Integrand size = 31, antiderivative size = 241 \[ \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {(e+f x)^3}{a d}+\frac {(e+f x)^4}{4 a f}-\frac {6 i f^2 (e+f x) \cosh (c+d x)}{a d^3}-\frac {i (e+f x)^3 \cosh (c+d x)}{a d}+\frac {6 f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {12 f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {12 f^3 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^4}+\frac {6 i f^3 \sinh (c+d x)}{a d^4}+\frac {3 i f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {(e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d} \] Output:
-(f*x+e)^3/a/d+1/4*(f*x+e)^4/a/f-6*I*f^2*(f*x+e)*cosh(d*x+c)/a/d^3-I*(f*x+ e)^3*cosh(d*x+c)/a/d+6*f*(f*x+e)^2*ln(1+I*exp(d*x+c))/a/d^2+12*f^2*(f*x+e) *polylog(2,-I*exp(d*x+c))/a/d^3-12*f^3*polylog(3,-I*exp(d*x+c))/a/d^4+6*I* f^3*sinh(d*x+c)/a/d^4+3*I*f*(f*x+e)^2*sinh(d*x+c)/a/d^2-(f*x+e)^3*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. \(2619\) vs. \(2(241)=482\).
Time = 3.34 (sec) , antiderivative size = 2619, normalized size of antiderivative = 10.87 \[ \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Result too large to show} \] Input:
Integrate[((e + f*x)^3*Sinh[c + d*x]^2)/(a + I*a*Sinh[c + d*x]),x]
Output:
((-10*I)*d^3*e^3*E^c*Cosh[(d*x)/2] - 2*d^3*e^3*E^(2*c)*Cosh[(d*x)/2] - (6* I)*d^2*e^2*E^c*f*Cosh[(d*x)/2] + 6*d^2*e^2*E^(2*c)*f*Cosh[(d*x)/2] - (12*I )*d*e*E^c*f^2*Cosh[(d*x)/2] - 12*d*e*E^(2*c)*f^2*Cosh[(d*x)/2] - (12*I)*E^ c*f^3*Cosh[(d*x)/2] + 12*E^(2*c)*f^3*Cosh[(d*x)/2] - (4*I)*d^4*e^3*E^c*x*C osh[(d*x)/2] + 4*d^4*e^3*E^(2*c)*x*Cosh[(d*x)/2] - (30*I)*d^3*e^2*E^c*f*x* Cosh[(d*x)/2] - 6*d^3*e^2*E^(2*c)*f*x*Cosh[(d*x)/2] - (12*I)*d^2*e*E^c*f^2 *x*Cosh[(d*x)/2] + 12*d^2*e*E^(2*c)*f^2*x*Cosh[(d*x)/2] - (12*I)*d*E^c*f^3 *x*Cosh[(d*x)/2] - 12*d*E^(2*c)*f^3*x*Cosh[(d*x)/2] - (6*I)*d^4*e^2*E^c*f* x^2*Cosh[(d*x)/2] + 6*d^4*e^2*E^(2*c)*f*x^2*Cosh[(d*x)/2] - (30*I)*d^3*e*E ^c*f^2*x^2*Cosh[(d*x)/2] - 6*d^3*e*E^(2*c)*f^2*x^2*Cosh[(d*x)/2] - (6*I)*d ^2*E^c*f^3*x^2*Cosh[(d*x)/2] + 6*d^2*E^(2*c)*f^3*x^2*Cosh[(d*x)/2] - (4*I) *d^4*e*E^c*f^2*x^3*Cosh[(d*x)/2] + 4*d^4*e*E^(2*c)*f^2*x^3*Cosh[(d*x)/2] - (10*I)*d^3*E^c*f^3*x^3*Cosh[(d*x)/2] - 2*d^3*E^(2*c)*f^3*x^3*Cosh[(d*x)/2 ] - I*d^4*E^c*f^3*x^4*Cosh[(d*x)/2] + d^4*E^(2*c)*f^3*x^4*Cosh[(d*x)/2] - 2*d^3*e^3*Cosh[(3*d*x)/2] - (2*I)*d^3*e^3*E^(3*c)*Cosh[(3*d*x)/2] - 6*d^2* e^2*f*Cosh[(3*d*x)/2] + (6*I)*d^2*e^2*E^(3*c)*f*Cosh[(3*d*x)/2] - 12*d*e*f ^2*Cosh[(3*d*x)/2] - (12*I)*d*e*E^(3*c)*f^2*Cosh[(3*d*x)/2] - 12*f^3*Cosh[ (3*d*x)/2] + (12*I)*E^(3*c)*f^3*Cosh[(3*d*x)/2] - 6*d^3*e^2*f*x*Cosh[(3*d* x)/2] - (6*I)*d^3*e^2*E^(3*c)*f*x*Cosh[(3*d*x)/2] - 12*d^2*e*f^2*x*Cosh[(3 *d*x)/2] + (12*I)*d^2*e*E^(3*c)*f^2*x*Cosh[(3*d*x)/2] - 12*d*f^3*x*Cosh...
