Integrand size = 31, antiderivative size = 287 \[ \int \frac {(e+f x)^2 \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {i f^2 x}{4 a d^2}-\frac {i (e+f x)^2}{a d}+\frac {i (e+f x)^3}{2 a f}+\frac {2 f^2 \cosh (c+d x)}{a d^3}+\frac {(e+f x)^2 \cosh (c+d x)}{a d}+\frac {4 i f (e+f x) \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {4 i f^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {2 f (e+f x) \sinh (c+d x)}{a d^2}-\frac {i f^2 \cosh (c+d x) \sinh (c+d x)}{4 a d^3}-\frac {i (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {i f (e+f x) \sinh ^2(c+d x)}{2 a d^2}-\frac {i (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d} \]
1/4*I*f^2*x/a/d^2-I*(f*x+e)^2/a/d+1/2*I*(f*x+e)^3/a/f+2*f^2*cosh(d*x+c)/a/ d^3+(f*x+e)^2*cosh(d*x+c)/a/d+4*I*f*(f*x+e)*ln(1+I*exp(d*x+c))/a/d^2+4*I*f ^2*polylog(2,-I*exp(d*x+c))/a/d^3-2*f*(f*x+e)*sinh(d*x+c)/a/d^2-1/4*I*f^2* cosh(d*x+c)*sinh(d*x+c)/a/d^3-1/2*I*(f*x+e)^2*cosh(d*x+c)*sinh(d*x+c)/a/d+ 1/2*I*f*(f*x+e)*sinh(d*x+c)^2/a/d^2-I*(f*x+e)^2*tanh(1/2*c+1/4*I*Pi+1/2*d* x)/a/d
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1661\) vs. \(2(287)=574\).
Time = 3.72 (sec) , antiderivative size = 1661, normalized size of antiderivative = 5.79 \[ \int \frac {(e+f x)^2 \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx =\text {Too large to display} \]
((-6*I)*d^2*e^2*E^c*Cosh[(3*d*x)/2] + 6*d^2*e^2*E^(4*c)*Cosh[(3*d*x)/2] - (14*I)*d*e*E^c*f*Cosh[(3*d*x)/2] - 14*d*e*E^(4*c)*f*Cosh[(3*d*x)/2] - (15* I)*E^c*f^2*Cosh[(3*d*x)/2] + 15*E^(4*c)*f^2*Cosh[(3*d*x)/2] - (12*I)*d^2*e *E^c*f*x*Cosh[(3*d*x)/2] + 12*d^2*e*E^(4*c)*f*x*Cosh[(3*d*x)/2] - (14*I)*d *E^c*f^2*x*Cosh[(3*d*x)/2] - 14*d*E^(4*c)*f^2*x*Cosh[(3*d*x)/2] - (6*I)*d^ 2*E^c*f^2*x^2*Cosh[(3*d*x)/2] + 6*d^2*E^(4*c)*f^2*x^2*Cosh[(3*d*x)/2] + 2* d^2*e^2*Cosh[(5*d*x)/2] - (2*I)*d^2*e^2*E^(5*c)*Cosh[(5*d*x)/2] + 2*d*e*f* Cosh[(5*d*x)/2] + (2*I)*d*e*E^(5*c)*f*Cosh[(5*d*x)/2] + f^2*Cosh[(5*d*x)/2 ] - I*E^(5*c)*f^2*Cosh[(5*d*x)/2] + 4*d^2*e*f*x*Cosh[(5*d*x)/2] - (4*I)*d^ 2*e*E^(5*c)*f*x*Cosh[(5*d*x)/2] + 2*d*f^2*x*Cosh[(5*d*x)/2] + (2*I)*d*E^(5 *c)*f^2*x*Cosh[(5*d*x)/2] + 2*d^2*f^2*x^2*Cosh[(5*d*x)/2] - (2*I)*d^2*E^(5 *c)*f^2*x^2*Cosh[(5*d*x)/2] + 8*E^(2*c)*Cosh[(d*x)/2]*(2*(1 - I*E^c)*f^2 + 2*d*(1 + I*E^c)*f*(e + f*x) + d^2*(5 - I*E^c)*(e + f*x)^2 + d^3*(1 + I*E^ c)*x*(3*e^2 + 3*e*f*x + f^2*x^2) + 8*d*(1 + I*E^c)*f*(e + f*x)*Log[1 - I*E ^(-c - d*x)]) - 40*d^2*e^2*E^(2*c)*Sinh[(d*x)/2] - (8*I)*d^2*e^2*E^(3*c)*S inh[(d*x)/2] - 16*d*e*E^(2*c)*f*Sinh[(d*x)/2] + (16*I)*d*e*E^(3*c)*f*Sinh[ (d*x)/2] - 16*E^(2*c)*f^2*Sinh[(d*x)/2] - (16*I)*E^(3*c)*f^2*Sinh[(d*x)/2] - 24*d^3*e^2*E^(2*c)*x*Sinh[(d*x)/2] + (24*I)*d^3*e^2*E^(3*c)*x*Sinh[(d*x )/2] - 80*d^2*e*E^(2*c)*f*x*Sinh[(d*x)/2] - (16*I)*d^2*e*E^(3*c)*f*x*Sinh[ (d*x)/2] - 16*d*E^(2*c)*f^2*x*Sinh[(d*x)/2] + (16*I)*d*E^(3*c)*f^2*x*Si...
