Integrand size = 34, antiderivative size = 510 \[ \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {f^2 x}{4 b d^2}+\frac {a^2 (e+f x)^3}{3 b^3 f}+\frac {(e+f x)^3}{6 b f}-\frac {2 a f^2 \cosh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^2 \cosh (c+d x)}{b^2 d}-\frac {f (e+f x) \cosh ^2(c+d x)}{2 b d^2}-\frac {a \sqrt {a^2+b^2} (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {a \sqrt {a^2+b^2} (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d}-\frac {2 a \sqrt {a^2+b^2} f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {2 a \sqrt {a^2+b^2} f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {2 a \sqrt {a^2+b^2} f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^3}-\frac {2 a \sqrt {a^2+b^2} f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^3}+\frac {2 a f (e+f x) \sinh (c+d x)}{b^2 d^2}+\frac {f^2 \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 b d} \] Output:
1/4*f^2*x/b/d^2+1/3*a^2*(f*x+e)^3/b^3/f+1/6*(f*x+e)^3/b/f-2*a*f^2*cosh(d*x +c)/b^2/d^3-a*(f*x+e)^2*cosh(d*x+c)/b^2/d-1/2*f*(f*x+e)*cosh(d*x+c)^2/b/d^ 2-a*(a^2+b^2)^(1/2)*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^3/d +a*(a^2+b^2)^(1/2)*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^3/d- 2*a*(a^2+b^2)^(1/2)*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2))) /b^3/d^2+2*a*(a^2+b^2)^(1/2)*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2 )^(1/2)))/b^3/d^2+2*a*(a^2+b^2)^(1/2)*f^2*polylog(3,-b*exp(d*x+c)/(a-(a^2+ b^2)^(1/2)))/b^3/d^3-2*a*(a^2+b^2)^(1/2)*f^2*polylog(3,-b*exp(d*x+c)/(a+(a ^2+b^2)^(1/2)))/b^3/d^3+2*a*f*(f*x+e)*sinh(d*x+c)/b^2/d^2+1/4*f^2*cosh(d*x +c)*sinh(d*x+c)/b/d^3+1/2*(f*x+e)^2*cosh(d*x+c)*sinh(d*x+c)/b/d
Leaf count is larger than twice the leaf count of optimal. \(1188\) vs. \(2(510)=1020\).
Time = 4.27 (sec) , antiderivative size = 1188, normalized size of antiderivative = 2.33 \[ \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Input:
Integrate[((e + f*x)^2*Cosh[c + d*x]^2*Sinh[c + d*x])/(a + b*Sinh[c + d*x] ),x]
Output:
(6*b^2*e^2*(c/d + x - (2*a*ArcTan[(b - a*Tanh[(c + d*x)/2])/Sqrt[-a^2 - b^ 2]])/(Sqrt[-a^2 - b^2]*d)) + 6*b^2*e*f*(x^2 - (2*a*(d*x*(Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] - Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^ 2])]) + PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - PolyLog[2, -( (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]))/(Sqrt[a^2 + b^2]*d^2)) + 2*b^2*f ^2*(x^3 - (3*a*(d^2*x^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] - d ^2*x^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + 2*d*x*PolyLog[2, ( b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - 2*d*x*PolyLog[2, -((b*E^(c + d*x) )/(a + Sqrt[a^2 + b^2]))] - 2*PolyLog[3, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + 2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]))/(Sqrt[a^ 2 + b^2]*d^3)) + f^2*(2*(4*a^2 + b^2)*x^3 - (6*a*(4*a^2 + 3*b^2)*(d^2*x^2* Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] - d^2*x^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + 2*d*x*PolyLog[2, (b*E^(c + d*x))/(-a + Sqr t[a^2 + b^2])] - 2*d*x*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])) ] - 2*PolyLog[3, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + 2*PolyLog[3, -( (b*E^(c + d*x))/(a + 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*b^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*Co sh[c] + (2 + d^2*x^2)*Sinh[c])*Sinh[d*x])/d^3 + (3*b^2*((1 + 2*d^2*x^2)*Co sh[2*c] - 2*d*x*Sinh[2*c])*Sinh[2*d*x])/d^3) + (6*e^2*((4*a^2 + b^2)*(c...
Result contains complex when optimal does not.
