Optimal. Leaf size=22 \[ \frac {1}{2} \text {ArcSin}(\tanh (x))+\frac {1}{2} \sqrt {\text {sech}^2(x)} \tanh (x) \]
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Rubi [A]
time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3738, 4207,
201, 222} \begin {gather*} \frac {1}{2} \text {ArcSin}(\tanh (x))+\frac {1}{2} \tanh (x) \sqrt {\text {sech}^2(x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 222
Rule 3738
Rule 4207
Rubi steps
\begin {align*} \int \left (1-\tanh ^2(x)\right )^{3/2} \, dx &=\int \text {sech}^2(x)^{3/2} \, dx\\ &=\text {Subst}\left (\int \sqrt {1-x^2} \, dx,x,\tanh (x)\right )\\ &=\frac {1}{2} \sqrt {\text {sech}^2(x)} \tanh (x)+\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,\tanh (x)\right )\\ &=\frac {1}{2} \sin ^{-1}(\tanh (x))+\frac {1}{2} \sqrt {\text {sech}^2(x)} \tanh (x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 29, normalized size = 1.32 \begin {gather*} \frac {\text {sech}(x) \left (2 \text {ArcTan}\left (\tanh \left (\frac {x}{2}\right )\right )+\text {sech}(x) \tanh (x)\right )}{2 \sqrt {\text {sech}^2(x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 21, normalized size = 0.95
method | result | size |
derivativedivides | \(\frac {\tanh \left (x \right ) \sqrt {1-\left (\tanh ^{2}\left (x \right )\right )}}{2}+\frac {\arcsin \left (\tanh \left (x \right )\right )}{2}\) | \(21\) |
default | \(\frac {\tanh \left (x \right ) \sqrt {1-\left (\tanh ^{2}\left (x \right )\right )}}{2}+\frac {\arcsin \left (\tanh \left (x \right )\right )}{2}\) | \(21\) |
risch | \(\frac {\sqrt {\frac {{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left ({\mathrm e}^{2 x}-1\right )}{1+{\mathrm e}^{2 x}}+\frac {i \sqrt {\frac {{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right ) \ln \left ({\mathrm e}^{x}+i\right )}{2}-\frac {i \sqrt {\frac {{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right ) \ln \left ({\mathrm e}^{x}-i\right )}{2}\) | \(100\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 28, normalized size = 1.27 \begin {gather*} \frac {e^{\left (3 \, x\right )} - e^{x}}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1} + \arctan \left (e^{x}\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. 140 vs.
\(2 (16) = 32\).
time = 0.34, size = 140, normalized size = 6.36 \begin {gather*} \frac {\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3} + {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) + {\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right ) - \cosh \left (x\right )}{\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1} \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 \left (1 - \tanh ^{2}{\left (x \right )}\right )^{\frac {3}{2}}\, 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. 45 vs.
\(2 (16) = 32\).
time = 0.40, size = 45, normalized size = 2.05 \begin {gather*} \frac {1}{4} \, \pi - \frac {e^{\left (-x\right )} - e^{x}}{{\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4} + \frac {1}{2} \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.10, size = 20, normalized size = 0.91 \begin {gather*} \frac {\mathrm {asin}\left (\mathrm {tanh}\left (x\right )\right )}{2}+\frac {\mathrm {tanh}\left (x\right )\,\sqrt {1-{\mathrm {tanh}\left (x\right )}^2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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