Integrand size = 32, antiderivative size = 638 \[ \int \frac {(e+f x)^3 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {(e+f x)^4}{4 b f}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b d}+\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b d}-\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}-\frac {3 \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b d^2}+\frac {3 \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b d^2}+\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}-\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}+\frac {6 \sqrt {a^2+b^2} f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b d^3}-\frac {6 \sqrt {a^2+b^2} f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b d^3}-\frac {6 f^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )}{a d^4}+\frac {6 f^3 \operatorname {PolyLog}\left (4,e^{c+d x}\right )}{a d^4}-\frac {6 \sqrt {a^2+b^2} f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b d^4}+\frac {6 \sqrt {a^2+b^2} f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b d^4} \] Output:
1/4*(f*x+e)^4/b/f-2*(f*x+e)^3*arctanh(exp(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/b/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/b/d-3*f*(f*x+e)^2*polylog(2,-e xp(d*x+c))/a/d^2+3*f*(f*x+e)^2*polylog(2,exp(d*x+c))/a/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/b/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/b/d ^2+6*f^2*(f*x+e)*polylog(3,-exp(d*x+c))/a/d^3-6*f^2*(f*x+e)*polylog(3,exp( d*x+c))/a/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/b/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/b/d^3-6*f^3*polylog(4,-exp(d*x+c))/a/d^4+6*f^3*p olylog(4,exp(d*x+c))/a/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/b/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/b/d^4
Time = 0.97 (sec) , antiderivative size = 781, normalized size of antiderivative = 1.22 \[ \int \frac {(e+f x)^3 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {x \left (4 e^3+6 e^2 f x+4 e f^2 x^2+f^3 x^3\right )}{4 b}+\frac {(e+f x)^3 \log \left (1-e^{c+d x}\right )-(e+f x)^3 \log \left (1+e^{c+d x}\right )-\frac {3 f \left (d^2 (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )-2 d f (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )+2 f^2 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )\right )}{d^3}+\frac {3 f \left (d^2 (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )-2 d f (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )+2 f^2 \operatorname {PolyLog}\left (4,e^{c+d x}\right )\right )}{d^3}}{a d}-\frac {\sqrt {a^2+b^2} \left (-2 d^3 e^3 \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+3 d^3 e^2 f x \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+3 d^3 e f^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+d^3 f^3 x^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-3 d^3 e^2 f x \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-3 d^3 e f^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-d^3 f^3 x^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+3 d^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-3 d^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-6 d e f^2 \operatorname {PolyLog}\left (3,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-6 d f^3 x \operatorname {PolyLog}\left (3,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+6 d e f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+6 d f^3 x \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+6 f^3 \operatorname {PolyLog}\left (4,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-6 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{a b d^4} \] Input:
Integrate[((e + f*x)^3*Cosh[c + d*x]*Coth[c + d*x])/(a + b*Sinh[c + d*x]), x]
Output:
(x*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3))/(4*b) + ((e + f*x)^3*Log[1 - E^(c + d*x)] - (e + f*x)^3*Log[1 + E^(c + d*x)] - (3*f*(d^2*(e + f*x)^2 *PolyLog[2, -E^(c + d*x)] - 2*d*f*(e + f*x)*PolyLog[3, -E^(c + d*x)] + 2*f ^2*PolyLog[4, -E^(c + d*x)]))/d^3 + (3*f*(d^2*(e + f*x)^2*PolyLog[2, E^(c + d*x)] - 2*d*f*(e + f*x)*PolyLog[3, E^(c + d*x)] + 2*f^2*PolyLog[4, E^(c + d*x)]))/d^3)/(a*d) - (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 - Sqrt[a ^2 + b^2])] + 3*d^3*e*f^2*x^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2] )] + d^3*f^3*x^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] - 3*d^3*e^ 2*f*x*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] - 3*d^3*e*f^2*x^2*Log [1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] - d^3*f^3*x^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + 3*d^2*f*(e + f*x)^2*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - 3*d^2*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] - 6*d*e*f^2*PolyLog[3, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - 6*d*f^3*x*PolyLog[3, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + 6*d*e*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] + 6*d*f^3*x*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] + 6*f^3*P olyLog[4, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - 6*f^3*PolyLog[4, -((b* E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]))/(a*b*d^4)
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.
