Optimal. Leaf size=15 \[ \frac {\text {ArcTan}\left (\frac {\tanh (x)}{\sqrt {3}}\right )}{\sqrt {3}} \]
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Rubi [A]
time = 0.02, 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 = {209}
\begin {gather*} \frac {\text {ArcTan}\left (\frac {\tanh (x)}{\sqrt {3}}\right )}{\sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rubi steps
\begin {align*} \int \text {csch}(3 x) \sinh (x) \, dx &=\text {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] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.02, size = 44, normalized size = 2.93 \begin {gather*} -\frac {1}{4} e^{2 x} \left (-2 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};e^{6 x}\right )+e^{2 x} \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};e^{6 x}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.88, size = 40, normalized size = 2.67
method | result | size |
risch | \(\frac {i \sqrt {3}\, \ln \left ({\mathrm e}^{2 x}+\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{6}-\frac {i \sqrt {3}\, \ln \left ({\mathrm e}^{2 x}+\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{6}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 39 vs.
\(2 (13) = 26\).
time = 0.48, size = 39, normalized size = 2.60 \begin {gather*} \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 {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. 31 vs.
\(2 (13) = 26\).
time = 0.36, size = 31, normalized size = 2.07 \begin {gather*} -\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 {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 (3 x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 19, normalized size = 1.27 \begin {gather*} \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{\left (2 \, x\right )} + 1\right )}\right ) \end {gather*}
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 \begin {gather*} \frac {\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,\left (2\,{\mathrm {e}}^{2\,x}+1\right )}{3}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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