Optimal. Leaf size=48 \[ -\frac {1}{4} i \tanh ^{-1}(\cosh (x))-\frac {2 \coth ^3(x)}{3}+\frac {\coth ^5(x)}{5}+\frac {1}{4} i \coth (x) \text {csch}(x)+\frac {1}{2} i \coth (x) \text {csch}^3(x) \]
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Rubi [A]
time = 0.06, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2788, 3852, 8,
3853, 3855} \begin {gather*} \frac {\coth ^5(x)}{5}-\frac {2 \coth ^3(x)}{3}-\frac {1}{4} i \tanh ^{-1}(\cosh (x))+\frac {1}{2} i \coth (x) \text {csch}^3(x)+\frac {1}{4} i \coth (x) \text {csch}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2788
Rule 3852
Rule 3853
Rule 3855
Rubi steps
\begin {align*} \int \frac {\coth ^6(x)}{(i+\sinh (x))^2} \, dx &=\int \left (\text {csch}^2(x)-2 i \text {csch}^3(x)-2 i \text {csch}^5(x)-\text {csch}^6(x)\right ) \, dx\\ &=-\left (2 i \int \text {csch}^3(x) \, dx\right )-2 i \int \text {csch}^5(x) \, dx+\int \text {csch}^2(x) \, dx-\int \text {csch}^6(x) \, dx\\ &=i \coth (x) \text {csch}(x)+\frac {1}{2} i \coth (x) \text {csch}^3(x)+i \int \text {csch}(x) \, dx-i \text {Subst}(\int 1 \, dx,x,-i \coth (x))+i \text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-i \coth (x)\right )+\frac {3}{2} i \int \text {csch}^3(x) \, dx\\ &=-i \tanh ^{-1}(\cosh (x))-\frac {2 \coth ^3(x)}{3}+\frac {\coth ^5(x)}{5}+\frac {1}{4} i \coth (x) \text {csch}(x)+\frac {1}{2} i \coth (x) \text {csch}^3(x)-\frac {3}{4} i \int \text {csch}(x) \, dx\\ &=-\frac {1}{4} i \tanh ^{-1}(\cosh (x))-\frac {2 \coth ^3(x)}{3}+\frac {\coth ^5(x)}{5}+\frac {1}{4} i \coth (x) \text {csch}(x)+\frac {1}{2} i \coth (x) \text {csch}^3(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. \(175\) vs. \(2(48)=96\).
time = 0.04, size = 175, normalized size = 3.65 \begin {gather*} -\frac {7}{30} \coth \left (\frac {x}{2}\right )+\frac {1}{16} i \text {csch}^2\left (\frac {x}{2}\right )-\frac {19}{480} \coth \left (\frac {x}{2}\right ) \text {csch}^2\left (\frac {x}{2}\right )+\frac {1}{32} i \text {csch}^4\left (\frac {x}{2}\right )+\frac {1}{160} \coth \left (\frac {x}{2}\right ) \text {csch}^4\left (\frac {x}{2}\right )-\frac {1}{4} i \log \left (\cosh \left (\frac {x}{2}\right )\right )+\frac {1}{4} i \log \left (\sinh \left (\frac {x}{2}\right )\right )+\frac {1}{16} i \text {sech}^2\left (\frac {x}{2}\right )-\frac {1}{32} i \text {sech}^4\left (\frac {x}{2}\right )-\frac {7}{30} \tanh \left (\frac {x}{2}\right )+\frac {19}{480} \text {sech}^2\left (\frac {x}{2}\right ) \tanh \left (\frac {x}{2}\right )+\frac {1}{160} \text {sech}^4\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. 73 vs. \(2 (35 ) = 70\).
