Integrand size = 21, antiderivative size = 377 \[ \int x^3 (a+i a \sinh (e+f x))^{3/2} \, dx=-\frac {1280 a \sqrt {a+i a \sinh (e+f x)}}{9 f^4}-\frac {16 a x^2 \sqrt {a+i a \sinh (e+f x)}}{f^2}-\frac {64 a \cosh ^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)}}{27 f^4}-\frac {8 a x^2 \cosh ^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)}}{3 f^2}+\frac {32 a x \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sinh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)}}{9 f^3}+\frac {4 a x^3 \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sinh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)}}{3 f}+\frac {640 a x \sqrt {a+i a \sinh (e+f x)} \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{9 f^3}+\frac {8 a x^3 \sqrt {a+i a \sinh (e+f x)} \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{3 f} \] Output:
-1280/9*a*(a+I*a*sinh(f*x+e))^(1/2)/f^4-16*a*x^2*(a+I*a*sinh(f*x+e))^(1/2) /f^2-64/27*a*cosh(1/2*e+1/4*I*Pi+1/2*f*x)^2*(a+I*a*sinh(f*x+e))^(1/2)/f^4- 8/3*a*x^2*cosh(1/2*e+1/4*I*Pi+1/2*f*x)^2*(a+I*a*sinh(f*x+e))^(1/2)/f^2+32/ 9*a*x*cosh(1/2*e+1/4*I*Pi+1/2*f*x)*sinh(1/2*e+1/4*I*Pi+1/2*f*x)*(a+I*a*sin h(f*x+e))^(1/2)/f^3+4/3*a*x^3*cosh(1/2*e+1/4*I*Pi+1/2*f*x)*sinh(1/2*e+1/4* I*Pi+1/2*f*x)*(a+I*a*sinh(f*x+e))^(1/2)/f+640/9*a*x*(a+I*a*sinh(f*x+e))^(1 /2)*tanh(1/2*e+1/4*I*Pi+1/2*f*x)/f^3+8/3*a*x^3*(a+I*a*sinh(f*x+e))^(1/2)*t anh(1/2*e+1/4*I*Pi+1/2*f*x)/f
Time = 7.56 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.71 \[ \int x^3 (a+i a \sinh (e+f x))^{3/2} \, dx=-\frac {a (-i+\sinh (e+f x)) \sqrt {a+i a \sinh (e+f x)} \left (81 \left (48 i+24 f x+6 i f^2 x^2+f^3 x^3\right ) \cosh \left (\frac {1}{2} (e+f x)\right )+\left (-16 i+24 f x-18 i f^2 x^2+9 f^3 x^3\right ) \cosh \left (\frac {3}{2} (e+f x)\right )-3888 \sinh \left (\frac {1}{2} (e+f x)\right )-1944 i f x \sinh \left (\frac {1}{2} (e+f x)\right )-486 f^2 x^2 \sinh \left (\frac {1}{2} (e+f x)\right )-81 i f^3 x^3 \sinh \left (\frac {1}{2} (e+f x)\right )-16 \sinh \left (\frac {3}{2} (e+f x)\right )+24 i f x \sinh \left (\frac {3}{2} (e+f x)\right )-18 f^2 x^2 \sinh \left (\frac {3}{2} (e+f x)\right )+9 i f^3 x^3 \sinh \left (\frac {3}{2} (e+f x)\right )\right )}{27 f^4 \left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right )^3} \] Input:
Integrate[x^3*(a + I*a*Sinh[e + f*x])^(3/2),x]
Output:
-1/27*(a*(-I + Sinh[e + f*x])*Sqrt[a + I*a*Sinh[e + f*x]]*(81*(48*I + 24*f *x + (6*I)*f^2*x^2 + f^3*x^3)*Cosh[(e + f*x)/2] + (-16*I + 24*f*x - (18*I) *f^2*x^2 + 9*f^3*x^3)*Cosh[(3*(e + f*x))/2] - 3888*Sinh[(e + f*x)/2] - (19 44*I)*f*x*Sinh[(e + f*x)/2] - 486*f^2*x^2*Sinh[(e + f*x)/2] - (81*I)*f^3*x ^3*Sinh[(e + f*x)/2] - 16*Sinh[(3*(e + f*x))/2] + (24*I)*f*x*Sinh[(3*(e + f*x))/2] - 18*f^2*x^2*Sinh[(3*(e + f*x))/2] + (9*I)*f^3*x^3*Sinh[(3*(e + f *x))/2]))/(f^4*(Cosh[(e + f*x)/2] + I*Sinh[(e + f*x)/2])^3)
Time = 1.63 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.06, number of steps used = 23, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.095, Rules used = {3042, 3800, 3042, 3792, 3042, 3777, 26, 3042, 26, 3777, 3042, 3777, 26, 3042, 26, 3118, 3791, 3042, 3777, 26, 3042, 26, 3118}
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.