| \(\displaystyle \int \frac {(e+f x)^3 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6119 |
| \(\displaystyle \frac {\int (e+f x)^3 \cosh (c+d x) \coth (c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 5973 |
| \(\displaystyle \frac {\int (e+f x)^3 \sinh (c+d x)dx+\int (e+f x)^3 \text {csch}(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int -i (e+f x)^3 \sin (i c+i d x)dx+\int i (e+f x)^3 \csc (i c+i d x)dx}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x)^3 \csc (i c+i d x)dx-i \int (e+f x)^3 \sin (i c+i d x)dx}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x)^3 \csc (i c+i d x)dx-i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \int (e+f x)^2 \cosh (c+d x)dx}{d}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x)^3 \csc (i c+i d x)dx-i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x)^3 \csc (i c+i d x)dx-i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 i f \int -i (e+f x) \sinh (c+d x)dx}{d}\right )}{d}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x)^3 \csc (i c+i d x)dx-i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int (e+f x) \sinh (c+d x)dx}{d}\right )}{d}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x)^3 \csc (i c+i d x)dx-i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int -i (e+f x) \sin (i c+i d x)dx}{d}\right )}{d}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x)^3 \csc (i c+i d x)dx-i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \int (e+f x) \sin (i c+i d x)dx}{d}\right )}{d}\right )}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x)^3 \csc (i c+i d x)dx-i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \cosh (c+d x)dx}{d}\right )}{d}\right )}{d}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x)^3 \csc (i c+i d x)dx-i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{d}\right )}{d}\right )}{a}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x)^3 \csc (i c+i d x)dx-i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{a}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (\frac {3 i f \int (e+f x)^2 \log \left (1-e^{c+d x}\right )dx}{d}-\frac {3 i f \int (e+f x)^2 \log \left (1+e^{c+d x}\right )dx}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{a}\) |
| \(\Big \downarrow \) 6099 |
| \(\displaystyle -\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \int (e+f x)^3dx}{b^2}+\frac {\int (e+f x)^3 \sinh (c+d x)dx}{b}\right )}{a}+\frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{a}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle -\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a+b \sinh (c+d x)}dx}{b^2}+\frac {\int (e+f x)^3 \sinh (c+d x)dx}{b}-\frac {a (e+f x)^4}{4 b^2 f}\right )}{a}+\frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a-i b \sin (i c+i d x)}dx}{b^2}+\frac {\int -i (e+f x)^3 \sin (i c+i d x)dx}{b}-\frac {a (e+f x)^4}{4 b^2 f}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \int (e+f x)^3 \sin (i c+i d x)dx}{b}-\frac {a (e+f x)^4}{4 b^2 f}\right )}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \int (e+f x)^2 \cosh (c+d x)dx}{d}\right )}{b}-\frac {a (e+f x)^4}{4 b^2 f}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{b}-\frac {a (e+f x)^4}{4 b^2 f}\right )}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 i f \int -i (e+f x) \sinh (c+d x)dx}{d}\right )}{d}\right )}{b}-\frac {a (e+f x)^4}{4 b^2 f}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int (e+f x) \sinh (c+d x)dx}{d}\right )}{d}\right )}{b}-\frac {a (e+f x)^4}{4 b^2 f}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int -i (e+f x) \sin (i c+i d x)dx}{d}\right )}{d}\right )}{b}-\frac {a (e+f x)^4}{4 b^2 f}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \int (e+f x) \sin (i c+i d x)dx}{d}\right )}{d}\right )}{b}-\frac {a (e+f x)^4}{4 b^2 f}\right )}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \cosh (c+d x)dx}{d}\right )}{d}\right )}{d}\right )}{b}-\frac {a (e+f x)^4}{4 b^2 f}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{d}\right )}{d}\right )}{b}-\frac {a (e+f x)^4}{4 b^2 f}\right )}{a}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle \frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {a (e+f x)^4}{4 b^2 f}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b}\right )}{a}\) |
| \(\Big \downarrow \) 3803 |
| \(\displaystyle \frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{a}-\frac {b \left (\frac {2 \left (a^2+b^2\right ) \int -\frac {e^{c+d x} (e+f x)^3}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{b^2}-\frac {a (e+f x)^4}{4 b^2 f}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{a}-\frac {b \left (-\frac {2 \left (a^2+b^2\right ) \int \frac {e^{c+d x} (e+f x)^3}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{b^2}-\frac {a (e+f x)^4}{4 b^2 f}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b}\right )}{a}\) |
Input:
Int[((e + f*x)^3*Cosh[c + d*x]*Coth[c + d*x])/(a + b*Sinh[c + d*x]),x]
Output:
$Aborted
\[\int \frac {\left (f x +e \right )^{3} \cosh \left (d x +c \right ) \coth \left (d x +c \right )}{a +b \sinh \left (d x +c \right )}d x\]
Input:
int((f*x+e)^3*cosh(d*x+c)*coth(d*x+c)/(a+b*sinh(d*x+c)),x)
Output:
int((f*x+e)^3*cosh(d*x+c)*coth(d*x+c)/(a+b*sinh(d*x+c)),x)
Leaf count of result is larger than twice the leaf count of optimal. 1470 vs. \(2 (585) = 1170\).
