Optimal. Leaf size=14 \[ \sinh (x)-2 i \log (\sinh (x)+i) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2667, 43} \[ \sinh (x)-2 i \log (\sinh (x)+i) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 2667
Rubi steps
\begin {align*} \int \frac {\cosh ^3(x)}{(i+\sinh (x))^2} \, dx &=-\operatorname {Subst}\left (\int \frac {i-x}{i+x} \, dx,x,\sinh (x)\right )\\ &=-\operatorname {Subst}\left (\int \left (-1+\frac {2 i}{i+x}\right ) \, dx,x,\sinh (x)\right )\\ &=-2 i \log (i+\sinh (x))+\sinh (x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 14, normalized size = 1.00 \[ \sinh (x)-2 i \log (\sinh (x)+i) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.82, size = 26, normalized size = 1.86 \[ \frac {1}{2} \, {\left (4 i \, x e^{x} - 8 i \, e^{x} \log \left (e^{x} + i\right ) + e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.16, size = 21, normalized size = 1.50 \[ 2 i \, x - \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} - 4 i \, \log \left (e^{x} + i\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.07, size = 53, normalized size = 3.79 \[ 2 i \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )-\frac {1}{\tanh \left (\frac {x}{2}\right )-1}+2 i \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )-\frac {1}{\tanh \left (\frac {x}{2}\right )+1}-4 i \ln \left (\tanh \left (\frac {x}{2}\right )+i\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.32, size = 23, normalized size = 1.64 \[ -2 i \, x - \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} - 4 i \, \log \left (e^{\left (-x\right )} - i\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.55, size = 24, normalized size = 1.71 \[ \frac {{\mathrm {e}}^x}{2}-\frac {{\mathrm {e}}^{-x}}{2}+x\,2{}\mathrm {i}-\ln \left ({\mathrm {e}}^x+1{}\mathrm {i}\right )\,4{}\mathrm {i} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.18, size = 26, normalized size = 1.86 \[ 2 i x + \frac {e^{x}}{2} - 4 i \log {\left (e^{x} + i \right )} - \frac {e^{- x}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________