Integrand size = 32, antiderivative size = 348 \[ \int \frac {(e+f x) \cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a f x}{4 b^2 d}+\frac {a^3 (e+f x)^2}{2 b^4 f}-\frac {a^2 f \cosh (c+d x)}{b^3 d^2}+\frac {f \cosh (c+d x)}{3 b d^2}-\frac {f \cosh ^3(c+d x)}{9 b d^2}-\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {a^3 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {a^3 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^2}+\frac {a^2 (e+f x) \sinh (c+d x)}{b^3 d}+\frac {a f \cosh (c+d x) \sinh (c+d x)}{4 b^2 d^2}-\frac {a (e+f x) \sinh ^2(c+d x)}{2 b^2 d}+\frac {(e+f x) \sinh ^3(c+d x)}{3 b d} \] Output:
-1/4*a*f*x/b^2/d+1/2*a^3*(f*x+e)^2/b^4/f-a^2*f*cosh(d*x+c)/b^3/d^2+1/3*f*c osh(d*x+c)/b/d^2-1/9*f*cosh(d*x+c)^3/b/d^2-a^3*(f*x+e)*ln(1+b*exp(d*x+c)/( a-(a^2+b^2)^(1/2)))/b^4/d-a^3*(f*x+e)*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2) ))/b^4/d-a^3*f*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^4/d^2-a^3*f* polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^4/d^2+a^2*(f*x+e)*sinh(d*x+ c)/b^3/d+1/4*a*f*cosh(d*x+c)*sinh(d*x+c)/b^2/d^2-1/2*a*(f*x+e)*sinh(d*x+c) ^2/b^2/d+1/3*(f*x+e)*sinh(d*x+c)^3/b/d
Time = 0.81 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.30 \[ \int \frac {(e+f x) \cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {36 a^3 c^2 f-36 a^3 d^2 f x^2+72 a^2 b f \cosh (c+d x)-18 b^3 f \cosh (c+d x)+18 a b^2 d f x \cosh (2 (c+d x))+2 b^3 f \cosh (3 (c+d x))+72 a^3 c f \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+72 a^3 d f x \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+72 a^3 c f \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+72 a^3 d f x \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-72 a^3 c f \log \left (b-2 a e^{c+d x}-b e^{2 (c+d x)}\right )+72 a^3 d e \log (a+b \sinh (c+d x))+72 a^3 f \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+72 a^3 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-72 a^2 b d e \sinh (c+d x)-72 a^2 b d f x \sinh (c+d x)+18 b^3 d f x \sinh (c+d x)+36 a b^2 d e \sinh ^2(c+d x)-24 b^3 d e \sinh ^3(c+d x)-9 a b^2 f \sinh (2 (c+d x))-6 b^3 d f x \sinh (3 (c+d x))}{72 b^4 d^2} \] Input:
Integrate[((e + f*x)*Cosh[c + d*x]*Sinh[c + d*x]^3)/(a + b*Sinh[c + d*x]), x]
Output:
-1/72*(36*a^3*c^2*f - 36*a^3*d^2*f*x^2 + 72*a^2*b*f*Cosh[c + d*x] - 18*b^3 *f*Cosh[c + d*x] + 18*a*b^2*d*f*x*Cosh[2*(c + d*x)] + 2*b^3*f*Cosh[3*(c + d*x)] + 72*a^3*c*f*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + 72*a^3 *d*f*x*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + 72*a^3*c*f*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + 72*a^3*d*f*x*Log[1 + (b*E^(c + d *x))/(a + Sqrt[a^2 + b^2])] - 72*a^3*c*f*Log[b - 2*a*E^(c + d*x) - b*E^(2* (c + d*x))] + 72*a^3*d*e*Log[a + b*Sinh[c + d*x]] + 72*a^3*f*PolyLog[2, (b *E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + 72*a^3*f*PolyLog[2, -((b*E^(c + d* x))/(a + Sqrt[a^2 + b^2]))] - 72*a^2*b*d*e*Sinh[c + d*x] - 72*a^2*b*d*f*x* Sinh[c + d*x] + 18*b^3*d*f*x*Sinh[c + d*x] + 36*a*b^2*d*e*Sinh[c + d*x]^2 - 24*b^3*d*e*Sinh[c + d*x]^3 - 9*a*b^2*f*Sinh[2*(c + d*x)] - 6*b^3*d*f*x*S inh[3*(c + d*x)])/(b^4*d^2)
Time = 2.29 (sec) , antiderivative size = 333, normalized size of antiderivative = 0.96, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.719, Rules used = {6113, 5969, 3042, 26, 3113, 2009, 6113, 5969, 3042, 25, 3115, 24, 6113, 3042, 3777, 26, 3042, 26, 3118, 6095, 2620, 2715, 2838}
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.
