Optimal. Leaf size=31 \[ -\coth (x) \sqrt {a \tanh ^4(x)}+x \coth ^2(x) \sqrt {a \tanh ^4(x)} \]
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Rubi [A]
time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3739, 3554, 8}
\begin {gather*} x \coth ^2(x) \sqrt {a \tanh ^4(x)}-\coth (x) \sqrt {a \tanh ^4(x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3554
Rule 3739
Rubi steps
\begin {align*} \int \sqrt {a \tanh ^4(x)} \, dx &=\left (\coth ^2(x) \sqrt {a \tanh ^4(x)}\right ) \int \tanh ^2(x) \, dx\\ &=-\coth (x) \sqrt {a \tanh ^4(x)}+\left (\coth ^2(x) \sqrt {a \tanh ^4(x)}\right ) \int 1 \, dx\\ &=-\coth (x) \sqrt {a \tanh ^4(x)}+x \coth ^2(x) \sqrt {a \tanh ^4(x)}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 19, normalized size = 0.61 \begin {gather*} \coth (x) (-1+x \coth (x)) \sqrt {a \tanh ^4(x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.72, size = 32, normalized size = 1.03
method | result | size |
derivativedivides | \(-\frac {\sqrt {a \left (\tanh ^{4}\left (x \right )\right )}\, \left (2 \tanh \left (x \right )+\ln \left (\tanh \left (x \right )-1\right )-\ln \left (1+\tanh \left (x \right )\right )\right )}{2 \tanh \left (x \right )^{2}}\) | \(32\) |
default | \(-\frac {\sqrt {a \left (\tanh ^{4}\left (x \right )\right )}\, \left (2 \tanh \left (x \right )+\ln \left (\tanh \left (x \right )-1\right )-\ln \left (1+\tanh \left (x \right )\right )\right )}{2 \tanh \left (x \right )^{2}}\) | \(32\) |
risch | \(\frac {\sqrt {\frac {a \left ({\mathrm e}^{2 x}-1\right )^{4}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, \left (1+{\mathrm e}^{2 x}\right )^{2} x}{\left ({\mathrm e}^{2 x}-1\right )^{2}}+\frac {2 \sqrt {\frac {a \left ({\mathrm e}^{2 x}-1\right )^{4}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, \left (1+{\mathrm e}^{2 x}\right )}{\left ({\mathrm e}^{2 x}-1\right )^{2}}\) | \(76\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 19, normalized size = 0.61 \begin {gather*} \sqrt {a} x - \frac {2 \, \sqrt {a}}{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. 213 vs.
\(2 (27) = 54\).
time = 0.37, size = 213, normalized size = 6.87 \begin {gather*} \frac {{\left (x \cosh \left (x\right )^{2} + {\left (x e^{\left (4 \, x\right )} + 2 \, x e^{\left (2 \, x\right )} + x\right )} \sinh \left (x\right )^{2} + {\left (x \cosh \left (x\right )^{2} + x + 2\right )} e^{\left (4 \, x\right )} + 2 \, {\left (x \cosh \left (x\right )^{2} + x + 2\right )} e^{\left (2 \, x\right )} + 2 \, {\left (x \cosh \left (x\right ) e^{\left (4 \, x\right )} + 2 \, x \cosh \left (x\right ) e^{\left (2 \, x\right )} + x \cosh \left (x\right )\right )} \sinh \left (x\right ) + x + 2\right )} \sqrt {\frac {a e^{\left (8 \, x\right )} - 4 \, a e^{\left (6 \, x\right )} + 6 \, a e^{\left (4 \, x\right )} - 4 \, a e^{\left (2 \, x\right )} + a}{e^{\left (8 \, x\right )} + 4 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} + 4 \, e^{\left (2 \, x\right )} + 1}}}{{\left (e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1\right )} \sinh \left (x\right )^{2} + \cosh \left (x\right )^{2} + {\left (\cosh \left (x\right )^{2} + 1\right )} e^{\left (4 \, x\right )} - 2 \, {\left (\cosh \left (x\right )^{2} + 1\right )} e^{\left (2 \, x\right )} + 2 \, {\left (\cosh \left (x\right ) e^{\left (4 \, x\right )} - 2 \, \cosh \left (x\right ) e^{\left (2 \, x\right )} + \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 \sqrt {a \tanh ^{4}{\left (x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 16, normalized size = 0.52 \begin {gather*} \sqrt {a} {\left (x + \frac {2}{e^{\left (2 \, x\right )} + 1}\right )} \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 = -0.03 \begin {gather*} \int \sqrt {a\,{\mathrm {tanh}\left (x\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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