Optimal. Leaf size=14 \[ x-\frac {\cosh (x)}{\sinh (x)+i} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2735, 2648} \[ x-\frac {\cosh (x)}{\sinh (x)+i} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2648
Rule 2735
Rubi steps
\begin {align*} \int \frac {\sinh (x)}{i+\sinh (x)} \, dx &=x-i \int \frac {1}{i+\sinh (x)} \, dx\\ &=x-\frac {\cosh (x)}{i+\sinh (x)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.06, size = 43, normalized size = 3.07 \[ i \text {sech}(x) \left (i \sinh (x)+2 \sqrt {\cosh ^2(x)} \sin ^{-1}\left (\frac {\sqrt {1-i \sinh (x)}}{\sqrt {2}}\right )+1\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.02, size = 16, normalized size = 1.14 \[ \frac {x e^{x} + i \, x + 2 i}{e^{x} + i} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.19, size = 10, normalized size = 0.71 \[ x + \frac {2 i}{e^{x} + i} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.04, size = 29, normalized size = 2.07 \[ -\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )-\frac {2}{\tanh \left (\frac {x}{2}\right )+i} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.59, size = 12, normalized size = 0.86 \[ x + \frac {2 i}{e^{\left (-x\right )} - i} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.41, size = 12, normalized size = 0.86 \[ x+\frac {2{}\mathrm {i}}{{\mathrm {e}}^x+1{}\mathrm {i}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.09, size = 8, normalized size = 0.57 \[ x + \frac {2}{- i e^{x} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________