Optimal. Leaf size=20 \[ -x-\frac {2 i \cosh (x)}{1-i \sinh (x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4391, 2680, 8} \[ -x-\frac {2 i \cosh (x)}{1-i \sinh (x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2680
Rule 4391
Rubi steps
\begin {align*} \int \frac {1}{(\text {sech}(x)-i \tanh (x))^2} \, dx &=\int \frac {\cosh ^2(x)}{(1-i \sinh (x))^2} \, dx\\ &=-\frac {2 i \cosh (x)}{1-i \sinh (x)}-\int 1 \, dx\\ &=-x-\frac {2 i \cosh (x)}{1-i \sinh (x)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 31, normalized size = 1.55 \[ -x+\frac {4 \sinh \left (\frac {x}{2}\right )}{\cosh \left (\frac {x}{2}\right )-i \sinh \left (\frac {x}{2}\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.42, size = 17, normalized size = 0.85 \[ -\frac {x e^{x} + i \, x + 4 i}{e^{x} + i} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.12, size = 12, normalized size = 0.60 \[ -x - \frac {4 i}{e^{x} + i} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.29, size = 29, normalized size = 1.45 \[ \frac {4}{\tanh \left (\frac {x}{2}\right )+i}+\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )-\ln \left (\tanh \left (\frac {x}{2}\right )+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.50, size = 14, normalized size = 0.70 \[ -x - \frac {4 i}{e^{\left (-x\right )} - i} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.51, size = 14, normalized size = 0.70 \[ -x-\frac {4{}\mathrm {i}}{{\mathrm {e}}^x+1{}\mathrm {i}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (- i \tanh {\relax (x )} + \operatorname {sech}{\relax (x )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________