Optimal. Leaf size=15 \[ \frac{\tan ^{-1}\left (\frac{\tanh (x)}{\sqrt{3}}\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.0362462, antiderivative size = 15, normalized size of antiderivative = 1., 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*}
Mathematica [C] time = 0.0241455, 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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Maple [C] time = 0.033, size = 40, normalized size = 2.7 \begin{align*}{\frac{i}{6}}\sqrt{3}\ln \left ({{\rm e}^{2\,x}}+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) -{\frac{i}{6}}\sqrt{3}\ln \left ({{\rm e}^{2\,x}}+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.53863, size = 53, normalized size = 3.53 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.12863, size = 115, normalized size = 7.67 \begin{align*} -\frac{1}{3} \, \sqrt{3} \arctan \left (-\frac{3 \, \sqrt{3} \cosh \left (x\right ) + \sqrt{3} \sinh \left (x\right )}{3 \,{\left (\cosh \left (x\right ) - \sinh \left (x\right )\right )}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sinh{\left (x \right )} \operatorname{csch}{\left (3 x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20302, size = 26, normalized size = 1.73 \begin{align*} \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, e^{\left (2 \, x\right )} + 1\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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