Time = 2.21 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.10, number of steps used = 30, number of rules used = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.935, Rules used = {6091, 3042, 26, 3777, 3042, 3777, 26, 3042, 26, 3777, 3042, 3117, 6091, 17, 3042, 3799, 25, 25, 3042, 4672, 26, 3042, 26, 4199, 26, 2620, 3011, 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.
| \(\displaystyle \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6091 |
| \(\displaystyle i \int \frac {(e+f x)^3 \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {i \int (e+f x)^3 \sinh (c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \int \frac {(e+f x)^3 \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {i \int -i (e+f x)^3 \sin (i c+i d x)dx}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {(e+f x)^3 \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {\int (e+f x)^3 \sin (i c+i d x)dx}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle i \int \frac {(e+f x)^3 \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \int (e+f x)^2 \cosh (c+d x)dx}{d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \int \frac {(e+f x)^3 \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle i \int \frac {(e+f x)^3 \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 i f \int -i (e+f x) \sinh (c+d x)dx}{d}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {(e+f x)^3 \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int (e+f x) \sinh (c+d x)dx}{d}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \int \frac {(e+f x)^3 \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int -i (e+f x) \sin (i c+i d x)dx}{d}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {(e+f x)^3 \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \int (e+f x) \sin (i c+i d x)dx}{d}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle i \int \frac {(e+f x)^3 \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \cosh (c+d x)dx}{d}\right )}{d}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \int \frac {(e+f x)^3 \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{d}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle i \int \frac {(e+f x)^3 \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 6091 |
| \(\displaystyle i \left (i \int \frac {(e+f x)^3}{i \sinh (c+d x) a+a}dx-\frac {i \int (e+f x)^3dx}{a}\right )-\frac {\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle i \left (i \int \frac {(e+f x)^3}{i \sinh (c+d x) a+a}dx-\frac {i (e+f x)^4}{4 a f}\right )-\frac {\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (i \int \frac {(e+f x)^3}{\sin (i c+i d x) a+a}dx-\frac {i (e+f x)^4}{4 a f}\right )-\frac {\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 3799 |
| \(\displaystyle i \left (\frac {i \int -(e+f x)^3 \text {csch}^2\left (\frac {c}{2}+\frac {d x}{2}-\frac {i \pi }{4}\right )dx}{2 a}-\frac {i (e+f x)^4}{4 a f}\right )-\frac {\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \left (-\frac {i \int -(e+f x)^3 \text {sech}^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{2 a}-\frac {i (e+f x)^4}{4 a f}\right )-\frac {\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \left (\frac {i \int (e+f x)^3 \text {sech}^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{2 a}-\frac {i (e+f x)^4}{4 a f}\right )-\frac {\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {i \int (e+f x)^3 \csc \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}-\frac {i (e+f x)^4}{4 a f}\right )-\frac {\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 4672 |
