Optimal. Leaf size=42 \[ \frac {10 \coth (x)}{3}+2 i \tanh ^{-1}(\cosh (x))-\frac {2 i \coth (x)}{\sinh (x)+i}+\frac {\coth (x)}{3 (\sinh (x)+i)^2} \]
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Rubi [A] time = 0.12, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2766, 2978, 2748, 3767, 8, 3770} \[ \frac {10 \coth (x)}{3}+2 i \tanh ^{-1}(\cosh (x))-\frac {2 i \coth (x)}{\sinh (x)+i}+\frac {\coth (x)}{3 (\sinh (x)+i)^2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2748
Rule 2766
Rule 2978
Rule 3767
Rule 3770
Rubi steps
\begin {align*} \int \frac {\text {csch}^2(x)}{(i+\sinh (x))^2} \, dx &=\frac {\coth (x)}{3 (i+\sinh (x))^2}-\frac {1}{3} \int \frac {\text {csch}^2(x) (4 i-2 \sinh (x))}{i+\sinh (x)} \, dx\\ &=\frac {\coth (x)}{3 (i+\sinh (x))^2}-\frac {2 i \coth (x)}{i+\sinh (x)}+\frac {1}{3} \int \text {csch}^2(x) (-10-6 i \sinh (x)) \, dx\\ &=\frac {\coth (x)}{3 (i+\sinh (x))^2}-\frac {2 i \coth (x)}{i+\sinh (x)}-2 i \int \text {csch}(x) \, dx-\frac {10}{3} \int \text {csch}^2(x) \, dx\\ &=2 i \tanh ^{-1}(\cosh (x))+\frac {\coth (x)}{3 (i+\sinh (x))^2}-\frac {2 i \coth (x)}{i+\sinh (x)}+\frac {10}{3} i \operatorname {Subst}(\int 1 \, dx,x,-i \coth (x))\\ &=2 i \tanh ^{-1}(\cosh (x))+\frac {10 \coth (x)}{3}+\frac {\coth (x)}{3 (i+\sinh (x))^2}-\frac {2 i \coth (x)}{i+\sinh (x)}\\ \end {align*}
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Mathematica [B] time = 0.39, size = 88, normalized size = 2.10 \[ \frac {1}{6} \left (\frac {2}{\sinh (x)+i}+3 \tanh \left (\frac {x}{2}\right )+3 \coth \left (\frac {x}{2}\right )-12 i \log \left (\sinh \left (\frac {x}{2}\right )\right )+12 i \log \left (\cosh \left (\frac {x}{2}\right )\right )-\frac {4 \sinh \left (\frac {x}{2}\right ) (7 \sinh (x)+8 i)}{\left (\sinh \left (\frac {x}{2}\right )+i \cosh \left (\frac {x}{2}\right )\right )^3}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 129, normalized size = 3.07 \[ \frac {{\left (6 i \, e^{\left (5 \, x\right )} - 18 \, e^{\left (4 \, x\right )} - 24 i \, e^{\left (3 \, x\right )} + 24 \, e^{\left (2 \, x\right )} + 18 i \, e^{x} - 6\right )} \log \left (e^{x} + 1\right ) + {\left (-6 i \, e^{\left (5 \, x\right )} + 18 \, e^{\left (4 \, x\right )} + 24 i \, e^{\left (3 \, x\right )} - 24 \, e^{\left (2 \, x\right )} - 18 i \, e^{x} + 6\right )} \log \left (e^{x} - 1\right ) - 12 i \, e^{\left (4 \, x\right )} + 36 \, e^{\left (3 \, x\right )} + 44 i \, e^{\left (2 \, x\right )} - 48 \, e^{x} - 20 i}{3 \, e^{\left (5 \, x\right )} + 9 i \, e^{\left (4 \, x\right )} - 12 \, e^{\left (3 \, x\right )} - 12 i \, e^{\left (2 \, x\right )} + 9 \, e^{x} + 3 i} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 46, normalized size = 1.10 \[ \frac {2}{e^{\left (2 \, x\right )} - 1} - \frac {2 \, {\left (6 i \, e^{\left (2 \, x\right )} - 15 \, e^{x} - 7 i\right )}}{3 \, {\left (e^{x} + i\right )}^{3}} + 2 i \, \log \left (e^{x} + 1\right ) - 2 i \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 58, normalized size = 1.38 \[ \frac {\tanh \left (\frac {x}{2}\right )}{2}-\frac {2 i}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}-\frac {4}{3 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{3}}+\frac {6}{\tanh \left (\frac {x}{2}\right )+i}-2 i \ln \left (\tanh \left (\frac {x}{2}\right )\right )+\frac {1}{2 \tanh \left (\frac {x}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 81, normalized size = 1.93 \[ \frac {4 \, {\left (12 \, e^{\left (-x\right )} + 11 i \, e^{\left (-2 \, x\right )} - 9 \, e^{\left (-3 \, x\right )} - 3 i \, e^{\left (-4 \, x\right )} - 5 i\right )}}{9 \, e^{\left (-x\right )} + 12 i \, e^{\left (-2 \, x\right )} - 12 \, e^{\left (-3 \, x\right )} - 9 i \, e^{\left (-4 \, x\right )} + 3 \, e^{\left (-5 \, x\right )} - 3 i} + 2 i \, \log \left (e^{\left (-x\right )} + 1\right ) - 2 i \, \log \left (e^{\left (-x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.71, size = 85, normalized size = 2.02 \[ \frac {2}{{\mathrm {e}}^{2\,x}-1+{\mathrm {e}}^x\,2{}\mathrm {i}}+\frac {2}{{\mathrm {e}}^{2\,x}-1}-\ln \left ({\mathrm {e}}^x\,4{}\mathrm {i}-4{}\mathrm {i}\right )\,2{}\mathrm {i}+\ln \left ({\mathrm {e}}^x\,4{}\mathrm {i}+4{}\mathrm {i}\right )\,2{}\mathrm {i}-\frac {4{}\mathrm {i}}{{\mathrm {e}}^x+1{}\mathrm {i}}-\frac {4{}\mathrm {i}}{3\,\left ({\mathrm {e}}^{2\,x}\,3{}\mathrm {i}+{\mathrm {e}}^{3\,x}-3\,{\mathrm {e}}^x-\mathrm {i}\right )} \]
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 \[ \int \frac {\operatorname {csch}^{2}{\relax (x )}}{\left (\sinh {\relax (x )} + i\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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