Integrand size = 28, antiderivative size = 551 \[ \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a (e+f x)^4}{4 b^2 f}+\frac {6 f^2 (e+f x) \cosh (c+d x)}{b d^3}+\frac {(e+f x)^3 \cosh (c+d x)}{b d}+\frac {a^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \sqrt {a^2+b^2} d}-\frac {a^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \sqrt {a^2+b^2} d}+\frac {3 a^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \sqrt {a^2+b^2} d^2}-\frac {3 a^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \sqrt {a^2+b^2} d^2}-\frac {6 a^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \sqrt {a^2+b^2} d^3}+\frac {6 a^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \sqrt {a^2+b^2} d^3}+\frac {6 a^2 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \sqrt {a^2+b^2} d^4}-\frac {6 a^2 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \sqrt {a^2+b^2} d^4}-\frac {6 f^3 \sinh (c+d x)}{b d^4}-\frac {3 f (e+f x)^2 \sinh (c+d x)}{b d^2} \] Output:
-1/4*a*(f*x+e)^4/b^2/f+6*f^2*(f*x+e)*cosh(d*x+c)/b/d^3+(f*x+e)^3*cosh(d*x+ c)/b/d+a^2*(f*x+e)^3*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^2/(a^2+b^2)^ (1/2)/d-a^2*(f*x+e)^3*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^2/(a^2+b^2) ^(1/2)/d+3*a^2*f*(f*x+e)^2*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^ 2/(a^2+b^2)^(1/2)/d^2-3*a^2*f*(f*x+e)^2*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^ 2)^(1/2)))/b^2/(a^2+b^2)^(1/2)/d^2-6*a^2*f^2*(f*x+e)*polylog(3,-b*exp(d*x+ c)/(a-(a^2+b^2)^(1/2)))/b^2/(a^2+b^2)^(1/2)/d^3+6*a^2*f^2*(f*x+e)*polylog( 3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^2/(a^2+b^2)^(1/2)/d^3+6*a^2*f^3*pol ylog(4,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^2/(a^2+b^2)^(1/2)/d^4-6*a^2*f^ 3*polylog(4,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^2/(a^2+b^2)^(1/2)/d^4-6*f ^3*sinh(d*x+c)/b/d^4-3*f*(f*x+e)^2*sinh(d*x+c)/b/d^2
Time = 1.64 (sec) , antiderivative size = 979, normalized size of antiderivative = 1.78 \[ \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {-4 a \sqrt {a^2+b^2} d^4 e^3 x-6 a \sqrt {a^2+b^2} d^4 e^2 f x^2-4 a \sqrt {a^2+b^2} d^4 e f^2 x^3-a \sqrt {a^2+b^2} d^4 f^3 x^4-8 a^2 d^3 e^3 \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+4 b \sqrt {a^2+b^2} d^3 e^3 \cosh (c+d x)+24 b \sqrt {a^2+b^2} d e f^2 \cosh (c+d x)+12 b \sqrt {a^2+b^2} d^3 e^2 f x \cosh (c+d x)+24 b \sqrt {a^2+b^2} d f^3 x \cosh (c+d x)+12 b \sqrt {a^2+b^2} d^3 e f^2 x^2 \cosh (c+d x)+4 b \sqrt {a^2+b^2} d^3 f^3 x^3 \cosh (c+d x)+12 a^2 d^3 e^2 f x \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+12 a^2 d^3 e f^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+4 a^2 d^3 f^3 x^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-12 a^2 d^3 e^2 f x \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-12 a^2 d^3 e f^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-4 a^2 d^3 f^3 x^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+12 a^2 d^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-12 a^2 d^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-24 a^2 d e f^2 \operatorname {PolyLog}\left (3,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-24 a^2 d f^3 x \operatorname {PolyLog}\left (3,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+24 a^2 d e f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+24 a^2 d f^3 x \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+24 a^2 f^3 \operatorname {PolyLog}\left (4,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-24 a^2 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-12 b \sqrt {a^2+b^2} d^2 e^2 f \sinh (c+d x)-24 b \sqrt {a^2+b^2} f^3 \sinh (c+d x)-24 b \sqrt {a^2+b^2} d^2 e f^2 x \sinh (c+d x)-12 b \sqrt {a^2+b^2} d^2 f^3 x^2 \sinh (c+d x)}{4 b^2 \sqrt {a^2+b^2} d^4} \] Input:
Integrate[((e + f*x)^3*Sinh[c + d*x]^2)/(a + b*Sinh[c + d*x]),x]
Output:
