Integrand size = 28, antiderivative size = 510 \[ \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\frac {f^2 x}{4 a d^2}+\frac {(e+f x)^3}{6 a f}+\frac {b^2 (e+f x)^3}{3 a^3 f}-\frac {2 b f^2 \cosh (c+d x)}{a^2 d^3}-\frac {b (e+f x)^2 \cosh (c+d x)}{a^2 d}-\frac {f (e+f x) \cosh ^2(c+d x)}{2 a d^2}-\frac {b \sqrt {a^2+b^2} (e+f x)^2 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^3 d}+\frac {b \sqrt {a^2+b^2} (e+f x)^2 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^3 d}-\frac {2 b \sqrt {a^2+b^2} f (e+f x) \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {2 b \sqrt {a^2+b^2} f (e+f x) \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {2 b \sqrt {a^2+b^2} f^2 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^3 d^3}-\frac {2 b \sqrt {a^2+b^2} f^2 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^3 d^3}+\frac {2 b f (e+f x) \sinh (c+d x)}{a^2 d^2}+\frac {f^2 \cosh (c+d x) \sinh (c+d x)}{4 a d^3}+\frac {(e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 a d} \] Output:
1/4*f^2*x/a/d^2+1/6*(f*x+e)^3/a/f+1/3*b^2*(f*x+e)^3/a^3/f-2*b*f^2*cosh(d*x +c)/a^2/d^3-b*(f*x+e)^2*cosh(d*x+c)/a^2/d-1/2*f*(f*x+e)*cosh(d*x+c)^2/a/d^ 2-b*(a^2+b^2)^(1/2)*(f*x+e)^2*ln(1+a*exp(d*x+c)/(b-(a^2+b^2)^(1/2)))/a^3/d +b*(a^2+b^2)^(1/2)*(f*x+e)^2*ln(1+a*exp(d*x+c)/(b+(a^2+b^2)^(1/2)))/a^3/d- 2*b*(a^2+b^2)^(1/2)*f*(f*x+e)*polylog(2,-a*exp(d*x+c)/(b-(a^2+b^2)^(1/2))) /a^3/d^2+2*b*(a^2+b^2)^(1/2)*f*(f*x+e)*polylog(2,-a*exp(d*x+c)/(b+(a^2+b^2 )^(1/2)))/a^3/d^2+2*b*(a^2+b^2)^(1/2)*f^2*polylog(3,-a*exp(d*x+c)/(b-(a^2+ b^2)^(1/2)))/a^3/d^3-2*b*(a^2+b^2)^(1/2)*f^2*polylog(3,-a*exp(d*x+c)/(b+(a ^2+b^2)^(1/2)))/a^3/d^3+2*b*f*(f*x+e)*sinh(d*x+c)/a^2/d^2+1/4*f^2*cosh(d*x +c)*sinh(d*x+c)/a/d^3+1/2*(f*x+e)^2*cosh(d*x+c)*sinh(d*x+c)/a/d
Leaf count is larger than twice the leaf count of optimal. \(1216\) vs. \(2(510)=1020\).
