Integrand size = 31, antiderivative size = 378 \[ \int \frac {(e+f x)^3 \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {3 i f (e+f x)^2}{8 a d^2}-\frac {i (e+f x)^3}{a d}+\frac {3 i (e+f x)^4}{8 a f}+\frac {6 f^2 (e+f x) \cosh (c+d x)}{a d^3}+\frac {(e+f x)^3 \cosh (c+d x)}{a d}+\frac {6 i f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {12 i f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {12 i f^3 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^4}-\frac {6 f^3 \sinh (c+d x)}{a d^4}-\frac {3 f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {3 i f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 a d^3}-\frac {i (e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {3 i f^3 \sinh ^2(c+d x)}{8 a d^4}+\frac {3 i f (e+f x)^2 \sinh ^2(c+d x)}{4 a d^2}-\frac {i (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d} \] Output:
3/8*I*f*(f*x+e)^2/a/d^2-I*(f*x+e)^3/a/d+3/8*I*(f*x+e)^4/a/f+6*f^2*(f*x+e)* cosh(d*x+c)/a/d^3+(f*x+e)^3*cosh(d*x+c)/a/d+6*I*f*(f*x+e)^2*ln(1+I*exp(d*x +c))/a/d^2+12*I*f^2*(f*x+e)*polylog(2,-I*exp(d*x+c))/a/d^3-12*I*f^3*polylo g(3,-I*exp(d*x+c))/a/d^4-6*f^3*sinh(d*x+c)/a/d^4-3*f*(f*x+e)^2*sinh(d*x+c) /a/d^2-3/4*I*f^2*(f*x+e)*cosh(d*x+c)*sinh(d*x+c)/a/d^3-1/2*I*(f*x+e)^3*cos h(d*x+c)*sinh(d*x+c)/a/d+3/8*I*f^3*sinh(d*x+c)^2/a/d^4+3/4*I*f*(f*x+e)^2*s inh(d*x+c)^2/a/d^2-I*(f*x+e)^3*tanh(1/2*c+1/4*I*Pi+1/2*d*x)/a/d
Time = 4.42 (sec) , antiderivative size = 376, normalized size of antiderivative = 0.99 \[ \int \frac {(e+f x)^3 \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {24 i e^3 x+36 i e^2 f x^2+24 i e f^2 x^3+6 i f^3 x^4+\frac {32 (e+f x)^3}{d \left (-i+e^c\right )}+\frac {96 f^2 (e+f x) \cosh (c+d x)}{d^3}+\frac {16 (e+f x)^3 \cosh (c+d x)}{d}+\frac {3 i f^3 \cosh (2 (c+d x))}{d^4}+\frac {6 i f (e+f x)^2 \cosh (2 (c+d x))}{d^2}+\frac {96 i f (e+f x)^2 \log \left (1-i e^{-c-d x}\right )}{d^2}-\frac {192 i f^2 \left (d (e+f x) \operatorname {PolyLog}\left (2,i e^{-c-d x}\right )+f \operatorname {PolyLog}\left (3,i e^{-c-d x}\right )\right )}{d^4}-\frac {32 i (e+f x)^3 \sinh \left (\frac {d x}{2}\right )}{d \left (\cosh \left (\frac {c}{2}\right )+i \sinh \left (\frac {c}{2}\right )\right ) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )}-\frac {96 f^3 \sinh (c+d x)}{d^4}-\frac {48 f (e+f x)^2 \sinh (c+d x)}{d^2}-\frac {6 i f^2 (e+f x) \sinh (2 (c+d x))}{d^3}-\frac {4 i (e+f x)^3 \sinh (2 (c+d x))}{d}}{16 a} \] Input:
Integrate[((e + f*x)^3*Sinh[c + d*x]^3)/(a + I*a*Sinh[c + d*x]),x]
Output:
((24*I)*e^3*x + (36*I)*e^2*f*x^2 + (24*I)*e*f^2*x^3 + (6*I)*f^3*x^4 + (32* (e + f*x)^3)/(d*(-I + E^c)) + (96*f^2*(e + f*x)*Cosh[c + d*x])/d^3 + (16*( e + f*x)^3*Cosh[c + d*x])/d + ((3*I)*f^3*Cosh[2*(c + d*x)])/d^4 + ((6*I)*f *(e + f*x)^2*Cosh[2*(c + d*x)])/d^2 + ((96*I)*f*(e + f*x)^2*Log[1 - I*E^(- c - d*x)])/d^2 - ((192*I)*f^2*(d*(e + f*x)*PolyLog[2, I*E^(-c - d*x)] + f* PolyLog[3, I*E^(-c - d*x)]))/d^4 - ((32*I)*(e + f*x)^3*Sinh[(d*x)/2])/(d*( Cosh[c/2] + I*Sinh[c/2])*(Cosh[(c + d*x)/2] + I*Sinh[(c + d*x)/2])) - (96* f^3*Sinh[c + d*x])/d^4 - (48*f*(e + f*x)^2*Sinh[c + d*x])/d^2 - ((6*I)*f^2 *(e + f*x)*Sinh[2*(c + d*x)])/d^3 - ((4*I)*(e + f*x)^3*Sinh[2*(c + d*x)])/ d)/(16*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.
| \(\displaystyle \int \frac {(e+f x)^3 \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6091 |
