Optimal. Leaf size=30 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {1+\tanh (x)}}{\sqrt {2}}\right )}{\sqrt {2}}+\frac {1}{\sqrt {1+\tanh (x)}} \]
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Rubi [A]
time = 0.03, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3607, 3561,
212} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {\tanh (x)+1}}{\sqrt {2}}\right )}{\sqrt {2}}+\frac {1}{\sqrt {\tanh (x)+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 3561
Rule 3607
Rubi steps
\begin {align*} \int \frac {\tanh (x)}{\sqrt {1+\tanh (x)}} \, dx &=\frac {1}{\sqrt {1+\tanh (x)}}+\frac {1}{2} \int \sqrt {1+\tanh (x)} \, dx\\ &=\frac {1}{\sqrt {1+\tanh (x)}}+\text {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1+\tanh (x)}\right )\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {1+\tanh (x)}}{\sqrt {2}}\right )}{\sqrt {2}}+\frac {1}{\sqrt {1+\tanh (x)}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 30, normalized size = 1.00 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {1+\tanh (x)}}{\sqrt {2}}\right )}{\sqrt {2}}+\frac {1}{\sqrt {1+\tanh (x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.67, size = 25, normalized size = 0.83
method | result | size |
derivativedivides | \(\frac {\arctanh \left (\frac {\sqrt {1+\tanh \left (x \right )}\, \sqrt {2}}{2}\right ) \sqrt {2}}{2}+\frac {1}{\sqrt {1+\tanh \left (x \right )}}\) | \(25\) |
default | \(\frac {\arctanh \left (\frac {\sqrt {1+\tanh \left (x \right )}\, \sqrt {2}}{2}\right ) \sqrt {2}}{2}+\frac {1}{\sqrt {1+\tanh \left (x \right )}}\) | \(25\) |
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. 85 vs.
\(2 (24) = 48\).
time = 0.34, size = 85, normalized size = 2.83 \begin {gather*} \frac {{\left (\sqrt {2} \cosh \left (x\right ) + \sqrt {2} \sinh \left (x\right )\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 ) + 4 \, \sqrt {\frac {\cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}}}{4 \, {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.56, size = 63, normalized size = 2.10 \begin {gather*} - \begin {cases} - \frac {\sqrt {2} \operatorname {acoth}{\left (\frac {\sqrt {2} \sqrt {\tanh {\left (x \right )} + 1}}{2} \right )}}{2} & \text {for}\: \tanh {\left (x \right )} > 1 \\- \frac {\sqrt {2} \operatorname {atanh}{\left (\frac {\sqrt {2} \sqrt {\tanh {\left (x \right )} + 1}}{2} \right )}}{2} & \text {for}\: \tanh {\left (x \right )} < 1 \end {cases} + \frac {1}{\sqrt {\tanh {\left (x \right )} + 1}} \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. 52 vs.
\(2 (24) = 48\).
time = 0.41, size = 52, normalized size = 1.73 \begin {gather*} \frac {1}{4} \, \sqrt {2} {\left (\frac {2}{\sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}} - \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 = 0.13, size = 24, normalized size = 0.80 \begin {gather*} \frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {\mathrm {tanh}\left (x\right )+1}}{2}\right )}{2}+\frac {1}{\sqrt {\mathrm {tanh}\left (x\right )+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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