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(e+f x)^2 \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6091 |
| \(\displaystyle i \int \frac {(e+f x)^2 \sinh ^2(c+d x)}{i \sinh (c+d x) a+a}dx-\frac {i \int (e+f x)^2 \sinh ^2(c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \int \frac {(e+f x)^2 \sinh ^2(c+d x)}{i \sinh (c+d x) a+a}dx-\frac {i \int -(e+f x)^2 \sin (i c+i d x)^2dx}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {i \int (e+f x)^2 \sin (i c+i d x)^2dx}{a}+i \int \frac {(e+f x)^2 \sinh ^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle \frac {i \left (\frac {f^2 \int -\sinh ^2(c+d x)dx}{2 d^2}+\frac {1}{2} \int (e+f x)^2dx+\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{a}+i \int \frac {(e+f x)^2 \sinh ^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle \frac {i \left (\frac {f^2 \int -\sinh ^2(c+d x)dx}{2 d^2}+\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{a}+i \int \frac {(e+f x)^2 \sinh ^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {i \left (-\frac {f^2 \int \sinh ^2(c+d x)dx}{2 d^2}+\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{a}+i \int \frac {(e+f x)^2 \sinh ^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {i \left (-\frac {f^2 \int -\sin (i c+i d x)^2dx}{2 d^2}+\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{a}+i \int \frac {(e+f x)^2 \sinh ^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {i \left (\frac {f^2 \int \sin (i c+i d x)^2dx}{2 d^2}+\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{a}+i \int \frac {(e+f x)^2 \sinh ^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {i \left (\frac {f^2 \left (\frac {\int 1dx}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}+\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{a}+i \int \frac {(e+f x)^2 \sinh ^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle i \int \frac {(e+f x)^2 \sinh ^2(c+d x)}{i \sinh (c+d x) a+a}dx+\frac {i \left (\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{a}\) |
| \(\Big \downarrow \) 6091 |
| \(\displaystyle i \left (i \int \frac {(e+f x)^2 \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {i \int (e+f x)^2 \sinh (c+d x)dx}{a}\right )+\frac {i \left (\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (i \int \frac {(e+f x)^2 \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {i \int -i (e+f x)^2 \sin (i c+i d x)dx}{a}\right )+\frac {i \left (\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (i \int \frac {(e+f x)^2 \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {\int (e+f x)^2 \sin (i c+i d x)dx}{a}\right )+\frac {i \left (\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle i \left (i \int \frac {(e+f x)^2 \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \int (e+f x) \cosh (c+d x)dx}{d}}{a}\right )+\frac {i \left (\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (i \int \frac {(e+f x)^2 \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}}{a}\right )+\frac {i \left (\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle i \left (i \int \frac {(e+f x)^2 \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {i f \int -i \sinh (c+d x)dx}{d}\right )}{d}}{a}\right )+\frac {i \left (\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (i \int \frac {(e+f x)^2 \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int \sinh (c+d x)dx}{d}\right )}{d}}{a}\right )+\frac {i \left (\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (i \int \frac {(e+f x)^2 \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int -i \sin (i c+i d x)dx}{d}\right )}{d}}{a}\right )+\frac {i \left (\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (i \int \frac {(e+f x)^2 \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}+\frac {i f \int \sin (i c+i d x)dx}{d}\right )}{d}}{a}\right )+\frac {i \left (\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{a}\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle i \left (i \int \frac {(e+f x)^2 \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}}{a}\right )+\frac {i \left (\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{a}\) |