\(\displaystyle \int x^3 (a+i a \sinh (e+f x))^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int x^3 (a+a \sin (i e+i f x))^{3/2}dx\) |
\(\Big \downarrow \) 3800 |
\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \int x^3 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \int x^3 \sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )^3dx\) |
\(\Big \downarrow \) 3792 |
\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 \int x \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx}{3 f^2}+\frac {2}{3} \int x^3 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx-\frac {4 x^2 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 \int x \sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )^3dx}{3 f^2}+\frac {2}{3} \int x^3 \sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )dx-\frac {4 x^2 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 \int x \sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )^3dx}{3 f^2}+\frac {2}{3} \left (\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {6 i \int -i x^2 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx}{f}\right )-\frac {4 x^2 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 \int x \sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )^3dx}{3 f^2}+\frac {2}{3} \left (\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {6 \int x^2 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx}{f}\right )-\frac {4 x^2 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 \int x \sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )^3dx}{3 f^2}+\frac {2}{3} \left (\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {6 \int -i x^2 \sin \left (\frac {i e}{2}+\frac {i f x}{2}-\frac {\pi }{4}\right )dx}{f}\right )-\frac {4 x^2 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 \int x \sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )^3dx}{3 f^2}+\frac {2}{3} \left (\frac {6 i \int x^2 \sin \left (\frac {i e}{2}+\frac {i f x}{2}-\frac {\pi }{4}\right )dx}{f}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )-\frac {4 x^2 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 \int x \sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )^3dx}{3 f^2}+\frac {2}{3} \left (\frac {6 i \left (\frac {2 i x^2 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 i \int x \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx}{f}\right )}{f}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )-\frac {4 x^2 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 \int x \sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )^3dx}{3 f^2}+\frac {2}{3} \left (\frac {6 i \left (\frac {2 i x^2 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 i \int x \sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )dx}{f}\right )}{f}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )-\frac {4 x^2 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 \int x \sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )^3dx}{3 f^2}+\frac {2}{3} \left (\frac {6 i \left (\frac {2 i x^2 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 i \left (\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {2 i \int -i \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx}{f}\right )}{f}\right )}{f}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )-\frac {4 x^2 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 \int x \sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )^3dx}{3 f^2}+\frac {2}{3} \left (\frac {6 i \left (\frac {2 i x^2 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 i \left (\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {2 \int \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx}{f}\right )}{f}\right )}{f}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )-\frac {4 x^2 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 \int x \sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )^3dx}{3 f^2}+\frac {2}{3} \left (\frac {6 i \left (\frac {2 i x^2 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 i \left (\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {2 \int -i \sin \left (\frac {i e}{2}+\frac {i f x}{2}-\frac {\pi }{4}\right )dx}{f}\right )}{f}\right )}{f}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )-\frac {4 x^2 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 \int x \sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )^3dx}{3 f^2}+\frac {2}{3} \left (\frac {6 i \left (\frac {2 i x^2 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 i \left (\frac {2 i \int \sin \left (\frac {i e}{2}+\frac {i f x}{2}-\frac {\pi }{4}\right )dx}{f}+\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )}{f}\right )}{f}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )-\frac {4 x^2 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\) |
\(\Big \downarrow \) 3118 |
