Optimal. Leaf size=40 \[ -\frac {5 x}{8}-\frac {5}{12} i \cosh ^3(x)+\frac {\cosh ^5(x)}{4 (\sinh (x)+i)}-\frac {5}{8} \sinh (x) \cosh (x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2679, 2682, 2635, 8} \[ -\frac {5 x}{8}-\frac {5}{12} i \cosh ^3(x)+\frac {\cosh ^5(x)}{4 (\sinh (x)+i)}-\frac {5}{8} \sinh (x) \cosh (x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2635
Rule 2679
Rule 2682
Rubi steps
\begin {align*} \int \frac {\cosh ^6(x)}{(i+\sinh (x))^2} \, dx &=\frac {\cosh ^5(x)}{4 (i+\sinh (x))}-\frac {5}{4} i \int \frac {\cosh ^4(x)}{i+\sinh (x)} \, dx\\ &=-\frac {5}{12} i \cosh ^3(x)+\frac {\cosh ^5(x)}{4 (i+\sinh (x))}-\frac {5}{4} \int \cosh ^2(x) \, dx\\ &=-\frac {5}{12} i \cosh ^3(x)-\frac {5}{8} \cosh (x) \sinh (x)+\frac {\cosh ^5(x)}{4 (i+\sinh (x))}-\frac {5 \int 1 \, dx}{8}\\ &=-\frac {5 x}{8}-\frac {5}{12} i \cosh ^3(x)-\frac {5}{8} \cosh (x) \sinh (x)+\frac {\cosh ^5(x)}{4 (i+\sinh (x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.19, size = 121, normalized size = 3.02 \[ -\frac {i \cosh ^7(x) \left (6 \sinh ^4(x)-10 i \sinh ^3(x)+7 \sinh ^2(x)-25 i \sinh (x)+\frac {30 \sqrt {1-i \sinh (x)} \sin ^{-1}\left (\frac {\sqrt {1-i \sinh (x)}}{\sqrt {2}}\right )}{\sqrt {1+i \sinh (x)}}+16\right )}{24 \left (\cosh \left (\frac {x}{2}\right )-i \sinh \left (\frac {x}{2}\right )\right )^8 \left (\cosh \left (\frac {x}{2}\right )+i \sinh \left (\frac {x}{2}\right )\right )^6} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.58, size = 55, normalized size = 1.38 \[ -\frac {1}{192} \, {\left (120 \, x e^{\left (4 \, x\right )} - 3 \, e^{\left (8 \, x\right )} + 16 i \, e^{\left (7 \, x\right )} + 24 \, e^{\left (6 \, x\right )} + 48 i \, e^{\left (5 \, x\right )} + 48 i \, e^{\left (3 \, x\right )} - 24 \, e^{\left (2 \, x\right )} + 16 i \, e^{x} + 3\right )} e^{\left (-4 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 50, normalized size = 1.25 \[ -\frac {1}{192} \, {\left (48 i \, e^{\left (3 \, x\right )} - 24 \, e^{\left (2 \, x\right )} + 16 i \, e^{x} + 3\right )} e^{\left (-4 \, x\right )} - \frac {5}{8} \, x + \frac {1}{64} \, e^{\left (4 \, x\right )} - \frac {1}{12} i \, e^{\left (3 \, x\right )} - \frac {1}{8} \, e^{\left (2 \, x\right )} - \frac {1}{4} i \, e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.07, size = 166, normalized size = 4.15 \[ \frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {i}{\tanh \left (\frac {x}{2}\right )+1}-\frac {1}{8 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {i}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {3}{8 \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {2 i}{3 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {1}{4 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{4}}+\frac {5 \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{8}+\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {i}{\left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {1}{8 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {2 i}{3 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {3}{8 \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {i}{\tanh \left (\frac {x}{2}\right )-1}-\frac {1}{4 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {5 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.35, size = 54, normalized size = 1.35 \[ -\frac {1}{192} \, {\left (16 i \, e^{\left (-x\right )} + 24 \, e^{\left (-2 \, x\right )} + 48 i \, e^{\left (-3 \, x\right )} - 3\right )} e^{\left (4 \, x\right )} - \frac {5}{8} \, x - \frac {1}{4} i \, e^{\left (-x\right )} + \frac {1}{8} \, e^{\left (-2 \, x\right )} - \frac {1}{12} i \, e^{\left (-3 \, x\right )} - \frac {1}{64} \, e^{\left (-4 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.14, size = 54, normalized size = 1.35 \[ \frac {{\mathrm {e}}^{-2\,x}}{8}-\frac {{\mathrm {e}}^{-x}\,1{}\mathrm {i}}{4}-\frac {5\,x}{8}-\frac {{\mathrm {e}}^{2\,x}}{8}-\frac {{\mathrm {e}}^{-3\,x}\,1{}\mathrm {i}}{12}-\frac {{\mathrm {e}}^{3\,x}\,1{}\mathrm {i}}{12}-\frac {{\mathrm {e}}^{-4\,x}}{64}+\frac {{\mathrm {e}}^{4\,x}}{64}-\frac {{\mathrm {e}}^x\,1{}\mathrm {i}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.24, size = 65, normalized size = 1.62 \[ - \frac {5 x}{8} + \frac {e^{4 x}}{64} - \frac {i e^{3 x}}{12} - \frac {e^{2 x}}{8} - \frac {i e^{x}}{4} - \frac {i e^{- x}}{4} + \frac {e^{- 2 x}}{8} - \frac {i e^{- 3 x}}{12} - \frac {e^{- 4 x}}{64} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________