| \(\Big \downarrow \) 6091 |
| \(\displaystyle i \left (i \left (i \int \frac {(e+f x)^2}{i \sinh (c+d x) a+a}dx-\frac {i \int (e+f x)^2dx}{a}\right )-\frac {\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}}{a}\right )+\frac {i \left (\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{a}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle i \left (i \left (i \int \frac {(e+f x)^2}{i \sinh (c+d x) a+a}dx-\frac {i (e+f x)^3}{3 a f}\right )-\frac {\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}}{a}\right )+\frac {i \left (\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (i \left (i \int \frac {(e+f x)^2}{\sin (i c+i d x) a+a}dx-\frac {i (e+f x)^3}{3 a f}\right )-\frac {\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}}{a}\right )+\frac {i \left (\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{a}\) |
| \(\Big \downarrow \) 3799 |
| \(\displaystyle i \left (i \left (\frac {i \int -(e+f x)^2 \text {csch}^2\left (\frac {c}{2}+\frac {d x}{2}-\frac {i \pi }{4}\right )dx}{2 a}-\frac {i (e+f x)^3}{3 a f}\right )-\frac {\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}}{a}\right )+\frac {i \left (\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \left (i \left (-\frac {i \int -(e+f x)^2 \text {sech}^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{2 a}-\frac {i (e+f x)^3}{3 a f}\right )-\frac {\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}}{a}\right )+\frac {i \left (\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \left (i \left (\frac {i \int (e+f x)^2 \text {sech}^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{2 a}-\frac {i (e+f x)^3}{3 a f}\right )-\frac {\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}}{a}\right )+\frac {i \left (\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (i \left (\frac {i \int (e+f x)^2 \csc \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}-\frac {i (e+f x)^3}{3 a f}\right )-\frac {\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}}{a}\right )+\frac {i \left (\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{a}\) |
| \(\Big \downarrow \) 4672 |
| \(\displaystyle i \left (i \left (\frac {i \left (\frac {2 (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 i f \int -i (e+f x) \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{d}\right )}{2 a}-\frac {i (e+f x)^3}{3 a f}\right )-\frac {\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}}{a}\right )+\frac {i \left (\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (i \left (\frac {i \left (\frac {2 (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 f \int (e+f x) \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{d}\right )}{2 a}-\frac {i (e+f x)^3}{3 a f}\right )-\frac {\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}}{a}\right )+\frac {i \left (\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (i \left (\frac {i \left (\frac {2 (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 f \int -i (e+f x) \tan \left (\frac {i c}{2}+\frac {i d x}{2}-\frac {\pi }{4}\right )dx}{d}\right )}{2 a}-\frac {i (e+f x)^3}{3 a f}\right )-\frac {\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}}{a}\right )+\frac {i \left (\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{a}\) |
3.2.100.3.1 Defintions of rubi rules used
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[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( -(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*C os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 2*((n - 1)/n) Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 *m*((m - 1)/(f^2*n^2)) Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.) , x_Symbol] :> Simp[(2*a)^n Int[(c + d*x)^m*Sin[(1/2)*(e + Pi*(a/(2*b))) + f*(x/2)]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2 - b^ 2, 0] && IntegerQ[n] && (GtQ[n, 0] || IGtQ[m, 0])
Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp [(-(c + d*x)^m)*(Cot[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1) *Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 559 vs. \(2 (257 ) = 514\).