| \(\displaystyle \int \frac {(e+f x) \sinh ^3(c+d x) \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6113 |
| \(\displaystyle \frac {\int (e+f x) \cosh (c+d x) \sinh ^2(c+d x)dx}{b}-\frac {a \int \frac {(e+f x) \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 5969 |
| \(\displaystyle \frac {\frac {(e+f x) \sinh ^3(c+d x)}{3 d}-\frac {f \int \sinh ^3(c+d x)dx}{3 d}}{b}-\frac {a \int \frac {(e+f x) \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \int \frac {(e+f x) \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x) \sinh ^3(c+d x)}{3 d}-\frac {f \int i \sin (i c+i d x)^3dx}{3 d}}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {a \int \frac {(e+f x) \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x) \sinh ^3(c+d x)}{3 d}-\frac {i f \int \sin (i c+i d x)^3dx}{3 d}}{b}\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle \frac {\frac {f \int \left (1-\cosh ^2(c+d x)\right )d\cosh (c+d x)}{3 d^2}+\frac {(e+f x) \sinh ^3(c+d x)}{3 d}}{b}-\frac {a \int \frac {(e+f x) \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {f \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{3 d^2}+\frac {(e+f x) \sinh ^3(c+d x)}{3 d}}{b}-\frac {a \int \frac {(e+f x) \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 6113 |
| \(\displaystyle \frac {\frac {f \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{3 d^2}+\frac {(e+f x) \sinh ^3(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx}{b}-\frac {a \int \frac {(e+f x) \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 5969 |
| \(\displaystyle \frac {\frac {f \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{3 d^2}+\frac {(e+f x) \sinh ^3(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \sinh ^2(c+d x)}{2 d}-\frac {f \int \sinh ^2(c+d x)dx}{2 d}}{b}-\frac {a \int \frac {(e+f x) \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {f \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{3 d^2}+\frac {(e+f x) \sinh ^3(c+d x)}{3 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x) \sinh ^2(c+d x)}{2 d}-\frac {f \int -\sin (i c+i d x)^2dx}{2 d}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {f \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{3 d^2}+\frac {(e+f x) \sinh ^3(c+d x)}{3 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \int \sin (i c+i d x)^2dx}{2 d}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {f \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{3 d^2}+\frac {(e+f x) \sinh ^3(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\frac {f \left (\frac {\int 1dx}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}+\frac {(e+f x) \sinh ^2(c+d x)}{2 d}}{b}-\frac {a \int \frac {(e+f x) \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {f \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{3 d^2}+\frac {(e+f x) \sinh ^3(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}}{b}-\frac {a \int \frac {(e+f x) \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 6113 |
| \(\displaystyle \frac {\frac {f \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{3 d^2}+\frac {(e+f x) \sinh ^3(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}}{b}-\frac {a \left (\frac {\int (e+f x) \cosh (c+d x)dx}{b}-\frac {a \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {f \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{3 d^2}+\frac {(e+f x) \sinh ^3(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {\frac {f \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{3 d^2}+\frac {(e+f x) \sinh ^3(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x) \sinh (c+d x)}{d}-\frac {i f \int -i \sinh (c+d x)dx}{d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {f \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{3 d^2}+\frac {(e+f x) \sinh ^3(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int \sinh (c+d x)dx}{d}}{b}-\frac {a \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {f \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{3 d^2}+\frac {(e+f x) \sinh ^3(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int -i \sin (i c+i d x)dx}{d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {f \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{3 d^2}+\frac {(e+f x) \sinh ^3(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x) \sinh (c+d x)}{d}+\frac {i f \int \sin (i c+i d x)dx}{d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle \frac {\frac {f \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{3 d^2}+\frac {(e+f x) \sinh ^3(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}}{b}-\frac {a \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 6095 |
| \(\displaystyle \frac {\frac {f \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{3 d^2}+\frac {(e+f x) \sinh ^3(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}}{b}-\frac {a \left (\int \frac {e^{c+d x} (e+f x)}{a+b e^{c+d x}-\sqrt {a^2+b^2}}dx+\int \frac {e^{c+d x} (e+f x)}{a+b e^{c+d x}+\sqrt {a^2+b^2}}dx-\frac {(e+f x)^2}{2 b f}\right )}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {\frac {f \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{3 d^2}+\frac {(e+f x) \sinh ^3(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}}{b}-\frac {a \left (-\frac {f \int \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )dx}{b d}-\frac {f \int \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )dx}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^2}{2 b f}\right )}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {\frac {f \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{3 d^2}+\frac {(e+f x) \sinh ^3(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}}{b}-\frac {a \left (-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2}-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^2}{2 b f}\right )}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {\frac {f \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{3 d^2}+\frac {(e+f x) \sinh ^3(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}}{b}-\frac {a \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^2}{2 b f}\right )}{b}\right )}{b}\right )}{b}\) |
Input:
Int[((e + f*x)*Cosh[c + d*x]*Sinh[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]
Output:
((f*(Cosh[c + d*x] - Cosh[c + d*x]^3/3))/(3*d^2) + ((e + f*x)*Sinh[c + d*x ]^3)/(3*d))/b - (a*(-((a*(-((a*(-1/2*(e + f*x)^2/(b*f) + ((e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*d) + ((e + f*x)*Log[1 + (b*E^ (c + d*x))/(a + Sqrt[a^2 + b^2])])/(b*d) + (f*PolyLog[2, -((b*E^(c + d*x)) /(a - Sqrt[a^2 + b^2]))])/(b*d^2) + (f*PolyLog[2, -((b*E^(c + d*x))/(a + S qrt[a^2 + b^2]))])/(b*d^2)))/b) + (-((f*Cosh[c + d*x])/d^2) + ((e + f*x)*S inh[c + d*x])/d)/b))/b) + (((e + f*x)*Sinh[c + d*x]^2)/(2*d) + (f*(x/2 - ( Cosh[c + d*x]*Sinh[c + d*x])/(2*d)))/(2*d))/b))/b
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[(((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[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 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[Cosh[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)* (x_)]^(n_.), x_Symbol] :> Simp[(c + d*x)^m*(Sinh[a + b*x]^(n + 1)/(b*(n + 1 ))), x] - Simp[d*(m/(b*(n + 1))) Int[(c + d*x)^(m - 1)*Sinh[a + b*x]^(n + 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin h[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[-(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[(e + f*x)^m*(E^(c + d*x)/(a - Rt[a^2 + b^2, 2] + b*E^(c + d*x))) , x] + Int[(e + f*x)^m*(E^(c + d*x)/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x))) , x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 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]
Leaf count of result is larger than twice the leaf count of optimal. \(670\) vs. \(2(322)=644\).
