Optimal. Leaf size=16 \[ \coth (x) \log (\cosh (x)) \sqrt {a \tanh ^2(x)} \]
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Rubi [A]
time = 0.01, antiderivative size = 16, 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 = {3739, 3556}
\begin {gather*} \coth (x) \sqrt {a \tanh ^2(x)} \log (\cosh (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 3739
Rubi steps
\begin {align*} \int \sqrt {a \tanh ^2(x)} \, dx &=\left (\coth (x) \sqrt {a \tanh ^2(x)}\right ) \int \tanh (x) \, dx\\ &=\coth (x) \log (\cosh (x)) \sqrt {a \tanh ^2(x)}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} \coth (x) \log (\cosh (x)) \sqrt {a \tanh ^2(x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.76, size = 26, normalized size = 1.62
method | result | size |
derivativedivides | \(-\frac {\sqrt {a \left (\tanh ^{2}\left (x \right )\right )}\, \left (\ln \left (\tanh \left (x \right )-1\right )+\ln \left (1+\tanh \left (x \right )\right )\right )}{2 \tanh \left (x \right )}\) | \(26\) |
default | \(-\frac {\sqrt {a \left (\tanh ^{2}\left (x \right )\right )}\, \left (\ln \left (\tanh \left (x \right )-1\right )+\ln \left (1+\tanh \left (x \right )\right )\right )}{2 \tanh \left (x \right )}\) | \(26\) |
risch | \(-\frac {\sqrt {\frac {a \left ({\mathrm e}^{2 x}-1\right )^{2}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right ) x}{{\mathrm e}^{2 x}-1}+\frac {\sqrt {\frac {a \left ({\mathrm e}^{2 x}-1\right )^{2}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right ) \ln \left (1+{\mathrm e}^{2 x}\right )}{{\mathrm e}^{2 x}-1}\) | \(81\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 19, normalized size = 1.19 \begin {gather*} -\sqrt {a} x - \sqrt {a} \log \left (e^{\left (-2 \, 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. 72 vs.
\(2 (14) = 28\).
time = 0.41, size = 72, normalized size = 4.50 \begin {gather*} -\frac {{\left (x e^{\left (2 \, x\right )} - {\left (e^{\left (2 \, x\right )} + 1\right )} \log \left (\frac {2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + x\right )} \sqrt {\frac {a e^{\left (4 \, x\right )} - 2 \, a e^{\left (2 \, x\right )} + a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}}}{e^{\left (2 \, 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 \sqrt {a \tanh ^{2}{\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. 31 vs.
\(2 (14) = 28\).
time = 0.41, size = 31, normalized size = 1.94 \begin {gather*} -{\left (x \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) - \log \left (e^{\left (2 \, x\right )} + 1\right ) \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right )\right )} \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 = -0.06 \begin {gather*} \int \sqrt {a\,{\mathrm {tanh}\left (x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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