Integrand size = 34, antiderivative size = 718 \[ \int \frac {(e+f x)^3 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b (e+f x)^4}{4 a^2 f}-\frac {\left (a^2+b^2\right ) (e+f x)^4}{4 a^2 b f}-\frac {6 f (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a d^2}-\frac {(e+f x)^3 \text {csch}(c+d x)}{a d}+\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 b d}+\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 b d}-\frac {b (e+f x)^3 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}-\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^3}+\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^3}+\frac {3 \left (a^2+b^2\right ) f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 b d^2}+\frac {3 \left (a^2+b^2\right ) f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 b d^2}-\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a^2 d^2}+\frac {6 f^3 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^4}-\frac {6 f^3 \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^4}-\frac {6 \left (a^2+b^2\right ) f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 b d^3}-\frac {6 \left (a^2+b^2\right ) f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 b d^3}+\frac {3 b f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a^2 d^3}+\frac {6 \left (a^2+b^2\right ) f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 b d^4}+\frac {6 \left (a^2+b^2\right ) f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 b d^4}-\frac {3 b f^3 \operatorname {PolyLog}\left (4,e^{2 (c+d x)}\right )}{4 a^2 d^4} \] Output:
1/4*b*(f*x+e)^4/a^2/f-1/4*(a^2+b^2)*(f*x+e)^4/a^2/b/f-6*f*(f*x+e)^2*arctan h(exp(d*x+c))/a/d^2-(f*x+e)^3*csch(d*x+c)/a/d+(a^2+b^2)*(f*x+e)^3*ln(1+b*e xp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^2/b/d+(a^2+b^2)*(f*x+e)^3*ln(1+b*exp(d*x+ c)/(a+(a^2+b^2)^(1/2)))/a^2/b/d-b*(f*x+e)^3*ln(1-exp(2*d*x+2*c))/a^2/d-6*f ^2*(f*x+e)*polylog(2,-exp(d*x+c))/a/d^3+6*f^2*(f*x+e)*polylog(2,exp(d*x+c) )/a/d^3+3*(a^2+b^2)*f*(f*x+e)^2*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2) ))/a^2/b/d^2+3*(a^2+b^2)*f*(f*x+e)^2*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^ (1/2)))/a^2/b/d^2-3/2*b*f*(f*x+e)^2*polylog(2,exp(2*d*x+2*c))/a^2/d^2+6*f^ 3*polylog(3,-exp(d*x+c))/a/d^4-6*f^3*polylog(3,exp(d*x+c))/a/d^4-6*(a^2+b^ 2)*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^2/b/d^3-6*(a ^2+b^2)*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^2/b/d^3 +3/2*b*f^2*(f*x+e)*polylog(3,exp(2*d*x+2*c))/a^2/d^3+6*(a^2+b^2)*f^3*polyl og(4,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^2/b/d^4+6*(a^2+b^2)*f^3*polylog( 4,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^2/b/d^4-3/4*b*f^3*polylog(4,exp(2*d *x+2*c))/a^2/d^4
Leaf count is larger than twice the leaf count of optimal. \(2696\) vs. \(2(718)=1436\).
Time = 10.18 (sec) , antiderivative size = 2696, normalized size of antiderivative = 3.75 \[ \int \frac {(e+f x)^3 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show} \] Input:
Integrate[((e + f*x)^3*Cosh[c + d*x]*Coth[c + d*x]^2)/(a + b*Sinh[c + d*x] ),x]
Output:
(2*d^3*e^2*(-1 + E^(2*c))*f*(b*d*e - 3*a*f)*x + 2*d^3*e^2*(-1 + E^(2*c))*f *(b*d*e + 3*a*f)*x + b*d^4*(e + f*x)^4 - 6*d^2*e*(-1 + E^(2*c))*f^2*(b*d*e - 2*a*f)*x*Log[1 - E^(-c - d*x)] - 6*d^2*(-1 + E^(2*c))*f^3*(b*d*e - a*f) *x^2*Log[1 - E^(-c - d*x)] - 2*b*d^3*(-1 + E^(2*c))*f^4*x^3*Log[1 - E^(-c - d*x)] - 6*d^2*e*(-1 + E^(2*c))*f^2*(b*d*e + 2*a*f)*x*Log[1 + E^(-c - d*x )] - 6*d^2*(-1 + E^(2*c))*f^3*(b*d*e + a*f)*x^2*Log[1 + E^(-c - d*x)] - 2* b*d^3*(-1 + E^(2*c))*f^4*x^3*Log[1 + E^(-c - d*x)] - 2*d^2*e^2*(-1 + E^(2* c))*f*(b*d*e - 3*a*f)*Log[1 - E^(c + d*x)] - 2*d^2*e^2*(-1 + E^(2*c))*f*(b *d*e + 3*a*f)*Log[1 + E^(c + d*x)] + 6*d*e*(-1 + E^(2*c))*f^2*(b*d*e + 2*a *f)*PolyLog[2, -E^(-c - d*x)] + 12*d*(-1 + E^(2*c))*f^3*(b*d*e + a*f)*x*Po lyLog[2, -E^(-c - d*x)] + 6*b*d^2*(-1 + E^(2*c))*f^4*x^2*PolyLog[2, -E^(-c - d*x)] + 6*d*e*(-1 + E^(2*c))*f^2*(b*d*e - 2*a*f)*PolyLog[2, E^(-c - d*x )] + 12*d*(-1 + E^(2*c))*f^3*(b*d*e - a*f)*x*PolyLog[2, E^(-c - d*x)] + 6* b*d^2*(-1 + E^(2*c))*f^4*x^2*PolyLog[2, E^(-c - d*x)] + 12*(-1 + E^(2*c))* f^3*(b*d*e + a*f)*PolyLog[3, -E^(-c - d*x)] + 12*b*d*(-1 + E^(2*c))*f^4*x* PolyLog[3, -E^(-c - d*x)] - 12*(-1 + E^(2*c))*f^3*(-(b*d*e) + a*f)*PolyLog [3, E^(-c - d*x)] + 12*b*d*(-1 + E^(2*c))*f^4*x*PolyLog[3, E^(-c - d*x)] + 12*b*(-1 + E^(2*c))*f^4*PolyLog[4, -E^(-c - d*x)] + 12*b*(-1 + E^(2*c))*f ^4*PolyLog[4, E^(-c - d*x)])/(2*a^2*d^4*(-1 + E^(2*c))*f) - ((a^2 + b^2)*( 4*e^3*E^(2*c)*x + 6*e^2*E^(2*c)*f*x^2 + 4*e*E^(2*c)*f^2*x^3 + E^(2*c)*f...