Time = 4.03 (sec) , antiderivative size = 1216, normalized size of antiderivative = 2.38 \[ \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx =\text {Too large to display} \] Input:
Integrate[((e + f*x)^2*Cosh[c + d*x]^2)/(a + b*Csch[c + d*x]),x]
Output:
(Csch[c + d*x]*(b + a*Sinh[c + d*x])*(6*a^2*e^2*(c/d + x - (2*b*ArcTan[(a - b*Tanh[(c + d*x)/2])/Sqrt[-a^2 - b^2]])/(Sqrt[-a^2 - b^2]*d)) + 6*a^2*e* f*(x^2 - (2*b*(d*x*(Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])] - Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])]) + PolyLog[2, (a*E^(c + d*x))/(- b + Sqrt[a^2 + b^2])] - PolyLog[2, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]) )]))/(Sqrt[a^2 + b^2]*d^2)) + 2*a^2*f^2*(x^3 - (3*b*(d^2*x^2*Log[1 + (a*E^ (c + d*x))/(b - Sqrt[a^2 + b^2])] - d^2*x^2*Log[1 + (a*E^(c + d*x))/(b + S qrt[a^2 + b^2])] + 2*d*x*PolyLog[2, (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])] + 2*PolyLog[3, -((a*E^(c + d*x ))/(b + Sqrt[a^2 + b^2]))]))/(Sqrt[a^2 + b^2]*d^3)) + f^2*(2*(a^2 + 4*b^2) *x^3 - (6*b*(3*a^2 + 4*b^2)*(d^2*x^2*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + 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*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])] + 2*PolyLog[3, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]) )]))/(Sqrt[a^2 + b^2]*d^3) - (24*a*b*Cosh[d*x]*((2 + d^2*x^2)*Cosh[c] - 2* d*x*Sinh[c]))/d^3 + (3*a^2*Cosh[2*d*x]*(-2*d*x*Cosh[2*c] + (1 + 2*d^2*x^2) *Sinh[2*c]))/d^3 - (24*a*b*(-2*d*x*Cosh[c] + (2 + d^2*x^2)*Sinh[c])*Sinh[d *x])/d^3 + (3*a^2*((1 + 2*d^2*x^2)*Cosh[2*c] - 2*d*x*Sinh[2*c])*Sinh[2*...
Result contains complex when optimal does not.
Time = 2.81 (sec) , antiderivative size = 484, normalized size of antiderivative = 0.95, number of steps used = 28, number of rules used = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.964, Rules used = {6128, 6113, 3042, 3792, 17, 3042, 3115, 24, 6099, 17, 3042, 26, 3777, 3042, 3777, 26, 3042, 26, 3118, 3803, 25, 2694, 27, 2620, 3011, 2720, 7143}
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.
| \(\displaystyle \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 6128 |
| \(\displaystyle \int \frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{a \sinh (c+d x)+b}dx\) |
| \(\Big \downarrow \) 6113 |
| \(\displaystyle \frac {\int (e+f x)^2 \cosh ^2(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{b+a \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )^2dx}{a}\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle \frac {\frac {f^2 \int \cosh ^2(c+d x)dx}{2 d^2}+\frac {1}{2} \int (e+f x)^2dx-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}}{a}-\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{b+a \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle \frac {\frac {f^2 \int \cosh ^2(c+d x)dx}{2 d^2}-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}-\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{b+a \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\frac {f^2 \int \sin \left (i c+i d x+\frac {\pi }{2}\right )^2dx}{2 d^2}-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {f^2 \left (\frac {\int 1dx}{2}+\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}-\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{b+a \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}-\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{b+a \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 6099 |