| \(\displaystyle i \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{i \sinh (c+d x) a+a}dx-\frac {i \int (e+f x)^3 \sinh ^2(c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{i \sinh (c+d x) a+a}dx-\frac {i \int -(e+f x)^3 \sin (i c+i d x)^2dx}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {i \int (e+f x)^3 \sin (i c+i d x)^2dx}{a}+i \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle \frac {i \left (\frac {3 f^2 \int -\left ((e+f x) \sinh ^2(c+d x)\right )dx}{2 d^2}+\frac {1}{2} \int (e+f x)^3dx+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{a}+i \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle \frac {i \left (\frac {3 f^2 \int -\left ((e+f x) \sinh ^2(c+d x)\right )dx}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}\right )}{a}+i \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {i \left (-\frac {3 f^2 \int (e+f x) \sinh ^2(c+d x)dx}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}\right )}{a}+i \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {i \left (-\frac {3 f^2 \int -\left ((e+f x) \sin (i c+i d x)^2\right )dx}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}\right )}{a}+i \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {i \left (\frac {3 f^2 \int (e+f x) \sin (i c+i d x)^2dx}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}\right )}{a}+i \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3791 |
| \(\displaystyle \frac {i \left (\frac {3 f^2 \left (\frac {1}{2} \int (e+f x)dx+\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}\right )}{a}+i \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle i \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{i \sinh (c+d x) a+a}dx+\frac {i \left (\frac {3 f^2 \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}\right )}{a}\) |
| \(\Big \downarrow \) 6091 |
| \(\displaystyle i \left (i \int \frac {(e+f x)^3 \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {i \int (e+f x)^3 \sinh (c+d x)dx}{a}\right )+\frac {i \left (\frac {3 f^2 \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (i \int \frac {(e+f x)^3 \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {i \int -i (e+f x)^3 \sin (i c+i d x)dx}{a}\right )+\frac {i \left (\frac {3 f^2 \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (i \int \frac {(e+f x)^3 \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {\int (e+f x)^3 \sin (i c+i d x)dx}{a}\right )+\frac {i \left (\frac {3 f^2 \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}\right )}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle i \left (i \int \frac {(e+f x)^3 \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \int (e+f x)^2 \cosh (c+d x)dx}{d}}{a}\right )+\frac {i \left (\frac {3 f^2 \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (i \int \frac {(e+f x)^3 \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}}{a}\right )+\frac {i \left (\frac {3 f^2 \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}\right )}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle i \left (i \int \frac {(e+f x)^3 \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 i f \int -i (e+f x) \sinh (c+d x)dx}{d}\right )}{d}}{a}\right )+\frac {i \left (\frac {3 f^2 \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (i \int \frac {(e+f x)^3 \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int (e+f x) \sinh (c+d x)dx}{d}\right )}{d}}{a}\right )+\frac {i \left (\frac {3 f^2 \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (i \int \frac {(e+f x)^3 \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int -i (e+f x) \sin (i c+i d x)dx}{d}\right )}{d}}{a}\right )+\frac {i \left (\frac {3 f^2 \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (i \int \frac {(e+f x)^3 \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \int (e+f x) \sin (i c+i d x)dx}{d}\right )}{d}}{a}\right )+\frac {i \left (\frac {3 f^2 \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}\right )}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle i \left (i \int \frac {(e+f x)^3 \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \cosh (c+d x)dx}{d}\right )}{d}\right )}{d}}{a}\right )+\frac {i \left (\frac {3 f^2 \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (i \int \frac {(e+f x)^3 \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{d}\right )}{d}}{a}\right )+\frac {i \left (\frac {3 f^2 \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}\right )}{a}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle i \left (i \int \frac {(e+f x)^3 \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}}{a}\right )+\frac {i \left (\frac {3 f^2 \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}\right )}{a}\) |
| \(\Big \downarrow \) 6091 |
| \(\displaystyle i \left (i \left (i \int \frac {(e+f x)^3}{i \sinh (c+d x) a+a}dx-\frac {i \int (e+f x)^3dx}{a}\right )-\frac {\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}}{a}\right )+\frac {i \left (\frac {3 f^2 \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}\right )}{a}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle i \left (i \left (i \int \frac {(e+f x)^3}{i \sinh (c+d x) a+a}dx-\frac {i (e+f x)^4}{4 a f}\right )-\frac {\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}}{a}\right )+\frac {i \left (\frac {3 f^2 \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (i \left (i \int \frac {(e+f x)^3}{\sin (i c+i d x) a+a}dx-\frac {i (e+f x)^4}{4 a f}\right )-\frac {\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}}{a}\right )+\frac {i \left (\frac {3 f^2 \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}\right )}{a}\) |
| \(\Big \downarrow \) 3799 |
| \(\displaystyle i \left (i \left (\frac {i \int -(e+f x)^3 \text {csch}^2\left (\frac {c}{2}+\frac {d x}{2}-\frac {i \pi }{4}\right )dx}{2 a}-\frac {i (e+f x)^4}{4 a f}\right )-\frac {\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}}{a}\right )+\frac {i \left (\frac {3 f^2 \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \left (i \left (-\frac {i \int -(e+f x)^3 \text {sech}^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{2 a}-\frac {i (e+f x)^4}{4 a f}\right )-\frac {\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}}{a}\right )+\frac {i \left (\frac {3 f^2 \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \left (i \left (\frac {i \int (e+f x)^3 \text {sech}^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{2 a}-\frac {i (e+f x)^4}{4 a f}\right )-\frac {\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}}{a}\right )+\frac {i \left (\frac {3 f^2 \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (i \left (\frac {i \int (e+f x)^3 \csc \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}-\frac {i (e+f x)^4}{4 a f}\right )-\frac {\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}}{a}\right )+\frac {i \left (\frac {3 f^2 \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}\right )}{a}\) |
| \(\Big \downarrow \) 4672 |
| \(\displaystyle i \left (i \left (\frac {i \left (\frac {2 (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {6 i f \int -i (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{d}\right )}{2 a}-\frac {i (e+f x)^4}{4 a f}\right )-\frac {\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}}{a}\right )+\frac {i \left (\frac {3 f^2 \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}\right )}{a}\) |
Input:
Int[((e + f*x)^3*Sinh[c + d*x]^3)/(a + I*a*Sinh[c + d*x]),x]
Output:
$Aborted
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1005 vs. \(2 (342 ) = 684\).
