Optimal. Leaf size=58 \[ -\frac {7}{2} \tanh ^{-1}(\cosh (x))+\frac {16}{3} i \coth (x)+\frac {7}{2} \coth (x) \text {csch}(x)+\frac {\coth (x) \text {csch}(x)}{3 (i+\sinh (x))^2}-\frac {8 i \coth (x) \text {csch}(x)}{3 (i+\sinh (x))} \]
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Rubi [A]
time = 0.10, 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 = {2845, 3057,
2827, 3853, 3855, 3852, 8} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2827
Rule 2845
Rule 3057
Rule 3852
Rule 3853
Rule 3855
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} \text {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] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(131\) vs. \(2(58)=116\).
time = 0.28, size = 131, normalized size = 2.26 \begin {gather*} \frac {1}{24} \left (24 i \coth \left (\frac {x}{2}\right )+3 \text {csch}^2\left (\frac {x}{2}\right )+84 \log \left (\tanh \left (\frac {x}{2}\right )\right )+3 \text {sech}^2\left (\frac {x}{2}\right )+\frac {8}{\left (\cosh \left (\frac {x}{2}\right )-i \sinh \left (\frac {x}{2}\right )\right )^2}+\frac {160 i \sinh \left (\frac {x}{2}\right )}{\cosh \left (\frac {x}{2}\right )-i \sinh \left (\frac {x}{2}\right )}+\frac {16 \sinh \left (\frac {x}{2}\right )}{\left (i \cosh \left (\frac {x}{2}\right )+\sinh \left (\frac {x}{2}\right )\right )^3}+24 i \tanh \left (\frac {x}{2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.69, size = 76, normalized size = 1.31
method | result | size |
risch | \(\frac {-98 \,{\mathrm e}^{4 x}+63 i {\mathrm e}^{5 x}+97 \,{\mathrm e}^{2 x}-126 i {\mathrm e}^{3 x}+21 \,{\mathrm e}^{6 x}-32+75 i {\mathrm e}^{x}}{3 \left ({\mathrm e}^{2 x}-1\right )^{2} \left ({\mathrm e}^{x}+i\right )^{3}}+\frac {7 \ln \left ({\mathrm e}^{x}-1\right )}{2}-\frac {7 \ln \left ({\mathrm e}^{x}+1\right )}{2}\) | \(72\) |
default | \(i \tanh \left (\frac {x}{2}\right )-\frac {\left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{8}+\frac {i}{\tanh \left (\frac {x}{2}\right )}+\frac {1}{8 \tanh \left (\frac {x}{2}\right )^{2}}+\frac {7 \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{2}+\frac {8 i}{\tanh \left (\frac {x}{2}\right )+i}-\frac {4 i}{3 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{3}}+\frac {2}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}\) | \(76\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 105 vs. \(2 (40) = 80\).
time = 0.28, size = 105, normalized size = 1.81 \begin {gather*} -\frac {-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}{3 \, {\left (3 \, e^{\left (-x\right )} + 5 i \, e^{\left (-2 \, x\right )} - 7 \, e^{\left (-3 \, x\right )} - 7 i \, e^{\left (-4 \, x\right )} + 5 \, e^{\left (-5 \, x\right )} + 3 i \, e^{\left (-6 \, x\right )} - e^{\left (-7 \, x\right )} - i\right )}} - \frac {7}{2} \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac {7}{2} \, \log \left (e^{\left (-x\right )} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 174 vs. \(2 (40) = 80\).
time = 0.48, size = 174, normalized size = 3.00 \begin {gather*} -\frac {21 \, {\left (e^{\left (7 \, x\right )} + 3 i \, e^{\left (6 \, x\right )} - 5 \, e^{\left (5 \, x\right )} - 7 i \, e^{\left (4 \, x\right )} + 7 \, e^{\left (3 \, x\right )} + 5 i \, e^{\left (2 \, x\right )} - 3 \, e^{x} - i\right )} \log \left (e^{x} + 1\right ) - 21 \, {\left (e^{\left (7 \, x\right )} + 3 i \, e^{\left (6 \, x\right )} - 5 \, e^{\left (5 \, x\right )} - 7 i \, e^{\left (4 \, x\right )} + 7 \, e^{\left (3 \, x\right )} + 5 i \, e^{\left (2 \, x\right )} - 3 \, e^{x} - 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 \, {\left (e^{\left (7 \, x\right )} + 3 i \, e^{\left (6 \, x\right )} - 5 \, e^{\left (5 \, x\right )} - 7 i \, e^{\left (4 \, x\right )} + 7 \, e^{\left (3 \, x\right )} + 5 i \, e^{\left (2 \, x\right )} - 3 \, e^{x} - i\right )}} \end {gather*}
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 \begin {gather*} \int \frac {\operatorname {csch}^{3}{\left (x \right )}}{\left (\sinh {\left (x \right )} + i\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 59, normalized size = 1.02 \begin {gather*} \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 ) \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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