(-4*a*Sqrt[a^2 + b^2]*d^4*e^3*x - 6*a*Sqrt[a^2 + b^2]*d^4*e^2*f*x^2 - 4*a* Sqrt[a^2 + b^2]*d^4*e*f^2*x^3 - a*Sqrt[a^2 + b^2]*d^4*f^3*x^4 - 8*a^2*d^3* e^3*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] + 4*b*Sqrt[a^2 + b^2]*d^3 *e^3*Cosh[c + d*x] + 24*b*Sqrt[a^2 + b^2]*d*e*f^2*Cosh[c + d*x] + 12*b*Sqr t[a^2 + b^2]*d^3*e^2*f*x*Cosh[c + d*x] + 24*b*Sqrt[a^2 + b^2]*d*f^3*x*Cosh [c + d*x] + 12*b*Sqrt[a^2 + b^2]*d^3*e*f^2*x^2*Cosh[c + d*x] + 4*b*Sqrt[a^ 2 + b^2]*d^3*f^3*x^3*Cosh[c + d*x] + 12*a^2*d^3*e^2*f*x*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + 12*a^2*d^3*e*f^2*x^2*Log[1 + (b*E^(c + d*x) )/(a - Sqrt[a^2 + b^2])] + 4*a^2*d^3*f^3*x^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] - 12*a^2*d^3*e^2*f*x*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a ^2 + b^2])] - 12*a^2*d^3*e*f^2*x^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] - 4*a^2*d^3*f^3*x^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + 12*a^2*d^2*f*(e + f*x)^2*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^ 2])] - 12*a^2*d^2*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] - 24*a^2*d*e*f^2*PolyLog[3, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^ 2])] - 24*a^2*d*f^3*x*PolyLog[3, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + 24*a^2*d*e*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] + 24* a^2*d*f^3*x*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] + 24*a^2* f^3*PolyLog[4, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - 24*a^2*f^3*PolyLo g[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] - 12*b*Sqrt[a^2 + b^2]*d...
Result contains complex when optimal does not.
Time = 2.97 (sec) , antiderivative size = 501, normalized size of antiderivative = 0.91, number of steps used = 25, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {6091, 3042, 26, 3777, 3042, 3777, 26, 3042, 26, 3777, 3042, 3117, 6091, 17, 3042, 3803, 25, 2694, 27, 2620, 3011, 7163, 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.
\(\displaystyle \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
\(\Big \downarrow \) 6091 |
\(\displaystyle \frac {\int (e+f x)^3 \sinh (c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^3 \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int -i (e+f x)^3 \sin (i c+i d x)dx}{b}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {i \int (e+f x)^3 \sin (i c+i d x)dx}{b}\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \int (e+f x)^2 \cosh (c+d x)dx}{d}\right )}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{b}\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 i f \int -i (e+f x) \sinh (c+d x)dx}{d}\right )}{d}\right )}{b}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int (e+f x) \sinh (c+d x)dx}{d}\right )}{d}\right )}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int -i (e+f x) \sin (i c+i d x)dx}{d}\right )}{d}\right )}{b}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \int (e+f x) \sin (i c+i d x)dx}{d}\right )}{d}\right )}{b}\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \cosh (c+d x)dx}{d}\right )}{d}\right )}{d}\right )}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{d}\right )}{d}\right )}{b}\) |
\(\Big \downarrow \) 3117 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b}\) |
\(\Big \downarrow \) 6091 |
\(\displaystyle -\frac {a \left (\frac {\int (e+f x)^3dx}{b}-\frac {a \int \frac {(e+f x)^3}{a+b \sinh (c+d x)}dx}{b}\right )}{b}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b}\) |
\(\Big \downarrow \) 17 |
\(\displaystyle -\frac {a \left (\frac {(e+f x)^4}{4 b f}-\frac {a \int \frac {(e+f x)^3}{a+b \sinh (c+d x)}dx}{b}\right )}{b}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a \left (\frac {(e+f x)^4}{4 b f}-\frac {a \int \frac {(e+f x)^3}{a-i b \sin (i c+i d x)}dx}{b}\right )}{b}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b}\) |
\(\Big \downarrow \) 3803 |
\(\displaystyle -\frac {a \left (\frac {(e+f x)^4}{4 b f}-\frac {2 a \int -\frac {e^{c+d x} (e+f x)^3}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{b}\right )}{b}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {a \left (\frac {2 a \int \frac {e^{c+d x} (e+f x)^3}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{b}+\frac {(e+f x)^4}{4 b f}\right )}{b}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b}\) |
\(\Big \downarrow \) 2694 |