| \(\displaystyle \frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{b+a \sinh (c+d x)}dx}{a^2}-\frac {b \int (e+f x)^2dx}{a^2}+\frac {\int (e+f x)^2 \sinh (c+d x)dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle \frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{b+a \sinh (c+d x)}dx}{a^2}+\frac {\int (e+f x)^2 \sinh (c+d x)dx}{a}-\frac {b (e+f x)^3}{3 a^2 f}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{b-i a \sin (i c+i d x)}dx}{a^2}+\frac {\int -i (e+f x)^2 \sin (i c+i d x)dx}{a}-\frac {b (e+f x)^3}{3 a^2 f}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{b-i a \sin (i c+i d x)}dx}{a^2}-\frac {i \int (e+f x)^2 \sin (i c+i d x)dx}{a}-\frac {b (e+f x)^3}{3 a^2 f}\right )}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{b-i a \sin (i c+i d x)}dx}{a^2}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \int (e+f x) \cosh (c+d x)dx}{d}\right )}{a}-\frac {b (e+f x)^3}{3 a^2 f}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{b-i a \sin (i c+i d x)}dx}{a^2}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{a}-\frac {b (e+f x)^3}{3 a^2 f}\right )}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{b-i a \sin (i c+i d x)}dx}{a^2}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {i f \int -i \sinh (c+d x)dx}{d}\right )}{d}\right )}{a}-\frac {b (e+f x)^3}{3 a^2 f}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{b-i a \sin (i c+i d x)}dx}{a^2}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int \sinh (c+d x)dx}{d}\right )}{d}\right )}{a}-\frac {b (e+f x)^3}{3 a^2 f}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{b-i a \sin (i c+i d x)}dx}{a^2}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int -i \sin (i c+i d x)dx}{d}\right )}{d}\right )}{a}-\frac {b (e+f x)^3}{3 a^2 f}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{b-i a \sin (i c+i d x)}dx}{a^2}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}+\frac {i f \int \sin (i c+i d x)dx}{d}\right )}{d}\right )}{a}-\frac {b (e+f x)^3}{3 a^2 f}\right )}{a}\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle \frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{b-i a \sin (i c+i d x)}dx}{a^2}-\frac {b (e+f x)^3}{3 a^2 f}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3803 |
| \(\displaystyle \frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}-\frac {b \left (\frac {2 \left (a^2+b^2\right ) \int -\frac {e^{c+d x} (e+f x)^2}{-e^{2 (c+d x)} a+a-2 b e^{c+d x}}dx}{a^2}-\frac {b (e+f x)^3}{3 a^2 f}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}-\frac {b \left (-\frac {2 \left (a^2+b^2\right ) \int \frac {e^{c+d x} (e+f x)^2}{-e^{2 (c+d x)} a+a-2 b e^{c+d x}}dx}{a^2}-\frac {b (e+f x)^3}{3 a^2 f}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2694 |
