Optimal. Leaf size=58 \[ \frac {16}{3} i \coth (x)-\frac {7}{2} \tanh ^{-1}(\cosh (x))+\frac {7}{2} \coth (x) \text {csch}(x)-\frac {8 i \coth (x) \text {csch}(x)}{3 (\sinh (x)+i)}+\frac {\coth (x) \text {csch}(x)}{3 (\sinh (x)+i)^2} \]
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Rubi [A] time = 0.14, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {2766, 2978, 2748, 3768, 3770, 3767, 8} \[ \frac {16}{3} i \coth (x)-\frac {7}{2} \tanh ^{-1}(\cosh (x))+\frac {7}{2} \coth (x) \text {csch}(x)-\frac {8 i \coth (x) \text {csch}(x)}{3 (\sinh (x)+i)}+\frac {\coth (x) \text {csch}(x)}{3 (\sinh (x)+i)^2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2748
Rule 2766
Rule 2978
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \frac {\text {csch}^3(x)}{(i+\sinh (x))^2} \, dx &=\frac {\coth (x) \text {csch}(x)}{3 (i+\sinh (x))^2}-\frac {1}{3} \int \frac {\text {csch}^3(x) (5 i-3 \sinh (x))}{i+\sinh (x)} \, dx\\ &=\frac {\coth (x) \text {csch}(x)}{3 (i+\sinh (x))^2}-\frac {8 i \coth (x) \text {csch}(x)}{3 (i+\sinh (x))}+\frac {1}{3} \int \text {csch}^3(x) (-21-16 i \sinh (x)) \, dx\\ &=\frac {\coth (x) \text {csch}(x)}{3 (i+\sinh (x))^2}-\frac {8 i \coth (x) \text {csch}(x)}{3 (i+\sinh (x))}-\frac {16}{3} i \int \text {csch}^2(x) \, dx-7 \int \text {csch}^3(x) \, dx\\ &=\frac {7}{2} \coth (x) \text {csch}(x)+\frac {\coth (x) \text {csch}(x)}{3 (i+\sinh (x))^2}-\frac {8 i \coth (x) \text {csch}(x)}{3 (i+\sinh (x))}+\frac {7}{2} \int \text {csch}(x) \, dx-\frac {16}{3} \operatorname {Subst}(\int 1 \, dx,x,-i \coth (x))\\ &=-\frac {7}{2} \tanh ^{-1}(\cosh (x))+\frac {16}{3} i \coth (x)+\frac {7}{2} \coth (x) \text {csch}(x)+\frac {\coth (x) \text {csch}(x)}{3 (i+\sinh (x))^2}-\frac {8 i \coth (x) \text {csch}(x)}{3 (i+\sinh (x))}\\ \end {align*}
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Mathematica [B] time = 0.37, size = 131, normalized size = 2.26 \[ \frac {1}{24} \left (24 i \tanh \left (\frac {x}{2}\right )+24 i \coth \left (\frac {x}{2}\right )+3 \text {csch}^2\left (\frac {x}{2}\right )+3 \text {sech}^2\left (\frac {x}{2}\right )+84 \log \left (\tanh \left (\frac {x}{2}\right )\right )+\frac {160 i \sinh \left (\frac {x}{2}\right )}{\cosh \left (\frac {x}{2}\right )-i \sinh \left (\frac {x}{2}\right )}+\frac {8}{\left (\cosh \left (\frac {x}{2}\right )-i \sinh \left (\frac {x}{2}\right )\right )^2}+\frac {16 \sinh \left (\frac {x}{2}\right )}{\left (\sinh \left (\frac {x}{2}\right )+i \cosh \left (\frac {x}{2}\right )\right )^3}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.61, size = 179, normalized size = 3.09 \[ -\frac {{\left (21 \, e^{\left (7 \, x\right )} + 63 i \, e^{\left (6 \, x\right )} - 105 \, e^{\left (5 \, x\right )} - 147 i \, e^{\left (4 \, x\right )} + 147 \, e^{\left (3 \, x\right )} + 105 i \, e^{\left (2 \, x\right )} - 63 \, e^{x} - 21 i\right )} \log \left (e^{x} + 1\right ) - {\left (21 \, e^{\left (7 \, x\right )} + 63 i \, e^{\left (6 \, x\right )} - 105 \, e^{\left (5 \, x\right )} - 147 i \, e^{\left (4 \, x\right )} + 147 \, e^{\left (3 \, x\right )} + 105 i \, e^{\left (2 \, x\right )} - 63 \, e^{x} - 21 i\right )} \log \left (e^{x} - 1\right ) - 42 \, e^{\left (6 \, x\right )} - 126 i \, e^{\left (5 \, x\right )} + 196 \, e^{\left (4 \, x\right )} + 252 i \, e^{\left (3 \, x\right )} - 194 \, e^{\left (2 \, x\right )} - 150 i \, e^{x} + 64}{6 \, e^{\left (7 \, x\right )} + 18 i \, e^{\left (6 \, x\right )} - 30 \, e^{\left (5 \, x\right )} - 42 i \, e^{\left (4 \, x\right )} + 42 \, e^{\left (3 \, x\right )} + 30 i \, e^{\left (2 \, x\right )} - 18 \, e^{x} - 6 i} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 59, normalized size = 1.02 \[ \frac {e^{\left (3 \, x\right )} + 4 i \, e^{\left (2 \, x\right )} + e^{x} - 4 i}{{\left (e^{\left (2 \, x\right )} - 1\right )}^{2}} + \frac {2 \, {\left (9 \, e^{\left (2 \, x\right )} + 21 i \, e^{x} - 10\right )}}{3 \, {\left (e^{x} + i\right )}^{3}} - \frac {7}{2} \, \log \left (e^{x} + 1\right ) + \frac {7}{2} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 76, normalized size = 1.31 \[ i \tanh \left (\frac {x}{2}\right )-\frac {\left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{8}+\frac {8 i}{\tanh \left (\frac {x}{2}\right )+i}-\frac {4 i}{3 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{3}}+\frac {2}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}+\frac {i}{\tanh \left (\frac {x}{2}\right )}+\frac {1}{8 \tanh \left (\frac {x}{2}\right )^{2}}+\frac {7 \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 105, normalized size = 1.81 \[ -\frac {8 \, {\left (-75 i \, e^{\left (-x\right )} + 97 \, e^{\left (-2 \, x\right )} + 126 i \, e^{\left (-3 \, x\right )} - 98 \, e^{\left (-4 \, x\right )} - 63 i \, e^{\left (-5 \, x\right )} + 21 \, e^{\left (-6 \, x\right )} - 32\right )}}{72 \, e^{\left (-x\right )} + 120 i \, e^{\left (-2 \, x\right )} - 168 \, e^{\left (-3 \, x\right )} - 168 i \, e^{\left (-4 \, x\right )} + 120 \, e^{\left (-5 \, x\right )} + 72 i \, e^{\left (-6 \, x\right )} - 24 \, e^{\left (-7 \, x\right )} - 24 i} - \frac {7}{2} \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac {7}{2} \, \log \left (e^{\left (-x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.77, size = 79, normalized size = 1.36 \[ \frac {{\mathrm {e}}^x}{{\mathrm {e}}^{2\,x}-1}-\frac {7\,\ln \left ({\mathrm {e}}^x+1\right )}{2}-\frac {7\,\ln \left (\frac {1}{{\mathrm {e}}^x-1}\right )}{2}+\frac {2\,{\mathrm {e}}^x}{{\left ({\mathrm {e}}^{2\,x}-1\right )}^2}+\frac {6}{{\mathrm {e}}^x+1{}\mathrm {i}}+\frac {2{}\mathrm {i}}{{\left ({\mathrm {e}}^x+1{}\mathrm {i}\right )}^2}+\frac {4}{3\,{\left ({\mathrm {e}}^x+1{}\mathrm {i}\right )}^3}+\frac {4{}\mathrm {i}}{{\mathrm {e}}^{2\,x}-1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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