Time = 3.07 (sec) , antiderivative size = 484, normalized size of antiderivative = 0.95, number of steps used = 27, number of rules used = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.765, Rules used = {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.
| \(\displaystyle \int \frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6113 |
| \(\displaystyle \frac {\int (e+f x)^2 \cosh ^2(c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )^2dx}{b}\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle \frac {\frac {f^2 \int \cosh ^2(c+d x)dx}{2 d^2}+\frac {1}{2} \int (e+f x)^2dx-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle \frac {\frac {f^2 \int \cosh ^2(c+d x)dx}{2 d^2}-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {f^2 \int \sin \left (i c+i d x+\frac {\pi }{2}\right )^2dx}{2 d^2}-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {f^2 \left (\frac {\int 1dx}{2}+\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 6099 |
| \(\displaystyle \frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \int (e+f x)^2dx}{b^2}+\frac {\int (e+f x)^2 \sinh (c+d x)dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle \frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a+b \sinh (c+d x)}dx}{b^2}+\frac {\int (e+f x)^2 \sinh (c+d x)dx}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{b^2}+\frac {\int -i (e+f x)^2 \sin (i c+i d x)dx}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \int (e+f x)^2 \sin (i c+i d x)dx}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \int (e+f x) \cosh (c+d x)dx}{d}\right )}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {i f \int -i \sinh (c+d x)dx}{d}\right )}{d}\right )}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int \sinh (c+d x)dx}{d}\right )}{d}\right )}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int -i \sin (i c+i d x)dx}{d}\right )}{d}\right )}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}+\frac {i f \int \sin (i c+i d x)dx}{d}\right )}{d}\right )}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{b}\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle \frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {a (e+f x)^3}{3 b^2 f}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3803 |
| \(\displaystyle \frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \left (\frac {2 \left (a^2+b^2\right ) \int -\frac {e^{c+d x} (e+f x)^2}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{b^2}-\frac {a (e+f x)^3}{3 b^2 f}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \left (-\frac {2 \left (a^2+b^2\right ) \int \frac {e^{c+d x} (e+f x)^2}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{b^2}-\frac {a (e+f x)^3}{3 b^2 f}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2694 |
| \(\displaystyle \frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \left (-\frac {2 \left (a^2+b^2\right ) \left (\frac {b \int -\frac {e^{c+d x} (e+f x)^2}{2 \left (a+b e^{c+d x}-\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}-\frac {b \int -\frac {e^{c+d x} (e+f x)^2}{2 \left (a+b e^{c+d x}+\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}\right )}{b^2}-\frac {a (e+f x)^3}{3 b^2 f}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \left (-\frac {2 \left (a^2+b^2\right ) \left (\frac {b \int \frac {e^{c+d x} (e+f x)^2}{a+b e^{c+d x}+\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}-\frac {b \int \frac {e^{c+d x} (e+f x)^2}{a+b e^{c+d x}-\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}\right )}{b^2}-\frac {a (e+f x)^3}{3 b^2 f}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \left (-\frac {2 \left (a^2+b^2\right ) \left (\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {2 f \int (e+f x) \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {2 f \int (e+f x) \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{b^2}-\frac {a (e+f x)^3}{3 b^2 f}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \left (-\frac {2 \left (a^2+b^2\right ) \left (\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {2 f \left (\frac {f \int \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )dx}{d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {2 f \left (\frac {f \int \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )dx}{d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{b^2}-\frac {a (e+f x)^3}{3 b^2 f}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \left (-\frac {2 \left (a^2+b^2\right ) \left (\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {2 f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {2 f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{b^2}-\frac {a (e+f x)^3}{3 b^2 f}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \left (-\frac {2 \left (a^2+b^2\right ) \left (\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {2 f \left (\frac {f \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {2 f \left (\frac {f \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{b^2}-\frac {a (e+f x)^3}{3 b^2 f}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{b}\right )}{b}\) |
Input:
Int[((e + f*x)^2*Cosh[c + d*x]^2*Sinh[c + d*x])/(a + b*Sinh[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))/b - (a*(-1/3*(a*(e + f*x)^3)/(b^2*f) - (2*(a^2 + b^2 )*(-1/2*(b*(((e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/( b*d) - (2*f*(-(((e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2 ]))])/d) + (f*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/d^2))/ (b*d)))/Sqrt[a^2 + b^2] + (b*(((e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sq rt[a^2 + b^2])])/(b*d) - (2*f*(-(((e + f*x)*PolyLog[2, -((b*E^(c + d*x))/( a + Sqrt[a^2 + b^2]))])/d) + (f*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/d^2))/(b*d)))/(2*Sqrt[a^2 + b^2])))/b^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))/b))/b
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[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]
\[\int \frac {\left (f x +e \right )^{2} \cosh \left (d x +c \right )^{2} \sinh \left (d x +c \right )}{a +b \sinh \left (d x +c \right )}d x\]
Input:
int((f*x+e)^2*cosh(d*x+c)^2*sinh(d*x+c)/(a+b*sinh(d*x+c)),x)
Output:
int((f*x+e)^2*cosh(d*x+c)^2*sinh(d*x+c)/(a+b*sinh(d*x+c)),x)
Leaf count of result is larger than twice the leaf count of optimal. 2410 vs. \(2 (466) = 932\).