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.
| \(\displaystyle \int \frac {(e+f x)^3 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6119 |
| \(\displaystyle \frac {\int (e+f x)^3 \cosh (c+d x) \coth ^2(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 5973 |
| \(\displaystyle \frac {\int (e+f x)^3 \cosh (c+d x)dx+\int (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)dx+\int (e+f x)^3 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-\frac {3 i f \int -i (e+f x)^2 \sinh (c+d x)dx}{d}+\int (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)dx+\frac {(e+f x)^3 \sinh (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {-\frac {3 f \int (e+f x)^2 \sinh (c+d x)dx}{d}+\int (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)dx+\frac {(e+f x)^3 \sinh (c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-\frac {3 f \int -i (e+f x)^2 \sin (i c+i d x)dx}{d}+\int (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)dx+\frac {(e+f x)^3 \sinh (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\frac {3 i f \int (e+f x)^2 \sin (i c+i d x)dx}{d}+\int (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)dx+\frac {(e+f x)^3 \sinh (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \int (e+f x) \cosh (c+d x)dx}{d}\right )}{d}+\int (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)dx+\frac {(e+f x)^3 \sinh (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{d}+\int (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)dx+\frac {(e+f x)^3 \sinh (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {i f \int -i \sinh (c+d x)dx}{d}\right )}{d}\right )}{d}+\int (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)dx+\frac {(e+f x)^3 \sinh (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int \sinh (c+d x)dx}{d}\right )}{d}\right )}{d}+\int (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)dx+\frac {(e+f x)^3 \sinh (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)dx+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int -i \sin (i c+i d x)dx}{d}\right )}{d}\right )}{d}+\frac {(e+f x)^3 \sinh (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)dx+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}+\frac {i f \int \sin (i c+i d x)dx}{d}\right )}{d}\right )}{d}+\frac {(e+f x)^3 \sinh (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)dx+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}+\frac {(e+f x)^3 \sinh (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 5975 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\frac {3 f \int (e+f x)^2 \text {csch}(c+d x)dx}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}+\frac {(e+f x)^3 \sinh (c+d x)}{d}-\frac {(e+f x)^3 \text {csch}(c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\frac {3 f \int i (e+f x)^2 \csc (i c+i d x)dx}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}+\frac {(e+f x)^3 \sinh (c+d x)}{d}-\frac {(e+f x)^3 \text {csch}(c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\frac {3 i f \int (e+f x)^2 \csc (i c+i d x)dx}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}+\frac {(e+f x)^3 \sinh (c+d x)}{d}-\frac {(e+f x)^3 \text {csch}(c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\frac {3 i f \left (\frac {2 i f \int (e+f x) \log \left (1-e^{c+d x}\right )dx}{d}-\frac {2 i f \int (e+f x) \log \left (1+e^{c+d x}\right )dx}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}+\frac {(e+f x)^3 \sinh (c+d x)}{d}-\frac {(e+f x)^3 \text {csch}(c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\frac {3 i f \left (-\frac {2 i f \left (\frac {f \int \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}+\frac {(e+f x)^3 \sinh (c+d x)}{d}-\frac {(e+f x)^3 \text {csch}(c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\frac {3 i f \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}+\frac {(e+f x)^3 \sinh (c+d x)}{d}-\frac {(e+f x)^3 \text {csch}(c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 6119 |