\(\displaystyle -\frac {a \left (\frac {2 a \left (\frac {b \int -\frac {e^{c+d x} (e+f x)^3}{2 \left (a+b e^{c+d x}-\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}-\frac {b \int -\frac {e^{c+d x} (e+f x)^3}{2 \left (a+b e^{c+d x}+\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}\right )}{b}+\frac {(e+f x)^4}{4 b f}\right )}{b}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {a \left (\frac {2 a \left (\frac {b \int \frac {e^{c+d x} (e+f x)^3}{a+b e^{c+d x}+\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}-\frac {b \int \frac {e^{c+d x} (e+f x)^3}{a+b e^{c+d x}-\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}\right )}{b}+\frac {(e+f x)^4}{4 b f}\right )}{b}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle -\frac {a \left (\frac {2 a \left (\frac {b \left (\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {3 f \int (e+f x)^2 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {3 f \int (e+f x)^2 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{b}+\frac {(e+f x)^4}{4 b f}\right )}{b}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b}\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle -\frac {a \left (\frac {2 a \left (\frac {b \left (\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {3 f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {3 f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{b}+\frac {(e+f x)^4}{4 b f}\right )}{b}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b}\) |
\(\Big \downarrow \) 7163 |
\(\displaystyle -\frac {a \left (\frac {2 a \left (\frac {b \left (\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}-\frac {f \int \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )dx}{d}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}-\frac {f \int \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )dx}{d}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{b}+\frac {(e+f x)^4}{4 b f}\right )}{b}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle -\frac {a \left (\frac {2 a \left (\frac {b \left (\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}-\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )de^{c+d x}}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}-\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )de^{c+d x}}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{b}+\frac {(e+f x)^4}{4 b f}\right )}{b}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b}\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle -\frac {a \left (\frac {2 a \left (\frac {b \left (\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}-\frac {f \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}-\frac {f \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{b}+\frac {(e+f x)^4}{4 b f}\right )}{b}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b}\) |
Input:
Int[((e + f*x)^3*Sinh[c + d*x]^2)/(a + b*Sinh[c + d*x]),x]
Output:
-((a*((e + f*x)^4/(4*b*f) + (2*a*(-1/2*(b*(((e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*d) - (3*f*(-(((e + f*x)^2*PolyLog[2, -((b *E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/d) + (2*f*(((e + f*x)*PolyLog[3, -( (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/d - (f*PolyLog[4, -((b*E^(c + d*x ))/(a - Sqrt[a^2 + b^2]))])/d^2))/d))/(b*d)))/Sqrt[a^2 + b^2] + (b*(((e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b*d) - (3*f*(-(((e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/d) + (2*f* (((e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/d - (f*P olyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/d^2))/d))/(b*d)))/(2* Sqrt[a^2 + b^2])))/b))/b) - (I*((I*(e + f*x)^3*Cosh[c + d*x])/d - ((3*I)*f *(((e + f*x)^2*Sinh[c + d*x])/d + ((2*I)*f*((I*(e + f*x)*Cosh[c + d*x])/d - (I*f*Sinh[c + d*x])/d^2))/d))/d))/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[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
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_.)/((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[(((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_ .)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[1/b Int[(e + f*x)^m*Sinh[ c + d*x]^(n - 1), x], x] - Simp[a/b Int[(e + f*x)^m*(Sinh[c + d*x]^(n - 1 )/(a + b*Sinh[c + d*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 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[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. )*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F])) Int[(e + f*x) ^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c , d, e, f, n, p}, x] && GtQ[m, 0]
\[\int \frac {\left (f x +e \right )^{3} \sinh \left (d x +c \right )^{2}}{a +b \sinh \left (d x +c \right )}d x\]
Input:
int((f*x+e)^3*sinh(d*x+c)^2/(a+b*sinh(d*x+c)),x)
Output:
int((f*x+e)^3*sinh(d*x+c)^2/(a+b*sinh(d*x+c)),x)
Leaf count of result is larger than twice the leaf count of optimal. 2612 vs. \(2 (507) = 1014\).
