Integrand size = 28, antiderivative size = 623 \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=-\frac {b (e+f x)^2}{4 a^2 d}+\frac {b \left (a^2+b^2\right ) (e+f x)^3}{3 a^4 f}-\frac {4 f (e+f x) \cosh (c+d x)}{3 a d^2}-\frac {2 b^2 f (e+f x) \cosh (c+d x)}{a^3 d^2}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 a d^2}-\frac {b \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {b \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {2 b \left (a^2+b^2\right ) f (e+f x) \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^2}-\frac {2 b \left (a^2+b^2\right ) f (e+f x) \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^2}+\frac {2 b \left (a^2+b^2\right ) f^2 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^3}+\frac {2 b \left (a^2+b^2\right ) f^2 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^3}+\frac {14 f^2 \sinh (c+d x)}{9 a d^3}+\frac {2 b^2 f^2 \sinh (c+d x)}{a^3 d^3}+\frac {2 (e+f x)^2 \sinh (c+d x)}{3 a d}+\frac {b^2 (e+f x)^2 \sinh (c+d x)}{a^3 d}+\frac {b f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 a^2 d^2}+\frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 a d}-\frac {b f^2 \sinh ^2(c+d x)}{4 a^2 d^3}-\frac {b (e+f x)^2 \sinh ^2(c+d x)}{2 a^2 d}+\frac {2 f^2 \sinh ^3(c+d x)}{27 a d^3} \] Output:
-1/4*b*(f*x+e)^2/a^2/d+1/3*b*(a^2+b^2)*(f*x+e)^3/a^4/f-4/3*f*(f*x+e)*cosh( d*x+c)/a/d^2-2*b^2*f*(f*x+e)*cosh(d*x+c)/a^3/d^2-2/9*f*(f*x+e)*cosh(d*x+c) ^3/a/d^2-b*(a^2+b^2)*(f*x+e)^2*ln(1+a*exp(d*x+c)/(b-(a^2+b^2)^(1/2)))/a^4/ d-b*(a^2+b^2)*(f*x+e)^2*ln(1+a*exp(d*x+c)/(b+(a^2+b^2)^(1/2)))/a^4/d-2*b*( a^2+b^2)*f*(f*x+e)*polylog(2,-a*exp(d*x+c)/(b-(a^2+b^2)^(1/2)))/a^4/d^2-2* b*(a^2+b^2)*f*(f*x+e)*polylog(2,-a*exp(d*x+c)/(b+(a^2+b^2)^(1/2)))/a^4/d^2 +2*b*(a^2+b^2)*f^2*polylog(3,-a*exp(d*x+c)/(b-(a^2+b^2)^(1/2)))/a^4/d^3+2* b*(a^2+b^2)*f^2*polylog(3,-a*exp(d*x+c)/(b+(a^2+b^2)^(1/2)))/a^4/d^3+14/9* f^2*sinh(d*x+c)/a/d^3+2*b^2*f^2*sinh(d*x+c)/a^3/d^3+2/3*(f*x+e)^2*sinh(d*x +c)/a/d+b^2*(f*x+e)^2*sinh(d*x+c)/a^3/d+1/2*b*f*(f*x+e)*cosh(d*x+c)*sinh(d *x+c)/a^2/d^2+1/3*(f*x+e)^2*cosh(d*x+c)^2*sinh(d*x+c)/a/d-1/4*b*f^2*sinh(d *x+c)^2/a^2/d^3-1/2*b*(f*x+e)^2*sinh(d*x+c)^2/a^2/d+2/27*f^2*sinh(d*x+c)^3 /a/d^3
Leaf count is larger than twice the leaf count of optimal. \(2129\) vs. \(2(623)=1246\).