Time = 2.42 (sec) , antiderivative size = 560, normalized size of antiderivative = 1.95
| method | result | size |
| risch | \(-\frac {2 i f^{2} x^{2}}{a d}-\frac {4 i f^{2} c x}{a \,d^{2}}+\frac {i \left (2 d^{2} x^{2} f^{2}+4 d^{2} e f x +2 d^{2} e^{2}+2 x d \,f^{2}+2 d e f +f^{2}\right ) {\mathrm e}^{-2 d x -2 c}}{16 d^{3} a}+\frac {2 i e f \ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{a \,d^{2}}-\frac {2 i c \,f^{2} \ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{a \,d^{3}}+\frac {\left (d^{2} x^{2} f^{2}+2 d^{2} e f x +d^{2} e^{2}-2 x d \,f^{2}-2 d e f +2 f^{2}\right ) {\mathrm e}^{d x +c}}{2 d^{3} a}+\frac {\left (d^{2} x^{2} f^{2}+2 d^{2} e f x +d^{2} e^{2}+2 x d \,f^{2}+2 d e f +2 f^{2}\right ) {\mathrm e}^{-d x -c}}{2 d^{3} a}+\frac {4 i f^{2} \ln \left (1+i {\mathrm e}^{d x +c}\right ) x}{a \,d^{2}}+\frac {2 x^{2} f^{2}+4 e f x +2 e^{2}}{d a \left ({\mathrm e}^{d x +c}-i\right )}+\frac {4 i c \,f^{2} \ln \left ({\mathrm e}^{d x +c}\right )}{a \,d^{3}}-\frac {4 e f \arctan \left ({\mathrm e}^{d x +c}\right )}{a \,d^{2}}+\frac {4 i f^{2} \operatorname {polylog}\left (2, -i {\mathrm e}^{d x +c}\right )}{a \,d^{3}}+\frac {4 i f^{2} \ln \left (1+i {\mathrm e}^{d x +c}\right ) c}{a \,d^{3}}+\frac {3 i f e \,x^{2}}{2 a}+\frac {i e^{3}}{2 a f}-\frac {i \left (2 d^{2} x^{2} f^{2}+4 d^{2} e f x +2 d^{2} e^{2}-2 x d \,f^{2}-2 d e f +f^{2}\right ) {\mathrm e}^{2 d x +2 c}}{16 d^{3} a}-\frac {2 i f^{2} c^{2}}{a \,d^{3}}+\frac {3 i e^{2} x}{2 a}-\frac {4 i e f \ln \left ({\mathrm e}^{d x +c}\right )}{a \,d^{2}}+\frac {4 c \,f^{2} \arctan \left ({\mathrm e}^{d x +c}\right )}{a \,d^{3}}+\frac {i f^{2} x^{3}}{2 a}\) | \(560\) |
-2*I/a/d*f^2*x^2-4*I/a/d^2*f^2*c*x+1/16*I*(2*d^2*f^2*x^2+4*d^2*e*f*x+2*d^2 *e^2+2*d*f^2*x+2*d*e*f+f^2)/d^3/a*exp(-2*d*x-2*c)+2*I/a/d^2*e*f*ln(1+exp(2 *d*x+2*c))-2*I/a/d^3*c*f^2*ln(1+exp(2*d*x+2*c))+1/2*(d^2*f^2*x^2+2*d^2*e*f *x+d^2*e^2-2*d*f^2*x-2*d*e*f+2*f^2)/d^3/a*exp(d*x+c)+1/2*(d^2*f^2*x^2+2*d^ 2*e*f*x+d^2*e^2+2*d*f^2*x+2*d*e*f+2*f^2)/d^3/a*exp(-d*x-c)+4*I/a/d^2*f^2*l n(1+I*exp(d*x+c))*x+2*(f^2*x^2+2*e*f*x+e^2)/d/a/(exp(d*x+c)-I)+4*I/a/d^3*c *f^2*ln(exp(d*x+c))-4/a/d^2*e*f*arctan(exp(d*x+c))+4*I*f^2*polylog(2,-I*ex p(d*x+c))/a/d^3+4*I/a/d^3*f^2*ln(1+I*exp(d*x+c))*c+3/2*I/a*f*e*x^2+1/2*I/a /f*e^3-1/16*I*(2*d^2*f^2*x^2+4*d^2*e*f*x+2*d^2*e^2-2*d*f^2*x-2*d*e*f+f^2)/ d^3/a*exp(2*d*x+2*c)-2*I/a/d^3*f^2*c^2+3/2*I/a*e^2*x-4*I/a/d^2*e*f*ln(exp( d*x+c))+4/a/d^3*c*f^2*arctan(exp(d*x+c))+1/2*I/a*f^2*x^3
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 595 vs. \(2 (244) = 488\).
