Optimal. Leaf size=21 \[ \frac{1}{3} \log (\sinh (x))-\frac{1}{6} \log \left (4 \sinh ^2(x)+3\right ) \]
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Rubi [A] time = 0.0305192, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.714, Rules used = {4356, 266, 36, 29, 31} \[ \frac{1}{3} \log (\sinh (x))-\frac{1}{6} \log \left (4 \sinh ^2(x)+3\right ) \]
Antiderivative was successfully verified.
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Rule 4356
Rule 266
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \cosh (x) \text{csch}(3 x) \, dx &=\operatorname{Subst}\left (\int \frac{1}{x \left (3+4 x^2\right )} \, dx,x,\sinh (x)\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x (3+4 x)} \, dx,x,\sinh ^2(x)\right )\\ &=\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\sinh ^2(x)\right )-\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{3+4 x} \, dx,x,\sinh ^2(x)\right )\\ &=\frac{1}{3} \log (\sinh (x))-\frac{1}{6} \log \left (3+4 \sinh ^2(x)\right )\\ \end{align*}
Mathematica [A] time = 0.0092672, size = 21, normalized size = 1. \[ \frac{1}{3} \log (\sinh (x))-\frac{1}{6} \log \left (4 \sinh ^2(x)+3\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.038, size = 24, normalized size = 1.1 \begin{align*}{\frac{\ln \left ({{\rm e}^{2\,x}}-1 \right ) }{3}}-{\frac{\ln \left ({{\rm e}^{4\,x}}+{{\rm e}^{2\,x}}+1 \right ) }{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.63466, size = 63, normalized size = 3. \begin{align*} -\frac{1}{6} \, \log \left (e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right ) + \frac{1}{3} \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac{1}{3} \, \log \left (e^{\left (-x\right )} - 1\right ) - \frac{1}{6} \, \log \left (-e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.0435, size = 173, normalized size = 8.24 \begin{align*} -\frac{1}{6} \, \log \left (\frac{2 \, \cosh \left (x\right )^{2} + 2 \, \sinh \left (x\right )^{2} + 1}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}\right ) + \frac{1}{3} \, \log \left (\frac{2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\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 \cosh{\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 [B] time = 1.15222, size = 54, normalized size = 2.57 \begin{align*} -\frac{1}{6} \, \log \left (e^{\left (2 \, x\right )} + e^{x} + 1\right ) - \frac{1}{6} \, \log \left (e^{\left (2 \, x\right )} - e^{x} + 1\right ) + \frac{1}{3} \, \log \left (e^{x} + 1\right ) + \frac{1}{3} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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