Optimal. Leaf size=31 \[ x \coth ^2(x) \sqrt {a \tanh ^4(x)}-\coth (x) \sqrt {a \tanh ^4(x)} \]
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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 = {3658, 3473, 8} \[ x \coth ^2(x) \sqrt {a \tanh ^4(x)}-\coth (x) \sqrt {a \tanh ^4(x)} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3473
Rule 3658
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 = 0.02, size = 19, normalized size = 0.61 \[ \coth (x) (x \coth (x)-1) \sqrt {a \tanh ^4(x)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 213, normalized size = 6.87 \[ \frac {{\left (x \cosh \relax (x)^{2} + {\left (x e^{\left (4 \, x\right )} + 2 \, x e^{\left (2 \, x\right )} + x\right )} \sinh \relax (x)^{2} + {\left (x \cosh \relax (x)^{2} + x + 2\right )} e^{\left (4 \, x\right )} + 2 \, {\left (x \cosh \relax (x)^{2} + x + 2\right )} e^{\left (2 \, x\right )} + 2 \, {\left (x \cosh \relax (x) e^{\left (4 \, x\right )} + 2 \, x \cosh \relax (x) e^{\left (2 \, x\right )} + x \cosh \relax (x)\right )} \sinh \relax (x) + x + 2\right )} \sqrt {\frac {a e^{\left (8 \, x\right )} - 4 \, a e^{\left (6 \, x\right )} + 6 \, a e^{\left (4 \, x\right )} - 4 \, a e^{\left (2 \, x\right )} + a}{e^{\left (8 \, x\right )} + 4 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} + 4 \, e^{\left (2 \, x\right )} + 1}}}{{\left (e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1\right )} \sinh \relax (x)^{2} + \cosh \relax (x)^{2} + {\left (\cosh \relax (x)^{2} + 1\right )} e^{\left (4 \, x\right )} - 2 \, {\left (\cosh \relax (x)^{2} + 1\right )} e^{\left (2 \, x\right )} + 2 \, {\left (\cosh \relax (x) e^{\left (4 \, x\right )} - 2 \, \cosh \relax (x) e^{\left (2 \, x\right )} + \cosh \relax (x)\right )} \sinh \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 16, normalized size = 0.52 \[ \sqrt {a} {\left (x + \frac {2}{e^{\left (2 \, x\right )} + 1}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 32, normalized size = 1.03 \[ -\frac {\sqrt {a \left (\tanh ^{4}\relax (x )\right )}\, \left (2 \tanh \relax (x )+\ln \left (\tanh \relax (x )-1\right )-\ln \left (1+\tanh \relax (x )\right )\right )}{2 \tanh \relax (x )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 19, normalized size = 0.61 \[ \sqrt {a} x - \frac {2 \, \sqrt {a}}{e^{\left (-2 \, x\right )} + 1} \]
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 \[ \int \sqrt {a\,{\mathrm {tanh}\relax (x)}^4} \,d x \]
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 \[ \int \sqrt {a \tanh ^{4}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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