| \(\displaystyle i \left (\frac {i \left (\frac {2 (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {6 i f \int -i (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{d}\right )}{2 a}-\frac {i (e+f x)^4}{4 a f}\right )-\frac {\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {i \left (\frac {2 (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {6 f \int (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{d}\right )}{2 a}-\frac {i (e+f x)^4}{4 a f}\right )-\frac {\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {i \left (\frac {2 (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {6 f \int -i (e+f x)^2 \tan \left (\frac {i c}{2}+\frac {i d x}{2}-\frac {\pi }{4}\right )dx}{d}\right )}{2 a}-\frac {i (e+f x)^4}{4 a f}\right )-\frac {\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {i \left (\frac {6 i f \int (e+f x)^2 \tan \left (\frac {i c}{2}+\frac {i d x}{2}-\frac {\pi }{4}\right )dx}{d}+\frac {2 (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )}{2 a}-\frac {i (e+f x)^4}{4 a f}\right )-\frac {\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 4199 |
| \(\displaystyle i \left (\frac {i \left (\frac {6 i f \left (2 i \int \frac {i e^{c+d x} (e+f x)^2}{1+i e^{c+d x}}dx-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {2 (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )}{2 a}-\frac {i (e+f x)^4}{4 a f}\right )-\frac {\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {i \left (\frac {6 i f \left (-2 \int \frac {e^{c+d x} (e+f x)^2}{1+i e^{c+d x}}dx-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {2 (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )}{2 a}-\frac {i (e+f x)^4}{4 a f}\right )-\frac {\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle i \left (\frac {i \left (\frac {6 i f \left (-2 \left (\frac {2 i f \int (e+f x) \log \left (1+i e^{c+d x}\right )dx}{d}-\frac {i (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {2 (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )}{2 a}-\frac {i (e+f x)^4}{4 a f}\right )-\frac {\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle i \left (\frac {i \left (\frac {6 i f \left (-2 \left (\frac {2 i f \left (\frac {f \int \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {2 (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )}{2 a}-\frac {i (e+f x)^4}{4 a f}\right )-\frac {\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle i \left (\frac {i \left (\frac {6 i f \left (-2 \left (\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {2 (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )}{2 a}-\frac {i (e+f x)^4}{4 a f}\right )-\frac {\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle i \left (\frac {i \left (\frac {6 i f \left (-2 \left (\frac {2 i f \left (\frac {f \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {2 (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )}{2 a}-\frac {i (e+f x)^4}{4 a f}\right )-\frac {\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}}{a}\) |
Input:
Int[((e + f*x)^3*Sinh[c + d*x]^2)/(a + I*a*Sinh[c + d*x]),x]
Output:
-(((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)/a) + I*(((-1/4*I)*(e + f*x)^4)/(a*f) + ((I/2)*(((6*I)*f*(((-1/3*I)*( e + f*x)^3)/f - 2*(((-I)*(e + f*x)^2*Log[1 + I*E^(c + d*x)])/d + ((2*I)*f* (-(((e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/d) + (f*PolyLog[3, (-I)*E^(c + d*x)])/d^2))/d)))/d + (2*(e + f*x)^3*Tanh[c/2 + (I/4)*Pi + (d*x)/2])/d))/ a)
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[(((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[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_.) + (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[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_ .)*(x_)], x_Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp [2*I Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x ))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && In tegerQ[4*k] && 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]
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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 698 vs. \(2 (222 ) = 444\).