Time = 0.26 (sec) , antiderivative size = 595, normalized size of antiderivative = 2.07 \[ \int \frac {(e+f x)^2 \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {2 \, d^{2} f^{2} x^{2} + 2 \, d^{2} e^{2} + 2 \, d e f + f^{2} + 2 \, {\left (2 \, d^{2} e f + d f^{2}\right )} x - 64 \, {\left (-i \, f^{2} e^{\left (3 \, d x + 3 \, c\right )} - f^{2} e^{\left (2 \, d x + 2 \, c\right )}\right )} {\rm Li}_2\left (-i \, e^{\left (d x + c\right )}\right ) + {\left (-2 i \, d^{2} f^{2} x^{2} - 2 i \, d^{2} e^{2} + 2 i \, d e f - i \, f^{2} - 2 \, {\left (2 i \, d^{2} e f - i \, d f^{2}\right )} x\right )} e^{\left (5 \, d x + 5 \, c\right )} + {\left (6 \, d^{2} f^{2} x^{2} + 6 \, d^{2} e^{2} - 14 \, d e f + 15 \, f^{2} + 2 \, {\left (6 \, d^{2} e f - 7 \, d f^{2}\right )} x\right )} e^{\left (4 \, d x + 4 \, c\right )} - 8 \, {\left (-i \, d^{3} f^{2} x^{3} + i \, d^{2} e^{2} + 2 \, {\left (4 i \, c - i\right )} d e f + 2 \, {\left (-2 i \, c^{2} + i\right )} f^{2} + {\left (-3 i \, d^{3} e f + 5 i \, d^{2} f^{2}\right )} x^{2} + {\left (-3 i \, d^{3} e^{2} + 10 i \, d^{2} e f - 2 i \, d f^{2}\right )} x\right )} e^{\left (3 \, d x + 3 \, c\right )} + 8 \, {\left (d^{3} f^{2} x^{3} + 5 \, d^{2} e^{2} - 2 \, {\left (4 \, c - 1\right )} d e f + 2 \, {\left (2 \, c^{2} + 1\right )} f^{2} + {\left (3 \, d^{3} e f + d^{2} f^{2}\right )} x^{2} + {\left (3 \, d^{3} e^{2} + 2 \, d^{2} e f + 2 \, d f^{2}\right )} x\right )} e^{\left (2 \, d x + 2 \, c\right )} + {\left (-6 i \, d^{2} f^{2} x^{2} - 6 i \, d^{2} e^{2} - 14 i \, d e f - 15 i \, f^{2} - 2 \, {\left (6 i \, d^{2} e f + 7 i \, d f^{2}\right )} x\right )} e^{\left (d x + c\right )} - 64 \, {\left ({\left (-i \, d e f + i \, c f^{2}\right )} e^{\left (3 \, d x + 3 \, c\right )} - {\left (d e f - c f^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )} \log \left (e^{\left (d x + c\right )} - i\right ) - 64 \, {\left ({\left (-i \, d f^{2} x - i \, c f^{2}\right )} e^{\left (3 \, d x + 3 \, c\right )} - {\left (d f^{2} x + c f^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )} \log \left (i \, e^{\left (d x + c\right )} + 1\right )}{16 \, {\left (a d^{3} e^{\left (3 \, d x + 3 \, c\right )} - i \, a d^{3} e^{\left (2 \, d x + 2 \, c\right )}\right )}} \]
1/16*(2*d^2*f^2*x^2 + 2*d^2*e^2 + 2*d*e*f + f^2 + 2*(2*d^2*e*f + d*f^2)*x - 64*(-I*f^2*e^(3*d*x + 3*c) - f^2*e^(2*d*x + 2*c))*dilog(-I*e^(d*x + c)) + (-2*I*d^2*f^2*x^2 - 2*I*d^2*e^2 + 2*I*d*e*f - I*f^2 - 2*(2*I*d^2*e*f - I *d*f^2)*x)*e^(5*d*x + 5*c) + (6*d^2*f^2*x^2 + 6*d^2*e^2 - 14*d*e*f + 15*f^ 2 + 2*(6*d^2*e*f - 7*d*f^2)*x)*e^(4*d*x + 4*c) - 8*(-I*d^3*f^2*x^3 + I*d^2 *e^2 + 2*(4*I*c - I)*d*e*f + 2*(-2*I*c^2 + I)*f^2 + (-3*I*d^3*e*f + 5*I*d^ 2*f^2)*x^2 + (-3*I*d^3*e^2 + 10*I*d^2*e*f - 2*I*d*f^2)*x)*e^(3*d*x + 3*c) + 8*(d^3*f^2*x^3 + 5*d^2*e^2 - 2*(4*c - 1)*d*e*f + 2*(2*c^2 + 1)*f^2 + (3* d^3*e*f + d^2*f^2)*x^2 + (3*d^3*e^2 + 2*d^2*e*f + 2*d*f^2)*x)*e^(2*d*x + 2 *c) + (-6*I*d^2*f^2*x^2 - 6*I*d^2*e^2 - 14*I*d*e*f - 15*I*f^2 - 2*(6*I*d^2 *e*f + 7*I*d*f^2)*x)*e^(d*x + c) - 64*((-I*d*e*f + I*c*f^2)*e^(3*d*x + 3*c ) - (d*e*f - c*f^2)*e^(2*d*x + 2*c))*log(e^(d*x + c) - I) - 64*((-I*d*f^2* x - I*c*f^2)*e^(3*d*x + 3*c) - (d*f^2*x + c*f^2)*e^(2*d*x + 2*c))*log(I*e^ (d*x + c) + 1))/(a*d^3*e^(3*d*x + 3*c) - I*a*d^3*e^(2*d*x + 2*c))