| \(\displaystyle \frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}-\frac {b \left (-\frac {2 \left (a^2+b^2\right ) \left (\frac {a \int -\frac {e^{c+d x} (e+f x)^2}{2 \left (e^{c+d x} a+b-\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}-\frac {a \int -\frac {e^{c+d x} (e+f x)^2}{2 \left (e^{c+d x} a+b+\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}\right )}{a^2}-\frac {b (e+f x)^3}{3 a^2 f}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}-\frac {b \left (-\frac {2 \left (a^2+b^2\right ) \left (\frac {a \int \frac {e^{c+d x} (e+f x)^2}{e^{c+d x} a+b+\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}-\frac {a \int \frac {e^{c+d x} (e+f x)^2}{e^{c+d x} a+b-\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}\right )}{a^2}-\frac {b (e+f x)^3}{3 a^2 f}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}-\frac {b \left (-\frac {2 \left (a^2+b^2\right ) \left (\frac {a \left (\frac {(e+f x)^2 \log \left (\frac {a e^{c+d x}}{\sqrt {a^2+b^2}+b}+1\right )}{a d}-\frac {2 f \int (e+f x) \log \left (\frac {e^{c+d x} a}{b+\sqrt {a^2+b^2}}+1\right )dx}{a d}\right )}{2 \sqrt {a^2+b^2}}-\frac {a \left (\frac {(e+f x)^2 \log \left (\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}+1\right )}{a d}-\frac {2 f \int (e+f x) \log \left (\frac {e^{c+d x} a}{b-\sqrt {a^2+b^2}}+1\right )dx}{a d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2}-\frac {b (e+f x)^3}{3 a^2 f}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}-\frac {b \left (-\frac {2 \left (a^2+b^2\right ) \left (\frac {a \left (\frac {(e+f x)^2 \log \left (\frac {a e^{c+d x}}{\sqrt {a^2+b^2}+b}+1\right )}{a d}-\frac {2 f \left (\frac {f \int \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )dx}{d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{d}\right )}{a d}\right )}{2 \sqrt {a^2+b^2}}-\frac {a \left (\frac {(e+f x)^2 \log \left (\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}+1\right )}{a d}-\frac {2 f \left (\frac {f \int \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )dx}{d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{d}\right )}{a d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2}-\frac {b (e+f x)^3}{3 a^2 f}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}-\frac {b \left (-\frac {2 \left (a^2+b^2\right ) \left (\frac {a \left (\frac {(e+f x)^2 \log \left (\frac {a e^{c+d x}}{\sqrt {a^2+b^2}+b}+1\right )}{a d}-\frac {2 f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{d}\right )}{a d}\right )}{2 \sqrt {a^2+b^2}}-\frac {a \left (\frac {(e+f x)^2 \log \left (\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}+1\right )}{a d}-\frac {2 f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{d}\right )}{a d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2}-\frac {b (e+f x)^3}{3 a^2 f}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}-\frac {b \left (-\frac {2 \left (a^2+b^2\right ) \left (\frac {a \left (\frac {(e+f x)^2 \log \left (\frac {a e^{c+d x}}{\sqrt {a^2+b^2}+b}+1\right )}{a d}-\frac {2 f \left (\frac {f \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{d}\right )}{a d}\right )}{2 \sqrt {a^2+b^2}}-\frac {a \left (\frac {(e+f x)^2 \log \left (\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}+1\right )}{a d}-\frac {2 f \left (\frac {f \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{d}\right )}{a d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2}-\frac {b (e+f x)^3}{3 a^2 f}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}\right )}{a}\) |