Time = 17.47 (sec) , antiderivative size = 2129, normalized size of antiderivative = 3.42 \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\text {Result too large to show} \] Input:
Integrate[((e + f*x)^2*Cosh[c + d*x]^3)/(a + b*Csch[c + d*x]),x]
Output:
(f^2*Csch[c + d*x]*((4*b*x^3)/(-1 + E^(2*c)) - 2*b*x^3*Coth[c] - (6*a^2*b* (d^2*x^2*Log[1 + ((b - Sqrt[a^2 + b^2])*E^(-c - d*x))/a] - 2*d*x*PolyLog[2 , ((-b + Sqrt[a^2 + b^2])*E^(-c - d*x))/a] - 2*PolyLog[3, ((-b + Sqrt[a^2 + b^2])*E^(-c - d*x))/a]))/(Sqrt[a^2 + b^2]*(-b + Sqrt[a^2 + b^2])*d^3) - (6*a^2*b*(d^2*x^2*Log[1 + ((b + Sqrt[a^2 + b^2])*E^(-c - d*x))/a] - 2*d*x* PolyLog[2, -(((b + Sqrt[a^2 + b^2])*E^(-c - d*x))/a)] - 2*PolyLog[3, -(((b + Sqrt[a^2 + b^2])*E^(-c - d*x))/a)]))/(Sqrt[a^2 + b^2]*(b + Sqrt[a^2 + b ^2])*d^3) + (6*b^2*(d^2*x^2*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])] + 2*d*x*PolyLog[2, (a*E^(c + d*x))/(-b + Sqrt[a^2 + b^2])] - 2*PolyLog[3, (a*E^(c + d*x))/(-b + Sqrt[a^2 + b^2])]))/(Sqrt[a^2 + b^2]*d^3) - (6*b^2* (d^2*x^2*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])] + 2*d*x*PolyLog[2, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))] - 2*PolyLog[3, -((a*E^(c + d*x) )/(b + Sqrt[a^2 + b^2]))]))/(Sqrt[a^2 + b^2]*d^3) + (6*a*Cosh[d*x]*(-2*d*x *Cosh[c] + (2 + d^2*x^2)*Sinh[c]))/d^3 + (6*a*((2 + d^2*x^2)*Cosh[c] - 2*d *x*Sinh[c])*Sinh[d*x])/d^3)*(b + a*Sinh[c + d*x]))/(12*a^2*(a + b*Csch[c + d*x])) - (e^2*Csch[c + d*x]*((b*Log[b + a*Sinh[c + d*x]])/a^2 - Sinh[c + d*x]/a)*(b + a*Sinh[c + d*x]))/(2*d*(a + b*Csch[c + d*x])) + (e*f*Csch[c + d*x]*(b + a*Sinh[c + d*x])*(-2*a*Cosh[c + d*x] - b*(2*c*(c + d*x) - (c + d*x)^2 + 2*(c + d*x)*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])] + 2*(c + d*x)*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])] - 2*c*Log[a - 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.
| \(\displaystyle \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 6128 |
| \(\displaystyle \int \frac {(e+f x)^2 \sinh (c+d x) \cosh ^3(c+d x)}{a \sinh (c+d x)+b}dx\) |
| \(\Big \downarrow \) 6113 |
| \(\displaystyle \frac {\int (e+f x)^2 \cosh ^3(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{b+a \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{a}\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle \frac {\frac {2 f^2 \int \cosh ^3(c+d x)dx}{9 d^2}+\frac {2}{3} \int (e+f x)^2 \cosh (c+d x)dx-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{b+a \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\frac {2 f^2 \int \sin \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{9 d^2}+\frac {2}{3} \int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\frac {2 i f^2 \int \left (\sinh ^2(c+d x)+1\right )d(-i \sinh (c+d x))}{9 d^3}+\frac {2}{3} \int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\frac {2}{3} \int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 i f \int -i (e+f x) \sinh (c+d x)dx}{d}\right )+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int (e+f x) \sinh (c+d x)dx}{d}\right )+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int -i (e+f x) \sin (i c+i d x)dx}{d}\right )+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \int (e+f x) \sin (i c+i d x)dx}{d}\right )+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \cosh (c+d x)dx}{d}\right )}{d}\right )+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{d}\right )+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 6099 |
| \(\displaystyle -\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{b+a \sinh (c+d x)}dx}{a^2}-\frac {b \int (e+f x)^2 \cosh (c+d x)dx}{a^2}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx}{a}\right )}{a}+\frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{b+a \sinh (c+d x)}dx}{a^2}-\frac {b \int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{a^2}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{b+a \sinh (c+d x)}dx}{a^2}-\frac {b \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 i f \int -i (e+f x) \sinh (c+d x)dx}{d}\right )}{a^2}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{b+a \sinh (c+d x)}dx}{a^2}-\frac {b \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int (e+f x) \sinh (c+d x)dx}{d}\right )}{a^2}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx}{a}\right )}{a}+\frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{b+a \sinh (c+d x)}dx}{a^2}-\frac {b \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int -i (e+f x) \sin (i c+i d x)dx}{d}\right )}{a^2}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{b+a \sinh (c+d x)}dx}{a^2}-\frac {b \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \int (e+f x) \sin (i c+i d x)dx}{d}\right )}{a^2}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{b+a \sinh (c+d x)}dx}{a^2}-\frac {b \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \cosh (c+d x)dx}{d}\right )}{d}\right )}{a^2}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{b+a \sinh (c+d x)}dx}{a^2}-\frac {b \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{d}\right )}{a^2}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle \frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{b+a \sinh (c+d x)}dx}{a^2}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx}{a}-\frac {b \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{a^2}\right )}{a}\) |
| \(\Big \downarrow \) 5969 |
| \(\displaystyle \frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{b+a \sinh (c+d x)}dx}{a^2}+\frac {\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 d}-\frac {f \int (e+f x) \sinh ^2(c+d x)dx}{d}}{a}-\frac {b \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{a^2}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{b+a \sinh (c+d x)}dx}{a^2}+\frac {\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 d}-\frac {f \int -\left ((e+f x) \sin (i c+i d x)^2\right )dx}{d}}{a}-\frac {b \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{a^2}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{b+a \sinh (c+d x)}dx}{a^2}+\frac {\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 d}+\frac {f \int (e+f x) \sin (i c+i d x)^2dx}{d}}{a}-\frac {b \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{a^2}\right )}{a}\) |
| \(\Big \downarrow \) 3791 |
| \(\displaystyle \frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{b+a \sinh (c+d x)}dx}{a^2}+\frac {\frac {f \left (\frac {1}{2} \int (e+f x)dx+\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{d}+\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 d}}{a}-\frac {b \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{a^2}\right )}{a}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle \frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{b+a \sinh (c+d x)}dx}{a^2}-\frac {b \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{a^2}+\frac {\frac {f \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{d}+\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 6095 |
| \(\displaystyle \frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \left (\int \frac {e^{c+d x} (e+f x)^2}{e^{c+d x} a+b-\sqrt {a^2+b^2}}dx+\int \frac {e^{c+d x} (e+f x)^2}{e^{c+d x} a+b+\sqrt {a^2+b^2}}dx-\frac {(e+f x)^3}{3 a f}\right )}{a^2}-\frac {b \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{a^2}+\frac {\frac {f \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{d}+\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \left (-\frac {2 f \int (e+f x) \log \left (\frac {e^{c+d x} a}{b-\sqrt {a^2+b^2}}+1\right )dx}{a d}-\frac {2 f \int (e+f x) \log \left (\frac {e^{c+d x} a}{b+\sqrt {a^2+b^2}}+1\right )dx}{a d}+\frac {(e+f x)^2 \log \left (\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}+1\right )}{a d}+\frac {(e+f x)^2 \log \left (\frac {a e^{c+d x}}{\sqrt {a^2+b^2}+b}+1\right )}{a d}-\frac {(e+f x)^3}{3 a f}\right )}{a^2}-\frac {b \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{a^2}+\frac {\frac {f \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{d}+\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 d}}{a}\right )}{a}\) |
Input:
Int[((e + f*x)^2*Cosh[c + d*x]^3)/(a + b*Csch[c + d*x]),x]
Output:
$Aborted
\[\int \frac {\left (f x +e \right )^{2} \cosh \left (d x +c \right )^{3}}{a +\operatorname {csch}\left (d x +c \right ) b}d x\]
Input:
int((f*x+e)^2*cosh(d*x+c)^3/(a+csch(d*x+c)*b),x)
Output:
int((f*x+e)^2*cosh(d*x+c)^3/(a+csch(d*x+c)*b),x)
Leaf count of result is larger than twice the leaf count of optimal. 4887 vs. \(2 (581) = 1162\).