Time = 1.10 (sec) , antiderivative size = 1006, normalized size of antiderivative = 2.66
| method | result | size |
| risch | \(\frac {3 i e^{2} f \ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{a \,d^{2}}+\frac {12 i e \,f^{2} \operatorname {polylog}\left (2, -i {\mathrm e}^{d x +c}\right )}{a \,d^{3}}+\frac {12 i f^{3} \operatorname {polylog}\left (2, -i {\mathrm e}^{d x +c}\right ) x}{a \,d^{3}}+\frac {12 c \,f^{2} e \arctan \left ({\mathrm e}^{d x +c}\right )}{a \,d^{3}}+\frac {12 i c \,f^{2} e \ln \left ({\mathrm e}^{d x +c}\right )}{a \,d^{3}}-\frac {12 i f^{2} e c x}{a \,d^{2}}-\frac {6 i c \,f^{2} e \ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{a \,d^{3}}+\frac {12 i f^{2} e \ln \left (1+i {\mathrm e}^{d x +c}\right ) c}{a \,d^{3}}-\frac {6 e^{2} f \arctan \left ({\mathrm e}^{d x +c}\right )}{a \,d^{2}}-\frac {2 i f^{3} x^{3}}{a d}+\frac {4 i f^{3} c^{3}}{a \,d^{4}}-\frac {6 c^{2} f^{3} \arctan \left ({\mathrm e}^{d x +c}\right )}{a \,d^{4}}+\frac {6 i f^{3} c^{2} x}{a \,d^{3}}+\frac {3 i c^{2} f^{3} \ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{a \,d^{4}}-\frac {6 i e \,f^{2} x^{2}}{a d}-\frac {6 i e \,f^{2} c^{2}}{a \,d^{3}}-\frac {6 i e^{2} f \ln \left ({\mathrm e}^{d x +c}\right )}{a \,d^{2}}+\frac {\left (d^{3} x^{3} f^{3}+3 d^{3} e \,f^{2} x^{2}+3 d^{3} e^{2} f x -3 d^{2} f^{3} x^{2}+d^{3} e^{3}-6 d^{2} e \,f^{2} x -3 d^{2} e^{2} f +6 d \,f^{3} x +6 d e \,f^{2}-6 f^{3}\right ) {\mathrm e}^{d x +c}}{2 d^{4} a}+\frac {6 i f^{3} \ln \left (1+i {\mathrm e}^{d x +c}\right ) x^{2}}{a \,d^{2}}-\frac {6 i f^{3} \ln \left (1+i {\mathrm e}^{d x +c}\right ) c^{2}}{a \,d^{4}}-\frac {6 i c^{2} f^{3} \ln \left ({\mathrm e}^{d x +c}\right )}{a \,d^{4}}+\frac {12 i e \,f^{2} \ln \left (1+i {\mathrm e}^{d x +c}\right ) x}{a \,d^{2}}+\frac {\left (d^{3} x^{3} f^{3}+3 d^{3} e \,f^{2} x^{2}+3 d^{3} e^{2} f x +3 d^{2} f^{3} x^{2}+d^{3} e^{3}+6 d^{2} e \,f^{2} x +3 d^{2} e^{2} f +6 d \,f^{3} x +6 d e \,f^{2}+6 f^{3}\right ) {\mathrm e}^{-d x -c}}{2 d^{4} a}+\frac {2 x^{3} f^{3}+6 e \,f^{2} x^{2}+6 e^{2} f x +2 e^{3}}{d a \left ({\mathrm e}^{d x +c}-i\right )}+\frac {3 i f^{2} e \,x^{3}}{2 a}+\frac {9 i f \,e^{2} x^{2}}{4 a}-\frac {i \left (4 d^{3} x^{3} f^{3}+12 d^{3} e \,f^{2} x^{2}+12 d^{3} e^{2} f x -6 d^{2} f^{3} x^{2}+4 d^{3} e^{3}-12 d^{2} e \,f^{2} x -6 d^{2} e^{2} f +6 d \,f^{3} x +6 d e \,f^{2}-3 f^{3}\right ) {\mathrm e}^{2 d x +2 c}}{32 d^{4} a}+\frac {i \left (4 d^{3} x^{3} f^{3}+12 d^{3} e \,f^{2} x^{2}+12 d^{3} e^{2} f x +6 d^{2} f^{3} x^{2}+4 d^{3} e^{3}+12 d^{2} e \,f^{2} x +6 d^{2} e^{2} f +6 d \,f^{3} x +6 d e \,f^{2}+3 f^{3}\right ) {\mathrm e}^{-2 d x -2 c}}{32 d^{4} a}-\frac {12 i f^{3} \operatorname {polylog}\left (3, -i {\mathrm e}^{d x +c}\right )}{a \,d^{4}}+\frac {3 i f^{3} x^{4}}{8 a}+\frac {3 i e^{3} x}{2 a}+\frac {3 i e^{4}}{8 a f}\) | \(1006\) |
Input:
int((f*x+e)^3*sinh(d*x+c)^3/(a+I*a*sinh(d*x+c)),x,method=_RETURNVERBOSE)
Output:
3/2*I/a*f^2*e*x^3+9/4*I/a*f*e^2*x^2-1/32*I*(4*d^3*f^3*x^3+12*d^3*e*f^2*x^2 +12*d^3*e^2*f*x-6*d^2*f^3*x^2+4*d^3*e^3-12*d^2*e*f^2*x-6*d^2*e^2*f+6*d*f^3 *x+6*d*e*f^2-3*f^3)/d^4/a*exp(2*d*x+2*c)+3/8*I/a*f^3*x^4+3/2*I/a*e^3*x+3/8 *I/a/f*e^4+12/a/d^3*c*f^2*e*arctan(exp(d*x+c))+6*I/a/d^3*f^3*c^2*x+3*I/a/d ^4*c^2*f^3*ln(1+exp(2*d*x+2*c))-6*I/a/d*e*f^2*x^2-6*I/a/d^3*e*f^2*c^2-6*I/ a/d^4*c^2*f^3*ln(exp(d*x+c))-6*I/a/d^2*e^2*f*ln(exp(d*x+c))-6*I/a/d^4*f^3* ln(1+I*exp(d*x+c))*c^2+3*I/a/d^2*e^2*f*ln(1+exp(2*d*x+2*c))+6*I/a/d^2*f^3* ln(1+I*exp(d*x+c))*x^2+12*I/a/d^3*e*f^2*polylog(2,-I*exp(d*x+c))+12*I/a/d^ 3*f^3*polylog(2,-I*exp(d*x+c))*x-12*I/a/d^2*e*f^2*c*x+12*I/a/d^3*c*f^2*e*l n(exp(d*x+c))+12*I/a/d^3*e*f^2*ln(1+I*exp(d*x+c))*c+2*(f^3*x^3+3*e*f^2*x^2 +3*e^2*f*x+e^3)/d/a/(exp(d*x+c)-I)+1/32*I*(4*d^3*f^3*x^3+12*d^3*e*f^2*x^2+ 12*d^3*e^2*f*x+6*d^2*f^3*x^2+4*d^3*e^3+12*d^2*e*f^2*x+6*d^2*e^2*f+6*d*f^3* x+6*d*e*f^2+3*f^3)/d^4/a*exp(-2*d*x-2*c)-12*I*f^3*polylog(3,-I*exp(d*x+c)) /a/d^4-6/a/d^2*e^2*f*arctan(exp(d*x+c))+4*I/a/d^4*f^3*c^3-2*I/a/d*f^3*x^3- 6*I/a/d^3*c*f^2*e*ln(1+exp(2*d*x+2*c))+12*I/a/d^2*e*f^2*ln(1+I*exp(d*x+c)) *x-6/a/d^4*c^2*f^3*arctan(exp(d*x+c))+1/2*(d^3*f^3*x^3+3*d^3*e*f^2*x^2+3*d ^3*e^2*f*x-3*d^2*f^3*x^2+d^3*e^3-6*d^2*e*f^2*x-3*d^2*e^2*f+6*d*f^3*x+6*d*e *f^2-6*f^3)/d^4/a*exp(d*x+c)+1/2*(d^3*f^3*x^3+3*d^3*e*f^2*x^2+3*d^3*e^2*f* x+3*d^2*f^3*x^2+d^3*e^3+6*d^2*e*f^2*x+3*d^2*e^2*f+6*d*f^3*x+6*d*e*f^2+6*f^ 3)/d^4/a*exp(-d*x-c)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1044 vs. \(2 (326) = 652\).
