Optimal. Leaf size=63 \[ -2 \coth (x) \sqrt {a \tanh ^3(x)}+\frac {\text {ArcTan}\left (\sqrt {\tanh (x)}\right ) \sqrt {a \tanh ^3(x)}}{\tanh ^{\frac {3}{2}}(x)}+\frac {\tanh ^{-1}\left (\sqrt {\tanh (x)}\right ) \sqrt {a \tanh ^3(x)}}{\tanh ^{\frac {3}{2}}(x)} \]
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Rubi [A]
time = 0.02, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {3739, 3554,
3557, 335, 218, 212, 209} \begin {gather*} \frac {\sqrt {a \tanh ^3(x)} \text {ArcTan}\left (\sqrt {\tanh (x)}\right )}{\tanh ^{\frac {3}{2}}(x)}+\frac {\tanh ^{-1}\left (\sqrt {\tanh (x)}\right ) \sqrt {a \tanh ^3(x)}}{\tanh ^{\frac {3}{2}}(x)}-2 \coth (x) \sqrt {a \tanh ^3(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 {a \tanh ^3(x)} \, dx &=\frac {\sqrt {a \tanh ^3(x)} \int \tanh ^{\frac {3}{2}}(x) \, dx}{\tanh ^{\frac {3}{2}}(x)}\\ &=-2 \coth (x) \sqrt {a \tanh ^3(x)}+\frac {\sqrt {a \tanh ^3(x)} \int \frac {1}{\sqrt {\tanh (x)}} \, dx}{\tanh ^{\frac {3}{2}}(x)}\\ &=-2 \coth (x) \sqrt {a \tanh ^3(x)}-\frac {\sqrt {a \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 {a \tanh ^3(x)}-\frac {\left (2 \sqrt {a \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 {a \tanh ^3(x)}+\frac {\sqrt {a \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 {a \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 {a \tanh ^3(x)}+\frac {\tan ^{-1}\left (\sqrt {\tanh (x)}\right ) \sqrt {a \tanh ^3(x)}}{\tanh ^{\frac {3}{2}}(x)}+\frac {\tanh ^{-1}\left (\sqrt {\tanh (x)}\right ) \sqrt {a \tanh ^3(x)}}{\tanh ^{\frac {3}{2}}(x)}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 40, normalized size = 0.63 \begin {gather*} \frac {\left (\text {ArcTan}\left (\sqrt {\tanh (x)}\right )+\tanh ^{-1}\left (\sqrt {\tanh (x)}\right )-2 \sqrt {\tanh (x)}\right ) \sqrt {a \tanh ^3(x)}}{\tanh ^{\frac {3}{2}}(x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.15, size = 59, normalized size = 0.94
method | result | size |
derivativedivides | \(\frac {\sqrt {a \left (\tanh ^{3}\left (x \right )\right )}\, \left (-2 \sqrt {a \tanh \left (x \right )}+\sqrt {a}\, \arctan \left (\frac {\sqrt {a \tanh \left (x \right )}}{\sqrt {a}}\right )+\sqrt {a}\, \arctanh \left (\frac {\sqrt {a \tanh \left (x \right )}}{\sqrt {a}}\right )\right )}{\tanh \left (x \right ) \sqrt {a \tanh \left (x \right )}}\) | \(59\) |
default | \(\frac {\sqrt {a \left (\tanh ^{3}\left (x \right )\right )}\, \left (-2 \sqrt {a \tanh \left (x \right )}+\sqrt {a}\, \arctan \left (\frac {\sqrt {a \tanh \left (x \right )}}{\sqrt {a}}\right )+\sqrt {a}\, \arctanh \left (\frac {\sqrt {a \tanh \left (x \right )}}{\sqrt {a}}\right )\right )}{\tanh \left (x \right ) \sqrt {a \tanh \left (x \right )}}\) | \(59\) |
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. 173 vs.
\(2 (49) = 98\).
time = 0.40, size = 376, normalized size = 5.97 \begin {gather*} \left [-\frac {1}{2} \, \sqrt {-a} \arctan \left (\frac {{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\right )} \sqrt {-a} \sqrt {\frac {a \sinh \left (x\right )}{\cosh \left (x\right )}}}{a \cosh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) \sinh \left (x\right ) + a \sinh \left (x\right )^{2} - a}\right ) + \frac {1}{4} \, \sqrt {-a} \log \left (-\frac {a \cosh \left (x\right )^{4} + 4 \, a \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, a \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, a \cosh \left (x\right ) \sinh \left (x\right )^{3} + a \sinh \left (x\right )^{4} + 2 \, {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1\right )} \sqrt {-a} \sqrt {\frac {a \sinh \left (x\right )}{\cosh \left (x\right )}} - 2 \, a}{\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4}}\right ) - 2 \, \sqrt {\frac {a \sinh \left (x\right )}{\cosh \left (x\right )}}, -\frac {1}{2} \, \sqrt {a} \arctan \left (\frac {\sqrt {a} \sqrt {\frac {a \sinh \left (x\right )}{\cosh \left (x\right )}}}{a \cosh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) \sinh \left (x\right ) + a \sinh \left (x\right )^{2} - a}\right ) + \frac {1}{4} \, \sqrt {a} \log \left (2 \, a \cosh \left (x\right )^{4} + 8 \, a \cosh \left (x\right )^{3} \sinh \left (x\right ) + 12 \, a \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 8 \, a \cosh \left (x\right ) \sinh \left (x\right )^{3} + 2 \, a \sinh \left (x\right )^{4} + 2 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + {\left (6 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + \cosh \left (x\right )^{2} + 2 \, {\left (2 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \sqrt {a} \sqrt {\frac {a \sinh \left (x\right )}{\cosh \left (x\right )}} - a\right ) - 2 \, \sqrt {\frac {a \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 {a \tanh ^{3}{\left (x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 115 vs.
\(2 (49) = 98\).
time = 0.43, size = 115, normalized size = 1.83 \begin {gather*} \sqrt {a} \arctan \left (-\frac {\sqrt {a} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} - a}}{\sqrt {a}}\right ) \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) - \frac {1}{2} \, \sqrt {a} \log \left ({\left | -\sqrt {a} e^{\left (2 \, x\right )} + \sqrt {a e^{\left (4 \, x\right )} - a} \right |}\right ) \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) - \frac {4 \, a \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right )}{\sqrt {a} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} - a} + \sqrt {a}} \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 {a\,{\mathrm {tanh}\left (x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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