Time = 0.16 (sec) , antiderivative size = 4887, normalized size of antiderivative = 7.84 \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^2*cosh(d*x+c)^3/(a+b*csch(d*x+c)),x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)**2*cosh(d*x+c)**3/(a+b*csch(d*x+c)),x)
Output:
Timed out
\[ \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \cosh \left (d x + c\right )^{3}}{b \operatorname {csch}\left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)^2*cosh(d*x+c)^3/(a+b*csch(d*x+c)),x, algorithm="maxima")
Output:
-1/24*e^2*((3*a*b*e^(-d*x - c) - a^2 - 3*(3*a^2 + 4*b^2)*e^(-2*d*x - 2*c)) *e^(3*d*x + 3*c)/(a^3*d) + 24*(a^2*b + b^3)*(d*x + c)/(a^4*d) + (3*a*b*e^( -2*d*x - 2*c) + a^2*e^(-3*d*x - 3*c) + 3*(3*a^2 + 4*b^2)*e^(-d*x - c))/(a^ 3*d) + 24*(a^2*b + b^3)*log(-2*b*e^(-d*x - c) + a*e^(-2*d*x - 2*c) - a)/(a ^4*d)) - 1/432*(144*(a^2*b*d^3*f^2*e^(3*c) + b^3*d^3*f^2*e^(3*c))*x^3 + 43 2*(a^2*b*d^3*e*f*e^(3*c) + b^3*d^3*e*f*e^(3*c))*x^2 - 2*(9*a^3*d^2*f^2*x^2 *e^(6*c) + 6*(3*d^2*e*f - d*f^2)*a^3*x*e^(6*c) - 2*(3*d*e*f - f^2)*a^3*e^( 6*c))*e^(3*d*x) + 27*(2*a^2*b*d^2*f^2*x^2*e^(5*c) + 2*(2*d^2*e*f - d*f^2)* a^2*b*x*e^(5*c) - (2*d*e*f - f^2)*a^2*b*e^(5*c))*e^(2*d*x) + 54*(6*(d*e*f - f^2)*a^3*e^(4*c) + 8*(d*e*f - f^2)*a*b^2*e^(4*c) - (3*a^3*d^2*f^2*e^(4*c ) + 4*a*b^2*d^2*f^2*e^(4*c))*x^2 - 2*(3*(d^2*e*f - d*f^2)*a^3*e^(4*c) + 4* (d^2*e*f - d*f^2)*a*b^2*e^(4*c))*x)*e^(d*x) + 54*(6*(d*e*f + f^2)*a^3*e^(2 *c) + 8*(d*e*f + f^2)*a*b^2*e^(2*c) + (3*a^3*d^2*f^2*e^(2*c) + 4*a*b^2*d^2 *f^2*e^(2*c))*x^2 + 2*(3*(d^2*e*f + d*f^2)*a^3*e^(2*c) + 4*(d^2*e*f + d*f^ 2)*a*b^2*e^(2*c))*x)*e^(-d*x) + 27*(2*a^2*b*d^2*f^2*x^2*e^c + 2*(2*d^2*e*f + d*f^2)*a^2*b*x*e^c + (2*d*e*f + f^2)*a^2*b*e^c)*e^(-2*d*x) + 2*(9*a^3*d ^2*f^2*x^2 + 6*(3*d^2*e*f + d*f^2)*a^3*x + 2*(3*d*e*f + f^2)*a^3)*e^(-3*d* x))*e^(-3*c)/(a^4*d^3) + integrate(-2*((a^3*b*f^2 + a*b^3*f^2)*x^2 + 2*(a^ 3*b*e*f + a*b^3*e*f)*x - ((a^2*b^2*f^2*e^c + b^4*f^2*e^c)*x^2 + 2*(a^2*b^2 *e*f*e^c + b^4*e*f*e^c)*x)*e^(d*x))/(a^5*e^(2*d*x + 2*c) + 2*a^4*b*e^(d...
\[ \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \cosh \left (d x + c\right )^{3}}{b \operatorname {csch}\left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)^2*cosh(d*x+c)^3/(a+b*csch(d*x+c)),x, algorithm="giac")
Output:
integrate((f*x + e)^2*cosh(d*x + c)^3/(b*csch(d*x + c) + a), x)
Timed out. \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^3\,{\left (e+f\,x\right )}^2}{a+\frac {b}{\mathrm {sinh}\left (c+d\,x\right )}} \,d x \] Input:
int((cosh(c + d*x)^3*(e + f*x)^2)/(a + b/sinh(c + d*x)),x)
Output:
int((cosh(c + d*x)^3*(e + f*x)^2)/(a + b/sinh(c + d*x)), x)
\[ \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int \frac {\left (f x +e \right )^{2} \cosh \left (d x +c \right )^{3}}{a +b \,\mathrm {csch}\left (d x +c \right )}d x \] Input:
int((f*x+e)^2*cosh(d*x+c)^3/(a+b*csch(d*x+c)),x)
Output:
int((f*x+e)^2*cosh(d*x+c)^3/(a+b*csch(d*x+c)),x)