Time = 28.71 (sec) , antiderivative size = 671, normalized size of antiderivative = 1.93
| method | result | size |
| risch | \(\frac {a^{3} f \,x^{2}}{2 b^{4}}-\frac {a^{3} e x}{b^{4}}+\frac {\left (3 d x f +3 d e -f \right ) {\mathrm e}^{3 d x +3 c}}{72 b \,d^{2}}-\frac {a \left (2 d x f +2 d e -f \right ) {\mathrm e}^{2 d x +2 c}}{16 b^{2} d^{2}}+\frac {\left (4 a^{2} d f x -b^{2} d f x +4 a^{2} d e -b^{2} d e -4 a^{2} f +b^{2} f \right ) {\mathrm e}^{d x +c}}{8 b^{3} d^{2}}-\frac {\left (4 a^{2}-b^{2}\right ) \left (d x f +d e +f \right ) {\mathrm e}^{-d x -c}}{8 b^{3} d^{2}}-\frac {a \left (2 d x f +2 d e +f \right ) {\mathrm e}^{-2 d x -2 c}}{16 b^{2} d^{2}}-\frac {\left (3 d x f +3 d e +f \right ) {\mathrm e}^{-3 d x -3 c}}{72 b \,d^{2}}+\frac {2 a^{3} e \ln \left ({\mathrm e}^{d x +c}\right )}{d \,b^{4}}+\frac {2 a^{3} f c x}{d \,b^{4}}-\frac {a^{3} f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,b^{4}}-\frac {a^{3} f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,b^{4}}-\frac {a^{3} f \operatorname {dilog}\left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} b^{4}}-\frac {a^{3} f \operatorname {dilog}\left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} b^{4}}+\frac {a^{3} f \,c^{2}}{d^{2} b^{4}}-\frac {a^{3} f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} b^{4}}-\frac {a^{3} f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} b^{4}}+\frac {a^{3} c f \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{d^{2} b^{4}}-\frac {2 a^{3} c f \ln \left ({\mathrm e}^{d x +c}\right )}{d^{2} b^{4}}-\frac {a^{3} e \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{d \,b^{4}}\) | \(671\) |
Input:
int((f*x+e)*cosh(d*x+c)*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x,method=_RETURNVE RBOSE)
Output:
1/2*a^3/b^4*f*x^2-a^3/b^4*e*x+1/72*(3*d*f*x+3*d*e-f)/b/d^2*exp(3*d*x+3*c)- 1/16*a*(2*d*f*x+2*d*e-f)/b^2/d^2*exp(2*d*x+2*c)+1/8*(4*a^2*d*f*x-b^2*d*f*x +4*a^2*d*e-b^2*d*e-4*a^2*f+b^2*f)/b^3/d^2*exp(d*x+c)-1/8*(4*a^2-b^2)*(d*f* x+d*e+f)/b^3/d^2*exp(-d*x-c)-1/16*a*(2*d*f*x+2*d*e+f)/b^2/d^2*exp(-2*d*x-2 *c)-1/72*(3*d*f*x+3*d*e+f)/b/d^2*exp(-3*d*x-3*c)+2/d*a^3/b^4*e*ln(exp(d*x+ c))+2/d*a^3/b^4*f*c*x-1/d*a^3/b^4*f*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/( -a+(a^2+b^2)^(1/2)))*x-1/d*a^3/b^4*f*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/( a+(a^2+b^2)^(1/2)))*x-1/d^2*a^3/b^4*f*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2) -a)/(-a+(a^2+b^2)^(1/2)))-1/d^2*a^3/b^4*f*dilog((b*exp(d*x+c)+(a^2+b^2)^(1 /2)+a)/(a+(a^2+b^2)^(1/2)))+1/d^2*a^3/b^4*f*c^2-1/d^2*a^3/b^4*f*ln((-b*exp (d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c-1/d^2*a^3/b^4*f*ln((b*e xp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c+1/d^2*a^3/b^4*c*f*ln(b *exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)-2/d^2*a^3/b^4*c*f*ln(exp(d*x+c))-1/d*a^3 /b^4*e*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)
Leaf count of result is larger than twice the leaf count of optimal. 2129 vs. \(2 (320) = 640\).
