Optimal. Leaf size=13 \[ -i \log (-\sinh (x)+i) \]
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Rubi [A] time = 0.03, antiderivative size = 13, 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 = {3159, 2667, 31} \[ -i \log (-\sinh (x)+i) \]
Antiderivative was successfully verified.
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Rule 31
Rule 2667
Rule 3159
Rubi steps
\begin {align*} \int \frac {1}{\text {sech}(x)+i \tanh (x)} \, dx &=\int \frac {\cosh (x)}{1+i \sinh (x)} \, dx\\ &=-\left (i \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,i \sinh (x)\right )\right )\\ &=-i \log (i-\sinh (x))\\ \end {align*}
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Mathematica [A] time = 0.02, size = 17, normalized size = 1.31 \[ 2 \tan ^{-1}\left (\tanh \left (\frac {x}{2}\right )\right )-i \log (\cosh (x)) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 11, normalized size = 0.85 \[ i \, x - 2 i \, \log \left (e^{x} - i\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 13, normalized size = 1.00 \[ i \, x - 2 i \, \log \left (i \, e^{x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.24, size = 33, normalized size = 2.54 \[ i \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+i \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )-2 i \ln \left (\tanh \left (\frac {x}{2}\right )-i\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 15, normalized size = 1.15 \[ -i \, x - 2 i \, \log \left (i \, e^{\left (-x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.56, size = 14, normalized size = 1.08 \[ x\,1{}\mathrm {i}-\ln \left ({\mathrm {e}}^x-\mathrm {i}\right )\,2{}\mathrm {i} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.21, size = 22, normalized size = 1.69 \[ - i x - i \log {\left (i \tanh {\relax (x )} + \operatorname {sech}{\relax (x )} \right )} + i \log {\left (\tanh {\relax (x )} + 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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