time = 1.03, size = 74, normalized size = 1.54
method | result | size |
default | \(-\frac {3 \tanh \left (\frac {x}{2}\right )}{16}+\frac {\left (\tanh ^{5}\left (\frac {x}{2}\right )\right )}{160}-\frac {i \left (\tanh ^{4}\left (\frac {x}{2}\right )\right )}{32}-\frac {5 \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{96}+\frac {i \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{4}-\frac {3}{16 \tanh \left (\frac {x}{2}\right )}+\frac {i}{32 \tanh \left (\frac {x}{2}\right )^{4}}-\frac {5}{96 \tanh \left (\frac {x}{2}\right )^{3}}+\frac {1}{160 \tanh \left (\frac {x}{2}\right )^{5}}\) | \(74\) |
risch | \(\frac {i \left (60 i {\mathrm e}^{8 x}+15 \,{\mathrm e}^{9 x}-240 i {\mathrm e}^{6 x}+90 \,{\mathrm e}^{7 x}+40 i {\mathrm e}^{4 x}-80 i {\mathrm e}^{2 x}-90 \,{\mathrm e}^{3 x}+28 i-15 \,{\mathrm e}^{x}\right )}{30 \left ({\mathrm e}^{2 x}-1\right )^{5}}+\frac {i \ln \left ({\mathrm e}^{x}-1\right )}{4}-\frac {i \ln \left ({\mathrm e}^{x}+1\right )}{4}\) | \(82\) |
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. 103 vs. \(2 (32) = 64\).
time = 0.28, size = 103, normalized size = 2.15 \begin {gather*} \frac {-15 i \, e^{\left (-x\right )} - 80 \, e^{\left (-2 \, x\right )} - 90 i \, e^{\left (-3 \, x\right )} + 40 \, e^{\left (-4 \, x\right )} - 240 \, e^{\left (-6 \, x\right )} + 90 i \, e^{\left (-7 \, x\right )} + 60 \, e^{\left (-8 \, x\right )} + 15 i \, e^{\left (-9 \, x\right )} + 28}{30 \, {\left (5 \, e^{\left (-2 \, x\right )} - 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} - 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} - 1\right )}} - \frac {1}{4} i \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac {1}{4} i \, \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. 160 vs. \(2 (32) = 64\).
time = 0.43, size = 160, normalized size = 3.33 \begin {gather*} -\frac {15 \, {\left (i \, e^{\left (10 \, x\right )} - 5 i \, e^{\left (8 \, x\right )} + 10 i \, e^{\left (6 \, x\right )} - 10 i \, e^{\left (4 \, x\right )} + 5 i \, e^{\left (2 \, x\right )} - i\right )} \log \left (e^{x} + 1\right ) + 15 \, {\left (-i \, e^{\left (10 \, x\right )} + 5 i \, e^{\left (8 \, x\right )} - 10 i \, e^{\left (6 \, x\right )} + 10 i \, e^{\left (4 \, x\right )} - 5 i \, e^{\left (2 \, x\right )} + i\right )} \log \left (e^{x} - 1\right ) - 30 i \, e^{\left (9 \, x\right )} + 120 \, e^{\left (8 \, x\right )} - 180 i \, e^{\left (7 \, x\right )} - 480 \, e^{\left (6 \, x\right )} + 80 \, e^{\left (4 \, x\right )} + 180 i \, e^{\left (3 \, x\right )} - 160 \, e^{\left (2 \, x\right )} + 30 i \, e^{x} + 56}{60 \, {\left (e^{\left (10 \, x\right )} - 5 \, e^{\left (8 \, x\right )} + 10 \, e^{\left (6 \, x\right )} - 10 \, e^{\left (4 \, x\right )} + 5 \, 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. 114 vs. \(2 (44) = 88\).
time = 0.15, size = 114, normalized size = 2.38 \begin {gather*} \operatorname {RootSum} {\left (16 z^{2} + 1, \left ( i \mapsto i \log {\left (4 i i + e^{x} \right )} \right )\right )} + \frac {15 i e^{9 x} - 60 e^{8 x} + 90 i e^{7 x} + 240 e^{6 x} - 40 e^{4 x} - 90 i e^{3 x} + 80 e^{2 x} - 15 i e^{x} - 28}{30 e^{10 x} - 150 e^{8 x} + 300 e^{6 x} - 300 e^{4 x} + 150 e^{2 x} - 30} \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. 74 vs. \(2 (32) = 64\).
