Optimal. Leaf size=75 \[ \frac{1}{5} \sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \tan ^{-1}\left (\frac{\tanh (x)}{\sqrt{5-2 \sqrt{5}}}\right )-\frac{1}{5} \sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{\tanh (x)}{\sqrt{5+2 \sqrt{5}}}\right ) \]
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Rubi [A] time = 0.108097, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {1166, 203} \[ \frac{1}{5} \sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \tan ^{-1}\left (\frac{\tanh (x)}{\sqrt{5-2 \sqrt{5}}}\right )-\frac{1}{5} \sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{\tanh (x)}{\sqrt{5+2 \sqrt{5}}}\right ) \]
Antiderivative was successfully verified.
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Rule 1166
Rule 203
Rubi steps
\begin{align*} \int \text{csch}(5 x) \sinh (x) \, dx &=\operatorname{Subst}\left (\int \frac{1-x^2}{5+10 x^2+x^4} \, dx,x,\tanh (x)\right )\\ &=\frac{1}{10} \left (-5+3 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{5-2 \sqrt{5}+x^2} \, dx,x,\tanh (x)\right )-\frac{1}{10} \left (5+3 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{5+2 \sqrt{5}+x^2} \, dx,x,\tanh (x)\right )\\ &=\frac{1}{5} \sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \tan ^{-1}\left (\frac{\tanh (x)}{\sqrt{5-2 \sqrt{5}}}\right )-\frac{1}{5} \sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{\tanh (x)}{\sqrt{5+2 \sqrt{5}}}\right )\\ \end{align*}
Mathematica [A] time = 0.103277, size = 84, normalized size = 1.12 \[ \frac{\sqrt{5+\sqrt{5}} \tan ^{-1}\left (\frac{\left (\sqrt{5}-3\right ) \tanh (x)}{\sqrt{10-2 \sqrt{5}}}\right )+\sqrt{5-\sqrt{5}} \tan ^{-1}\left (\frac{\left (3+\sqrt{5}\right ) \tanh (x)}{\sqrt{2 \left (5+\sqrt{5}\right )}}\right )}{5 \sqrt{2}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.038, size = 41, normalized size = 0.6 \begin{align*} 2\,\sum _{{\it \_R}={\it RootOf} \left ( 32000\,{{\it \_Z}}^{4}+400\,{{\it \_Z}}^{2}+1 \right ) }{\it \_R}\,\ln \left ( 4000\,{{\it \_R}}^{3}-200\,{{\it \_R}}^{2}+{{\rm e}^{2\,x}}+30\,{\it \_R}-1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.35452, size = 558, normalized size = 7.44 \begin{align*} -\frac{1}{5} \, \sqrt{2} \sqrt{-\sqrt{5} + 5} \arctan \left (\frac{1}{40} \, \sqrt{5} \sqrt{2} \sqrt{-32 \,{\left (\sqrt{5} - 1\right )} e^{\left (2 \, x\right )} + 64 \, e^{\left (4 \, x\right )} + 64} \sqrt{-\sqrt{5} + 5} - \frac{1}{20} \,{\left (4 \, \sqrt{5} \sqrt{2} e^{\left (2 \, x\right )} + \sqrt{5} \sqrt{2} - 5 \, \sqrt{2}\right )} \sqrt{-\sqrt{5} + 5}\right ) + \frac{1}{5} \, \sqrt{2} \sqrt{\sqrt{5} + 5} \arctan \left (-\frac{1}{20} \,{\left (4 \, \sqrt{5} \sqrt{2} e^{\left (2 \, x\right )} + \sqrt{5} \sqrt{2} + 5 \, \sqrt{2}\right )} \sqrt{\sqrt{5} + 5} + \frac{1}{5} \, \sqrt{5} \sqrt{{\left (\sqrt{5} + 1\right )} e^{\left (2 \, x\right )} + 2 \, e^{\left (4 \, x\right )} + 2} \sqrt{\sqrt{5} + 5}\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 (5 x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18576, size = 92, normalized size = 1.23 \begin{align*} \frac{1}{10} \, \sqrt{-2 \, \sqrt{5} + 10} \arctan \left (-\frac{\sqrt{5} - 4 \, e^{\left (2 \, x\right )} - 1}{\sqrt{2 \, \sqrt{5} + 10}}\right ) - \frac{1}{10} \, \sqrt{2 \, \sqrt{5} + 10} \arctan \left (\frac{\sqrt{5} + 4 \, e^{\left (2 \, x\right )} + 1}{\sqrt{-2 \, \sqrt{5} + 10}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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