Optimal. Leaf size=20 \[ -x-\frac {2 i \cosh (x)}{1-i \sinh (x)} \]
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Rubi [A] time = 0.08, antiderivative size = 20, normalized size of antiderivative = 1.00, 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 8
Rule 2670
Rule 2680
Rule 4391
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*}
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Mathematica [A] time = 0.01, size = 14, normalized size = 0.70 \[ -x+2 \tanh (x)-2 i \text {sech}(x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, 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.
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giac [A] time = 0.14, size = 12, normalized size = 0.60 \[ -x - \frac {4 i}{e^{x} + i} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.36, size = 16, normalized size = 0.80 \[ 2 \tanh \relax (x )-\frac {2 i}{\cosh \relax (x )}-x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 25, normalized size = 1.25 \[ -x - \frac {4 i}{e^{\left (-x\right )} + e^{x}} + \frac {4}{e^{\left (-2 \, x\right )} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.64, 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.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (i \tanh {\relax (x )} + \operatorname {sech}{\relax (x )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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