Integrand size = 28, antiderivative size = 699 \[ \int \frac {(e+f x)^3 \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {3 f (e+f x)^2}{8 b d^2}+\frac {a^2 (e+f x)^4}{4 b^3 f}-\frac {(e+f x)^4}{8 b f}-\frac {6 a f^2 (e+f x) \cosh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^3 \cosh (c+d x)}{b^2 d}-\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d}+\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d}-\frac {3 a^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^2}+\frac {3 a^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^2}+\frac {6 a^3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^3}-\frac {6 a^3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^3}-\frac {6 a^3 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^4}+\frac {6 a^3 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^4}+\frac {6 a f^3 \sinh (c+d x)}{b^2 d^4}+\frac {3 a f (e+f x)^2 \sinh (c+d x)}{b^2 d^2}+\frac {3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {3 f^3 \sinh ^2(c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 b d^2} \] Output:
-3/8*f*(f*x+e)^2/b/d^2+1/4*a^2*(f*x+e)^4/b^3/f-1/8*(f*x+e)^4/b/f-6*a*f^2*( f*x+e)*cosh(d*x+c)/b^2/d^3-a*(f*x+e)^3*cosh(d*x+c)/b^2/d-a^3*(f*x+e)^3*ln( 1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^3/(a^2+b^2)^(1/2)/d+a^3*(f*x+e)^3*ln (1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^3/(a^2+b^2)^(1/2)/d-3*a^3*f*(f*x+e) ^2*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^3/(a^2+b^2)^(1/2)/d^2+3* a^3*f*(f*x+e)^2*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^3/(a^2+b^2) ^(1/2)/d^2+6*a^3*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/ b^3/(a^2+b^2)^(1/2)/d^3-6*a^3*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a+(a^2+ b^2)^(1/2)))/b^3/(a^2+b^2)^(1/2)/d^3-6*a^3*f^3*polylog(4,-b*exp(d*x+c)/(a- (a^2+b^2)^(1/2)))/b^3/(a^2+b^2)^(1/2)/d^4+6*a^3*f^3*polylog(4,-b*exp(d*x+c )/(a+(a^2+b^2)^(1/2)))/b^3/(a^2+b^2)^(1/2)/d^4+6*a*f^3*sinh(d*x+c)/b^2/d^4 +3*a*f*(f*x+e)^2*sinh(d*x+c)/b^2/d^2+3/4*f^2*(f*x+e)*cosh(d*x+c)*sinh(d*x+ c)/b/d^3+1/2*(f*x+e)^3*cosh(d*x+c)*sinh(d*x+c)/b/d-3/8*f^3*sinh(d*x+c)^2/b /d^4-3/4*f*(f*x+e)^2*sinh(d*x+c)^2/b/d^2
Leaf count is larger than twice the leaf count of optimal. \(1407\) vs. \(2(699)=1398\).
Time = 3.38 (sec) , antiderivative size = 1407, normalized size of antiderivative = 2.01 \[ \int \frac {(e+f x)^3 \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Input:
Integrate[((e + f*x)^3*Sinh[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]
Output:
(16*a^2*Sqrt[a^2 + b^2]*d^4*e^3*x - 8*b^2*Sqrt[a^2 + b^2]*d^4*e^3*x + 24*a ^2*Sqrt[a^2 + b^2]*d^4*e^2*f*x^2 - 12*b^2*Sqrt[a^2 + b^2]*d^4*e^2*f*x^2 + 16*a^2*Sqrt[a^2 + b^2]*d^4*e*f^2*x^3 - 8*b^2*Sqrt[a^2 + b^2]*d^4*e*f^2*x^3 + 4*a^2*Sqrt[a^2 + b^2]*d^4*f^3*x^4 - 2*b^2*Sqrt[a^2 + b^2]*d^4*f^3*x^4 + 32*a^3*d^3*e^3*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] - 16*a*b*Sqrt [a^2 + b^2]*d^3*e^3*Cosh[c + d*x] - 96*a*b*Sqrt[a^2 + b^2]*d*e*f^2*Cosh[c + d*x] - 48*a*b*Sqrt[a^2 + b^2]*d^3*e^2*f*x*Cosh[c + d*x] - 96*a*b*Sqrt[a^ 2 + b^2]*d*f^3*x*Cosh[c + d*x] - 48*a*b*Sqrt[a^2 + b^2]*d^3*e*f^2*x^2*Cosh [c + d*x] - 16*a*b*Sqrt[a^2 + b^2]*d^3*f^3*x^3*Cosh[c + d*x] - 6*b^2*Sqrt[ a^2 + b^2]*d^2*e^2*f*Cosh[2*(c + d*x)] - 3*b^2*Sqrt[a^2 + b^2]*f^3*Cosh[2* (c + d*x)] - 12*b^2*Sqrt[a^2 + b^2]*d^2*e*f^2*x*Cosh[2*(c + d*x)] - 6*b^2* Sqrt[a^2 + b^2]*d^2*f^3*x^2*Cosh[2*(c + d*x)] - 48*a^3*d^3*e^2*f*x*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] - 48*a^3*d^3*e*f^2*x^2*Log[1 + (b* E^(c + d*x))/(a - Sqrt[a^2 + b^2])] - 16*a^3*d^3*f^3*x^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + 48*a^3*d^3*e^2*f*x*Log[1 + (b*E^(c + d*x)) /(a + Sqrt[a^2 + b^2])] + 48*a^3*d^3*e*f^2*x^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + 16*a^3*d^3*f^3*x^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt [a^2 + b^2])] - 48*a^3*d^2*f*(e + f*x)^2*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + 48*a^3*d^2*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/ (a + Sqrt[a^2 + b^2]))] + 96*a^3*d*e*f^2*PolyLog[3, (b*E^(c + d*x))/(-a...
