Optimal. Leaf size=115 \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a^2 x^2}}}+\frac{2 \sqrt{1-\frac{1}{a^2 x^2}}}{a (1-a x) \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{3 \sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{a \sqrt{c-\frac{c}{a^2 x^2}}} \]
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Rubi [A] time = 0.123677, antiderivative size = 115, 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} \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a^2 x^2}}}+\frac{2 \sqrt{1-\frac{1}{a^2 x^2}}}{a (1-a x) \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{3 \sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{a \sqrt{c-\frac{c}{a^2 x^2}}} \]
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 = 0.0475806, size = 56, normalized size = 0.49 \[ \frac{\sqrt{1-\frac{1}{a^2 x^2}} \left (a x+\frac{2}{1-a x}+3 \log (1-a x)\right )}{a \sqrt{c-\frac{c}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.243, size = 85, normalized size = 0.7 \begin{align*}{\frac{ \left ( ax-1 \right ) \left ({a}^{2}{x}^{2}+3\,\ln \left ( ax-1 \right ) xa-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{1}{\sqrt{c - \frac{c}{a^{2} x^{2}}} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54377, size = 105, normalized size = 0.91 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{c - \frac{c}{a^{2} x^{2}}} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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