Optimal. Leaf size=16 \[ \frac{\tanh (x) \log (\sinh (x))}{\sqrt{a \tanh ^2(x)}} \]
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Rubi [A] time = 0.0154868, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3658, 3475} \[ \frac{\tanh (x) \log (\sinh (x))}{\sqrt{a \tanh ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 3658
Rule 3475
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a \tanh ^2(x)}} \, dx &=\frac{\tanh (x) \int \coth (x) \, dx}{\sqrt{a \tanh ^2(x)}}\\ &=\frac{\log (\sinh (x)) \tanh (x)}{\sqrt{a \tanh ^2(x)}}\\ \end{align*}
Mathematica [A] time = 0.0077392, size = 16, normalized size = 1. \[ \frac{\tanh (x) \log (\sinh (x))}{\sqrt{a \tanh ^2(x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 29, normalized size = 1.8 \begin{align*} -{\frac{\tanh \left ( x \right ) \left ( \ln \left ( 1+\tanh \left ( x \right ) \right ) -2\,\ln \left ( \tanh \left ( x \right ) \right ) +\ln \left ( \tanh \left ( x \right ) -1 \right ) \right ) }{2}{\frac{1}{\sqrt{a \left ( \tanh \left ( x \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.59866, size = 42, normalized size = 2.62 \begin{align*} -\frac{x}{\sqrt{a}} - \frac{\log \left (e^{\left (-x\right )} + 1\right )}{\sqrt{a}} - \frac{\log \left (e^{\left (-x\right )} - 1\right )}{\sqrt{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.27023, size = 198, normalized size = 12.38 \begin{align*} -\frac{{\left (x e^{\left (2 \, x\right )} -{\left (e^{\left (2 \, x\right )} + 1\right )} \log \left (\frac{2 \, \sinh \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}}}{a e^{\left (2 \, 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 ^{2}{\left (x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19243, size = 50, normalized size = 3.12 \begin{align*} -\frac{x}{\sqrt{a} \mathrm{sgn}\left (e^{\left (4 \, x\right )} - 1\right )} + \frac{\log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right )}{\sqrt{a} \mathrm{sgn}\left (e^{\left (4 \, x\right )} - 1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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