Optimal. Leaf size=75 \[ \frac {1}{5} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \text {ArcTan}\left (\frac {\tanh (x)}{\sqrt {5-2 \sqrt {5}}}\right )-\frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \text {ArcTan}\left (\frac {\tanh (x)}{\sqrt {5+2 \sqrt {5}}}\right ) \]
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Rubi [A]
time = 0.09, antiderivative size = 75, normalized size of antiderivative = 1.00, 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 = {1180, 209}
\begin {gather*} \frac {1}{5} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \text {ArcTan}\left (\frac {\tanh (x)}{\sqrt {5-2 \sqrt {5}}}\right )-\frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \text {ArcTan}\left (\frac {\tanh (x)}{\sqrt {5+2 \sqrt {5}}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 1180
Rubi steps
\begin {align*} \int \text {csch}(5 x) \sinh (x) \, dx &=\text {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 ) \text {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 ) \text {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*}
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Mathematica [A]
time = 0.08, size = 84, normalized size = 1.12 \begin {gather*} \frac {\sqrt {5+\sqrt {5}} \text {ArcTan}\left (\frac {\left (-3+\sqrt {5}\right ) \tanh (x)}{\sqrt {10-2 \sqrt {5}}}\right )+\sqrt {5-\sqrt {5}} \text {ArcTan}\left (\frac {\left (3+\sqrt {5}\right ) \tanh (x)}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )}{5 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 1.06, size = 41, normalized size = 0.55
method | result | size |
risch | \(2 \left (\munderset {\textit {\_R} =\RootOf \left (32000 \textit {\_Z}^{4}+400 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (4000 \textit {\_R}^{3}-200 \textit {\_R}^{2}+{\mathrm e}^{2 x}+30 \textit {\_R} -1\right )\right )\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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. 171 vs.
\(2 (49) = 98\).
time = 0.43, size = 171, normalized size = 2.28 \begin {gather*} -\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 {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 \sinh {\left (x \right )} \operatorname {csch}{\left (5 x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 68, normalized size = 0.91 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.26, size = 282, normalized size = 3.76 \begin {gather*} 2\,\mathrm {atan}\left (\frac {\frac {{\mathrm {e}}^{2\,x}}{5}+\frac {9\,\sqrt {5}}{25}+\frac {6\,\sqrt {5}\,{\mathrm {e}}^{2\,x}}{25}+\frac {4}{5}}{5\,{\mathrm {e}}^{2\,x}\,\sqrt {\frac {\sqrt {5}}{200}+\frac {1}{40}}+\frac {9\,\sqrt {5}\,\sqrt {\frac {\sqrt {5}}{200}+\frac {1}{40}}}{5}+\sqrt {\frac {\sqrt {5}}{200}+\frac {1}{40}}+\frac {9\,\sqrt {5}\,{\mathrm {e}}^{2\,x}\,\sqrt {\frac {\sqrt {5}}{200}+\frac {1}{40}}}{5}}\right )\,\sqrt {\frac {\sqrt {5}}{200}+\frac {1}{40}}+\sqrt {\frac {1}{40}-\frac {\sqrt {5}}{200}}\,\left (\ln \left (\frac {9\,\sqrt {5}}{25}-\frac {{\mathrm {e}}^{2\,x}}{5}+\frac {6\,\sqrt {5}\,{\mathrm {e}}^{2\,x}}{25}-\frac {4}{5}-{\mathrm {e}}^{2\,x}\,\sqrt {\frac {1}{40}-\frac {\sqrt {5}}{200}}\,5{}\mathrm {i}+\frac {\sqrt {5}\,\sqrt {\frac {1}{40}-\frac {\sqrt {5}}{200}}\,9{}\mathrm {i}}{5}-\sqrt {\frac {1}{40}-\frac {\sqrt {5}}{200}}\,1{}\mathrm {i}+\frac {\sqrt {5}\,{\mathrm {e}}^{2\,x}\,\sqrt {\frac {1}{40}-\frac {\sqrt {5}}{200}}\,9{}\mathrm {i}}{5}\right )\,1{}\mathrm {i}-\ln \left (\frac {9\,\sqrt {5}}{25}-\frac {{\mathrm {e}}^{2\,x}}{5}+\frac {6\,\sqrt {5}\,{\mathrm {e}}^{2\,x}}{25}-\frac {4}{5}+{\mathrm {e}}^{2\,x}\,\sqrt {\frac {1}{40}-\frac {\sqrt {5}}{200}}\,5{}\mathrm {i}-\frac {\sqrt {5}\,\sqrt {\frac {1}{40}-\frac {\sqrt {5}}{200}}\,9{}\mathrm {i}}{5}+\sqrt {\frac {1}{40}-\frac {\sqrt {5}}{200}}\,1{}\mathrm {i}-\frac {\sqrt {5}\,{\mathrm {e}}^{2\,x}\,\sqrt {\frac {1}{40}-\frac {\sqrt {5}}{200}}\,9{}\mathrm {i}}{5}\right )\,1{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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