| \(\displaystyle -\frac {b \left (\frac {\int (e+f x)^3 \cosh ^2(c+d x) \coth (c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}+\frac {\frac {3 i f \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}+\frac {(e+f x)^3 \sinh (c+d x)}{d}-\frac {(e+f x)^3 \text {csch}(c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 5973 |
| \(\displaystyle -\frac {b \left (\frac {\int (e+f x)^3 \coth (c+d x)dx+\int (e+f x)^3 \cosh (c+d x) \sinh (c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}+\frac {\frac {3 i f \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}+\frac {(e+f x)^3 \sinh (c+d x)}{d}-\frac {(e+f x)^3 \text {csch}(c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 i f \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}+\frac {(e+f x)^3 \sinh (c+d x)}{d}-\frac {(e+f x)^3 \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int (e+f x)^3 \cosh (c+d x) \sinh (c+d x)dx+\int -i (e+f x)^3 \tan \left (i c+i d x+\frac {\pi }{2}\right )dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {3 i f \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}+\frac {(e+f x)^3 \sinh (c+d x)}{d}-\frac {(e+f x)^3 \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int (e+f x)^3 \cosh (c+d x) \sinh (c+d x)dx-i \int (e+f x)^3 \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle \frac {\frac {3 i f \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}+\frac {(e+f x)^3 \sinh (c+d x)}{d}-\frac {(e+f x)^3 \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int (e+f x)^3 \cosh (c+d x) \sinh (c+d x)dx-i \left (2 i \int \frac {e^{2 c+2 d x-i \pi } (e+f x)^3}{1+e^{2 c+2 d x-i \pi }}dx-\frac {i (e+f x)^4}{4 f}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {\frac {3 i f \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}+\frac {(e+f x)^3 \sinh (c+d x)}{d}-\frac {(e+f x)^3 \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int (e+f x)^3 \cosh (c+d x) \sinh (c+d x)dx-i \left (2 i \left (\frac {(e+f x)^3 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {3 f \int (e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )dx}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {\frac {3 i f \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}+\frac {(e+f x)^3 \sinh (c+d x)}{d}-\frac {(e+f x)^3 \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int (e+f x)^3 \cosh (c+d x) \sinh (c+d x)dx-i \left (2 i \left (\frac {(e+f x)^3 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 5969 |
| \(\displaystyle \frac {\frac {3 i f \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}+\frac {(e+f x)^3 \sinh (c+d x)}{d}-\frac {(e+f x)^3 \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-i \left (2 i \left (\frac {(e+f x)^3 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )-\frac {3 f \int (e+f x)^2 \sinh ^2(c+d x)dx}{2 d}+\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 i f \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}+\frac {(e+f x)^3 \sinh (c+d x)}{d}-\frac {(e+f x)^3 \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-i \left (2 i \left (\frac {(e+f x)^3 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )-\frac {3 f \int -(e+f x)^2 \sin (i c+i d x)^2dx}{2 d}+\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {3 i f \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}+\frac {(e+f x)^3 \sinh (c+d x)}{d}-\frac {(e+f x)^3 \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-i \left (2 i \left (\frac {(e+f x)^3 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )+\frac {3 f \int (e+f x)^2 \sin (i c+i d x)^2dx}{2 d}+\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 d}}{a}\right )}{a}\) |
Input:
Int[((e + f*x)^3*Cosh[c + d*x]*Coth[c + d*x]^2)/(a + b*Sinh[c + d*x]),x]
Output:
$Aborted
\[\int \frac {\left (f x +e \right )^{3} \cosh \left (d x +c \right ) \coth \left (d x +c \right )^{2}}{a +b \sinh \left (d x +c \right )}d x\]
Input:
int((f*x+e)^3*cosh(d*x+c)*coth(d*x+c)^2/(a+b*sinh(d*x+c)),x)
Output:
int((f*x+e)^3*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. 5829 vs. \(2 (674) = 1348\).
