Optimal. Leaf size=26 \[ -\frac {1}{2} \tanh ^{-1}(\cosh (x))+\frac {1}{3} i \coth ^3(x)-\frac {1}{2} \coth (x) \text {csch}(x) \]
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Rubi [A]
time = 0.05, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2785, 2687, 30,
2691, 3855} \begin {gather*} \frac {1}{3} i \coth ^3(x)-\frac {1}{2} \tanh ^{-1}(\cosh (x))-\frac {1}{2} \coth (x) \text {csch}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2687
Rule 2691
Rule 2785
Rule 3855
Rubi steps
\begin {align*} \int \frac {\coth ^4(x)}{i+\sinh (x)} \, dx &=-\left (i \int \coth ^2(x) \text {csch}^2(x) \, dx\right )+\int \coth ^2(x) \text {csch}(x) \, dx\\ &=-\frac {1}{2} \coth (x) \text {csch}(x)+\frac {1}{2} \int \text {csch}(x) \, dx-\text {Subst}\left (\int x^2 \, dx,x,i \coth (x)\right )\\ &=-\frac {1}{2} \tanh ^{-1}(\cosh (x))+\frac {1}{3} i \coth ^3(x)-\frac {1}{2} \coth (x) \text {csch}(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. \(100\) vs. \(2(26)=52\).
time = 0.03, size = 100, normalized size = 3.85 \begin {gather*} \frac {1}{6} i \coth \left (\frac {x}{2}\right )-\frac {1}{8} \text {csch}^2\left (\frac {x}{2}\right )+\frac {1}{24} i \coth \left (\frac {x}{2}\right ) \text {csch}^2\left (\frac {x}{2}\right )+\frac {1}{2} \log \left (\tanh \left (\frac {x}{2}\right )\right )-\frac {1}{8} \text {sech}^2\left (\frac {x}{2}\right )+\frac {1}{6} i \tanh \left (\frac {x}{2}\right )-\frac {1}{24} i \text {sech}^2\left (\frac {x}{2}\right ) \tanh \left (\frac {x}{2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 58 vs. \(2 (19 ) = 38\).
time = 0.75, size = 59, normalized size = 2.27
method | result | size |
risch | \(-\frac {-6 i {\mathrm e}^{4 x}+3 \,{\mathrm e}^{5 x}-2 i-3 \,{\mathrm e}^{x}}{3 \left ({\mathrm e}^{2 x}-1\right )^{3}}-\frac {\ln \left ({\mathrm e}^{x}+1\right )}{2}+\frac {\ln \left ({\mathrm e}^{x}-1\right )}{2}\) | \(46\) |
default | \(\frac {i \tanh \left (\frac {x}{2}\right )}{8}+\frac {i \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{24}+\frac {\left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{8}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )\right )}{2}-\frac {1}{8 \tanh \left (\frac {x}{2}\right )^{2}}+\frac {i}{24 \tanh \left (\frac {x}{2}\right )^{3}}+\frac {i}{8 \tanh \left (\frac {x}{2}\right )}\) | \(59\) |
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. 61 vs. \(2 (18) = 36\).
time = 0.28, size = 61, normalized size = 2.35 \begin {gather*} \frac {3 \, e^{\left (-x\right )} - 6 i \, e^{\left (-4 \, x\right )} - 3 \, e^{\left (-5 \, x\right )} - 2 i}{3 \, {\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} - \frac {1}{2} \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac {1}{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. 90 vs. \(2 (18) = 36\).
time = 0.37, size = 90, normalized size = 3.46 \begin {gather*} -\frac {3 \, {\left (e^{\left (6 \, x\right )} - 3 \, e^{\left (4 \, x\right )} + 3 \, e^{\left (2 \, x\right )} - 1\right )} \log \left (e^{x} + 1\right ) - 3 \, {\left (e^{\left (6 \, x\right )} - 3 \, e^{\left (4 \, x\right )} + 3 \, e^{\left (2 \, x\right )} - 1\right )} \log \left (e^{x} - 1\right ) + 6 \, e^{\left (5 \, x\right )} - 12 i \, e^{\left (4 \, x\right )} - 6 \, e^{x} - 4 i}{6 \, {\left (e^{\left (6 \, x\right )} - 3 \, e^{\left (4 \, x\right )} + 3 \, e^{\left (2 \, x\right )} - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 61 vs. \(2 (22) = 44\).
time = 0.09, size = 61, normalized size = 2.35 \begin {gather*} \frac {- 3 e^{5 x} + 6 i e^{4 x} + 3 e^{x} + 2 i}{3 e^{6 x} - 9 e^{4 x} + 9 e^{2 x} - 3} + \frac {\log {\left (e^{x} - 1 \right )}}{2} - \frac {\log {\left (e^{x} + 1 \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 44 vs. \(2 (18) = 36\).
time = 0.40, size = 44, normalized size = 1.69 \begin {gather*} -\frac {3 \, e^{\left (5 \, x\right )} - 6 i \, e^{\left (4 \, x\right )} - 3 \, e^{x} - 2 i}{3 \, {\left (e^{\left (2 \, x\right )} - 1\right )}^{3}} - \frac {1}{2} \, \log \left (e^{x} + 1\right ) + \frac {1}{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 = 0.31, size = 74, normalized size = 2.85 \begin {gather*} \frac {\ln \left (1-{\mathrm {e}}^x\right )}{2}-\frac {\ln \left ({\mathrm {e}}^x+1\right )}{2}-\frac {{\mathrm {e}}^x}{{\mathrm {e}}^{2\,x}-1}-\frac {2\,{\mathrm {e}}^x}{{\left ({\mathrm {e}}^{2\,x}-1\right )}^2}+\frac {2{}\mathrm {i}}{{\mathrm {e}}^{2\,x}-1}+\frac {4{}\mathrm {i}}{{\left ({\mathrm {e}}^{2\,x}-1\right )}^2}+\frac {8{}\mathrm {i}}{3\,{\left ({\mathrm {e}}^{2\,x}-1\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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