Optimal. Leaf size=33 \[ 2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1+\tanh (x)}}{\sqrt {2}}\right )-2 \sqrt {1+\tanh (x)} \]
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Rubi [A]
time = 0.02, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3559, 3561, 212}
\begin {gather*} 2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {\tanh (x)+1}}{\sqrt {2}}\right )-2 \sqrt {\tanh (x)+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 3559
Rule 3561
Rubi steps
\begin {align*} \int (1+\tanh (x))^{3/2} \, dx &=-2 \sqrt {1+\tanh (x)}+2 \int \sqrt {1+\tanh (x)} \, dx\\ &=-2 \sqrt {1+\tanh (x)}+4 \text {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1+\tanh (x)}\right )\\ &=2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1+\tanh (x)}}{\sqrt {2}}\right )-2 \sqrt {1+\tanh (x)}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 33, normalized size = 1.00 \begin {gather*} 2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1+\tanh (x)}}{\sqrt {2}}\right )-2 \sqrt {1+\tanh (x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.56, size = 27, normalized size = 0.82
method | result | size |
derivativedivides | \(2 \arctanh \left (\frac {\sqrt {1+\tanh \left (x \right )}\, \sqrt {2}}{2}\right ) \sqrt {2}-2 \sqrt {1+\tanh \left (x \right )}\) | \(27\) |
default | \(2 \arctanh \left (\frac {\sqrt {1+\tanh \left (x \right )}\, \sqrt {2}}{2}\right ) \sqrt {2}-2 \sqrt {1+\tanh \left (x \right )}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 57 vs.
\(2 (26) = 52\).
time = 0.47, size = 57, normalized size = 1.73 \begin {gather*} -\sqrt {2} \log \left (-\frac {\sqrt {2} - \frac {\sqrt {2}}{\sqrt {e^{\left (-2 \, x\right )} + 1}}}{\sqrt {2} + \frac {\sqrt {2}}{\sqrt {e^{\left (-2 \, x\right )} + 1}}}\right ) - \frac {2 \, \sqrt {2}}{\sqrt {e^{\left (-2 \, x\right )} + 1}} \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. 129 vs.
\(2 (26) = 52\).
time = 0.35, size = 129, normalized size = 3.91 \begin {gather*} -\frac {2 \, \sqrt {2} {\left (\sqrt {2} \cosh \left (x\right ) + \sqrt {2} \sinh \left (x\right )\right )} \sqrt {\frac {\cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}} - {\left (\sqrt {2} \cosh \left (x\right )^{2} + 2 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt {2} \sinh \left (x\right )^{2} + \sqrt {2}\right )} \log \left (-2 \, \sqrt {2} \sqrt {\frac {\cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} - 2 \, \cosh \left (x\right )^{2} - 4 \, \cosh \left (x\right ) \sinh \left (x\right ) - 2 \, \sinh \left (x\right )^{2} - 1\right )}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 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 (\tanh {\left (x \right )} + 1\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 52, normalized size = 1.58 \begin {gather*} \sqrt {2} {\left (\frac {2}{\sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )} - 1} - \log \left (-2 \, \sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} + 2 \, e^{\left (2 \, x\right )} + 1\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.12, size = 26, normalized size = 0.79 \begin {gather*} 2\,\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {\mathrm {tanh}\left (x\right )+1}}{2}\right )-2\,\sqrt {\mathrm {tanh}\left (x\right )+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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