Time = 0.12 (sec) , antiderivative size = 1044, normalized size of antiderivative = 2.76 \[ \int \frac {(e+f x)^3 \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^3*sinh(d*x+c)^3/(a+I*a*sinh(d*x+c)),x, algorithm="fricas ")
Output:
1/32*(4*d^3*f^3*x^3 + 4*d^3*e^3 + 6*d^2*e^2*f + 6*d*e*f^2 + 3*f^3 + 6*(2*d ^3*e*f^2 + d^2*f^3)*x^2 + 6*(2*d^3*e^2*f + 2*d^2*e*f^2 + d*f^3)*x - 384*(( -I*d*f^3*x - I*d*e*f^2)*e^(3*d*x + 3*c) - (d*f^3*x + d*e*f^2)*e^(2*d*x + 2 *c))*dilog(-I*e^(d*x + c)) + (-4*I*d^3*f^3*x^3 - 4*I*d^3*e^3 + 6*I*d^2*e^2 *f - 6*I*d*e*f^2 + 3*I*f^3 - 6*(2*I*d^3*e*f^2 - I*d^2*f^3)*x^2 - 6*(2*I*d^ 3*e^2*f - 2*I*d^2*e*f^2 + I*d*f^3)*x)*e^(5*d*x + 5*c) + 3*(4*d^3*f^3*x^3 + 4*d^3*e^3 - 14*d^2*e^2*f + 30*d*e*f^2 - 31*f^3 + 2*(6*d^3*e*f^2 - 7*d^2*f ^3)*x^2 + 2*(6*d^3*e^2*f - 14*d^2*e*f^2 + 15*d*f^3)*x)*e^(4*d*x + 4*c) - 4 *(-3*I*d^4*f^3*x^4 + 4*I*d^3*e^3 + 12*(4*I*c - I)*d^2*e^2*f + 24*(-2*I*c^2 + I)*d*e*f^2 + 8*(2*I*c^3 - 3*I)*f^3 + 4*(-3*I*d^4*e*f^2 + 5*I*d^3*f^3)*x ^3 + 6*(-3*I*d^4*e^2*f + 10*I*d^3*e*f^2 - 2*I*d^2*f^3)*x^2 + 12*(-I*d^4*e^ 3 + 5*I*d^3*e^2*f - 2*I*d^2*e*f^2 + 2*I*d*f^3)*x)*e^(3*d*x + 3*c) + 4*(3*d ^4*f^3*x^4 + 20*d^3*e^3 - 12*(4*c - 1)*d^2*e^2*f + 24*(2*c^2 + 1)*d*e*f^2 - 8*(2*c^3 - 3)*f^3 + 4*(3*d^4*e*f^2 + d^3*f^3)*x^3 + 6*(3*d^4*e^2*f + 2*d ^3*e*f^2 + 2*d^2*f^3)*x^2 + 12*(d^4*e^3 + d^3*e^2*f + 2*d^2*e*f^2 + 2*d*f^ 3)*x)*e^(2*d*x + 2*c) - 3*(4*I*d^3*f^3*x^3 + 4*I*d^3*e^3 + 14*I*d^2*e^2*f + 30*I*d*e*f^2 + 31*I*f^3 + 2*(6*I*d^3*e*f^2 + 7*I*d^2*f^3)*x^2 + 2*(6*I*d ^3*e^2*f + 14*I*d^2*e*f^2 + 15*I*d*f^3)*x)*e^(d*x + c) - 192*((-I*d^2*e^2* f + 2*I*c*d*e*f^2 - I*c^2*f^3)*e^(3*d*x + 3*c) - (d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*e^(2*d*x + 2*c))*log(e^(d*x + c) - I) - 192*((-I*d^2*f^3*x^2...