\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 \int x \sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )^3dx}{3 f^2}-\frac {4 x^2 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2}{3} \left (\frac {6 i \left (\frac {2 i x^2 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 i \left (\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}\right )}{f}\right )}{f}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\) |
\(\Big \downarrow \) 3791 |
\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 \left (\frac {2}{3} \int x \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx-\frac {4 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{9 f^2}+\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )}{3 f^2}-\frac {4 x^2 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2}{3} \left (\frac {6 i \left (\frac {2 i x^2 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 i \left (\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}\right )}{f}\right )}{f}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 \left (\frac {2}{3} \int x \sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )dx-\frac {4 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{9 f^2}+\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )}{3 f^2}-\frac {4 x^2 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2}{3} \left (\frac {6 i \left (\frac {2 i x^2 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 i \left (\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}\right )}{f}\right )}{f}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 \left (\frac {2}{3} \left (\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {2 i \int -i \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx}{f}\right )-\frac {4 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{9 f^2}+\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )}{3 f^2}-\frac {4 x^2 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2}{3} \left (\frac {6 i \left (\frac {2 i x^2 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 i \left (\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}\right )}{f}\right )}{f}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 \left (\frac {2}{3} \left (\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {2 \int \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx}{f}\right )-\frac {4 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{9 f^2}+\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )}{3 f^2}-\frac {4 x^2 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2}{3} \left (\frac {6 i \left (\frac {2 i x^2 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 i \left (\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}\right )}{f}\right )}{f}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 \left (\frac {2}{3} \left (\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {2 \int -i \sin \left (\frac {i e}{2}+\frac {i f x}{2}-\frac {\pi }{4}\right )dx}{f}\right )-\frac {4 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{9 f^2}+\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )}{3 f^2}-\frac {4 x^2 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2}{3} \left (\frac {6 i \left (\frac {2 i x^2 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 i \left (\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}\right )}{f}\right )}{f}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 \left (\frac {2}{3} \left (\frac {2 i \int \sin \left (\frac {i e}{2}+\frac {i f x}{2}-\frac {\pi }{4}\right )dx}{f}+\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )-\frac {4 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{9 f^2}+\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )}{3 f^2}-\frac {4 x^2 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2}{3} \left (\frac {6 i \left (\frac {2 i x^2 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 i \left (\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}\right )}{f}\right )}{f}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\) |
\(\Big \downarrow \) 3118 |
\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (-\frac {4 x^2 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2}{3} \left (\frac {6 i \left (\frac {2 i x^2 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 i \left (\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}\right )}{f}\right )}{f}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )+\frac {8 \left (-\frac {4 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{9 f^2}+\frac {2}{3} \left (\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}\right )+\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )}{3 f^2}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\) |
Input:
Int[x^3*(a + I*a*Sinh[e + f*x])^(3/2),x]
Output:
