Optimal. Leaf size=57 \[ \frac {\tanh ^{-1}\left (\sqrt {\tanh (x)}\right ) \sqrt {\tanh ^3(x)}}{\tanh ^{\frac {3}{2}}(x)}+\frac {\sqrt {\tanh ^3(x)} \tan ^{-1}\left (\sqrt {\tanh (x)}\right )}{\tanh ^{\frac {3}{2}}(x)}-2 \sqrt {\tanh ^3(x)} \coth (x) \]
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Rubi [A] time = 0.04, 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 = {3658, 3473, 3476, 329, 212, 206, 203} \[ \frac {\tanh ^{-1}\left (\sqrt {\tanh (x)}\right ) \sqrt {\tanh ^3(x)}}{\tanh ^{\frac {3}{2}}(x)}+\frac {\sqrt {\tanh ^3(x)} \tan ^{-1}\left (\sqrt {\tanh (x)}\right )}{\tanh ^{\frac {3}{2}}(x)}-2 \sqrt {\tanh ^3(x)} \coth (x) \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 329
Rule 3473
Rule 3476
Rule 3658
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)} \operatorname {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 ) \operatorname {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)} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {\tanh (x)}\right )}{\tanh ^{\frac {3}{2}}(x)}+\frac {\sqrt {\tanh ^3(x)} \operatorname {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 = 0.04, size = 38, normalized size = 0.67 \[ \frac {\sqrt {\tanh ^3(x)} \left (\tanh ^{-1}\left (\sqrt {\tanh (x)}\right )-2 \sqrt {\tanh (x)}+\tan ^{-1}\left (\sqrt {\tanh (x)}\right )\right )}{\tanh ^{\frac {3}{2}}(x)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.51, size = 106, normalized size = 1.86 \[ -2 \, \sqrt {\frac {\sinh \relax (x)}{\cosh \relax (x)}} + \arctan \left (-\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) - \sinh \relax (x)^{2} + {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + 1\right )} \sqrt {\frac {\sinh \relax (x)}{\cosh \relax (x)}}\right ) - \frac {1}{2} \, \log \left (-\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) - \sinh \relax (x)^{2} + {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + 1\right )} \sqrt {\frac {\sinh \relax (x)}{\cosh \relax (x)}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 55, normalized size = 0.96 \[ \frac {4}{\sqrt {e^{\left (4 \, x\right )} - 1} - e^{\left (2 \, x\right )} - 1} + \arctan \left (\sqrt {e^{\left (4 \, x\right )} - 1} - e^{\left (2 \, x\right )}\right ) - \frac {1}{2} \, \log \left (-\sqrt {e^{\left (4 \, x\right )} - 1} + e^{\left (2 \, x\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 43, normalized size = 0.75 \[ -\frac {\sqrt {\tanh ^{3}\relax (x )}\, \left (4 \left (\sqrt {\tanh }\relax (x )\right )+\ln \left (\sqrt {\tanh }\relax (x )-1\right )-\ln \left (\sqrt {\tanh }\relax (x )+1\right )-2 \arctan \left (\sqrt {\tanh }\relax (x )\right )\right )}{2 \tanh \relax (x )^{\frac {3}{2}}} \]
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 \[ \int \sqrt {\tanh \relax (x)^{3}}\,{d x} \]
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 \[ \int \sqrt {{\mathrm {tanh}\relax (x)}^3} \,d x \]
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 \[ \int \sqrt {\tanh ^{3}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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