\[ \int \frac {(e+f x)^2 \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {2 e^{2} + 4 e f x + 2 f^{2} x^{2}}{a d e^{c} e^{d x} - i a d} - \frac {i \left (\int \left (- \frac {i d e^{2}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \left (- \frac {i d f^{2} x^{2}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \left (- \frac {d e^{2} e^{c} e^{d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \left (- \frac {4 d e^{2} e^{3 c} e^{3 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \frac {d e^{2} e^{5 c} e^{5 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \left (- \frac {2 i d e f x}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \frac {4 i d e^{2} e^{2 c} e^{2 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \frac {i d e^{2} e^{4 c} e^{4 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \left (- \frac {16 i e f e^{2 c} e^{2 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \left (- \frac {16 i f^{2} x e^{2 c} e^{2 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \left (- \frac {d f^{2} x^{2} e^{c} e^{d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \left (- \frac {4 d f^{2} x^{2} e^{3 c} e^{3 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \frac {d f^{2} x^{2} e^{5 c} e^{5 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \frac {4 i d f^{2} x^{2} e^{2 c} e^{2 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \frac {i d f^{2} x^{2} e^{4 c} e^{4 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \left (- \frac {2 d e f x e^{c} e^{d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \left (- \frac {8 d e f x e^{3 c} e^{3 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \frac {2 d e f x e^{5 c} e^{5 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \frac {8 i d e f x e^{2 c} e^{2 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \frac {2 i d e f x e^{4 c} e^{4 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx\right ) e^{- 2 c}}{4 a d} \]
(2*e**2 + 4*e*f*x + 2*f**2*x**2)/(a*d*exp(c)*exp(d*x) - I*a*d) - I*(Integr al(-I*d*e**2/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(-I*d*f**2*x **2/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(-d*e**2*exp(c)*exp(d *x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(-4*d*e**2*exp(3*c)*e xp(3*d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(d*e**2*exp(5*c )*exp(5*d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(-2*I*d*e*f* x/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(4*I*d*e**2*exp(2*c)*ex p(2*d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(I*d*e**2*exp(4* c)*exp(4*d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(-16*I*e*f* exp(2*c)*exp(2*d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(-16* I*f**2*x*exp(2*c)*exp(2*d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Inte gral(-d*f**2*x**2*exp(c)*exp(d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(-4*d*f**2*x**2*exp(3*c)*exp(3*d*x)/(exp(c)*exp(3*d*x) - I*exp(2* d*x)), x) + Integral(d*f**2*x**2*exp(5*c)*exp(5*d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(4*I*d*f**2*x**2*exp(2*c)*exp(2*d*x)/(exp(c)*e xp(3*d*x) - I*exp(2*d*x)), x) + Integral(I*d*f**2*x**2*exp(4*c)*exp(4*d*x) /(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(-2*d*e*f*x*exp(c)*exp(d *x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(-8*d*e*f*x*exp(3*c)* exp(3*d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(2*d*e*f*x*exp (5*c)*exp(5*d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(8*I*...
Exception generated. \[ \int \frac {(e+f x)^2 \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Exception raised: RuntimeError} \]
\[ \int \frac {(e+f x)^2 \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \sinh \left (d x + c\right )^{3}}{i \, a \sinh \left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int \frac {(e+f x)^2 \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^3\,{\left (e+f\,x\right )}^2}{a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}} \,d x \]