time = 0.44, size = 74, normalized size = 1.54 \begin {gather*} -\frac {-15 i \, e^{\left (9 \, x\right )} + 60 \, e^{\left (8 \, x\right )} - 90 i \, e^{\left (7 \, x\right )} - 240 \, e^{\left (6 \, x\right )} + 40 \, e^{\left (4 \, x\right )} + 90 i \, e^{\left (3 \, x\right )} - 80 \, e^{\left (2 \, x\right )} + 15 i \, e^{x} + 28}{30 \, {\left (e^{\left (2 \, x\right )} - 1\right )}^{5}} - \frac {1}{4} i \, \log \left (e^{x} + 1\right ) + \frac {1}{4} i \, \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 = 246, normalized size = 5.12 \begin {gather*} -\frac {80\,{\mathrm {e}}^{4\,x}-160\,{\mathrm {e}}^{2\,x}-480\,{\mathrm {e}}^{6\,x}+120\,{\mathrm {e}}^{8\,x}+56-\ln \left (-\frac {{\mathrm {e}}^x\,1{}\mathrm {i}}{2}-\frac {1}{2}{}\mathrm {i}\right )\,15{}\mathrm {i}+\ln \left (-\frac {{\mathrm {e}}^x\,1{}\mathrm {i}}{2}+\frac {1}{2}{}\mathrm {i}\right )\,15{}\mathrm {i}+{\mathrm {e}}^{3\,x}\,180{}\mathrm {i}-{\mathrm {e}}^{7\,x}\,180{}\mathrm {i}-{\mathrm {e}}^{9\,x}\,30{}\mathrm {i}+{\mathrm {e}}^x\,30{}\mathrm {i}+\ln \left (-\frac {{\mathrm {e}}^x\,1{}\mathrm {i}}{2}-\frac {1}{2}{}\mathrm {i}\right )\,{\mathrm {e}}^{2\,x}\,75{}\mathrm {i}-\ln \left (-\frac {{\mathrm {e}}^x\,1{}\mathrm {i}}{2}+\frac {1}{2}{}\mathrm {i}\right )\,{\mathrm {e}}^{2\,x}\,75{}\mathrm {i}-\ln \left (-\frac {{\mathrm {e}}^x\,1{}\mathrm {i}}{2}-\frac {1}{2}{}\mathrm {i}\right )\,{\mathrm {e}}^{4\,x}\,150{}\mathrm {i}+\ln \left (-\frac {{\mathrm {e}}^x\,1{}\mathrm {i}}{2}+\frac {1}{2}{}\mathrm {i}\right )\,{\mathrm {e}}^{4\,x}\,150{}\mathrm {i}+\ln \left (-\frac {{\mathrm {e}}^x\,1{}\mathrm {i}}{2}-\frac {1}{2}{}\mathrm {i}\right )\,{\mathrm {e}}^{6\,x}\,150{}\mathrm {i}-\ln \left (-\frac {{\mathrm {e}}^x\,1{}\mathrm {i}}{2}+\frac {1}{2}{}\mathrm {i}\right )\,{\mathrm {e}}^{6\,x}\,150{}\mathrm {i}-\ln \left (-\frac {{\mathrm {e}}^x\,1{}\mathrm {i}}{2}-\frac {1}{2}{}\mathrm {i}\right )\,{\mathrm {e}}^{8\,x}\,75{}\mathrm {i}+\ln \left (-\frac {{\mathrm {e}}^x\,1{}\mathrm {i}}{2}+\frac {1}{2}{}\mathrm {i}\right )\,{\mathrm {e}}^{8\,x}\,75{}\mathrm {i}+\ln \left (-\frac {{\mathrm {e}}^x\,1{}\mathrm {i}}{2}-\frac {1}{2}{}\mathrm {i}\right )\,{\mathrm {e}}^{10\,x}\,15{}\mathrm {i}-\ln \left (-\frac {{\mathrm {e}}^x\,1{}\mathrm {i}}{2}+\frac {1}{2}{}\mathrm {i}\right )\,{\mathrm {e}}^{10\,x}\,15{}\mathrm {i}}{60\,{\left ({\mathrm {e}}^{2\,x}-1\right )}^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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