Time = 0.22 (sec) , antiderivative size = 5829, normalized size of antiderivative = 8.12 \[ \int \frac {(e+f x)^3 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^3*cosh(d*x+c)*coth(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorit hm="fricas")
Output:
Too large to include
\[ \int \frac {(e+f x)^3 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (e + f x\right )^{3} \cosh {\left (c + d x \right )} \coth ^{2}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \] Input:
integrate((f*x+e)**3*cosh(d*x+c)*coth(d*x+c)**2/(a+b*sinh(d*x+c)),x)
Output:
Integral((e + f*x)**3*cosh(c + d*x)*coth(c + d*x)**2/(a + b*sinh(c + d*x)) , x)
\[ \int \frac {(e+f x)^3 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \cosh \left (d x + c\right ) \coth \left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)^3*cosh(d*x+c)*coth(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorit hm="maxima")
Output:
e^3*((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)) - 3*e^2*f*log( e^(d*x + c) + 1)/(a*d^2) + 3*e^2*f*log(e^(d*x + c) - 1)/(a*d^2) - 1/4*(a*d *f^3*x^4 + 4*a*d*e*f^2*x^3 + 6*a*d*e^2*f*x^2 - (a*d*f^3*x^4*e^(2*c) + 4*a* d*e*f^2*x^3*e^(2*c) + 6*a*d*e^2*f*x^2*e^(2*c))*e^(2*d*x) + 8*(b*f^3*x^3*e^ c + 3*b*e*f^2*x^2*e^c + 3*b*e^2*f*x*e^c)*e^(d*x))/(a*b*d*e^(2*d*x + 2*c) - a*b*d) - (d^3*x^3*log(e^(d*x + c) + 1) + 3*d^2*x^2*dilog(-e^(d*x + c)) - 6*d*x*polylog(3, -e^(d*x + c)) + 6*polylog(4, -e^(d*x + c)))*b*f^3/(a^2*d^ 4) - (d^3*x^3*log(-e^(d*x + c) + 1) + 3*d^2*x^2*dilog(e^(d*x + c)) - 6*d*x *polylog(3, e^(d*x + c)) + 6*polylog(4, e^(d*x + c)))*b*f^3/(a^2*d^4) - 3* (b*d*e^2*f + 2*a*e*f^2)*(d*x*log(e^(d*x + c) + 1) + dilog(-e^(d*x + c)))/( a^2*d^3) - 3*(b*d*e^2*f - 2*a*e*f^2)*(d*x*log(-e^(d*x + c) + 1) + dilog(e^ (d*x + c)))/(a^2*d^3) - 3*(b*d*e*f^2 + a*f^3)*(d^2*x^2*log(e^(d*x + c) + 1 ) + 2*d*x*dilog(-e^(d*x + c)) - 2*polylog(3, -e^(d*x + c)))/(a^2*d^4) - 3* (b*d*e*f^2 - a*f^3)*(d^2*x^2*log(-e^(d*x + c) + 1) + 2*d*x*dilog(e^(d*x + c)) - 2*polylog(3, e^(d*x + c)))/(a^2*d^4) + 1/4*(b*d^4*f^3*x^4 + 4*(b*d*e *f^2 + a*f^3)*d^3*x^3 + 6*(b*d^2*e^2*f + 2*a*d*e*f^2)*d^2*x^2)/(a^2*d^4) + 1/4*(b*d^4*f^3*x^4 + 4*(b*d*e*f^2 - a*f^3)*d^3*x^3 + 6*(b*d^2*e^2*f - 2*a *d*e*f^2)*d^2*x^2)/(a^2*d^4) - integrate(-2*((a^2*b*f^3 + b^3*f^3)*x^3 ...
Timed out. \[ \int \frac {(e+f x)^3 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)^3*cosh(d*x+c)*coth(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorit hm="giac")
Output:
Timed out
Timed out. \[ \int \frac {(e+f x)^3 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )\,{\mathrm {coth}\left (c+d\,x\right )}^2\,{\left (e+f\,x\right )}^3}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \] Input:
int((cosh(c + d*x)*coth(c + d*x)^2*(e + f*x)^3)/(a + b*sinh(c + d*x)),x)
Output:
int((cosh(c + d*x)*coth(c + d*x)^2*(e + f*x)^3)/(a + b*sinh(c + d*x)), x)
\[ \int \frac {(e+f x)^3 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (f x +e \right )^{3} \cosh \left (d x +c \right ) \coth \left (d x +c \right )^{2}}{a +b \sinh \left (d x +c \right )}d x \] Input:
int((f*x+e)^3*cosh(d*x+c)*coth(d*x+c)^2/(a+b*sinh(d*x+c)),x)
Output:
int((f*x+e)^3*cosh(d*x+c)*coth(d*x+c)^2/(a+b*sinh(d*x+c)),x)