Optimal. Leaf size=57 \[ -2 \coth (x) \sqrt {\tanh ^3(x)}+\frac {\text {ArcTan}\left (\sqrt {\tanh (x)}\right ) \sqrt {\tanh ^3(x)}}{\tanh ^{\frac {3}{2}}(x)}+\frac {\tanh ^{-1}\left (\sqrt {\tanh (x)}\right ) \sqrt {\tanh ^3(x)}}{\tanh ^{\frac {3}{2}}(x)} \]
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Rubi [A]
time = 0.03, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {3739, 3554,
3557, 335, 218, 212, 209} \begin {gather*} \frac {\sqrt {\tanh ^3(x)} \text {ArcTan}\left (\sqrt {\tanh (x)}\right )}{\tanh ^{\frac {3}{2}}(x)}+\frac {\tanh ^{-1}\left (\sqrt {\tanh (x)}\right ) \sqrt {\tanh ^3(x)}}{\tanh ^{\frac {3}{2}}(x)}-2 \sqrt {\tanh ^3(x)} \coth (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 218
Rule 335
Rule 3554
Rule 3557
Rule 3739
Rubi steps
\begin {align*} \int \sqrt {\tanh ^3(x)} \, dx &=\frac {\sqrt {\tanh ^3(x)} \int \tanh ^{\frac {3}{2}}(x) \, dx}{\tanh ^{\frac {3}{2}}(x)}\\ &=-2 \coth (x) \sqrt {\tanh ^3(x)}+\frac {\sqrt {\tanh ^3(x)} \int \frac {1}{\sqrt {\tanh (x)}} \, dx}{\tanh ^{\frac {3}{2}}(x)}\\ &=-2 \coth (x) \sqrt {\tanh ^3(x)}-\frac {\sqrt {\tanh ^3(x)} \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (-1+x^2\right )} \, dx,x,\tanh (x)\right )}{\tanh ^{\frac {3}{2}}(x)}\\ &=-2 \coth (x) \sqrt {\tanh ^3(x)}-\frac {\left (2 \sqrt {\tanh ^3(x)}\right ) \text {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\sqrt {\tanh (x)}\right )}{\tanh ^{\frac {3}{2}}(x)}\\ &=-2 \coth (x) \sqrt {\tanh ^3(x)}+\frac {\sqrt {\tanh ^3(x)} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {\tanh (x)}\right )}{\tanh ^{\frac {3}{2}}(x)}+\frac {\sqrt {\tanh ^3(x)} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {\tanh (x)}\right )}{\tanh ^{\frac {3}{2}}(x)}\\ &=-2 \coth (x) \sqrt {\tanh ^3(x)}+\frac {\tan ^{-1}\left (\sqrt {\tanh (x)}\right ) \sqrt {\tanh ^3(x)}}{\tanh ^{\frac {3}{2}}(x)}+\frac {\tanh ^{-1}\left (\sqrt {\tanh (x)}\right ) \sqrt {\tanh ^3(x)}}{\tanh ^{\frac {3}{2}}(x)}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 38, normalized size = 0.67 \begin {gather*} \frac {\left (\text {ArcTan}\left (\sqrt {\tanh (x)}\right )+\tanh ^{-1}\left (\sqrt {\tanh (x)}\right )-2 \sqrt {\tanh (x)}\right ) \sqrt {\tanh ^3(x)}}{\tanh ^{\frac {3}{2}}(x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.87, size = 43, normalized size = 0.75
method | result | size |
derivativedivides | \(-\frac {\sqrt {\tanh ^{3}\left (x \right )}\, \left (4 \left (\sqrt {\tanh }\left (x \right )\right )+\ln \left (\sqrt {\tanh }\left (x \right )-1\right )-\ln \left (\sqrt {\tanh }\left (x \right )+1\right )-2 \arctan \left (\sqrt {\tanh }\left (x \right )\right )\right )}{2 \tanh \left (x \right )^{\frac {3}{2}}}\) | \(43\) |
default | \(-\frac {\sqrt {\tanh ^{3}\left (x \right )}\, \left (4 \left (\sqrt {\tanh }\left (x \right )\right )+\ln \left (\sqrt {\tanh }\left (x \right )-1\right )-\ln \left (\sqrt {\tanh }\left (x \right )+1\right )-2 \arctan \left (\sqrt {\tanh }\left (x \right )\right )\right )}{2 \tanh \left (x \right )^{\frac {3}{2}}}\) | \(43\) |
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. 106 vs.
\(2 (43) = 86\).
time = 0.44, size = 106, normalized size = 1.86 \begin {gather*} -2 \, \sqrt {\frac {\sinh \left (x\right )}{\cosh \left (x\right )}} + \arctan \left (-\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) - \sinh \left (x\right )^{2} + {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1\right )} \sqrt {\frac {\sinh \left (x\right )}{\cosh \left (x\right )}}\right ) - \frac {1}{2} \, \log \left (-\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) - \sinh \left (x\right )^{2} + {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1\right )} \sqrt {\frac {\sinh \left (x\right )}{\cosh \left (x\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 \sqrt {\tanh ^{3}{\left (x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 55, normalized size = 0.96 \begin {gather*} \frac {4}{\sqrt {e^{\left (4 \, x\right )} - 1} - e^{\left (2 \, x\right )} - 1} + \arctan \left (\sqrt {e^{\left (4 \, x\right )} - 1} - e^{\left (2 \, x\right )}\right ) - \frac {1}{2} \, \log \left (-\sqrt {e^{\left (4 \, x\right )} - 1} + e^{\left (2 \, x\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \sqrt {{\mathrm {tanh}\left (x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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