Optimal. Leaf size=113 \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a^2 x^2}}}-\frac{2 \sqrt{1-\frac{1}{a^2 x^2}}}{a (a x+1) \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{3 \sqrt{1-\frac{1}{a^2 x^2}} \log (a x+1)}{a \sqrt{c-\frac{c}{a^2 x^2}}} \]
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Rubi [A] time = 0.124986, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {6197, 6193, 77} \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a^2 x^2}}}-\frac{2 \sqrt{1-\frac{1}{a^2 x^2}}}{a (a x+1) \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{3 \sqrt{1-\frac{1}{a^2 x^2}} \log (a x+1)}{a \sqrt{c-\frac{c}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 6197
Rule 6193
Rule 77
Rubi steps
\begin{align*} \int \frac{e^{-3 \coth ^{-1}(a x)}}{\sqrt{c-\frac{c}{a^2 x^2}}} \, dx &=\frac{\sqrt{1-\frac{1}{a^2 x^2}} \int \frac{e^{-3 \coth ^{-1}(a x)}}{\sqrt{1-\frac{1}{a^2 x^2}}} \, dx}{\sqrt{c-\frac{c}{a^2 x^2}}}\\ &=\frac{\left (a \sqrt{1-\frac{1}{a^2 x^2}}\right ) \int \frac{x (-1+a x)}{(1+a x)^2} \, dx}{\sqrt{c-\frac{c}{a^2 x^2}}}\\ &=\frac{\left (a \sqrt{1-\frac{1}{a^2 x^2}}\right ) \int \left (\frac{1}{a}+\frac{2}{a (1+a x)^2}-\frac{3}{a (1+a x)}\right ) \, dx}{\sqrt{c-\frac{c}{a^2 x^2}}}\\ &=\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{\sqrt{c-\frac{c}{a^2 x^2}}}-\frac{2 \sqrt{1-\frac{1}{a^2 x^2}}}{a \sqrt{c-\frac{c}{a^2 x^2}} (1+a x)}-\frac{3 \sqrt{1-\frac{1}{a^2 x^2}} \log (1+a x)}{a \sqrt{c-\frac{c}{a^2 x^2}}}\\ \end{align*}
Mathematica [A] time = 0.048142, size = 54, normalized size = 0.48 \[ \frac{\sqrt{1-\frac{1}{a^2 x^2}} \left (a x-\frac{2}{a x+1}-3 \log (a x+1)\right )}{a \sqrt{c-\frac{c}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.256, size = 87, normalized size = 0.8 \begin{align*} -{\frac{ \left ( ax+1 \right ) \left ( -{a}^{2}{x}^{2}+3\,ax\ln \left ( ax+1 \right ) -ax+3\,\ln \left ( ax+1 \right ) +2 \right ) }{ \left ( ax-1 \right ) x{a}^{2}} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{\sqrt{c - \frac{c}{a^{2} x^{2}}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63462, size = 105, normalized size = 0.93 \begin{align*} \frac{{\left (a^{2} x^{2} + a x - 3 \,{\left (a x + 1\right )} \log \left (a x + 1\right ) - 2\right )} \sqrt{a^{2} c}}{a^{3} c x + a^{2} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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