Time = 0.17 (sec) , antiderivative size = 2612, normalized size of antiderivative = 4.74 \[ \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^3*sinh(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="fricas")
Output:
1/4*(2*(a^2*b + b^3)*d^3*f^3*x^3 + 2*(a^2*b + b^3)*d^3*e^3 + 6*(a^2*b + b^ 3)*d^2*e^2*f + 12*(a^2*b + b^3)*d*e*f^2 + 12*(a^2*b + b^3)*f^3 + 6*((a^2*b + b^3)*d^3*e*f^2 + (a^2*b + b^3)*d^2*f^3)*x^2 + 2*((a^2*b + b^3)*d^3*f^3* x^3 + (a^2*b + b^3)*d^3*e^3 - 3*(a^2*b + b^3)*d^2*e^2*f + 6*(a^2*b + b^3)* d*e*f^2 - 6*(a^2*b + b^3)*f^3 + 3*((a^2*b + b^3)*d^3*e*f^2 - (a^2*b + b^3) *d^2*f^3)*x^2 + 3*((a^2*b + b^3)*d^3*e^2*f - 2*(a^2*b + b^3)*d^2*e*f^2 + 2 *(a^2*b + b^3)*d*f^3)*x)*cosh(d*x + c)^2 + 2*((a^2*b + b^3)*d^3*f^3*x^3 + (a^2*b + b^3)*d^3*e^3 - 3*(a^2*b + b^3)*d^2*e^2*f + 6*(a^2*b + b^3)*d*e*f^ 2 - 6*(a^2*b + b^3)*f^3 + 3*((a^2*b + b^3)*d^3*e*f^2 - (a^2*b + b^3)*d^2*f ^3)*x^2 + 3*((a^2*b + b^3)*d^3*e^2*f - 2*(a^2*b + b^3)*d^2*e*f^2 + 2*(a^2* b + b^3)*d*f^3)*x)*sinh(d*x + c)^2 + 12*((a^2*b*d^2*f^3*x^2 + 2*a^2*b*d^2* e*f^2*x + a^2*b*d^2*e^2*f)*cosh(d*x + c) + (a^2*b*d^2*f^3*x^2 + 2*a^2*b*d^ 2*e*f^2*x + a^2*b*d^2*e^2*f)*sinh(d*x + c))*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))*sqr t((a^2 + b^2)/b^2) - b)/b + 1) - 12*((a^2*b*d^2*f^3*x^2 + 2*a^2*b*d^2*e*f^ 2*x + a^2*b*d^2*e^2*f)*cosh(d*x + c) + (a^2*b*d^2*f^3*x^2 + 2*a^2*b*d^2*e* f^2*x + a^2*b*d^2*e^2*f)*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2)*dilog((a*cos h(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) - 4*((a^2*b*d^3*e^3 - 3*a^2*b*c*d^2*e^2*f + 3*a ^2*b*c^2*d*e*f^2 - a^2*b*c^3*f^3)*cosh(d*x + c) + (a^2*b*d^3*e^3 - 3*a^...