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.
| \(\displaystyle \int \frac {(e+f x)^3 \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6091 |
| \(\displaystyle \frac {\int (e+f x)^3 \sinh ^2(c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int -(e+f x)^3 \sin (i c+i d x)^2dx}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {\int (e+f x)^3 \sin (i c+i d x)^2dx}{b}\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {\frac {3 f^2 \int -\left ((e+f x) \sinh ^2(c+d x)\right )dx}{2 d^2}+\frac {1}{2} \int (e+f x)^3dx+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}}{b}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {\frac {3 f^2 \int -\left ((e+f x) \sinh ^2(c+d x)\right )dx}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {-\frac {3 f^2 \int (e+f x) \sinh ^2(c+d x)dx}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {-\frac {3 f^2 \int -\left ((e+f x) \sin (i c+i d x)^2\right )dx}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {\frac {3 f^2 \int (e+f x) \sin (i c+i d x)^2dx}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}\) |
| \(\Big \downarrow \) 3791 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {\frac {3 f^2 \left (\frac {1}{2} \int (e+f x)dx+\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {\frac {3 f^2 \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}\) |
| \(\Big \downarrow \) 6091 |
| \(\displaystyle -\frac {a \left (\frac {\int (e+f x)^3 \sinh (c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^3 \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}-\frac {\frac {3 f^2 \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {3 f^2 \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int -i (e+f x)^3 \sin (i c+i d x)dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {\frac {3 f^2 \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {i \int (e+f x)^3 \sin (i c+i d x)dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {\frac {3 f^2 \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \int (e+f x)^2 \cosh (c+d x)dx}{d}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {3 f^2 \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {\frac {3 f^2 \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 i f \int -i (e+f x) \sinh (c+d x)dx}{d}\right )}{d}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {\frac {3 f^2 \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int (e+f x) \sinh (c+d x)dx}{d}\right )}{d}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {3 f^2 \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int -i (e+f x) \sin (i c+i d x)dx}{d}\right )}{d}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {\frac {3 f^2 \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \int (e+f x) \sin (i c+i d x)dx}{d}\right )}{d}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {\frac {3 f^2 \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \cosh (c+d x)dx}{d}\right )}{d}\right )}{d}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {3 f^2 \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{d}\right )}{d}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle -\frac {\frac {3 f^2 \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 6091 |
| \(\displaystyle -\frac {\frac {3 f^2 \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (-\frac {a \left (\frac {\int (e+f x)^3dx}{b}-\frac {a \int \frac {(e+f x)^3}{a+b \sinh (c+d x)}dx}{b}\right )}{b}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle -\frac {\frac {3 f^2 \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (-\frac {a \left (\frac {(e+f x)^4}{4 b f}-\frac {a \int \frac {(e+f x)^3}{a+b \sinh (c+d x)}dx}{b}\right )}{b}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {3 f^2 \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (-\frac {a \left (\frac {(e+f x)^4}{4 b f}-\frac {a \int \frac {(e+f x)^3}{a-i b \sin (i c+i d x)}dx}{b}\right )}{b}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3803 |
| \(\displaystyle -\frac {\frac {3 f^2 \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (-\frac {a \left (\frac {(e+f x)^4}{4 b f}-\frac {2 a \int -\frac {e^{c+d x} (e+f x)^3}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{b}\right )}{b}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {3 f^2 \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (-\frac {a \left (\frac {2 a \int \frac {e^{c+d x} (e+f x)^3}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{b}+\frac {(e+f x)^4}{4 b f}\right )}{b}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2694 |
| \(\displaystyle -\frac {\frac {3 f^2 \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (-\frac {a \left (\frac {2 a \left (\frac {b \int -\frac {e^{c+d x} (e+f x)^3}{2 \left (a+b e^{c+d x}-\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}-\frac {b \int -\frac {e^{c+d x} (e+f x)^3}{2 \left (a+b e^{c+d x}+\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}\right )}{b}+\frac {(e+f x)^4}{4 b f}\right )}{b}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {3 f^2 \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (-\frac {a \left (\frac {2 a \left (\frac {b \int \frac {e^{c+d x} (e+f x)^3}{a+b e^{c+d x}+\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}-\frac {b \int \frac {e^{c+d x} (e+f x)^3}{a+b e^{c+d x}-\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}\right )}{b}+\frac {(e+f x)^4}{4 b f}\right )}{b}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {\frac {3 f^2 \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (-\frac {a \left (\frac {2 a \left (\frac {b \left (\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {3 f \int (e+f x)^2 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {3 f \int (e+f x)^2 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{b}+\frac {(e+f x)^4}{4 b f}\right )}{b}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b}\right )}{b}\) |
Input:
Int[((e + f*x)^3*Sinh[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]
Output:
$Aborted
\[\int \frac {\left (f x +e \right )^{3} \sinh \left (d x +c \right )^{3}}{a +b \sinh \left (d x +c \right )}d x\]
Input:
int((f*x+e)^3*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x)
Output:
int((f*x+e)^3*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x)
Leaf count of result is larger than twice the leaf count of optimal. 5191 vs. \(2 (643) = 1286\).
