Optimal. Leaf size=20 \[ -x+\frac{2 i \cosh (x)}{1+i \sinh (x)} \]
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Rubi [A] time = 0.0743424, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {4391, 2670, 2680, 8} \[ -x+\frac{2 i \cosh (x)}{1+i \sinh (x)} \]
Antiderivative was successfully verified.
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Rule 4391
Rule 2670
Rule 2680
Rule 8
Rubi steps
\begin{align*} \int (\text{sech}(x)-i \tanh (x))^2 \, dx &=\int \text{sech}^2(x) (1-i \sinh (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 = 0.0048979, size = 14, normalized size = 0.7 \[ -x+2 \tanh (x)+2 i \text{sech}(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 26, normalized size = 1.3 \begin{align*} 2\,\tanh \left ( x \right ) -2\,i \left ({\frac{ \left ( \sinh \left ( x \right ) \right ) ^{2}}{\cosh \left ( x \right ) }}-\cosh \left ( x \right ) \right ) -x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.997069, size = 34, normalized size = 1.7 \begin{align*} -x + \frac{4 i}{e^{\left (-x\right )} + e^{x}} + \frac{4}{e^{\left (-2 \, x\right )} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01498, size = 43, normalized size = 2.15 \begin{align*} -\frac{x e^{x} - i \, x - 4 i}{e^{x} - i} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (- i \tanh{\left (x \right )} + \operatorname{sech}{\left (x \right )}\right )^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12289, size = 16, normalized size = 0.8 \begin{align*} -x + \frac{4 i}{e^{x} - i} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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