Optimal. Leaf size=26 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {1+\tanh (4 x)}}{\sqrt {2}}\right )}{2 \sqrt {2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3561, 212}
\begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {\tanh (4 x)+1}}{\sqrt {2}}\right )}{2 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 3561
Rubi steps
\begin {align*} \int \sqrt {1+\tanh (4 x)} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1+\tanh (4 x)}\right )\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {1+\tanh (4 x)}}{\sqrt {2}}\right )}{2 \sqrt {2}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 26, normalized size = 1.00 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {1+\tanh (4 x)}}{\sqrt {2}}\right )}{2 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded while calling a Python object} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.06, size = 20, normalized size = 0.77
method | result | size |
derivativedivides | \(\frac {\arctanh \left (\frac {\sqrt {1+\tanh \left (4 x \right )}\, \sqrt {2}}{2}\right ) \sqrt {2}}{4}\) | \(20\) |
default | \(\frac {\arctanh \left (\frac {\sqrt {1+\tanh \left (4 x \right )}\, \sqrt {2}}{2}\right ) \sqrt {2}}{4}\) | \(20\) |
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. 43 vs.
\(2 (19) = 38\).
time = 0.36, size = 43, normalized size = 1.65 \begin {gather*} -\frac {1}{8} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \frac {\sqrt {2}}{\sqrt {e^{\left (-8 \, x\right )} + 1}}}{\sqrt {2} + \frac {\sqrt {2}}{\sqrt {e^{\left (-8 \, x\right )} + 1}}}\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. 68 vs.
\(2 (19) = 38\).
time = 0.34, size = 68, normalized size = 2.62 \begin {gather*} \frac {1}{8} \, \sqrt {2} \log \left (-2 \, \sqrt {2} \sqrt {\frac {\cosh \left (4 \, x\right )}{\cosh \left (4 \, x\right ) - \sinh \left (4 \, x\right )}} {\left (\cosh \left (4 \, x\right ) + \sinh \left (4 \, x\right )\right )} - 2 \, \cosh \left (4 \, x\right )^{2} - 4 \, \cosh \left (4 \, x\right ) \sinh \left (4 \, x\right ) - 2 \, \sinh \left (4 \, x\right )^{2} - 1\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 {\tanh {\left (4 x \right )} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.01, size = 26, normalized size = 1.00 \begin {gather*} -\frac {1}{4} \sqrt {2} \ln \left (\sqrt {\left (\mathrm {e}^{x}\right )^{8}+1}-\left (\mathrm {e}^{x}\right )^{4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.16, size = 19, normalized size = 0.73 \begin {gather*} \frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {\mathrm {tanh}\left (4\,x\right )+1}}{2}\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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