Time = 0.14 (sec) , antiderivative size = 2129, normalized size of antiderivative = 6.12 \[ \int \frac {(e+f x) \cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)*cosh(d*x+c)*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm ="fricas")
Output:
1/144*(2*(3*b^3*d*f*x + 3*b^3*d*e - b^3*f)*cosh(d*x + c)^6 + 2*(3*b^3*d*f* x + 3*b^3*d*e - b^3*f)*sinh(d*x + c)^6 - 6*b^3*d*f*x - 9*(2*a*b^2*d*f*x + 2*a*b^2*d*e - a*b^2*f)*cosh(d*x + c)^5 - 3*(6*a*b^2*d*f*x + 6*a*b^2*d*e - 3*a*b^2*f - 4*(3*b^3*d*f*x + 3*b^3*d*e - b^3*f)*cosh(d*x + c))*sinh(d*x + c)^5 - 6*b^3*d*e + 18*((4*a^2*b - b^3)*d*f*x + (4*a^2*b - b^3)*d*e - (4*a^ 2*b - b^3)*f)*cosh(d*x + c)^4 + 3*(6*(4*a^2*b - b^3)*d*f*x + 6*(4*a^2*b - b^3)*d*e + 10*(3*b^3*d*f*x + 3*b^3*d*e - b^3*f)*cosh(d*x + c)^2 - 6*(4*a^2 *b - b^3)*f - 15*(2*a*b^2*d*f*x + 2*a*b^2*d*e - a*b^2*f)*cosh(d*x + c))*si nh(d*x + c)^4 - 2*b^3*f + 72*(a^3*d^2*f*x^2 + 2*a^3*d^2*e*x + 4*a^3*c*d*e - 2*a^3*c^2*f)*cosh(d*x + c)^3 + 2*(36*a^3*d^2*f*x^2 + 72*a^3*d^2*e*x + 14 4*a^3*c*d*e - 72*a^3*c^2*f + 20*(3*b^3*d*f*x + 3*b^3*d*e - b^3*f)*cosh(d*x + c)^3 - 45*(2*a*b^2*d*f*x + 2*a*b^2*d*e - a*b^2*f)*cosh(d*x + c)^2 + 36* ((4*a^2*b - b^3)*d*f*x + (4*a^2*b - b^3)*d*e - (4*a^2*b - b^3)*f)*cosh(d*x + c))*sinh(d*x + c)^3 - 18*((4*a^2*b - b^3)*d*f*x + (4*a^2*b - b^3)*d*e + (4*a^2*b - b^3)*f)*cosh(d*x + c)^2 + 6*(5*(3*b^3*d*f*x + 3*b^3*d*e - b^3* f)*cosh(d*x + c)^4 - 3*(4*a^2*b - b^3)*d*f*x - 15*(2*a*b^2*d*f*x + 2*a*b^2 *d*e - a*b^2*f)*cosh(d*x + c)^3 - 3*(4*a^2*b - b^3)*d*e + 18*((4*a^2*b - b ^3)*d*f*x + (4*a^2*b - b^3)*d*e - (4*a^2*b - b^3)*f)*cosh(d*x + c)^2 - 3*( 4*a^2*b - b^3)*f + 36*(a^3*d^2*f*x^2 + 2*a^3*d^2*e*x + 4*a^3*c*d*e - 2*a^3 *c^2*f)*cosh(d*x + c))*sinh(d*x + c)^2 - 9*(2*a*b^2*d*f*x + 2*a*b^2*d*e...
