Optimal. Leaf size=105 \[ \frac {2 e^x}{3 \left (1-e^{6 x}\right )}+\frac {\text {ArcTan}\left (\frac {1-2 e^x}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\text {ArcTan}\left (\frac {1+2 e^x}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {2}{9} \tanh ^{-1}\left (e^x\right )+\frac {1}{18} \log \left (1-e^x+e^{2 x}\right )-\frac {1}{18} \log \left (1+e^x+e^{2 x}\right ) \]
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Rubi [A]
time = 0.10, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 9, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {2320, 12, 294,
216, 648, 632, 210, 642, 212} \begin {gather*} \frac {\text {ArcTan}\left (\frac {1-2 e^x}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\text {ArcTan}\left (\frac {2 e^x+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2 e^x}{3 \left (1-e^{6 x}\right )}+\frac {1}{18} \log \left (-e^x+e^{2 x}+1\right )-\frac {1}{18} \log \left (e^x+e^{2 x}+1\right )-\frac {2}{9} \tanh ^{-1}\left (e^x\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 210
Rule 212
Rule 216
Rule 294
Rule 632
Rule 642
Rule 648
Rule 2320
Rubi steps
\begin {align*} \int e^x \text {csch}^2(3 x) \, dx &=\text {Subst}\left (\int \frac {4 x^6}{\left (1-x^6\right )^2} \, dx,x,e^x\right )\\ &=4 \text {Subst}\left (\int \frac {x^6}{\left (1-x^6\right )^2} \, dx,x,e^x\right )\\ &=\frac {2 e^x}{3 \left (1-e^{6 x}\right )}-\frac {2}{3} \text {Subst}\left (\int \frac {1}{1-x^6} \, dx,x,e^x\right )\\ &=\frac {2 e^x}{3 \left (1-e^{6 x}\right )}-\frac {2}{9} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,e^x\right )-\frac {2}{9} \text {Subst}\left (\int \frac {1-\frac {x}{2}}{1-x+x^2} \, dx,x,e^x\right )-\frac {2}{9} \text {Subst}\left (\int \frac {1+\frac {x}{2}}{1+x+x^2} \, dx,x,e^x\right )\\ &=\frac {2 e^x}{3 \left (1-e^{6 x}\right )}-\frac {2}{9} \tanh ^{-1}\left (e^x\right )+\frac {1}{18} \text {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,e^x\right )-\frac {1}{18} \text {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,e^x\right )-\frac {1}{6} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,e^x\right )-\frac {1}{6} \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,e^x\right )\\ &=\frac {2 e^x}{3 \left (1-e^{6 x}\right )}-\frac {2}{9} \tanh ^{-1}\left (e^x\right )+\frac {1}{18} \log \left (1-e^x+e^{2 x}\right )-\frac {1}{18} \log \left (1+e^x+e^{2 x}\right )+\frac {1}{3} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 e^x\right )+\frac {1}{3} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 e^x\right )\\ &=\frac {2 e^x}{3 \left (1-e^{6 x}\right )}+\frac {\tan ^{-1}\left (\frac {1-2 e^x}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {1+2 e^x}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {2}{9} \tanh ^{-1}\left (e^x\right )+\frac {1}{18} \log \left (1-e^x+e^{2 x}\right )-\frac {1}{18} \log \left (1+e^x+e^{2 x}\right )\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.02, size = 34, normalized size = 0.32 \begin {gather*} \frac {2}{3} e^x \left (\frac {1}{1-e^{6 x}}-\, _2F_1\left (\frac {1}{6},1;\frac {7}{6};e^{6 x}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.66, size = 148, normalized size = 1.41
method | result | size |
risch | \(-\frac {2 \,{\mathrm e}^{x}}{3 \left ({\mathrm e}^{6 x}-1\right )}+\frac {\ln \left ({\mathrm e}^{x}-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{18}+\frac {i \ln \left ({\mathrm e}^{x}-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{18}+\frac {\ln \left ({\mathrm e}^{x}-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{18}-\frac {i \ln \left ({\mathrm e}^{x}-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{18}+\frac {\ln \left ({\mathrm e}^{x}-1\right )}{9}-\frac {\ln \left ({\mathrm e}^{x}+1\right )}{9}-\frac {\ln \left ({\mathrm e}^{x}+\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{18}+\frac {i \ln \left ({\mathrm e}^{x}+\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{18}-\frac {\ln \left ({\mathrm e}^{x}+\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{18}-\frac {i \ln \left ({\mathrm e}^{x}+\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{18}\) | \(148\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 85, normalized size = 0.81 \begin {gather*} -\frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{x} + 1\right )}\right ) - \frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{x} - 1\right )}\right ) - \frac {2 \, e^{x}}{3 \, {\left (e^{\left (6 \, x\right )} - 1\right )}} - \frac {1}{18} \, \log \left (e^{\left (2 \, x\right )} + e^{x} + 1\right ) + \frac {1}{18} \, \log \left (e^{\left (2 \, x\right )} - e^{x} + 1\right ) - \frac {1}{9} \, \log \left (e^{x} + 1\right ) + \frac {1}{9} \, \log \left (e^{x} - 1\right ) \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. 560 vs.
