Optimal. Leaf size=45 \[ -\frac {i}{2 (1+i \sinh (x))}-\frac {3}{4} i \log (-\sinh (x)+i)-\frac {1}{4} i \log (\sinh (x)+i) \]
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Rubi [A] time = 0.05, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3879, 88} \[ -\frac {i}{2 (1+i \sinh (x))}-\frac {3}{4} i \log (-\sinh (x)+i)-\frac {1}{4} i \log (\sinh (x)+i) \]
Antiderivative was successfully verified.
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Rule 88
Rule 3879
Rubi steps
\begin {align*} \int \frac {\tanh (x)}{i+\text {csch}(x)} \, dx &=\operatorname {Subst}\left (\int \frac {x^2}{(i-i x) (i+i x)^2} \, dx,x,i \sinh (x)\right )\\ &=\operatorname {Subst}\left (\int \left (-\frac {i}{4 (-1+x)}+\frac {i}{2 (1+x)^2}-\frac {3 i}{4 (1+x)}\right ) \, dx,x,i \sinh (x)\right )\\ &=-\frac {3}{4} i \log (i-\sinh (x))-\frac {1}{4} i \log (i+\sinh (x))-\frac {i}{2 (1+i \sinh (x))}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 39, normalized size = 0.87 \[ \frac {1}{4} \left (-\frac {2}{\sinh (x)-i}-3 i \log (-\sinh (x)+i)-i \log (\sinh (x)+i)\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 1.22, size = 71, normalized size = 1.58 \[ \frac {2 i \, x e^{\left (2 \, x\right )} + 2 \, {\left (2 \, x - 1\right )} e^{x} + {\left (-i \, e^{\left (2 \, x\right )} - 2 \, e^{x} + i\right )} \log \left (e^{x} + i\right ) + {\left (-3 i \, e^{\left (2 \, x\right )} - 6 \, e^{x} + 3 i\right )} \log \left (e^{x} - i\right ) - 2 i \, x}{2 \, e^{\left (2 \, x\right )} - 4 i \, e^{x} - 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.12, size = 55, normalized size = 1.22 \[ \frac {3 i \, e^{\left (-x\right )} - 3 i \, e^{x} - 2}{4 \, {\left (e^{\left (-x\right )} - e^{x} + 2 i\right )}} - \frac {1}{4} i \, \log \left (-i \, e^{\left (-x\right )} + i \, e^{x} - 2\right ) - \frac {3}{4} i \, \log \left (-e^{\left (-x\right )} + e^{x} - 2 i\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.21, size = 65, normalized size = 1.44 \[ i \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+i \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )-\frac {i \ln \left (\tanh \left (\frac {x}{2}\right )+i\right )}{2}-\frac {3 i \ln \left (\tanh \left (\frac {x}{2}\right )-i\right )}{2}+\frac {i}{\left (\tanh \left (\frac {x}{2}\right )-i\right )^{2}}+\frac {1}{\tanh \left (\frac {x}{2}\right )-i} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 45, normalized size = 1.00 \[ -i \, x + \frac {e^{\left (-x\right )}}{2 i \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1} - \frac {1}{2} i \, \log \left (i \, e^{\left (-x\right )} + 1\right ) - \frac {3}{2} i \, \log \left (i \, e^{\left (-x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.23, size = 50, normalized size = 1.11 \[ x\,1{}\mathrm {i}+\mathrm {atan}\left ({\mathrm {e}}^x\right )-\ln \left (\left ({\mathrm {e}}^x-\mathrm {i}\right )\,\left ({\mathrm {e}}^x+1{}\mathrm {i}\right )\right )\,1{}\mathrm {i}+\frac {1{}\mathrm {i}}{1-{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,2{}\mathrm {i}}-\frac {1}{{\mathrm {e}}^x-\mathrm {i}} \]
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 \[ \int \frac {\tanh {\relax (x )}}{\operatorname {csch}{\relax (x )} + i}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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