Time = 0.16 (sec) , antiderivative size = 1470, normalized size of antiderivative = 2.30 \[ \int \frac {(e+f x)^3 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^3*cosh(d*x+c)*coth(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm ="fricas")
Output:
1/4*(a*d^4*f^3*x^4 + 4*a*d^4*e*f^2*x^3 + 6*a*d^4*e^2*f*x^2 + 4*a*d^4*e^3*x - 24*b*f^3*sqrt((a^2 + b^2)/b^2)*polylog(4, (a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2))/b) + 24 *b*f^3*sqrt((a^2 + b^2)/b^2)*polylog(4, (a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2))/b) + 24*b*f^ 3*polylog(4, cosh(d*x + c) + sinh(d*x + c)) - 24*b*f^3*polylog(4, -cosh(d* x + c) - sinh(d*x + c)) - 12*(b*d^2*f^3*x^2 + 2*b*d^2*e*f^2*x + b*d^2*e^2* f)*sqrt((a^2 + b^2)/b^2)*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cos h(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) + 12*(b*d^ 2*f^3*x^2 + 2*b*d^2*e*f^2*x + b*d^2*e^2*f)*sqrt((a^2 + b^2)/b^2)*dilog((a* cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt ((a^2 + b^2)/b^2) - b)/b + 1) + 4*(b*d^3*e^3 - 3*b*c*d^2*e^2*f + 3*b*c^2*d *e*f^2 - b*c^3*f^3)*sqrt((a^2 + b^2)/b^2)*log(2*b*cosh(d*x + c) + 2*b*sinh (d*x + c) + 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) - 4*(b*d^3*e^3 - 3*b*c*d^2*e^ 2*f + 3*b*c^2*d*e*f^2 - b*c^3*f^3)*sqrt((a^2 + b^2)/b^2)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) - 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) - 4*(b*d^3*f^3 *x^3 + 3*b*d^3*e*f^2*x^2 + 3*b*d^3*e^2*f*x + 3*b*c*d^2*e^2*f - 3*b*c^2*d*e *f^2 + b*c^3*f^3)*sqrt((a^2 + b^2)/b^2)*log(-(a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b) + 4*(b*d^3*f^3*x^3 + 3*b*d^3*e*f^2*x^2 + 3*b*d^3*e^2*f*x + 3*b*c*d^2*e^...
\[ \int \frac {(e+f x)^3 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (e + f x\right )^{3} \cosh {\left (c + d x \right )} \coth {\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \] Input:
integrate((f*x+e)**3*cosh(d*x+c)*coth(d*x+c)/(a+b*sinh(d*x+c)),x)
Output:
Integral((e + f*x)**3*cosh(c + d*x)*coth(c + d*x)/(a + b*sinh(c + d*x)), x )
\[ \int \frac {(e+f x)^3 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \cosh \left (d x + c\right ) \coth \left (d x + c\right )}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)^3*cosh(d*x+c)*coth(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm ="maxima")
Output:
e^3*((d*x + c)/(b*d) - log(e^(-d*x - c) + 1)/(a*d) + log(e^(-d*x - c) - 1) /(a*d) - sqrt(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*b*d)) - 3*(d*x*log(e^(d*x + c) + 1) + dilog(-e^(d*x + c)))*e^2*f/(a*d^2) + 3*(d*x*log(-e^(d*x + c) + 1) + dilog (e^(d*x + c)))*e^2*f/(a*d^2) + 1/4*(f^3*x^4 + 4*e*f^2*x^3 + 6*e^2*f*x^2)/b - 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)))*e*f^2/(a*d^3) + 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)))*e*f^2/(a*d^3) - (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)))*f^3/(a*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)))*f^3/(a*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*b^2*e^(2*d*x + 2*c) + 2*a^2*b*e^(d*x + c) - a*b^2), x)
Timed out. \[ \int \frac {(e+f x)^3 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)^3*cosh(d*x+c)*coth(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm ="giac")
Output:
Timed out
Timed out. \[ \int \frac {(e+f x)^3 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )\,\mathrm {coth}\left (c+d\,x\right )\,{\left (e+f\,x\right )}^3}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \] Input:
int((cosh(c + d*x)*coth(c + d*x)*(e + f*x)^3)/(a + b*sinh(c + d*x)),x)
Output:
int((cosh(c + d*x)*coth(c + d*x)*(e + f*x)^3)/(a + b*sinh(c + d*x)), x)
\[ \int \frac {(e+f x)^3 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (f x +e \right )^{3} \cosh \left (d x +c \right ) \coth \left (d x +c \right )}{a +b \sinh \left (d x +c \right )}d x \] Input:
int((f*x+e)^3*cosh(d*x+c)*coth(d*x+c)/(a+b*sinh(d*x+c)),x)
Output:
int((f*x+e)^3*cosh(d*x+c)*coth(d*x+c)/(a+b*sinh(d*x+c)),x)