Optimal. Leaf size=31 \[ \frac{x \tanh ^2(x)}{\sqrt{a \tanh ^4(x)}}-\frac{\tanh (x)}{\sqrt{a \tanh ^4(x)}} \]
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Rubi [A] time = 0.0138879, 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} \[ \frac{x \tanh ^2(x)}{\sqrt{a \tanh ^4(x)}}-\frac{\tanh (x)}{\sqrt{a \tanh ^4(x)}} \]
Antiderivative was successfully verified.
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Rule 3658
Rule 3473
Rule 8
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a \tanh ^4(x)}} \, dx &=\frac{\tanh ^2(x) \int \coth ^2(x) \, dx}{\sqrt{a \tanh ^4(x)}}\\ &=-\frac{\tanh (x)}{\sqrt{a \tanh ^4(x)}}+\frac{\tanh ^2(x) \int 1 \, dx}{\sqrt{a \tanh ^4(x)}}\\ &=-\frac{\tanh (x)}{\sqrt{a \tanh ^4(x)}}+\frac{x \tanh ^2(x)}{\sqrt{a \tanh ^4(x)}}\\ \end{align*}
Mathematica [A] time = 0.0332763, size = 19, normalized size = 0.61 \[ \frac{\tanh (x) (x \tanh (x)-1)}{\sqrt{a \tanh ^4(x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 32, normalized size = 1. \begin{align*}{\frac{\tanh \left ( x \right ) \left ( \ln \left ( 1+\tanh \left ( x \right ) \right ) \tanh \left ( x \right ) -\ln \left ( \tanh \left ( x \right ) -1 \right ) \tanh \left ( x \right ) -2 \right ) }{2}{\frac{1}{\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 = 1.64758, size = 31, normalized size = 1. \begin{align*} \frac{x}{\sqrt{a}} + \frac{2 \, \sqrt{a}}{a e^{\left (-2 \, x\right )} - a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.13279, size = 668, normalized size = 21.55 \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}}}{a \cosh \left (x\right )^{2} +{\left (a e^{\left (4 \, x\right )} - 2 \, a e^{\left (2 \, x\right )} + a\right )} \sinh \left (x\right )^{2} +{\left (a \cosh \left (x\right )^{2} - a\right )} e^{\left (4 \, x\right )} - 2 \,{\left (a \cosh \left (x\right )^{2} - a\right )} e^{\left (2 \, x\right )} + 2 \,{\left (a \cosh \left (x\right ) e^{\left (4 \, x\right )} - 2 \, a \cosh \left (x\right ) e^{\left (2 \, x\right )} + a \cosh \left (x\right )\right )} \sinh \left (x\right ) - a} \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 \frac{1}{\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 = 1.26143, size = 26, normalized size = 0.84 \begin{align*} \frac{x}{\sqrt{a}} - \frac{2}{\sqrt{a}{\left (e^{\left (2 \, x\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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