Time = 0.17 (sec) , antiderivative size = 2410, normalized size of antiderivative = 4.73 \[ \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^2*cosh(d*x+c)^2*sinh(d*x+c)/(a+b*sinh(d*x+c)),x, algorit hm="fricas")
Output:
-1/48*(6*b^2*d^2*f^2*x^2 + 6*b^2*d^2*e^2 + 6*b^2*d*e*f - 3*(2*b^2*d^2*f^2* x^2 + 2*b^2*d^2*e^2 - 2*b^2*d*e*f + b^2*f^2 + 2*(2*b^2*d^2*e*f - b^2*d*f^2 )*x)*cosh(d*x + c)^4 - 3*(2*b^2*d^2*f^2*x^2 + 2*b^2*d^2*e^2 - 2*b^2*d*e*f + b^2*f^2 + 2*(2*b^2*d^2*e*f - b^2*d*f^2)*x)*sinh(d*x + c)^4 + 3*b^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*b^2*d^2*f ^2*x^2 + 2*b^2*d^2*e^2 - 2*b^2*d*e*f + b^2*f^2 + 2*(2*b^2*d^2*e*f - b^2*d* f^2)*x)*cosh(d*x + c))*sinh(d*x + c)^3 - 8*((2*a^2 + b^2)*d^3*f^2*x^3 + 3* (2*a^2 + b^2)*d^3*e*f*x^2 + 3*(2*a^2 + b^2)*d^3*e^2*x)*cosh(d*x + c)^2 - 2 *(4*(2*a^2 + b^2)*d^3*f^2*x^3 + 12*(2*a^2 + b^2)*d^3*e*f*x^2 + 12*(2*a^2 + b^2)*d^3*e^2*x + 9*(2*b^2*d^2*f^2*x^2 + 2*b^2*d^2*e^2 - 2*b^2*d*e*f + b^2 *f^2 + 2*(2*b^2*d^2*e*f - b^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)/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) - 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 ...