\(2 (76) = 152\).
time = 0.48, size = 560, normalized size = 5.33 \begin {gather*} -\frac {2 \, {\left (\sqrt {3} \cosh \left (x\right )^{6} + 6 \, \sqrt {3} \cosh \left (x\right )^{5} \sinh \left (x\right ) + 15 \, \sqrt {3} \cosh \left (x\right )^{4} \sinh \left (x\right )^{2} + 20 \, \sqrt {3} \cosh \left (x\right )^{3} \sinh \left (x\right )^{3} + 15 \, \sqrt {3} \cosh \left (x\right )^{2} \sinh \left (x\right )^{4} + 6 \, \sqrt {3} \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sqrt {3} \sinh \left (x\right )^{6} - \sqrt {3}\right )} \arctan \left (\frac {2}{3} \, \sqrt {3} \cosh \left (x\right ) + \frac {2}{3} \, \sqrt {3} \sinh \left (x\right ) + \frac {1}{3} \, \sqrt {3}\right ) + 2 \, {\left (\sqrt {3} \cosh \left (x\right )^{6} + 6 \, \sqrt {3} \cosh \left (x\right )^{5} \sinh \left (x\right ) + 15 \, \sqrt {3} \cosh \left (x\right )^{4} \sinh \left (x\right )^{2} + 20 \, \sqrt {3} \cosh \left (x\right )^{3} \sinh \left (x\right )^{3} + 15 \, \sqrt {3} \cosh \left (x\right )^{2} \sinh \left (x\right )^{4} + 6 \, \sqrt {3} \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sqrt {3} \sinh \left (x\right )^{6} - \sqrt {3}\right )} \arctan \left (\frac {2}{3} \, \sqrt {3} \cosh \left (x\right ) + \frac {2}{3} \, \sqrt {3} \sinh \left (x\right ) - \frac {1}{3} \, \sqrt {3}\right ) + {\left (\cosh \left (x\right )^{6} + 6 \, \cosh \left (x\right )^{5} \sinh \left (x\right ) + 15 \, \cosh \left (x\right )^{4} \sinh \left (x\right )^{2} + 20 \, \cosh \left (x\right )^{3} \sinh \left (x\right )^{3} + 15 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{4} + 6 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6} - 1\right )} \log \left (\frac {2 \, \cosh \left (x\right ) + 1}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) - {\left (\cosh \left (x\right )^{6} + 6 \, \cosh \left (x\right )^{5} \sinh \left (x\right ) + 15 \, \cosh \left (x\right )^{4} \sinh \left (x\right )^{2} + 20 \, \cosh \left (x\right )^{3} \sinh \left (x\right )^{3} + 15 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{4} + 6 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6} - 1\right )} \log \left (\frac {2 \, \cosh \left (x\right ) - 1}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 2 \, {\left (\cosh \left (x\right )^{6} + 6 \, \cosh \left (x\right )^{5} \sinh \left (x\right ) + 15 \, \cosh \left (x\right )^{4} \sinh \left (x\right )^{2} + 20 \, \cosh \left (x\right )^{3} \sinh \left (x\right )^{3} + 15 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{4} + 6 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6} - 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) - 2 \, {\left (\cosh \left (x\right )^{6} + 6 \, \cosh \left (x\right )^{5} \sinh \left (x\right ) + 15 \, \cosh \left (x\right )^{4} \sinh \left (x\right )^{2} + 20 \, \cosh \left (x\right )^{3} \sinh \left (x\right )^{3} + 15 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{4} + 6 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6} - 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) + 12 \, \cosh \left (x\right ) + 12 \, \sinh \left (x\right )}{18 \, {\left (\cosh \left (x\right )^{6} + 6 \, \cosh \left (x\right )^{5} \sinh \left (x\right ) + 15 \, \cosh \left (x\right )^{4} \sinh \left (x\right )^{2} + 20 \, \cosh \left (x\right )^{3} \sinh \left (x\right )^{3} + 15 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{4} + 6 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6} - 1\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 e^{x} \operatorname {csch}^{2}{\left (3 x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 86, normalized size = 0.82 \begin {gather*} -\frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{x} + 1\right )}\right ) - \frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{x} - 1\right )}\right ) - \frac {2 \, e^{x}}{3 \, {\left (e^{\left (6 \, x\right )} - 1\right )}} - \frac {1}{18} \, \log \left (e^{\left (2 \, x\right )} + e^{x} + 1\right ) + \frac {1}{18} \, \log \left (e^{\left (2 \, x\right )} - e^{x} + 1\right ) - \frac {1}{9} \, \log \left (e^{x} + 1\right ) + \frac {1}{9} \, \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 = 0.40, size = 91, normalized size = 0.87 \begin {gather*} \frac {\ln \left (\frac {2}{3}-\frac {2\,{\mathrm {e}}^x}{3}\right )}{9}-\frac {\ln \left (-\frac {2\,{\mathrm {e}}^x}{3}-\frac {2}{3}\right )}{9}+\frac {\ln \left ({\left (\frac {2\,{\mathrm {e}}^x}{3}-\frac {1}{3}\right )}^2+\frac {1}{3}\right )}{18}-\frac {\ln \left ({\left (\frac {2\,{\mathrm {e}}^x}{3}+\frac {1}{3}\right )}^2+\frac {1}{3}\right )}{18}-\frac {2\,{\mathrm {e}}^x}{3\,\left ({\mathrm {e}}^{6\,x}-1\right )}-\frac {\sqrt {3}\,\mathrm {atan}\left (\sqrt {3}\,\left (\frac {2\,{\mathrm {e}}^x}{3}-\frac {1}{3}\right )\right )}{9}-\frac {\sqrt {3}\,\mathrm {atan}\left (\sqrt {3}\,\left (\frac {2\,{\mathrm {e}}^x}{3}+\frac {1}{3}\right )\right )}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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