Optimal. Leaf size=83 \[ \frac {\tanh ^{-1}\left (\sqrt {2} \tanh (x)\right )}{3 \sqrt {2}}+\frac {\tanh ^{-1}\left (\sqrt {1-\sqrt [3]{-1}} \tanh (x)\right )}{3 \sqrt {1-\sqrt [3]{-1}}}+\frac {\tanh ^{-1}\left (\sqrt {1+(-1)^{2/3}} \tanh (x)\right )}{3 \sqrt {1+(-1)^{2/3}}} \]
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Rubi [A]
time = 0.08, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3290, 3260,
212} \begin {gather*} \frac {\tanh ^{-1}\left (\sqrt {2} \tanh (x)\right )}{3 \sqrt {2}}+\frac {\tanh ^{-1}\left (\sqrt {1-\sqrt [3]{-1}} \tanh (x)\right )}{3 \sqrt {1-\sqrt [3]{-1}}}+\frac {\tanh ^{-1}\left (\sqrt {1+(-1)^{2/3}} \tanh (x)\right )}{3 \sqrt {1+(-1)^{2/3}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 3260
Rule 3290
Rubi steps
\begin {align*} \int \frac {1}{1-\sinh ^6(x)} \, dx &=\frac {1}{3} \int \frac {1}{1-\sinh ^2(x)} \, dx+\frac {1}{3} \int \frac {1}{1+\sqrt [3]{-1} \sinh ^2(x)} \, dx+\frac {1}{3} \int \frac {1}{1-(-1)^{2/3} \sinh ^2(x)} \, dx\\ &=\frac {1}{3} \text {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\tanh (x)\right )+\frac {1}{3} \text {Subst}\left (\int \frac {1}{1-\left (1-\sqrt [3]{-1}\right ) x^2} \, dx,x,\tanh (x)\right )+\frac {1}{3} \text {Subst}\left (\int \frac {1}{1-\left (1+(-1)^{2/3}\right ) x^2} \, dx,x,\tanh (x)\right )\\ &=\frac {\tanh ^{-1}\left (\sqrt {2} \tanh (x)\right )}{3 \sqrt {2}}+\frac {\tanh ^{-1}\left (\sqrt {1-\sqrt [3]{-1}} \tanh (x)\right )}{3 \sqrt {1-\sqrt [3]{-1}}}+\frac {\tanh ^{-1}\left (\sqrt {1+(-1)^{2/3}} \tanh (x)\right )}{3 \sqrt {1+(-1)^{2/3}}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.34, size = 70, normalized size = 0.84 \begin {gather*} \frac {1}{6} \left (-\text {ArcTan}(\text {csch}(x) \text {sech}(x))+i \sqrt {3} \left (\text {ArcTan}\left (\frac {1-2 i \tanh (x)}{\sqrt {3}}\right )-\text {ArcTan}\left (\frac {1+2 i \tanh (x)}{\sqrt {3}}\right )\right )+\sqrt {2} \tanh ^{-1}\left (\sqrt {2} \tanh (x)\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.62, size = 160, normalized size = 1.93
method | result | size |
risch | \(\left (\munderset {\textit {\_R} =\RootOf \left (1296 \textit {\_Z}^{4}-36 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (432 \textit {\_R}^{3}-72 \textit {\_R}^{2}+{\mathrm e}^{2 x}+1\right )\right )+\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 x}-3+2 \sqrt {2}\right )}{12}-\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 x}-3-2 \sqrt {2}\right )}{12}\) | \(71\) |
default | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}+2 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )}{\sum }\frac {\left (-\textit {\_R}^{2}+\textit {\_R} +1\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-\textit {\_R} \right )}{2 \textit {\_R}^{3}-3 \textit {\_R}^{2}+2 \textit {\_R} +1}\right )}{3}+\frac {\sqrt {2}\, \arctanh \left (\frac {\left (2 \tanh \left (\frac {x}{2}\right )+2\right ) \sqrt {2}}{4}\right )}{6}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+2 \textit {\_Z}^{3}+2 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )}{\sum }\frac {\left (-\textit {\_R}^{2}-\textit {\_R} +1\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-\textit {\_R} \right )}{2 \textit {\_R}^{3}+3 \textit {\_R}^{2}+2 \textit {\_R} -1}\right )}{3}+\frac {\sqrt {2}\, \arctanh \left (\frac {\left (2 \tanh \left (\frac {x}{2}\right )-2\right ) \sqrt {2}}{4}\right )}{6}\) | \(160\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 155 vs.
\(2 (57) = 114\).
time = 0.41, size = 155, normalized size = 1.87 \begin {gather*} -\frac {1}{12} \, \sqrt {3} \log \left (16 \, \sqrt {3} + 4 \, e^{\left (4 \, x\right )} + 28\right ) + \frac {1}{12} \, \sqrt {3} \log \left (-16 \, \sqrt {3} + 4 \, e^{\left (4 \, x\right )} + 28\right ) + \frac {1}{12} \, \sqrt {2} \log \left (\frac {2 \, {\left (2 \, \sqrt {2} - 3\right )} e^{\left (2 \, x\right )} - 12 \, \sqrt {2} + e^{\left (4 \, x\right )} + 17}{e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1}\right ) - \frac {1}{3} \, \arctan \left (-{\left (\sqrt {3} + 2\right )} e^{\left (2 \, x\right )} + \frac {1}{2} \, {\left (\sqrt {3} + 2\right )} \sqrt {-16 \, \sqrt {3} + 4 \, e^{\left (4 \, x\right )} + 28}\right ) + \frac {1}{3} \, \arctan \left (-{\left (\sqrt {3} - 2\right )} e^{\left (2 \, x\right )} + \sqrt {4 \, \sqrt {3} + e^{\left (4 \, x\right )} + 7} {\left (\sqrt {3} - 2\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 143 vs.