Timed out. \[ \int \frac {(e+f x) \cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)*cosh(d*x+c)*sinh(d*x+c)**3/(a+b*sinh(d*x+c)),x)
Output:
Timed out
\[ \int \frac {(e+f x) \cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)*cosh(d*x+c)*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm ="maxima")
Output:
-1/24*e*(24*(d*x + c)*a^3/(b^4*d) + 24*a^3*log(-2*a*e^(-d*x - c) + b*e^(-2 *d*x - 2*c) - b)/(b^4*d) + (3*a*b*e^(-d*x - c) - b^2 - 3*(4*a^2 - b^2)*e^( -2*d*x - 2*c))*e^(3*d*x + 3*c)/(b^3*d) + (3*a*b*e^(-2*d*x - 2*c) + b^2*e^( -3*d*x - 3*c) + 3*(4*a^2 - b^2)*e^(-d*x - c))/(b^3*d)) - 1/144*f*((72*a^3* d^2*x^2*e^(3*c) - 2*(3*b^3*d*x*e^(6*c) - b^3*e^(6*c))*e^(3*d*x) + 9*(2*a*b ^2*d*x*e^(5*c) - a*b^2*e^(5*c))*e^(2*d*x) + 18*(4*a^2*b*e^(4*c) - b^3*e^(4 *c) - (4*a^2*b*d*e^(4*c) - b^3*d*e^(4*c))*x)*e^(d*x) + 18*(4*a^2*b*e^(2*c) - b^3*e^(2*c) + (4*a^2*b*d*e^(2*c) - b^3*d*e^(2*c))*x)*e^(-d*x) + 9*(2*a* b^2*d*x*e^c + a*b^2*e^c)*e^(-2*d*x) + 2*(3*b^3*d*x + b^3)*e^(-3*d*x))*e^(- 3*c)/(b^4*d^2) - 9*integrate(32*(a^4*x*e^(d*x + c) - a^3*b*x)/(b^5*e^(2*d* x + 2*c) + 2*a*b^4*e^(d*x + c) - b^5), x))
\[ \int \frac {(e+f x) \cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)*cosh(d*x+c)*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm ="giac")
Output:
integrate((f*x + e)*cosh(d*x + c)*sinh(d*x + c)^3/(b*sinh(d*x + c) + a), x )
Timed out. \[ \int \frac {(e+f x) \cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )\,{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (e+f\,x\right )}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \] Input:
int((cosh(c + d*x)*sinh(c + d*x)^3*(e + f*x))/(a + b*sinh(c + d*x)),x)
Output:
int((cosh(c + d*x)*sinh(c + d*x)^3*(e + f*x))/(a + b*sinh(c + d*x)), x)
\[ \int \frac {(e+f x) \cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {-2 e^{6 d x +6 c} b^{6} f +18 e^{2 d x +2 c} b^{6} f -96 a^{4} b^{2} f -6 b^{6} d e +6 e^{6 d x +6 c} b^{6} d f x -18 e^{5 d x +5 c} a \,b^{5} d e -72 e^{2 d x +2 c} a^{2} b^{4} d e +18 e^{2 d x +2 c} b^{6} d f x +1152 e^{3 d x +3 c} \left (\int \frac {x}{e^{5 d x +5 c} b +2 e^{4 d x +4 c} a -e^{3 d x +3 c} b}d x \right ) a^{6} b \,d^{2} f +864 e^{3 d x +3 c} \left (\int \frac {x}{e^{5 d x +5 c} b +2 e^{4 d x +4 c} a -e^{3 d x +3 c} b}d x \right ) a^{4} b^{3} d^{2} f -2304 e^{3 d x +c} \left (\int \frac {x}{e^{4 d x +2 c} b +2 e^{3 d x +c} a -e^{2 d x} b}d x \right ) a^{5} b^{2} d^{2} f -288 e^{3 d x +c} \left (\int \frac {x}{e^{4 d x +2 c} b +2 e^{3 d x +c} a -e^{2 d x} b}d x \right ) a^{3} b^{4} d^{2} f +6 e^{6 d x +6 c} b^{6} d e +9 e^{5 d x +5 c} a \,b^{5} f -288 e^{2 d x +2 c} a^{4} b^{2} f -72 e^{2 d x +2 c} a^{2} b^{4} f +18 e^{2 d x +2 c} b^{6} d e -18 e^{4 d x +4 c} b^{6} d e +144 e^{d x +c} a^{5} b f +72 e^{d x +c} a^{3} b^{3} f -9 e^{d x +c} a \,b^{5} f -384 a^{6} d f x -6 b^{6} d f x -2 b^{6} f -128 a^{6} f -2304 e^{3 d x +c} \left (\int \frac {x}{e^{4 d x +2 c} b +2 e^{3 d x +c} a -e^{2 d x} b}d x \right ) a^{7} d^{2} f +72 e^{4 d x +4 c} a^{2} b^{4} d e -18 e^{4 d x +4 c} b^{6} d f x -18 e^{d x +c} a \,b^{5} d e -288 a^{4} b^{2} d f x -72 e^{4 d x +4 c} a^{2} b^{4} f +18 e^{4 d x +4 c} b^{6} f +72 e^{4 d x +4 c} a^{2} b^{4} d f x +288 e^{d x +c} a^{5} b d f x +144 e^{d x +c} a^{3} b^{3} d f x -18 e^{d x +c} a \,b^{5} d f x -18 e^{5 d x +5 c} a \,b^{5} d f x -144 e^{3 d x +3 c} \mathrm {log}\left (e^{2 d x +2 c} b +2 e^{d x +c} a -b \right ) a^{3} b^{3} d e +144 e^{3 d x +3 c} a^{3} b^{3} d^{2} e x -72 e^{3 d x +3 c} a^{3} b^{3} d^{2} f \,x^{2}-288 e^{2 d x +2 c} a^{4} b^{2} d f x -72 e^{2 d x +2 c} a^{2} b^{4} d f x}{144 e^{3 d x +3 c} b^{7} d^{2}} \] Input:
int((f*x+e)*cosh(d*x+c)*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x)
Output:
(6*e**(6*c + 6*d*x)*b**6*d*e + 6*e**(6*c + 6*d*x)*b**6*d*f*x - 2*e**(6*c + 6*d*x)*b**6*f - 18*e**(5*c + 5*d*x)*a*b**5*d*e - 18*e**(5*c + 5*d*x)*a*b* *5*d*f*x + 9*e**(5*c + 5*d*x)*a*b**5*f + 72*e**(4*c + 4*d*x)*a**2*b**4*d*e + 72*e**(4*c + 4*d*x)*a**2*b**4*d*f*x - 72*e**(4*c + 4*d*x)*a**2*b**4*f - 18*e**(4*c + 4*d*x)*b**6*d*e - 18*e**(4*c + 4*d*x)*b**6*d*f*x + 18*e**(4* c + 4*d*x)*b**6*f + 1152*e**(3*c + 3*d*x)*int(x/(e**(5*c + 5*d*x)*b + 2*e* *(4*c + 4*d*x)*a - e**(3*c + 3*d*x)*b),x)*a**6*b*d**2*f + 864*e**(3*c + 3* d*x)*int(x/(e**(5*c + 5*d*x)*b + 2*e**(4*c + 4*d*x)*a - e**(3*c + 3*d*x)*b ),x)*a**4*b**3*d**2*f - 144*e**(3*c + 3*d*x)*log(e**(2*c + 2*d*x)*b + 2*e* *(c + d*x)*a - b)*a**3*b**3*d*e + 144*e**(3*c + 3*d*x)*a**3*b**3*d**2*e*x - 72*e**(3*c + 3*d*x)*a**3*b**3*d**2*f*x**2 - 2304*e**(c + 3*d*x)*int(x/(e **(2*c + 4*d*x)*b + 2*e**(c + 3*d*x)*a - e**(2*d*x)*b),x)*a**7*d**2*f - 23 04*e**(c + 3*d*x)*int(x/(e**(2*c + 4*d*x)*b + 2*e**(c + 3*d*x)*a - e**(2*d *x)*b),x)*a**5*b**2*d**2*f - 288*e**(c + 3*d*x)*int(x/(e**(2*c + 4*d*x)*b + 2*e**(c + 3*d*x)*a - e**(2*d*x)*b),x)*a**3*b**4*d**2*f - 288*e**(2*c + 2 *d*x)*a**4*b**2*d*f*x - 288*e**(2*c + 2*d*x)*a**4*b**2*f - 72*e**(2*c + 2* d*x)*a**2*b**4*d*e - 72*e**(2*c + 2*d*x)*a**2*b**4*d*f*x - 72*e**(2*c + 2* d*x)*a**2*b**4*f + 18*e**(2*c + 2*d*x)*b**6*d*e + 18*e**(2*c + 2*d*x)*b**6 *d*f*x + 18*e**(2*c + 2*d*x)*b**6*f + 288*e**(c + d*x)*a**5*b*d*f*x + 144* e**(c + d*x)*a**5*b*f + 144*e**(c + d*x)*a**3*b**3*d*f*x + 72*e**(c + d...