Optimal. Leaf size=71 \[ \frac {\tanh ^{-1}\left (\sqrt {1+\sqrt [3]{-1}} \tanh (x)\right )}{3 \sqrt {1+\sqrt [3]{-1}}}+\frac {\tanh ^{-1}\left (\sqrt {1-(-1)^{2/3}} \tanh (x)\right )}{3 \sqrt {1-(-1)^{2/3}}}+\frac {\tanh (x)}{3} \]
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Rubi [A]
time = 0.09, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3290, 3260, 212,
3254, 3852, 8} \begin {gather*} \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}}}+\frac {\tanh (x)}{3} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 212
Rule 3254
Rule 3260
Rule 3290
Rule 3852
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} \int \text {sech}^2(x) \, dx+\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 {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}}}+\frac {1}{3} i \text {Subst}(\int 1 \, dx,x,-i \tanh (x))\\ &=\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}}}+\frac {\tanh (x)}{3}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.15, size = 87, normalized size = 1.23 \begin {gather*} \frac {1}{18} \left (\sqrt [4]{-3} \left (\left (-3-i \sqrt {3}\right ) \text {ArcTan}\left (\frac {1}{2} \sqrt [4]{-3} \left (1+i \sqrt {3}\right ) \tanh (x)\right )-\left (3 i+\sqrt {3}\right ) \text {ArcTan}\left (\frac {1}{2} \sqrt [4]{-\frac {1}{3}} \left (3+i \sqrt {3}\right ) \tanh (x)\right )\right )+6 \tanh (x)\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 = 0.59, size = 61, normalized size = 0.86
method | result | size |
risch | \(-\frac {2}{3 \left (1+{\mathrm e}^{2 x}\right )}+\left (\munderset {\textit {\_R} =\RootOf \left (3888 \textit {\_Z}^{4}-108 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (1296 \textit {\_R}^{3}-216 \textit {\_R}^{2}+{\mathrm e}^{2 x}+1\right )\right )\) | \(47\) |
default | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (3 \textit {\_Z}^{4}-3 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (\tanh ^{2}\left (\frac {x}{2}\right )+\left (-6 \textit {\_R}^{3}+6 \textit {\_R} \right ) \tanh \left (\frac {x}{2}\right )+1\right )\right )}{6}+\frac {2 \tanh \left (\frac {x}{2}\right )}{3 \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )}\) | \(61\) |
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. 692 vs.
\(2 (49) = 98\).
time = 0.41, size = 692, normalized size = 9.75 \begin {gather*} -\frac {4 \, {\left (12^{\frac {1}{4}} \sqrt {6} e^{\left (2 \, x\right )} + 12^{\frac {1}{4}} \sqrt {6}\right )} \sqrt {-4 \, \sqrt {3} + 8} \arctan \left ({\left (\sqrt {3} + 2\right )} e^{\left (2 \, x\right )} + \frac {1}{216} \, \sqrt {-6 \, {\left (12^{\frac {1}{4}} \sqrt {6} {\left (\sqrt {3} + 3\right )} e^{\left (2 \, x\right )} - 12^{\frac {1}{4}} \sqrt {6} {\left (5 \, \sqrt {3} + 9\right )}\right )} \sqrt {-4 \, \sqrt {3} + 8} + 144 \, \sqrt {3} + 36 \, e^{\left (4 \, x\right )} - 144 \, e^{\left (2 \, x\right )} + 252} {\left ({\left (12^{\frac {3}{4}} \sqrt {6} {\left (\sqrt {3} + 3\right )} + 3 \cdot 12^{\frac {1}{4}} \sqrt {6} {\left (\sqrt {3} + 3\right )}\right )} \sqrt {-4 \, \sqrt {3} + 8} - 36 \, \sqrt {3} - 72\right )} - \frac {2}{3} \, \sqrt {3} {\left (2 \, \sqrt {3} - 3\right )} - \frac {1}{36} \, {\left (12^{\frac {3}{4}} \sqrt {6} {\left (\sqrt {3} - 3\right )} + {\left (12^{\frac {3}{4}} \sqrt {6} {\left (\sqrt {3} + 3\right )} + 3 \cdot 12^{\frac {1}{4}} \sqrt {6} {\left (\sqrt {3} + 3\right )}\right )} e^{\left (2 \, x\right )} + 3 \cdot 12^{\frac {1}{4}} \sqrt {6} {\left (\sqrt {3} - 3\right )}\right )} \sqrt {-4 \, \sqrt {3} + 8} - 2 \, \sqrt {3} + 4\right ) + 4 \, {\left (12^{\frac {1}{4}} \sqrt {6} e^{\left (2 \, x\right )} + 12^{\frac {1}{4}} \sqrt {6}\right )} \sqrt {-4 \, \sqrt {3} + 8} \arctan \left (-{\left (\sqrt {3} + 2\right )} e^{\left (2 \, x\right )} + \frac {1}{216} \, \sqrt {6 \, {\left (12^{\frac {1}{4}} \sqrt {6} {\left (\sqrt {3} + 3\right )} e^{\left (2 \, x\right )} - 12^{\frac {1}{4}} \sqrt {6} {\left (5 \, \sqrt {3} + 9\right )}\right )} \sqrt {-4 \, \sqrt {3} + 8} + 144 \, \sqrt {3} + 36 \, e^{\left (4 \, x\right )} - 144 \, e^{\left (2 \, x\right )} + 252} {\left ({\left (12^{\frac {3}{4}} \sqrt {6} {\left (\sqrt {3} + 3\right )} + 3 \cdot 12^{\frac {1}{4}} \sqrt {6} {\left (\sqrt {3} + 3\right )}\right )} \sqrt {-4 \, \sqrt {3} + 8} + 36 \, \sqrt {3} + 72\right )} + \frac {2}{3} \, \sqrt {3} {\left (2 \, \sqrt {3} - 3\right )} - \frac {1}{36} \, {\left (12^{\frac {3}{4}} \sqrt {6} {\left (\sqrt {3} - 3\right )} + {\left (12^{\frac {3}{4}} \sqrt {6} {\left (\sqrt {3} + 3\right )} + 3 \cdot 12^{\frac {1}{4}} \sqrt {6} {\left (\sqrt {3} + 3\right )}\right )} e^{\left (2 \, x\right )} + 3 \cdot 12^{\frac {1}{4}} \sqrt {6} {\left (\sqrt {3} - 3\right )}\right )} \sqrt {-4 \, \sqrt {3} + 8} + 2 \, \sqrt {3} - 4\right ) - {\left (12^{\frac {1}{4}} \sqrt {6} {\left (\sqrt {3} + 2\right )} e^{\left (2 \, x\right )} + 12^{\frac {1}{4}} \sqrt {6} {\left (\sqrt {3} + 2\right )}\right )} \sqrt {-4 \, \sqrt {3} + 8} \log \left (6 \, {\left (12^{\frac {1}{4}} \sqrt {6} {\left (\sqrt {3} + 3\right )} e^{\left (2 \, x\right )} - 12^{\frac {1}{4}} \sqrt {6} {\left (5 \, \sqrt {3} + 9\right )}\right )} \sqrt {-4 \, \sqrt {3} + 8} + 144 \, \sqrt {3} + 36 \, e^{\left (4 \, x\right )} - 144 \, e^{\left (2 \, x\right )} + 252\right ) + {\left (12^{\frac {1}{4}} \sqrt {6} {\left (\sqrt {3} + 2\right )} e^{\left (2 \, x\right )} + 12^{\frac {1}{4}} \sqrt {6} {\left (\sqrt {3} + 2\right )}\right )} \sqrt {-4 \, \sqrt {3} + 8} \log \left (-6 \, {\left (12^{\frac {1}{4}} \sqrt {6} {\left (\sqrt {3} + 3\right )} e^{\left (2 \, x\right )} - 12^{\frac {1}{4}} \sqrt {6} {\left (5 \, \sqrt {3} + 9\right )}\right )} \sqrt {-4 \, \sqrt {3} + 8} + 144 \, \sqrt {3} + 36 \, e^{\left (4 \, x\right )} - 144 \, e^{\left (2 \, x\right )} + 252\right ) + 96}{144 \, {\left (e^{\left (2 \, x\right )} + 1\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 [A]
time = 0.41, size = 10, normalized size = 0.14 \begin {gather*} -\frac {2}{3 \, {\left (e^{\left (2 \, x\right )} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.21, size = 325, normalized size = 4.58 \begin {gather*} -\ln \left (\frac {1061158912\,{\mathrm {e}}^{2\,x}}{27}+\sqrt {\frac {1}{72}-\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}\,\left (\frac {548405248}{27}+\sqrt {\frac {1}{72}-\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}\,\left (\frac {3870294016}{9}+\sqrt {\frac {1}{72}-\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}\,\left (19788726272\,{\mathrm {e}}^{2\,x}-2864709632\right )-\frac {21515730944\,{\mathrm {e}}^{2\,x}}{9}\right )-\frac {2539651072\,{\mathrm {e}}^{2\,x}}{9}\right )-\frac {351797248}{81}\right )\,\sqrt {\frac {1}{72}-\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}-\ln \left (\frac {1061158912\,{\mathrm {e}}^{2\,x}}{27}+\sqrt {\frac {1}{72}+\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}\,\left (\frac {548405248}{27}+\sqrt {\frac {1}{72}+\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}\,\left (\frac {3870294016}{9}+\sqrt {\frac {1}{72}+\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}\,\left (19788726272\,{\mathrm {e}}^{2\,x}-2864709632\right )-\frac {21515730944\,{\mathrm {e}}^{2\,x}}{9}\right )-\frac {2539651072\,{\mathrm {e}}^{2\,x}}{9}\right )-\frac {351797248}{81}\right )\,\sqrt {\frac {1}{72}+\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}+\ln \left (\frac {1061158912\,{\mathrm {e}}^{2\,x}}{27}-\sqrt {\frac {1}{72}-\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}\,\left (\frac {548405248}{27}+\sqrt {\frac {1}{72}-\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}\,\left (\frac {21515730944\,{\mathrm {e}}^{2\,x}}{9}+\sqrt {\frac {1}{72}-\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}\,\left (19788726272\,{\mathrm {e}}^{2\,x}-2864709632\right )-\frac {3870294016}{9}\right )-\frac {2539651072\,{\mathrm {e}}^{2\,x}}{9}\right )-\frac {351797248}{81}\right )\,\sqrt {\frac {1}{72}-\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}+\ln \left (\frac {1061158912\,{\mathrm {e}}^{2\,x}}{27}-\sqrt {\frac {1}{72}+\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}\,\left (\frac {548405248}{27}+\sqrt {\frac {1}{72}+\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}\,\left (\frac {21515730944\,{\mathrm {e}}^{2\,x}}{9}+\sqrt {\frac {1}{72}+\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}\,\left (19788726272\,{\mathrm {e}}^{2\,x}-2864709632\right )-\frac {3870294016}{9}\right )-\frac {2539651072\,{\mathrm {e}}^{2\,x}}{9}\right )-\frac {351797248}{81}\right )\,\sqrt {\frac {1}{72}+\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}-\frac {2}{3\,\left ({\mathrm {e}}^{2\,x}+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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