\[ \int \frac {(e+f x)^3 \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx =\text {Too large to display} \] Input:
integrate((f*x+e)**3*sinh(d*x+c)**3/(a+I*a*sinh(d*x+c)),x)
Output:
(2*e**3 + 6*e**2*f*x + 6*e*f**2*x**2 + 2*f**3*x**3)/(a*d*exp(c)*exp(d*x) - I*a*d) - I*(Integral(-I*d*e**3/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + I ntegral(-I*d*f**3*x**3/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(- d*e**3*exp(c)*exp(d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(- 4*d*e**3*exp(3*c)*exp(3*d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Inte gral(d*e**3*exp(5*c)*exp(5*d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + I ntegral(-3*I*d*e*f**2*x**2/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integr al(-3*I*d*e**2*f*x/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(4*I*d *e**3*exp(2*c)*exp(2*d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integra l(I*d*e**3*exp(4*c)*exp(4*d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + In tegral(-24*I*e**2*f*exp(2*c)*exp(2*d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)) , x) + Integral(-24*I*f**3*x**2*exp(2*c)*exp(2*d*x)/(exp(c)*exp(3*d*x) - I *exp(2*d*x)), x) + Integral(-d*f**3*x**3*exp(c)*exp(d*x)/(exp(c)*exp(3*d*x ) - I*exp(2*d*x)), x) + Integral(-4*d*f**3*x**3*exp(3*c)*exp(3*d*x)/(exp(c )*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(d*f**3*x**3*exp(5*c)*exp(5*d*x )/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(4*I*d*f**3*x**3*exp(2* c)*exp(2*d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(I*d*f**3*x **3*exp(4*c)*exp(4*d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral( -48*I*e*f**2*x*exp(2*c)*exp(2*d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(-3*d*e*f**2*x**2*exp(c)*exp(d*x)/(exp(c)*exp(3*d*x) - I*exp(...
Exception generated. \[ \int \frac {(e+f x)^3 \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((f*x+e)^3*sinh(d*x+c)^3/(a+I*a*sinh(d*x+c)),x, algorithm="maxima ")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
\[ \int \frac {(e+f x)^3 \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \sinh \left (d x + c\right )^{3}}{i \, a \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)^3*sinh(d*x+c)^3/(a+I*a*sinh(d*x+c)),x, algorithm="giac")
Output:
integrate((f*x + e)^3*sinh(d*x + c)^3/(I*a*sinh(d*x + c) + a), x)
Timed out. \[ \int \frac {(e+f x)^3 \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^3\,{\left (e+f\,x\right )}^3}{a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}} \,d x \] Input:
int((sinh(c + d*x)^3*(e + f*x)^3)/(a + a*sinh(c + d*x)*1i),x)
