \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx-i \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )dx}{2 d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {\frac {2 i f \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {(e+f x)^2 \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx-i \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 5969 |
| \(\displaystyle \frac {\frac {2 i f \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {(e+f x)^2 \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-i \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )-\frac {f \int (e+f x) \sinh ^2(c+d x)dx}{d}+\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 i f \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {(e+f x)^2 \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-i \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )-\frac {f \int -\left ((e+f x) \sin (i c+i d x)^2\right )dx}{d}+\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 i f \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {(e+f x)^2 \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-i \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )+\frac {f \int (e+f x) \sin (i c+i d x)^2dx}{d}+\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3791 |
| \(\displaystyle \frac {\frac {2 i f \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {(e+f x)^2 \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-i \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )+\frac {f \left (\frac {1}{2} \int (e+f x)dx+\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{d}+\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle \frac {\frac {2 i f \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {(e+f x)^2 \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-i \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )+\frac {f \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{d}+\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 6099 |
| \(\displaystyle \frac {\frac {2 i f \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {(e+f x)^2 \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \int (e+f x)^2 \cosh (c+d x)dx}{b^2}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{a}+\frac {-i \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )+\frac {f \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{d}+\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 d}}{a}\right )}{a}\) |
Input:
Int[((e + f*x)^2*Cosh[c + d*x]*Coth[c + d*x]^2)/(a + b*Sinh[c + d*x]),x]
Output:
$Aborted
\[\int \frac {\left (f x +e \right )^{2} \cosh \left (d x +c \right ) \coth \left (d x +c \right )^{2}}{a +b \sinh \left (d x +c \right )}d x\]
Input:
int((f*x+e)^2*cosh(d*x+c)*coth(d*x+c)^2/(a+b*sinh(d*x+c)),x)
Output:
int((f*x+e)^2*cosh(d*x+c)*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. 3506 vs. \(2 (486) = 972\).
Time = 0.18 (sec) , antiderivative size = 3506, normalized size of antiderivative = 6.77 \[ \int \frac {(e+f x)^2 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^2*cosh(d*x+c)*coth(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorit hm="fricas")
Output:
Too large to include
\[ \int \frac {(e+f x)^2 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (e + f x\right )^{2} \cosh {\left (c + d x \right )} \coth ^{2}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \] Input:
integrate((f*x+e)**2*cosh(d*x+c)*coth(d*x+c)**2/(a+b*sinh(d*x+c)),x)
Output:
Integral((e + f*x)**2*cosh(c + d*x)*coth(c + d*x)**2/(a + b*sinh(c + d*x)) , x)
\[ \int \frac {(e+f x)^2 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \cosh \left (d x + c\right ) \coth \left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)^2*cosh(d*x+c)*coth(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorit hm="maxima")
Output:
e^2*((d*x + c)/(b*d) + 2*e^(-d*x - c)/((a*e^(-2*d*x - 2*c) - a)*d) - b*log (e^(-d*x - c) + 1)/(a^2*d) - b*log(e^(-d*x - c) - 1)/(a^2*d) + (a^2 + b^2) *log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(a^2*b*d)) - 1/3*(a*d*f^2 *x^3 + 3*a*d*e*f*x^2 - (a*d*f^2*x^3*e^(2*c) + 3*a*d*e*f*x^2*e^(2*c))*e^(2* d*x) + 6*(b*f^2*x^2*e^c + 2*b*e*f*x*e^c)*e^(d*x))/(a*b*d*e^(2*d*x + 2*c) - a*b*d) - 2*e*f*log(e^(d*x + c) + 1)/(a*d^2) + 2*e*f*log(e^(d*x + c) - 1)/ (a*d^2) - (d^2*x^2*log(e^(d*x + c) + 1) + 2*d*x*dilog(-e^(d*x + c)) - 2*po lylog(3, -e^(d*x + c)))*b*f^2/(a^2*d^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)))*b*f^2/(a^2*d^3) - 2 *(b*d*e*f + a*f^2)*(d*x*log(e^(d*x + c) + 1) + dilog(-e^(d*x + c)))/(a^2*d ^3) - 2*(b*d*e*f - a*f^2)*(d*x*log(-e^(d*x + c) + 1) + dilog(e^(d*x + c))) /(a^2*d^3) + 1/3*(b*d^3*f^2*x^3 + 3*(b*d*e*f + a*f^2)*d^2*x^2)/(a^2*d^3) + 1/3*(b*d^3*f^2*x^3 + 3*(b*d*e*f - a*f^2)*d^2*x^2)/(a^2*d^3) - integrate(- 2*((a^2*b*f^2 + b^3*f^2)*x^2 + 2*(a^2*b*e*f + b^3*e*f)*x - ((a^3*f^2*e^c + a*b^2*f^2*e^c)*x^2 + 2*(a^3*e*f*e^c + a*b^2*e*f*e^c)*x)*e^(d*x))/(a^2*b^2 *e^(2*d*x + 2*c) + 2*a^3*b*e^(d*x + c) - a^2*b^2), x)
Timed out. \[ \int \frac {(e+f x)^2 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)^2*cosh(d*x+c)*coth(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorit hm="giac")
Output:
Timed out
Timed out. \[ \int \frac {(e+f x)^2 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )\,{\mathrm {coth}\left (c+d\,x\right )}^2\,{\left (e+f\,x\right )}^2}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \] Input:
int((cosh(c + d*x)*coth(c + d*x)^2*(e + f*x)^2)/(a + b*sinh(c + d*x)),x)
Output:
int((cosh(c + d*x)*coth(c + d*x)^2*(e + f*x)^2)/(a + b*sinh(c + d*x)), x)
\[ \int \frac {(e+f x)^2 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (f x +e \right )^{2} \cosh \left (d x +c \right ) \coth \left (d x +c \right )^{2}}{a +b \sinh \left (d x +c \right )}d x \] Input:
int((f*x+e)^2*cosh(d*x+c)*coth(d*x+c)^2/(a+b*sinh(d*x+c)),x)
Output:
int((f*x+e)^2*cosh(d*x+c)*coth(d*x+c)^2/(a+b*sinh(d*x+c)),x)