Optimal. Leaf size=15 \[ \frac {\tan ^{-1}\left (\frac {\tanh (x)}{\sqrt {3}}\right )}{\sqrt {3}} \]
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Rubi [A] time = 0.04, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {203} \[ \frac {\tan ^{-1}\left (\frac {\tanh (x)}{\sqrt {3}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 203
Rubi steps
\begin {align*} \int \text {csch}(3 x) \sinh (x) \, dx &=\operatorname {Subst}\left (\int \frac {1}{3+x^2} \, dx,x,\tanh (x)\right )\\ &=\frac {\tan ^{-1}\left (\frac {\tanh (x)}{\sqrt {3}}\right )}{\sqrt {3}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 44, normalized size = 2.93 \[ -\frac {1}{4} e^{2 x} \left (e^{2 x} \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};e^{6 x}\right )-2 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};e^{6 x}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 31, normalized size = 2.07 \[ -\frac {1}{3} \, \sqrt {3} \arctan \left (-\frac {3 \, \sqrt {3} \cosh \relax (x) + \sqrt {3} \sinh \relax (x)}{3 \, {\left (\cosh \relax (x) - \sinh \relax (x)\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 19, normalized size = 1.27 \[ \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{\left (2 \, x\right )} + 1\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.23, size = 40, normalized size = 2.67 \[ \frac {i \ln \left ({\mathrm e}^{2 x}+\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{6}-\frac {i \ln \left ({\mathrm e}^{2 x}+\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 39, normalized size = 2.60 \[ \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{\left (-x\right )} + 1\right )}\right ) - \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{\left (-x\right )} - 1\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.46, size = 19, normalized size = 1.27 \[ \frac {\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,\left (2\,{\mathrm {e}}^{2\,x}+1\right )}{3}\right )}{3} \]
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 \[ \int \sinh {\relax (x )} \operatorname {csch}{\left (3 x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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