Time = 0.84 (sec) , antiderivative size = 699, normalized size of antiderivative = 2.90
| method | result | size |
| risch | \(-\frac {12 f^{2} e c \ln \left ({\mathrm e}^{d x +c}-i\right )}{a \,d^{3}}+\frac {12 f^{2} e \ln \left (1+i {\mathrm e}^{d x +c}\right ) c}{a \,d^{3}}+\frac {12 f^{2} e \ln \left (1+i {\mathrm e}^{d x +c}\right ) x}{a \,d^{2}}-\frac {i \left (d^{3} x^{3} f^{3}+3 d^{3} e \,f^{2} x^{2}+3 d^{3} e^{2} f x +3 d^{2} f^{3} x^{2}+d^{3} e^{3}+6 d^{2} e \,f^{2} x +3 d^{2} e^{2} f +6 d \,f^{3} x +6 d e \,f^{2}+6 f^{3}\right ) {\mathrm e}^{-d x -c}}{2 d^{4} a}+\frac {12 f^{3} \operatorname {polylog}\left (2, -i {\mathrm e}^{d x +c}\right ) x}{a \,d^{3}}+\frac {6 f \ln \left ({\mathrm e}^{d x +c}-i\right ) e^{2}}{a \,d^{2}}+\frac {6 f^{3} \ln \left (1+i {\mathrm e}^{d x +c}\right ) x^{2}}{a \,d^{2}}+\frac {12 f^{2} e \operatorname {polylog}\left (2, -i {\mathrm e}^{d x +c}\right )}{a \,d^{3}}+\frac {6 f^{3} c^{2} \ln \left ({\mathrm e}^{d x +c}-i\right )}{a \,d^{4}}+\frac {12 f^{2} e c \ln \left ({\mathrm e}^{d x +c}\right )}{a \,d^{3}}-\frac {12 f^{2} e c x}{a \,d^{2}}-\frac {12 f^{3} \operatorname {polylog}\left (3, -i {\mathrm e}^{d x +c}\right )}{a \,d^{4}}-\frac {6 f^{3} \ln \left (1+i {\mathrm e}^{d x +c}\right ) c^{2}}{a \,d^{4}}-\frac {i \left (d^{3} x^{3} f^{3}+3 d^{3} e \,f^{2} x^{2}+3 d^{3} e^{2} f x -3 d^{2} f^{3} x^{2}+d^{3} e^{3}-6 d^{2} e \,f^{2} x -3 d^{2} e^{2} f +6 d \,f^{3} x +6 d e \,f^{2}-6 f^{3}\right ) {\mathrm e}^{d x +c}}{2 d^{4} a}-\frac {2 i \left (x^{3} f^{3}+3 e \,f^{2} x^{2}+3 e^{2} f x +e^{3}\right )}{d a \left ({\mathrm e}^{d x +c}-i\right )}-\frac {6 f^{2} e \,x^{2}}{a d}-\frac {6 f \ln \left ({\mathrm e}^{d x +c}\right ) e^{2}}{a \,d^{2}}+\frac {6 f^{3} c^{2} x}{a \,d^{3}}-\frac {6 f^{3} c^{2} \ln \left ({\mathrm e}^{d x +c}\right )}{a \,d^{4}}-\frac {6 f^{2} e \,c^{2}}{a \,d^{3}}+\frac {f^{2} e \,x^{3}}{a}+\frac {3 f \,e^{2} x^{2}}{2 a}+\frac {e^{3} x}{a}+\frac {4 f^{3} c^{3}}{a \,d^{4}}-\frac {2 f^{3} x^{3}}{a d}+\frac {f^{3} x^{4}}{4 a}+\frac {e^{4}}{4 a f}\) | \(699\) |
Input:
int((f*x+e)^3*sinh(d*x+c)^2/(a+I*a*sinh(d*x+c)),x,method=_RETURNVERBOSE)
Output:
-1/2*I*(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)+12/a/ d^3*f^2*e*c*ln(exp(d*x+c))-12/a/d^3*f^2*e*c*ln(exp(d*x+c)-I)-12/a/d^2*f^2* e*c*x+12/a/d^3*f^2*e*ln(1+I*exp(d*x+c))*c+12/a/d^2*f^2*e*ln(1+I*exp(d*x+c) )*x-1/2*I*(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)-2*I *(f^3*x^3+3*e*f^2*x^2+3*e^2*f*x+e^3)/d/a/(exp(d*x+c)-I)-6/a/d*f^2*e*x^2-6/ a/d^2*f*ln(exp(d*x+c))*e^2+6/a/d^3*f^3*c^2*x-6/a/d^4*f^3*ln(1+I*exp(d*x+c) )*c^2+12/a/d^3*f^3*polylog(2,-I*exp(d*x+c))*x+6/a/d^2*f*ln(exp(d*x+c)-I)*e ^2+6/a/d^2*f^3*ln(1+I*exp(d*x+c))*x^2-6/a/d^4*f^3*c^2*ln(exp(d*x+c))-6/a/d ^3*f^2*e*c^2+12/a/d^3*f^2*e*polylog(2,-I*exp(d*x+c))+6/a/d^4*f^3*c^2*ln(ex p(d*x+c)-I)+1/a*f^2*e*x^3+3/2/a*f*e^2*x^2+1/a*e^3*x-12*f^3*polylog(3,-I*ex p(d*x+c))/a/d^4+4/a/d^4*f^3*c^3-2/a/d*f^3*x^3+1/4/a*f^3*x^4+1/4/a/f*e^4
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 823 vs. \(2 (213) = 426\).