Time = 0.22 (sec) , antiderivative size = 5191, normalized size of antiderivative = 7.43 \[ \int \frac {(e+f x)^3 \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^3*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {(e+f x)^3 \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)**3*sinh(d*x+c)**3/(a+b*sinh(d*x+c)),x)
Output:
Timed out
\[ \int \frac {(e+f x)^3 \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \sinh \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)^3*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="maxima")
Output:
-1/8*e^3*(8*a^3*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^3*d) + (4*a*e^(-d*x - c) - b)* e^(2*d*x + 2*c)/(b^2*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)) + 1/32*(4*(2*a^2*d^4*f^3*e^(2*c) - b^ 2*d^4*f^3*e^(2*c))*x^4 + 16*(2*a^2*d^4*e*f^2*e^(2*c) - b^2*d^4*e*f^2*e^(2* c))*x^3 + 24*(2*a^2*d^4*e^2*f*e^(2*c) - b^2*d^4*e^2*f*e^(2*c))*x^2 + (4*b^ 2*d^3*f^3*x^3*e^(4*c) + 6*(2*d^3*e*f^2 - d^2*f^3)*b^2*x^2*e^(4*c) + 6*(2*d ^3*e^2*f - 2*d^2*e*f^2 + d*f^3)*b^2*x*e^(4*c) - 3*(2*d^2*e^2*f - 2*d*e*f^2 + f^3)*b^2*e^(4*c))*e^(2*d*x) - 16*(a*b*d^3*f^3*x^3*e^(3*c) + 3*(d^3*e*f^ 2 - d^2*f^3)*a*b*x^2*e^(3*c) + 3*(d^3*e^2*f - 2*d^2*e*f^2 + 2*d*f^3)*a*b*x *e^(3*c) - 3*(d^2*e^2*f - 2*d*e*f^2 + 2*f^3)*a*b*e^(3*c))*e^(d*x) - 16*(a* b*d^3*f^3*x^3*e^c + 3*(d^3*e*f^2 + d^2*f^3)*a*b*x^2*e^c + 3*(d^3*e^2*f + 2 *d^2*e*f^2 + 2*d*f^3)*a*b*x*e^c + 3*(d^2*e^2*f + 2*d*e*f^2 + 2*f^3)*a*b*e^ c)*e^(-d*x) - (4*b^2*d^3*f^3*x^3 + 6*(2*d^3*e*f^2 + d^2*f^3)*b^2*x^2 + 6*( 2*d^3*e^2*f + 2*d^2*e*f^2 + d*f^3)*b^2*x + 3*(2*d^2*e^2*f + 2*d*e*f^2 + f^ 3)*b^2)*e^(-2*d*x))*e^(-2*c)/(b^3*d^4) - integrate(2*(a^3*f^3*x^3*e^c + 3* a^3*e*f^2*x^2*e^c + 3*a^3*e^2*f*x*e^c)*e^(d*x)/(b^4*e^(2*d*x + 2*c) + 2*a* b^3*e^(d*x + c) - b^4), x)
\[ \int \frac {(e+f x)^3 \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \sinh \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)^3*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="giac")
Output:
integrate((f*x + e)^3*sinh(d*x + c)^3/(b*sinh(d*x + c) + a), x)
Timed out. \[ \int \frac {(e+f x)^3 \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^3\,{\left (e+f\,x\right )}^3}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \] Input:
int((sinh(c + d*x)^3*(e + f*x)^3)/(a + b*sinh(c + d*x)),x)
Output:
int((sinh(c + d*x)^3*(e + f*x)^3)/(a + b*sinh(c + d*x)), x)
\[ \int \frac {(e+f x)^3 \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (f x +e \right )^{3} \sinh \left (d x +c \right )^{3}}{a +b \sinh \left (d x +c \right )}d x \] Input:
int((f*x+e)^3*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x)
Output:
int((f*x+e)^3*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x)