2*a*Sech[e/2 + (I/4)*Pi + (f*x)/2]*((-4*x^2*Cosh[e/2 + (I/4)*Pi + (f*x)/2] ^3)/(3*f^2) + (2*x^3*Cosh[e/2 + (I/4)*Pi + (f*x)/2]^2*Sinh[e/2 + (I/4)*Pi + (f*x)/2])/(3*f) + (8*((-4*Cosh[e/2 + (I/4)*Pi + (f*x)/2]^3)/(9*f^2) + (2 *x*Cosh[e/2 + (I/4)*Pi + (f*x)/2]^2*Sinh[e/2 + (I/4)*Pi + (f*x)/2])/(3*f) + (2*((-4*Cosh[e/2 + (I/4)*Pi + (f*x)/2])/f^2 + (2*x*Sinh[e/2 + (I/4)*Pi + (f*x)/2])/f))/3))/(3*f^2) + (2*((2*x^3*Sinh[e/2 + (I/4)*Pi + (f*x)/2])/f + ((6*I)*(((2*I)*x^2*Cosh[e/2 + (I/4)*Pi + (f*x)/2])/f - ((4*I)*((-4*Cosh[ e/2 + (I/4)*Pi + (f*x)/2])/f^2 + (2*x*Sinh[e/2 + (I/4)*Pi + (f*x)/2])/f))/ f))/f))/3)*Sqrt[a + I*a*Sinh[e + f*x]]
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[((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_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Simp[b*(c + d*x)*Cos[e + f*x ]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^2*((n - 1)/n) Int[(c + d* x)*(b*Sin[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]
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)^IntPart[n]*((a + b*Sin[e + f*x])^FracPart[n]/Sin[e /2 + a*(Pi/(4*b)) + f*(x/2)]^(2*FracPart[n])) Int[(c + d*x)^m*Sin[e/2 + a *(Pi/(4*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 + 1/2] && (GtQ[n, 0] || IGtQ[m, 0])
\[\int x^{3} \left (a +i a \sinh \left (f x +e \right )\right )^{\frac {3}{2}}d x\]
Input:
int(x^3*(a+I*a*sinh(f*x+e))^(3/2),x)
Output:
int(x^3*(a+I*a*sinh(f*x+e))^(3/2),x)
Exception generated. \[ \int x^3 (a+i a \sinh (e+f x))^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^3*(a+I*a*sinh(f*x+e))^(3/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (has polynomial part)
Timed out. \[ \int x^3 (a+i a \sinh (e+f x))^{3/2} \, dx=\text {Timed out} \] Input:
integrate(x**3*(a+I*a*sinh(f*x+e))**(3/2),x)
Output:
Timed out
\[ \int x^3 (a+i a \sinh (e+f x))^{3/2} \, dx=\int { {\left (i \, a \sinh \left (f x + e\right ) + a\right )}^{\frac {3}{2}} x^{3} \,d x } \] Input:
integrate(x^3*(a+I*a*sinh(f*x+e))^(3/2),x, algorithm="maxima")
Output:
integrate((I*a*sinh(f*x + e) + a)^(3/2)*x^3, x)
Time = 0.20 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.66 \[ \int x^3 (a+i a \sinh (e+f x))^{3/2} \, dx=-\frac {{\left (-\left (9 i - 9\right ) \, a^{\frac {3}{2}} f^{3} x^{3} e^{\left (3 \, f x + 3 \, e\right )} - \left (81 i + 81\right ) \, a^{\frac {3}{2}} f^{3} x^{3} e^{\left (2 \, f x + 2 \, e\right )} - \left (81 i - 81\right ) \, a^{\frac {3}{2}} f^{3} x^{3} e^{\left (f x + e\right )} - \left (9 i + 9\right ) \, a^{\frac {3}{2}} f^{3} x^{3} + \left (18 i - 18\right ) \, a^{\frac {3}{2}} f^{2} x^{2} e^{\left (3 \, f x + 3 \, e\right )} + \left (486 i + 486\right ) \, a^{\frac {3}{2}} f^{2} x^{2} e^{\left (2 \, f x + 2 \, e\right )} - \left (486 i - 486\right ) \, a^{\frac {3}{2}} f^{2} x^{2} e^{\left (f x + e\right )} - \left (18 i + 18\right ) \, a^{\frac {3}{2}} f^{2} x^{2} - \left (24 i - 24\right ) \, a^{\frac {3}{2}} f x e^{\left (3 \, f x + 3 \, e\right )} - \left (1944 i + 1944\right ) \, a^{\frac {3}{2}} f x e^{\left (2 \, f x + 2 \, e\right )} - \left (1944 i - 1944\right ) \, a^{\frac {3}{2}} f x e^{\left (f x + e\right )} - \left (24 i + 24\right ) \, a^{\frac {3}{2}} f x + \left (16 i - 16\right ) \, a^{\frac {3}{2}} e^{\left (3 \, f x + 3 \, e\right )} + \left (3888 i + 3888\right ) \, a^{\frac {3}{2}} e^{\left (2 \, f x + 2 \, e\right )} - \left (3888 i - 3888\right ) \, a^{\frac {3}{2}} e^{\left (f x + e\right )} - \left (16 i + 16\right ) \, a^{\frac {3}{2}}\right )} e^{\left (-\frac {3}{2} \, f x - \frac {3}{2} \, e\right )}}{54 \, f^{4}} \] Input:
integrate(x^3*(a+I*a*sinh(f*x+e))^(3/2),x, algorithm="giac")
Output:
-1/54*(-(9*I - 9)*a^(3/2)*f^3*x^3*e^(3*f*x + 3*e) - (81*I + 81)*a^(3/2)*f^ 3*x^3*e^(2*f*x + 2*e) - (81*I - 81)*a^(3/2)*f^3*x^3*e^(f*x + e) - (9*I + 9 )*a^(3/2)*f^3*x^3 + (18*I - 18)*a^(3/2)*f^2*x^2*e^(3*f*x + 3*e) + (486*I + 486)*a^(3/2)*f^2*x^2*e^(2*f*x + 2*e) - (486*I - 486)*a^(3/2)*f^2*x^2*e^(f *x + e) - (18*I + 18)*a^(3/2)*f^2*x^2 - (24*I - 24)*a^(3/2)*f*x*e^(3*f*x + 3*e) - (1944*I + 1944)*a^(3/2)*f*x*e^(2*f*x + 2*e) - (1944*I - 1944)*a^(3 /2)*f*x*e^(f*x + e) - (24*I + 24)*a^(3/2)*f*x + (16*I - 16)*a^(3/2)*e^(3*f *x + 3*e) + (3888*I + 3888)*a^(3/2)*e^(2*f*x + 2*e) - (3888*I - 3888)*a^(3 /2)*e^(f*x + e) - (16*I + 16)*a^(3/2))*e^(-3/2*f*x - 3/2*e)/f^4
Timed out. \[ \int x^3 (a+i a \sinh (e+f x))^{3/2} \, dx=\int x^3\,{\left (a+a\,\mathrm {sinh}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2} \,d x \] Input:
int(x^3*(a + a*sinh(e + f*x)*1i)^(3/2),x)
Output:
int(x^3*(a + a*sinh(e + f*x)*1i)^(3/2), x)
\[ \int x^3 (a+i a \sinh (e+f x))^{3/2} \, dx=\sqrt {a}\, a \left (\int \sqrt {\sinh \left (f x +e \right ) i +1}\, x^{3}d x +\left (\int \sqrt {\sinh \left (f x +e \right ) i +1}\, \sinh \left (f x +e \right ) x^{3}d x \right ) i \right ) \] Input:
int(x^3*(a+I*a*sinh(f*x+e))^(3/2),x)
Output:
sqrt(a)*a*(int(sqrt(sinh(e + f*x)*i + 1)*x**3,x) + int(sqrt(sinh(e + f*x)* i + 1)*sinh(e + f*x)*x**3,x)*i)