\(2 (57) = 114\).
time = 0.43, size = 143, normalized size = 1.72 \begin {gather*} -\frac {1}{36} \, {\left ({\left (2 \, \sqrt {3} - 3\right )} e^{\left (4 \, x\right )} + 2 \, \sqrt {3} - 3\right )} \arctan \left (\frac {e^{\left (2 \, x\right )}}{\sqrt {3} + 2}\right ) + \frac {1}{36} \, {\left ({\left (2 \, \sqrt {3} + 3\right )} e^{\left (4 \, x\right )} + 2 \, \sqrt {3} + 3\right )} \arctan \left (-\frac {e^{\left (2 \, x\right )}}{\sqrt {3} - 2}\right ) - \frac {1}{12} \, \sqrt {3} \log \left ({\left (\sqrt {3} + 2\right )}^{2} + e^{\left (4 \, x\right )}\right ) + \frac {1}{12} \, \sqrt {3} \log \left ({\left (\sqrt {3} - 2\right )}^{2} + e^{\left (4 \, x\right )}\right ) - \frac {1}{12} \, \sqrt {2} \log \left (\frac {{\left | -4 \, \sqrt {2} + 2 \, e^{\left (2 \, x\right )} - 6 \right |}}{{\left | 4 \, \sqrt {2} + 2 \, e^{\left (2 \, x\right )} - 6 \right |}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.71, size = 285, normalized size = 3.43 \begin {gather*} \frac {\mathrm {atan}\left (\frac {14009449395540459520\,{\mathrm {e}}^{2\,x}-955607545932677120\,\sqrt {3}+8088359377641144320\,\sqrt {3}\,{\mathrm {e}}^{2\,x}-1655160823988879360}{6177144285775790080\,{\mathrm {e}}^{2\,x}+2167269359741829120\,\sqrt {3}+3566375915854233600\,\sqrt {3}\,{\mathrm {e}}^{2\,x}+3753820658157486080}\right )}{6}-\frac {\sqrt {3}\,\ln \left ({\left (6177144285775790080\,{\mathrm {e}}^{2\,x}-2167269359741829120\,\sqrt {3}-3566375915854233600\,\sqrt {3}\,{\mathrm {e}}^{2\,x}+3753820658157486080\right )}^2+{\left (14009449395540459520\,{\mathrm {e}}^{2\,x}+955607545932677120\,\sqrt {3}-8088359377641144320\,\sqrt {3}\,{\mathrm {e}}^{2\,x}-1655160823988879360\right )}^2\right )}{12}+\frac {\sqrt {3}\,\ln \left ({\left (6177144285775790080\,{\mathrm {e}}^{2\,x}+2167269359741829120\,\sqrt {3}+3566375915854233600\,\sqrt {3}\,{\mathrm {e}}^{2\,x}+3753820658157486080\right )}^2+{\left (14009449395540459520\,{\mathrm {e}}^{2\,x}-955607545932677120\,\sqrt {3}+8088359377641144320\,\sqrt {3}\,{\mathrm {e}}^{2\,x}-1655160823988879360\right )}^2\right )}{12}+\frac {\sqrt {2}\,\ln \left (17674880313941032960\,{\mathrm {e}}^{2\,x}-2144322552070144000\,\sqrt {2}+12498027726650736640\,\sqrt {2}\,{\mathrm {e}}^{2\,x}-3032530035220152320\right )}{12}-\frac {\sqrt {2}\,\ln \left (17674880313941032960\,{\mathrm {e}}^{2\,x}+2144322552070144000\,\sqrt {2}-12498027726650736640\,\sqrt {2}\,{\mathrm {e}}^{2\,x}-3032530035220152320\right )}{12}-\frac {\ln \left ({\mathrm {e}}^{2\,x}\,\left (14009449395540459520-6177144285775790080{}\mathrm {i}\right )+\sqrt {3}\,\left (955607545932677120+2167269359741829120{}\mathrm {i}\right )+\sqrt {3}\,{\mathrm {e}}^{2\,x}\,\left (-8088359377641144320+3566375915854233600{}\mathrm {i}\right )-1655160823988879360-3753820658157486080{}\mathrm {i}\right )\,1{}\mathrm {i}}{12}+\frac {\ln \left ({\mathrm {e}}^{2\,x}\,\left (14009449395540459520+6177144285775790080{}\mathrm {i}\right )+\sqrt {3}\,\left (955607545932677120-2167269359741829120{}\mathrm {i}\right )+\sqrt {3}\,{\mathrm {e}}^{2\,x}\,\left (-8088359377641144320-3566375915854233600{}\mathrm {i}\right )-1655160823988879360+3753820658157486080{}\mathrm {i}\right )\,1{}\mathrm {i}}{12} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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