Input:
Int[((e + f*x)^2*Cosh[c + d*x]^2)/(a + b*Csch[c + d*x]),x]
Output:
((e + f*x)^3/(6*f) - (f*(e + f*x)*Cosh[c + d*x]^2)/(2*d^2) + ((e + f*x)^2* Cosh[c + d*x]*Sinh[c + d*x])/(2*d) + (f^2*(x/2 + (Cosh[c + d*x]*Sinh[c + d *x])/(2*d)))/(2*d^2))/a - (b*(-1/3*(b*(e + f*x)^3)/(a^2*f) - (2*(a^2 + b^2 )*(-1/2*(a*(((e + f*x)^2*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])])/( a*d) - (2*f*(-(((e + f*x)*PolyLog[2, -((a*E^(c + d*x))/(b - Sqrt[a^2 + b^2 ]))])/d) + (f*PolyLog[3, -((a*E^(c + d*x))/(b - Sqrt[a^2 + b^2]))])/d^2))/ (a*d)))/Sqrt[a^2 + b^2] + (a*(((e + f*x)^2*Log[1 + (a*E^(c + d*x))/(b + Sq rt[a^2 + b^2])])/(a*d) - (2*f*(-(((e + f*x)*PolyLog[2, -((a*E^(c + d*x))/( b + Sqrt[a^2 + b^2]))])/d) + (f*PolyLog[3, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))])/d^2))/(a*d)))/(2*Sqrt[a^2 + b^2])))/a^2 - (I*((I*(e + f*x)^2*C osh[c + d*x])/d - ((2*I)*f*(-((f*Cosh[c + d*x])/d^2) + ((e + f*x)*Sinh[c + d*x])/d))/d))/a))/a
Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 )/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.) *(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/q) Int [(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[2*(c/q) Int[(f + g*x) ^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[ v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m, 0]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( -(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*C os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 2*((n - 1)/n) Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 *m*((m - 1)/(f^2*n^2)) Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])* (f_.)*(x_)]), x_Symbol] :> Simp[2 Int[(c + d*x)^m*(E^((-I)*e + f*fz*x)/(( -I)*b + 2*a*E^((-I)*e + f*fz*x) + I*b*E^(2*((-I)*e + f*fz*x)))), x], x] /; FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
Int[(Cosh[(c_.) + (d_.)*(x_)]^(n_)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_. )*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[-a/b^2 Int[(e + f*x)^m*Cos h[c + d*x]^(n - 2), x], x] + (Simp[1/b Int[(e + f*x)^m*Cosh[c + d*x]^(n - 2)*Sinh[c + d*x], x], x] + Simp[(a^2 + b^2)/b^2 Int[(e + f*x)^m*(Cosh[c + d*x]^(n - 2)/(a + b*Sinh[c + d*x])), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 1] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0]
Int[(Cosh[(c_.) + (d_.)*(x_)]^(p_.)*((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> S imp[1/b Int[(e + f*x)^m*Cosh[c + d*x]^p*Sinh[c + d*x]^(n - 1), x], x] - S imp[a/b Int[(e + f*x)^m*Cosh[c + d*x]^p*(Sinh[c + d*x]^(n - 1)/(a + b*Sin h[c + d*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[ n, 0] && IGtQ[p, 0]