Timed out. \[ \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)**3*sinh(d*x+c)**2/(a+b*sinh(d*x+c)),x)
Output:
Timed out
\[ \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \sinh \left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)^3*sinh(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="maxima")
Output:
1/2*e^3*(2*a^2*log((b*e^(-d*x - c) - a - sqrt(a^2 + b^2))/(b*e^(-d*x - c) - a + sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*b^2*d) - 2*(d*x + c)*a/(b^2*d) + e^(d*x + c)/(b*d) + e^(-d*x - c)/(b*d)) - 1/4*(a*d^4*f^3*x^4*e^c + 4*a*d^4 *e*f^2*x^3*e^c + 6*a*d^4*e^2*f*x^2*e^c - 2*(b*d^3*f^3*x^3*e^(2*c) + 3*(d^3 *e*f^2 - d^2*f^3)*b*x^2*e^(2*c) + 3*(d^3*e^2*f - 2*d^2*e*f^2 + 2*d*f^3)*b* x*e^(2*c) - 3*(d^2*e^2*f - 2*d*e*f^2 + 2*f^3)*b*e^(2*c))*e^(d*x) - 2*(b*d^ 3*f^3*x^3 + 3*(d^3*e*f^2 + d^2*f^3)*b*x^2 + 3*(d^3*e^2*f + 2*d^2*e*f^2 + 2 *d*f^3)*b*x + 3*(d^2*e^2*f + 2*d*e*f^2 + 2*f^3)*b)*e^(-d*x))*e^(-c)/(b^2*d ^4) + integrate(2*(a^2*f^3*x^3*e^c + 3*a^2*e*f^2*x^2*e^c + 3*a^2*e^2*f*x*e ^c)*e^(d*x)/(b^3*e^(2*d*x + 2*c) + 2*a*b^2*e^(d*x + c) - b^3), x)
\[ \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \sinh \left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)^3*sinh(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="giac")
Output:
integrate((f*x + e)^3*sinh(d*x + c)^2/(b*sinh(d*x + c) + a), x)
Timed out. \[ \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^2\,{\left (e+f\,x\right )}^3}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \] Input:
int((sinh(c + d*x)^2*(e + f*x)^3)/(a + b*sinh(c + d*x)),x)
Output:
int((sinh(c + d*x)^2*(e + f*x)^3)/(a + b*sinh(c + d*x)), x)
\[ \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Input:
int((f*x+e)^3*sinh(d*x+c)^2/(a+b*sinh(d*x+c)),x)
Output:
(8*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqrt(a**2 + b**2))*a**2 *d**3*e**3*i + 4*cosh(c + d*x)*a**2*b*d**3*e**3 + 12*cosh(c + d*x)*a**2*b* d**3*e**2*f*x + 12*cosh(c + d*x)*a**2*b*d**3*e*f**2*x**2 + 4*cosh(c + d*x) *a**2*b*d**3*f**3*x**3 + 24*cosh(c + d*x)*a**2*b*d*e*f**2 + 24*cosh(c + d* x)*a**2*b*d*f**3*x + 4*cosh(c + d*x)*b**3*d**3*e**3 + 12*cosh(c + d*x)*b** 3*d**3*e**2*f*x + 12*cosh(c + d*x)*b**3*d**3*e*f**2*x**2 + 4*cosh(c + d*x) *b**3*d**3*f**3*x**3 + 24*cosh(c + d*x)*b**3*d*e*f**2 + 24*cosh(c + d*x)*b **3*d*f**3*x + 8*e**c*int((e**(d*x)*x**3)/(e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b),x)*a**4*d**4*f**3 + 8*e**c*int((e**(d*x)*x**3)/(e**(2*c + 2*d* x)*b + 2*e**(c + d*x)*a - b),x)*a**2*b**2*d**4*f**3 + 24*e**c*int((e**(d*x )*x**2)/(e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b),x)*a**4*d**4*e*f**2 + 24*e**c*int((e**(d*x)*x**2)/(e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b),x) *a**2*b**2*d**4*e*f**2 + 24*e**c*int((e**(d*x)*x)/(e**(2*c + 2*d*x)*b + 2* e**(c + d*x)*a - b),x)*a**4*d**4*e**2*f + 24*e**c*int((e**(d*x)*x)/(e**(2* c + 2*d*x)*b + 2*e**(c + d*x)*a - b),x)*a**2*b**2*d**4*e**2*f - 12*sinh(c + d*x)*a**2*b*d**2*e**2*f - 24*sinh(c + d*x)*a**2*b*d**2*e*f**2*x - 12*sin h(c + d*x)*a**2*b*d**2*f**3*x**2 - 24*sinh(c + d*x)*a**2*b*f**3 - 12*sinh( c + d*x)*b**3*d**2*e**2*f - 24*sinh(c + d*x)*b**3*d**2*e*f**2*x - 12*sinh( c + d*x)*b**3*d**2*f**3*x**2 - 24*sinh(c + d*x)*b**3*f**3 - 4*a**3*d**4*e* *3*x - 6*a**3*d**4*e**2*f*x**2 - 4*a**3*d**4*e*f**2*x**3 - a**3*d**4*f*...