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 = 0.0137952, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, 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 3658
Rule 3473
Rule 8
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*}
Mathematica [A] time = 0.0168269, 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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Maple [A] time = 0.032, size = 32, normalized size = 1. \begin{align*} -{\frac{2\,\tanh \left ( x \right ) +\ln \left ( \tanh \left ( x \right ) -1 \right ) -\ln \left ( 1+\tanh \left ( x \right ) \right ) }{2\, \left ( \tanh \left ( x \right ) \right ) ^{2}}\sqrt{a \left ( \tanh \left ( x \right ) \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.60266, size = 26, normalized size = 0.84 \begin{align*} \sqrt{a} x - \frac{2 \, \sqrt{a}}{e^{\left (-2 \, x\right )} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.32083, size = 647, normalized size = 20.87 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \tanh ^{4}{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19429, size = 22, normalized size = 0.71 \begin{align*} \sqrt{a}{\left (x + \frac{2}{e^{\left (2 \, x\right )} + 1}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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