Time = 0.12 (sec) , antiderivative size = 823, normalized size of antiderivative = 3.41 \[ \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx =\text {Too large to display} \] Input:
integrate((f*x+e)^3*sinh(d*x+c)^2/(a+I*a*sinh(d*x+c)),x, algorithm="fricas ")
Output:
-1/4*(2*d^3*f^3*x^3 + 2*d^3*e^3 + 6*d^2*e^2*f + 12*d*e*f^2 + 12*f^3 + 6*(d ^3*e*f^2 + d^2*f^3)*x^2 + 6*(d^3*e^2*f + 2*d^2*e*f^2 + 2*d*f^3)*x - 48*((d *f^3*x + d*e*f^2)*e^(2*d*x + 2*c) - (I*d*f^3*x + I*d*e*f^2)*e^(d*x + c))*d ilog(-I*e^(d*x + c)) + 2*(I*d^3*f^3*x^3 + I*d^3*e^3 - 3*I*d^2*e^2*f + 6*I* d*e*f^2 - 6*I*f^3 + 3*(I*d^3*e*f^2 - I*d^2*f^3)*x^2 + 3*(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) - (d^4*f^3*x^4 - 2*d^3*e^3 - 6* (4*c - 1)*d^2*e^2*f + 12*(2*c^2 - 1)*d*e*f^2 - 4*(2*c^3 - 3)*f^3 + 2*(2*d^ 4*e*f^2 - 5*d^3*f^3)*x^3 + 6*(d^4*e^2*f - 5*d^3*e*f^2 + d^2*f^3)*x^2 + 2*( 2*d^4*e^3 - 15*d^3*e^2*f + 6*d^2*e*f^2 - 6*d*f^3)*x)*e^(2*d*x + 2*c) - (-I *d^4*f^3*x^4 - 10*I*d^3*e^3 - 6*(-4*I*c + I)*d^2*e^2*f - 12*(2*I*c^2 + I)* d*e*f^2 - 4*(-2*I*c^3 + 3*I)*f^3 - 2*(2*I*d^4*e*f^2 + I*d^3*f^3)*x^3 - 6*( I*d^4*e^2*f + I*d^3*e*f^2 + I*d^2*f^3)*x^2 - 2*(2*I*d^4*e^3 + 3*I*d^3*e^2* f + 6*I*d^2*e*f^2 + 6*I*d*f^3)*x)*e^(d*x + c) - 24*((d^2*e^2*f - 2*c*d*e*f ^2 + c^2*f^3)*e^(2*d*x + 2*c) - (I*d^2*e^2*f - 2*I*c*d*e*f^2 + I*c^2*f^3)* e^(d*x + c))*log(e^(d*x + c) - I) - 24*((d^2*f^3*x^2 + 2*d^2*e*f^2*x + 2*c *d*e*f^2 - c^2*f^3)*e^(2*d*x + 2*c) - (I*d^2*f^3*x^2 + 2*I*d^2*e*f^2*x + 2 *I*c*d*e*f^2 - I*c^2*f^3)*e^(d*x + c))*log(I*e^(d*x + c) + 1) + 48*(f^3*e^ (2*d*x + 2*c) - I*f^3*e^(d*x + c))*polylog(3, -I*e^(d*x + c)))/(a*d^4*e^(2 *d*x + 2*c) - I*a*d^4*e^(d*x + c))
\[ \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {- 2 i e^{3} - 6 i e^{2} f x - 6 i e f^{2} x^{2} - 2 i f^{3} x^{3}}{a d e^{c} e^{d x} - i a d} - \frac {i \left (\int \frac {i d e^{3}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {i d f^{3} x^{3}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {d e^{3} e^{c} e^{d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {d e^{3} e^{3 c} e^{3 d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \left (- \frac {12 e^{2} f e^{c} e^{d x}}{e^{c} e^{2 d x} - i e^{d x}}\right )\, dx + \int \left (- \frac {12 f^{3} x^{2} e^{c} e^{d x}}{e^{c} e^{2 d x} - i e^{d x}}\right )\, dx + \int \frac {3 i d e f^{2} x^{2}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {3 i d e^{2} f x}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {i d e^{3} e^{2 c} e^{2 d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {d f^{3} x^{3} e^{c} e^{d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {d f^{3} x^{3} e^{3 c} e^{3 d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \left (- \frac {24 e f^{2} x e^{c} e^{d x}}{e^{c} e^{2 d x} - i e^{d x}}\right )\, dx + \int \frac {i d f^{3} x^{3} e^{2 c} e^{2 d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {3 d e f^{2} x^{2} e^{c} e^{d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {3 d e f^{2} x^{2} e^{3 c} e^{3 d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {3 d e^{2} f x e^{c} e^{d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {3 d e^{2} f x e^{3 c} e^{3 d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {3 i d e f^{2} x^{2} e^{2 c} e^{2 d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {3 i d e^{2} f x e^{2 c} e^{2 d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx\right ) e^{- c}}{2 a d} \] Input:
integrate((f*x+e)**3*sinh(d*x+c)**2/(a+I*a*sinh(d*x+c)),x)
Output:
(-2*I*e**3 - 6*I*e**2*f*x - 6*I*e*f**2*x**2 - 2*I*f**3*x**3)/(a*d*exp(c)*e xp(d*x) - I*a*d) - I*(Integral(I*d*e**3/(exp(c)*exp(2*d*x) - I*exp(d*x)), x) + Integral(I*d*f**3*x**3/(exp(c)*exp(2*d*x) - I*exp(d*x)), x) + Integra l(d*e**3*exp(c)*exp(d*x)/(exp(c)*exp(2*d*x) - I*exp(d*x)), x) + Integral(d *e**3*exp(3*c)*exp(3*d*x)/(exp(c)*exp(2*d*x) - I*exp(d*x)), x) + Integral( -12*e**2*f*exp(c)*exp(d*x)/(exp(c)*exp(2*d*x) - I*exp(d*x)), x) + Integral (-12*f**3*x**2*exp(c)*exp(d*x)/(exp(c)*exp(2*d*x) - I*exp(d*x)), x) + Inte gral(3*I*d*e*f**2*x**2/(exp(c)*exp(2*d*x) - I*exp(d*x)), x) + Integral(3*I *d*e**2*f*x/(exp(c)*exp(2*d*x) - I*exp(d*x)), x) + Integral(I*d*e**3*exp(2 *c)*exp(2*d*x)/(exp(c)*exp(2*d*x) - I*exp(d*x)), x) + Integral(d*f**3*x**3 *exp(c)*exp(d*x)/(exp(c)*exp(2*d*x) - I*exp(d*x)), x) + Integral(d*f**3*x* *3*exp(3*c)*exp(3*d*x)/(exp(c)*exp(2*d*x) - I*exp(d*x)), x) + Integral(-24 *e*f**2*x*exp(c)*exp(d*x)/(exp(c)*exp(2*d*x) - I*exp(d*x)), x) + Integral( I*d*f**3*x**3*exp(2*c)*exp(2*d*x)/(exp(c)*exp(2*d*x) - I*exp(d*x)), x) + I ntegral(3*d*e*f**2*x**2*exp(c)*exp(d*x)/(exp(c)*exp(2*d*x) - I*exp(d*x)), x) + Integral(3*d*e*f**2*x**2*exp(3*c)*exp(3*d*x)/(exp(c)*exp(2*d*x) - I*e xp(d*x)), x) + Integral(3*d*e**2*f*x*exp(c)*exp(d*x)/(exp(c)*exp(2*d*x) - I*exp(d*x)), x) + Integral(3*d*e**2*f*x*exp(3*c)*exp(3*d*x)/(exp(c)*exp(2* d*x) - I*exp(d*x)), x) + Integral(3*I*d*e*f**2*x**2*exp(2*c)*exp(2*d*x)/(e xp(c)*exp(2*d*x) - I*exp(d*x)), x) + Integral(3*I*d*e**2*f*x*exp(2*c)*e...
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 671 vs. \(2 (213) = 426\).
Time = 0.25 (sec) , antiderivative size = 671, normalized size of antiderivative = 2.78 \[ \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx =\text {Too large to display} \] Input:
integrate((f*x+e)^3*sinh(d*x+c)^2/(a+I*a*sinh(d*x+c)),x, algorithm="maxima ")
Output:
-3/2*e^2*f*(2*x*e^(d*x + c)/(a*d*e^(d*x + c) - I*a*d) + (I*d^2*x^2*e^c + I *d*x*e^c - (-I*d*x*e^(3*c) + I*e^(3*c))*e^(2*d*x) - (d^2*x^2*e^(2*c) - 3*d *x*e^(2*c) + e^(2*c))*e^(d*x) + (d*x + 1)*e^(-d*x) + I*e^c)/(a*d^2*e^(d*x + 2*c) - I*a*d^2*e^c) - 4*log((e^(d*x + c) - I)*e^(-c))/(a*d^2)) + 1/2*e^3 *(2*(d*x + c)/(a*d) + (-5*I*e^(-d*x - c) + 1)/((I*a*e^(-d*x - c) + a*e^(-2 *d*x - 2*c))*d) - I*e^(-d*x - c)/(a*d)) + 1/4*(-I*d^4*f^3*x^4 - 12*I*d*e*f ^2 + 2*(-2*I*d^4*e*f^2 - 5*I*d^3*f^3)*x^3 - 12*I*f^3 + 6*(-5*I*d^3*e*f^2 - I*d^2*f^3)*x^2 + 12*(-I*d^2*e*f^2 - I*d*f^3)*x + 2*(-I*d^3*f^3*x^3*e^(2*c ) + 3*(-I*d^3*e*f^2 + I*d^2*f^3)*x^2*e^(2*c) + 6*(I*d^2*e*f^2 - I*d*f^3)*x *e^(2*c) + 6*(-I*d*e*f^2 + I*f^3)*e^(2*c))*e^(2*d*x) + (d^4*f^3*x^4*e^c + 2*(2*d^4*e*f^2 - d^3*f^3)*x^3*e^c - 6*(d^3*e*f^2 - d^2*f^3)*x^2*e^c + 12*( d^2*e*f^2 - d*f^3)*x*e^c - 12*(d*e*f^2 - f^3)*e^c)*e^(d*x))/(a*d^4*e^(d*x + c) - I*a*d^4) + 12*(d*x*log(I*e^(d*x + c) + 1) + dilog(-I*e^(d*x + c)))* e*f^2/(a*d^3) + 6*(d^2*x^2*log(I*e^(d*x + c) + 1) + 2*d*x*dilog(-I*e^(d*x + c)) - 2*polylog(3, -I*e^(d*x + c)))*f^3/(a*d^4) - 2*(d^3*f^3*x^3 + 3*d^3 *e*f^2*x^2)/(a*d^4)