Int[(((e_.) + (f_.)*(x_))^(m_.)*(F_)[(c_.) + (d_.)*(x_)]^(n_.))/(Csch[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Int[(e + f*x)^m*Sinh[c + d*x]*(F [c + d*x]^n/(b + a*Sinh[c + d*x])), x] /; FreeQ[{a, b, c, d, e, f}, x] && H yperbolicQ[F] && IntegersQ[m, n]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
\[\int \frac {\left (f x +e \right )^{2} \cosh \left (d x +c \right )^{2}}{a +\operatorname {csch}\left (d x +c \right ) b}d x\]
Input:
int((f*x+e)^2*cosh(d*x+c)^2/(a+csch(d*x+c)*b),x)
Output:
int((f*x+e)^2*cosh(d*x+c)^2/(a+csch(d*x+c)*b),x)
Leaf count of result is larger than twice the leaf count of optimal. 2410 vs. \(2 (466) = 932\).
Time = 0.13 (sec) , antiderivative size = 2410, normalized size of antiderivative = 4.73 \[ \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^2*cosh(d*x+c)^2/(a+b*csch(d*x+c)),x, algorithm="fricas")
Output:
-1/48*(6*a^2*d^2*f^2*x^2 + 6*a^2*d^2*e^2 + 6*a^2*d*e*f - 3*(2*a^2*d^2*f^2* x^2 + 2*a^2*d^2*e^2 - 2*a^2*d*e*f + a^2*f^2 + 2*(2*a^2*d^2*e*f - a^2*d*f^2 )*x)*cosh(d*x + c)^4 - 3*(2*a^2*d^2*f^2*x^2 + 2*a^2*d^2*e^2 - 2*a^2*d*e*f + a^2*f^2 + 2*(2*a^2*d^2*e*f - a^2*d*f^2)*x)*sinh(d*x + c)^4 + 3*a^2*f^2 + 24*(a*b*d^2*f^2*x^2 + a*b*d^2*e^2 - 2*a*b*d*e*f + 2*a*b*f^2 + 2*(a*b*d^2* e*f - a*b*d*f^2)*x)*cosh(d*x + c)^3 + 12*(2*a*b*d^2*f^2*x^2 + 2*a*b*d^2*e^ 2 - 4*a*b*d*e*f + 4*a*b*f^2 + 4*(a*b*d^2*e*f - a*b*d*f^2)*x - (2*a^2*d^2*f ^2*x^2 + 2*a^2*d^2*e^2 - 2*a^2*d*e*f + a^2*f^2 + 2*(2*a^2*d^2*e*f - a^2*d* f^2)*x)*cosh(d*x + c))*sinh(d*x + c)^3 - 8*((a^2 + 2*b^2)*d^3*f^2*x^3 + 3* (a^2 + 2*b^2)*d^3*e*f*x^2 + 3*(a^2 + 2*b^2)*d^3*e^2*x)*cosh(d*x + c)^2 - 2 *(4*(a^2 + 2*b^2)*d^3*f^2*x^3 + 12*(a^2 + 2*b^2)*d^3*e*f*x^2 + 12*(a^2 + 2 *b^2)*d^3*e^2*x + 9*(2*a^2*d^2*f^2*x^2 + 2*a^2*d^2*e^2 - 2*a^2*d*e*f + a^2 *f^2 + 2*(2*a^2*d^2*e*f - a^2*d*f^2)*x)*cosh(d*x + c)^2 - 36*(a*b*d^2*f^2* x^2 + a*b*d^2*e^2 - 2*a*b*d*e*f + 2*a*b*f^2 + 2*(a*b*d^2*e*f - a*b*d*f^2)* x)*cosh(d*x + c))*sinh(d*x + c)^2 + 96*((a*b*d*f^2*x + a*b*d*e*f)*cosh(d*x + c)^2 + 2*(a*b*d*f^2*x + a*b*d*e*f)*cosh(d*x + c)*sinh(d*x + c) + (a*b*d *f^2*x + a*b*d*e*f)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/a^2)*dilog((b*cosh(d *x + c) + b*sinh(d*x + c) + (a*cosh(d*x + c) + a*sinh(d*x + c))*sqrt((a^2 + b^2)/a^2) - a)/a + 1) - 96*((a*b*d*f^2*x + a*b*d*e*f)*cosh(d*x + c)^2 + 2*(a*b*d*f^2*x + a*b*d*e*f)*cosh(d*x + c)*sinh(d*x + c) + (a*b*d*f^2*x ...
\[ \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int \frac {\left (e + f x\right )^{2} \cosh ^{2}{\left (c + d x \right )}}{a + b \operatorname {csch}{\left (c + d x \right )}}\, dx \] Input:
integrate((f*x+e)**2*cosh(d*x+c)**2/(a+b*csch(d*x+c)),x)
Output:
Integral((e + f*x)**2*cosh(c + d*x)**2/(a + b*csch(c + d*x)), x)
\[ \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \cosh \left (d x + c\right )^{2}}{b \operatorname {csch}\left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)^2*cosh(d*x+c)^2/(a+b*csch(d*x+c)),x, algorithm="maxima")
Output:
