Optimal. Leaf size=25 \[ \frac {\tanh ^{-1}\left (\sqrt {2} \tanh (x)\right )}{2 \sqrt {2}}+\frac {\tanh (x)}{2} \]
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Rubi [A] time = 0.02, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3209, 388, 206} \[ \frac {\tanh ^{-1}\left (\sqrt {2} \tanh (x)\right )}{2 \sqrt {2}}+\frac {\tanh (x)}{2} \]
Antiderivative was successfully verified.
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Rule 206
Rule 388
Rule 3209
Rubi steps
\begin {align*} \int \frac {1}{1-\sinh ^4(x)} \, dx &=\operatorname {Subst}\left (\int \frac {1-x^2}{1-2 x^2} \, dx,x,\tanh (x)\right )\\ &=\frac {\tanh (x)}{2}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\tanh (x)\right )\\ &=\frac {\tanh ^{-1}\left (\sqrt {2} \tanh (x)\right )}{2 \sqrt {2}}+\frac {\tanh (x)}{2}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 24, normalized size = 0.96 \[ \frac {1}{4} \left (\sqrt {2} \tanh ^{-1}\left (\sqrt {2} \tanh (x)\right )+2 \tanh (x)\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{1-\sinh ^4(x)} \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 1.32, size = 113, normalized size = 4.52 \[ \frac {{\left (\sqrt {2} \cosh \relax (x)^{2} + 2 \, \sqrt {2} \cosh \relax (x) \sinh \relax (x) + \sqrt {2} \sinh \relax (x)^{2} + \sqrt {2}\right )} \log \left (-\frac {3 \, {\left (2 \, \sqrt {2} - 3\right )} \cosh \relax (x)^{2} - 4 \, {\left (3 \, \sqrt {2} - 4\right )} \cosh \relax (x) \sinh \relax (x) + 3 \, {\left (2 \, \sqrt {2} - 3\right )} \sinh \relax (x)^{2} - 2 \, \sqrt {2} + 3}{\cosh \relax (x)^{2} + \sinh \relax (x)^{2} - 3}\right ) - 8}{8 \, {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.59, size = 48, normalized size = 1.92 \[ -\frac {1}{8} \, \sqrt {2} \log \left (\frac {{\left | -4 \, \sqrt {2} + 2 \, e^{\left (2 \, x\right )} - 6 \right |}}{{\left | 4 \, \sqrt {2} + 2 \, e^{\left (2 \, x\right )} - 6 \right |}}\right ) - \frac {1}{e^{\left (2 \, x\right )} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 46, normalized size = 1.84
method | result | size |
risch | \(-\frac {1}{1+{\mathrm e}^{2 x}}+\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 x}-3+2 \sqrt {2}\right )}{8}-\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 x}-3-2 \sqrt {2}\right )}{8}\) | \(46\) |
default | \(\frac {\sqrt {2}\, \arctanh \left (\frac {\left (2 \tanh \left (\frac {x}{2}\right )+2\right ) \sqrt {2}}{4}\right )}{4}+\frac {\tanh \left (\frac {x}{2}\right )}{\tanh ^{2}\left (\frac {x}{2}\right )+1}+\frac {\sqrt {2}\, \arctanh \left (\frac {\left (2 \tanh \left (\frac {x}{2}\right )-2\right ) \sqrt {2}}{4}\right )}{4}\) | \(55\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.96, size = 69, normalized size = 2.76 \[ \frac {1}{8} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - e^{\left (-x\right )} + 1}{\sqrt {2} + e^{\left (-x\right )} - 1}\right ) - \frac {1}{8} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - e^{\left (-x\right )} - 1}{\sqrt {2} + e^{\left (-x\right )} + 1}\right ) + \frac {1}{e^{\left (-2 \, x\right )} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.40, size = 63, normalized size = 2.52 \[ \frac {\sqrt {2}\,\ln \left (2\,{\mathrm {e}}^{2\,x}+\frac {\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}-4\right )}{8}\right )}{8}-\frac {\sqrt {2}\,\ln \left (2\,{\mathrm {e}}^{2\,x}-\frac {\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}-4\right )}{8}\right )}{8}-\frac {1}{{\mathrm {e}}^{2\,x}+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 7.44, size = 908, normalized size = 36.32 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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