Optimal. Leaf size=26 \[ -\frac {i x}{2}+\frac {\cosh ^3(x)}{3}-\frac {1}{2} i \sinh (x) \cosh (x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2682, 2635, 8} \[ -\frac {i x}{2}+\frac {\cosh ^3(x)}{3}-\frac {1}{2} i \sinh (x) \cosh (x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2635
Rule 2682
Rubi steps
\begin {align*} \int \frac {\cosh ^4(x)}{i+\sinh (x)} \, dx &=\frac {\cosh ^3(x)}{3}-i \int \cosh ^2(x) \, dx\\ &=\frac {\cosh ^3(x)}{3}-\frac {1}{2} i \cosh (x) \sinh (x)-\frac {1}{2} i \int 1 \, dx\\ &=-\frac {i x}{2}+\frac {\cosh ^3(x)}{3}-\frac {1}{2} i \cosh (x) \sinh (x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.16, size = 93, normalized size = 3.58 \[ \frac {\cosh ^5(x) \left (2 \sinh ^3(x)-i \sinh ^2(x)+5 \sinh (x)+\frac {6 i \sqrt {1-i \sinh (x)} \sin ^{-1}\left (\frac {\sqrt {1-i \sinh (x)}}{\sqrt {2}}\right )}{\sqrt {1+i \sinh (x)}}+2 i\right )}{6 (\sinh (x)-i)^2 (\sinh (x)+i)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.57, size = 41, normalized size = 1.58 \[ \frac {1}{24} \, {\left (-12 i \, x e^{\left (3 \, x\right )} + e^{\left (6 \, x\right )} - 3 i \, e^{\left (5 \, x\right )} + 3 \, e^{\left (4 \, x\right )} + 3 \, e^{\left (2 \, x\right )} + 3 i \, e^{x} + 1\right )} e^{\left (-3 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.44, size = 38, normalized size = 1.46 \[ \frac {1}{24} \, {\left (3 \, e^{\left (2 \, x\right )} + 3 i \, e^{x} + 1\right )} e^{\left (-3 \, x\right )} - \frac {1}{2} i \, x + \frac {1}{24} \, e^{\left (3 \, x\right )} - \frac {1}{8} i \, e^{\left (2 \, x\right )} + \frac {1}{8} \, e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.06, size = 126, normalized size = 4.85 \[ \frac {i \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2}-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {i}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {i}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {1}{3 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {i \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2}+\frac {1}{2 \tanh \left (\frac {x}{2}\right )+2}-\frac {i}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {i}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {1}{3 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.31, size = 42, normalized size = 1.62 \[ -\frac {1}{48} \, {\left (6 i \, e^{\left (-x\right )} - 6 \, e^{\left (-2 \, x\right )} - 2\right )} e^{\left (3 \, x\right )} - \frac {1}{2} i \, x + \frac {1}{8} \, e^{\left (-x\right )} + \frac {1}{8} i \, e^{\left (-2 \, x\right )} + \frac {1}{24} \, e^{\left (-3 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.12, size = 41, normalized size = 1.58 \[ \frac {{\mathrm {e}}^{-x}}{8}+\frac {{\mathrm {e}}^{-3\,x}}{24}+\frac {{\mathrm {e}}^{3\,x}}{24}+\frac {{\mathrm {e}}^x}{8}-\frac {x\,1{}\mathrm {i}}{2}+\frac {{\mathrm {e}}^{-2\,x}\,1{}\mathrm {i}}{8}-\frac {{\mathrm {e}}^{2\,x}\,1{}\mathrm {i}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.18, size = 48, normalized size = 1.85 \[ - \frac {i x}{2} + \frac {e^{3 x}}{24} - \frac {i e^{2 x}}{8} + \frac {e^{x}}{8} + \frac {e^{- x}}{8} + \frac {i e^{- 2 x}}{8} + \frac {e^{- 3 x}}{24} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________