-1/8*e^2*((4*b*e^(-d*x - c) - a)*e^(2*d*x + 2*c)/(a^2*d) - 4*(a^2 + 2*b^2) *(d*x + c)/(a^3*d) + (4*b*e^(-d*x - c) + a*e^(-2*d*x - 2*c))/(a^2*d) + 8*( a^2*b + b^3)*log((a*e^(-d*x - c) - b - sqrt(a^2 + b^2))/(a*e^(-d*x - c) - b + sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*a^3*d)) + 1/48*(8*(a^2*d^3*f^2*e^(2 *c) + 2*b^2*d^3*f^2*e^(2*c))*x^3 + 24*(a^2*d^3*e*f*e^(2*c) + 2*b^2*d^3*e*f *e^(2*c))*x^2 + 3*(2*a^2*d^2*f^2*x^2*e^(4*c) + 2*(2*d^2*e*f - d*f^2)*a^2*x *e^(4*c) - (2*d*e*f - f^2)*a^2*e^(4*c))*e^(2*d*x) - 24*(a*b*d^2*f^2*x^2*e^ (3*c) + 2*(d^2*e*f - d*f^2)*a*b*x*e^(3*c) - 2*(d*e*f - f^2)*a*b*e^(3*c))*e ^(d*x) - 24*(a*b*d^2*f^2*x^2*e^c + 2*(d^2*e*f + d*f^2)*a*b*x*e^c + 2*(d*e* f + f^2)*a*b*e^c)*e^(-d*x) - 3*(2*a^2*d^2*f^2*x^2 + 2*(2*d^2*e*f + d*f^2)* a^2*x + (2*d*e*f + f^2)*a^2)*e^(-2*d*x))*e^(-2*c)/(a^3*d^3) - integrate(2* ((a^2*b*f^2*e^c + b^3*f^2*e^c)*x^2 + 2*(a^2*b*e*f*e^c + b^3*e*f*e^c)*x)*e^ (d*x)/(a^4*e^(2*d*x + 2*c) + 2*a^3*b*e^(d*x + c) - a^4), x)
\[ \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \cosh \left (d x + c\right )^{2}}{b \operatorname {csch}\left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)^2*cosh(d*x+c)^2/(a+b*csch(d*x+c)),x, algorithm="giac")
Output:
integrate((f*x + e)^2*cosh(d*x + c)^2/(b*csch(d*x + c) + a), x)
Timed out. \[ \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^2\,{\left (e+f\,x\right )}^2}{a+\frac {b}{\mathrm {sinh}\left (c+d\,x\right )}} \,d x \] Input:
int((cosh(c + d*x)^2*(e + f*x)^2)/(a + b/sinh(c + d*x)),x)
Output:
int((cosh(c + d*x)^2*(e + f*x)^2)/(a + b/sinh(c + d*x)), x)
\[ \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx =\text {Too large to display} \] Input:
int((f*x+e)^2*cosh(d*x+c)^2/(a+b*csch(d*x+c)),x)
Output:
( - 96*e**(2*c + 2*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*a*i + b*i)/sq rt(a**2 + b**2))*a**2*b*d**2*e**2*i + 6*e**(4*c + 4*d*x)*a**4*d**2*e**2 + 12*e**(4*c + 4*d*x)*a**4*d**2*e*f*x + 6*e**(4*c + 4*d*x)*a**4*d**2*f**2*x* *2 - 6*e**(4*c + 4*d*x)*a**4*d*e*f - 6*e**(4*c + 4*d*x)*a**4*d*f**2*x + 3* e**(4*c + 4*d*x)*a**4*f**2 - 24*e**(3*c + 3*d*x)*a**3*b*d**2*e**2 - 48*e** (3*c + 3*d*x)*a**3*b*d**2*e*f*x - 24*e**(3*c + 3*d*x)*a**3*b*d**2*f**2*x** 2 + 48*e**(3*c + 3*d*x)*a**3*b*d*e*f + 48*e**(3*c + 3*d*x)*a**3*b*d*f**2*x - 48*e**(3*c + 3*d*x)*a**3*b*f**2 + 192*e**(2*c + 2*d*x)*int(x**2/(e**(4* c + 4*d*x)*a + 2*e**(3*c + 3*d*x)*b - e**(2*c + 2*d*x)*a),x)*a**3*b**2*d** 3*f**2 + 192*e**(2*c + 2*d*x)*int(x**2/(e**(4*c + 4*d*x)*a + 2*e**(3*c + 3 *d*x)*b - e**(2*c + 2*d*x)*a),x)*a*b**4*d**3*f**2 + 384*e**(2*c + 2*d*x)*i nt(x/(e**(4*c + 4*d*x)*a + 2*e**(3*c + 3*d*x)*b - e**(2*c + 2*d*x)*a),x)*a **3*b**2*d**3*e*f + 384*e**(2*c + 2*d*x)*int(x/(e**(4*c + 4*d*x)*a + 2*e** (3*c + 3*d*x)*b - e**(2*c + 2*d*x)*a),x)*a*b**4*d**3*e*f + 24*e**(2*c + 2* d*x)*a**4*d**3*e**2*x + 24*e**(2*c + 2*d*x)*a**4*d**3*e*f*x**2 + 8*e**(2*c + 2*d*x)*a**4*d**3*f**2*x**3 + 48*e**(2*c + 2*d*x)*a**2*b**2*d**3*e**2*x + 48*e**(2*c + 2*d*x)*a**2*b**2*d**3*e*f*x**2 + 16*e**(2*c + 2*d*x)*a**2*b **2*d**3*f**2*x**3 - 96*e**(c + 2*d*x)*int(x**2/(e**(2*c + 3*d*x)*a + 2*e* *(c + 2*d*x)*b - e**(d*x)*a),x)*a**4*b*d**3*f**2 - 480*e**(c + 2*d*x)*int( x**2/(e**(2*c + 3*d*x)*a + 2*e**(c + 2*d*x)*b - e**(d*x)*a),x)*a**2*b**...