Output:
int((sinh(c + d*x)^3*(e + f*x)^3)/(a + a*sinh(c + d*x)*1i), x)
\[ \int \frac {(e+f x)^3 \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {6 \cosh \left (d x +c \right )^{2} d^{4} e^{2} f i \,x^{2}+\cosh \left (d x +c \right )^{2} d^{4} f^{3} i \,x^{4}+6 \cosh \left (d x +c \right )^{2} d^{2} e^{2} f i +3 \cosh \left (d x +c \right )^{2} d^{2} f^{3} i \,x^{2}+3 \cosh \left (d x +c \right )^{2} f^{3} i -12 \cosh \left (d x +c \right ) \sinh \left (d x +c \right ) d^{3} e^{2} f i x -4 \cosh \left (d x +c \right ) \sinh \left (d x +c \right ) d^{3} f^{3} i \,x^{3}-6 \cosh \left (d x +c \right ) \sinh \left (d x +c \right ) d \,f^{3} i x +24 \cosh \left (d x +c \right ) d^{3} e^{2} f x +8 \cosh \left (d x +c \right ) d^{3} f^{3} x^{3}+48 \cosh \left (d x +c \right ) d \,f^{3} x +8 \left (\int \frac {\sinh \left (d x +c \right )^{3}}{\sinh \left (d x +c \right ) i +1}d x \right ) d^{4} e^{3}+8 \left (\int -\frac {x^{3}}{\sinh \left (d x +c \right )-i}d x \right ) d^{4} f^{3}+24 \left (\int -\frac {x}{\sinh \left (d x +c \right )-i}d x \right ) d^{4} e^{2} f +24 \left (\int \frac {\sinh \left (d x +c \right )^{3} x^{2}}{\sinh \left (d x +c \right ) i +1}d x \right ) d^{4} e \,f^{2}-6 \sinh \left (d x +c \right )^{2} d^{4} e^{2} f i \,x^{2}-\sinh \left (d x +c \right )^{2} d^{4} f^{3} i \,x^{4}+3 \sinh \left (d x +c \right )^{2} d^{2} f^{3} i \,x^{2}-24 \sinh \left (d x +c \right ) d^{2} e^{2} f -24 \sinh \left (d x +c \right ) d^{2} f^{3} x^{2}-48 \sinh \left (d x +c \right ) f^{3}+12 d^{4} e^{2} f i \,x^{2}+2 d^{4} f^{3} i \,x^{4}}{8 a \,d^{4}} \] Input:
int((f*x+e)^3*sinh(d*x+c)^3/(a+I*a*sinh(d*x+c)),x)
Output:
(6*cosh(c + d*x)**2*d**4*e**2*f*i*x**2 + cosh(c + d*x)**2*d**4*f**3*i*x**4 + 6*cosh(c + d*x)**2*d**2*e**2*f*i + 3*cosh(c + d*x)**2*d**2*f**3*i*x**2 + 3*cosh(c + d*x)**2*f**3*i - 12*cosh(c + d*x)*sinh(c + d*x)*d**3*e**2*f*i *x - 4*cosh(c + d*x)*sinh(c + d*x)*d**3*f**3*i*x**3 - 6*cosh(c + d*x)*sinh (c + d*x)*d*f**3*i*x + 24*cosh(c + d*x)*d**3*e**2*f*x + 8*cosh(c + d*x)*d* *3*f**3*x**3 + 48*cosh(c + d*x)*d*f**3*x + 8*int(sinh(c + d*x)**3/(sinh(c + d*x)*i + 1),x)*d**4*e**3 + 8*int(( - x**3)/(sinh(c + d*x) - i),x)*d**4*f **3 + 24*int(( - x)/(sinh(c + d*x) - i),x)*d**4*e**2*f + 24*int((sinh(c + d*x)**3*x**2)/(sinh(c + d*x)*i + 1),x)*d**4*e*f**2 - 6*sinh(c + d*x)**2*d* *4*e**2*f*i*x**2 - sinh(c + d*x)**2*d**4*f**3*i*x**4 + 3*sinh(c + d*x)**2* d**2*f**3*i*x**2 - 24*sinh(c + d*x)*d**2*e**2*f - 24*sinh(c + d*x)*d**2*f* *3*x**2 - 48*sinh(c + d*x)*f**3 + 12*d**4*e**2*f*i*x**2 + 2*d**4*f**3*i*x* *4)/(8*a*d**4)