\[ \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \sinh \left (d x + c\right )^{2}}{i \, a \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)^3*sinh(d*x+c)^2/(a+I*a*sinh(d*x+c)),x, algorithm="giac")
Output:
integrate((f*x + e)^3*sinh(d*x + c)^2/(I*a*sinh(d*x + c) + a), x)
Timed out. \[ \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^2\,{\left (e+f\,x\right )}^3}{a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}} \,d x \] Input:
int((sinh(c + d*x)^2*(e + f*x)^3)/(a + a*sinh(c + d*x)*1i),x)
Output:
int((sinh(c + d*x)^2*(e + f*x)^3)/(a + a*sinh(c + d*x)*1i), x)
\[ \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {-4 \cosh \left (d x +c \right ) d^{3} e^{3} i -12 \cosh \left (d x +c \right ) d^{3} e^{2} f i x -12 \cosh \left (d x +c \right ) d^{3} e \,f^{2} i \,x^{2}-4 \cosh \left (d x +c \right ) d^{3} f^{3} i \,x^{3}-24 \cosh \left (d x +c \right ) d e \,f^{2} i -24 \cosh \left (d x +c \right ) d \,f^{3} i x -4 \left (\int \frac {x^{3}}{\sinh \left (d x +c \right ) i +1}d x \right ) d^{4} f^{3}-12 \left (\int \frac {x^{2}}{\sinh \left (d x +c \right ) i +1}d x \right ) d^{4} e \,f^{2}-12 \left (\int \frac {x}{\sinh \left (d x +c \right ) i +1}d x \right ) d^{4} e^{2} f -4 \left (\int \frac {1}{\sinh \left (d x +c \right ) i +1}d x \right ) d^{4} e^{3}+12 \sinh \left (d x +c \right ) d^{2} e^{2} f i +24 \sinh \left (d x +c \right ) d^{2} e \,f^{2} i x +12 \sinh \left (d x +c \right ) d^{2} f^{3} i \,x^{2}+24 \sinh \left (d x +c \right ) f^{3} i +4 d^{4} e^{3} x +6 d^{4} e^{2} f \,x^{2}+4 d^{4} e \,f^{2} x^{3}+d^{4} f^{3} x^{4}}{4 a \,d^{4}} \] Input:
int((f*x+e)^3*sinh(d*x+c)^2/(a+I*a*sinh(d*x+c)),x)
Output:
( - 4*cosh(c + d*x)*d**3*e**3*i - 12*cosh(c + d*x)*d**3*e**2*f*i*x - 12*co sh(c + d*x)*d**3*e*f**2*i*x**2 - 4*cosh(c + d*x)*d**3*f**3*i*x**3 - 24*cos h(c + d*x)*d*e*f**2*i - 24*cosh(c + d*x)*d*f**3*i*x - 4*int(x**3/(sinh(c + d*x)*i + 1),x)*d**4*f**3 - 12*int(x**2/(sinh(c + d*x)*i + 1),x)*d**4*e*f* *2 - 12*int(x/(sinh(c + d*x)*i + 1),x)*d**4*e**2*f - 4*int(1/(sinh(c + d*x )*i + 1),x)*d**4*e**3 + 12*sinh(c + d*x)*d**2*e**2*f*i + 24*sinh(c + d*x)* d**2*e*f**2*i*x + 12*sinh(c + d*x)*d**2*f**3*i*x**2 + 24*sinh(c + d*x)*f** 3*i + 4*d**4*e**3*x + 6*d**4*e**2*f*x**2 + 4*d**4*e*f**2*x**3 + d**4*f**3* x**4)/(4*a*d**4)