Timed out. \[ \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)**2*cosh(d*x+c)**2*sinh(d*x+c)/(a+b*sinh(d*x+c)),x)
Output:
Timed out
\[ \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)^2*cosh(d*x+c)^2*sinh(d*x+c)/(a+b*sinh(d*x+c)),x, algorit hm="maxima")
Output:
-1/8*e^2*((4*a*e^(-d*x - c) - b)*e^(2*d*x + 2*c)/(b^2*d) + 8*sqrt(a^2 + b^ 2)*a*log((b*e^(-d*x - c) - a - sqrt(a^2 + b^2))/(b*e^(-d*x - c) - a + sqrt (a^2 + b^2)))/(b^3*d) - 4*(2*a^2 + b^2)*(d*x + c)/(b^3*d) + (4*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c))/(b^2*d)) + 1/48*(8*(2*a^2*d^3*f^2*e^(2*c) + b^2* d^3*f^2*e^(2*c))*x^3 + 24*(2*a^2*d^3*e*f*e^(2*c) + b^2*d^3*e*f*e^(2*c))*x^ 2 + 3*(2*b^2*d^2*f^2*x^2*e^(4*c) + 2*(2*d^2*e*f - d*f^2)*b^2*x*e^(4*c) - ( 2*d*e*f - f^2)*b^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*b^2*d^2*f^2*x^2 + 2*(2*d^2*e*f + d*f^2)*b^2*x + (2*d *e*f + f^2)*b^2)*e^(-2*d*x))*e^(-2*c)/(b^3*d^3) - integrate(2*((a^3*f^2*e^ c + a*b^2*f^2*e^c)*x^2 + 2*(a^3*e*f*e^c + a*b^2*e*f*e^c)*x)*e^(d*x)/(b^4*e ^(2*d*x + 2*c) + 2*a*b^3*e^(d*x + c) - b^4), x)
\[ \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)^2*cosh(d*x+c)^2*sinh(d*x+c)/(a+b*sinh(d*x+c)),x, algorit hm="giac")
Output:
integrate((f*x + e)^2*cosh(d*x + c)^2*sinh(d*x + c)/(b*sinh(d*x + c) + a), x)
Timed out. \[ \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^2\,\mathrm {sinh}\left (c+d\,x\right )\,{\left (e+f\,x\right )}^2}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \] Input:
int((cosh(c + d*x)^2*sinh(c + d*x)*(e + f*x)^2)/(a + b*sinh(c + d*x)),x)
Output:
int((cosh(c + d*x)^2*sinh(c + d*x)*(e + f*x)^2)/(a + b*sinh(c + d*x)), x)
\[ \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Input:
int((f*x+e)^2*cosh(d*x+c)^2*sinh(d*x+c)/(a+b*sinh(d*x+c)),x)
Output:
( - 96*e**(2*c + 2*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sq rt(a**2 + b**2))*a*b**2*d**2*e**2*i + 6*e**(4*c + 4*d*x)*b**4*d**2*e**2 + 12*e**(4*c + 4*d*x)*b**4*d**2*e*f*x + 6*e**(4*c + 4*d*x)*b**4*d**2*f**2*x* *2 - 6*e**(4*c + 4*d*x)*b**4*d*e*f - 6*e**(4*c + 4*d*x)*b**4*d*f**2*x + 3* e**(4*c + 4*d*x)*b**4*f**2 - 24*e**(3*c + 3*d*x)*a*b**3*d**2*e**2 - 48*e** (3*c + 3*d*x)*a*b**3*d**2*e*f*x - 24*e**(3*c + 3*d*x)*a*b**3*d**2*f**2*x** 2 + 48*e**(3*c + 3*d*x)*a*b**3*d*e*f + 48*e**(3*c + 3*d*x)*a*b**3*d*f**2*x - 48*e**(3*c + 3*d*x)*a*b**3*f**2 + 192*e**(2*c + 2*d*x)*int(x**2/(e**(4* c + 4*d*x)*b + 2*e**(3*c + 3*d*x)*a - e**(2*c + 2*d*x)*b),x)*a**4*b*d**3*f **2 + 192*e**(2*c + 2*d*x)*int(x**2/(e**(4*c + 4*d*x)*b + 2*e**(3*c + 3*d* x)*a - e**(2*c + 2*d*x)*b),x)*a**2*b**3*d**3*f**2 + 384*e**(2*c + 2*d*x)*i nt(x/(e**(4*c + 4*d*x)*b + 2*e**(3*c + 3*d*x)*a - e**(2*c + 2*d*x)*b),x)*a **4*b*d**3*e*f + 384*e**(2*c + 2*d*x)*int(x/(e**(4*c + 4*d*x)*b + 2*e**(3* c + 3*d*x)*a - e**(2*c + 2*d*x)*b),x)*a**2*b**3*d**3*e*f + 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 + 24*e**(2*c + 2*d*x)*b**4*d **3*e**2*x + 24*e**(2*c + 2*d*x)*b**4*d**3*e*f*x**2 + 8*e**(2*c + 2*d*x)*b **4*d**3*f**2*x**3 - 384*e**(c + 2*d*x)*int(x**2/(e**(2*c + 3*d*x)*b + 2*e **(c + 2*d*x)*a - e**(d*x)*b),x)*a**5*d**3*f**2 - 480*e**(c + 2*d*x)*int(x **2/(e**(2*c + 3*d*x)*b + 2*e**(c + 2*d*x)*a - e**(d*x)*b),x)*a**3*b**2...