Optimal. Leaf size=123 \[ \frac{\sqrt{1-a^2 x^2}}{a \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{2 \sqrt{1-a^2 x^2}}{a^2 x (1-a x) \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{3 \sqrt{1-a^2 x^2} \log (1-a x)}{a^2 x \sqrt{c-\frac{c}{a^2 x^2}}} \]
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Rubi [A] time = 0.139521, antiderivative size = 123, 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 = {6160, 6150, 77} \[ \frac{\sqrt{1-a^2 x^2}}{a \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{2 \sqrt{1-a^2 x^2}}{a^2 x (1-a x) \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{3 \sqrt{1-a^2 x^2} \log (1-a x)}{a^2 x \sqrt{c-\frac{c}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 6160
Rule 6150
Rule 77
Rubi steps
\begin{align*} \int \frac{e^{3 \tanh ^{-1}(a x)}}{\sqrt{c-\frac{c}{a^2 x^2}}} \, dx &=\frac{\sqrt{1-a^2 x^2} \int \frac{e^{3 \tanh ^{-1}(a x)} x}{\sqrt{1-a^2 x^2}} \, dx}{\sqrt{c-\frac{c}{a^2 x^2}} x}\\ &=\frac{\sqrt{1-a^2 x^2} \int \frac{x (1+a x)}{(1-a x)^2} \, dx}{\sqrt{c-\frac{c}{a^2 x^2}} x}\\ &=\frac{\sqrt{1-a^2 x^2} \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}} x}\\ &=\frac{\sqrt{1-a^2 x^2}}{a \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{2 \sqrt{1-a^2 x^2}}{a^2 \sqrt{c-\frac{c}{a^2 x^2}} x (1-a x)}+\frac{3 \sqrt{1-a^2 x^2} \log (1-a x)}{a^2 \sqrt{c-\frac{c}{a^2 x^2}} x}\\ \end{align*}
Mathematica [A] time = 0.0445779, size = 59, normalized size = 0.48 \[ \frac{\sqrt{1-a^2 x^2} \left (a x+\frac{2}{1-a x}+3 \log (1-a x)\right )}{a^2 x \sqrt{c-\frac{c}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.151, size = 77, normalized size = 0.6 \begin{align*}{\frac{{a}^{2}{x}^{2}+3\,\ln \left ( ax-1 \right ) xa-ax-3\,\ln \left ( ax-1 \right ) -2}{{a}^{2}x \left ( ax-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1}{\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*} -\frac{i}{2 \,{\left (a x + 1\right )}{\left (a x - 1\right )} a \sqrt{c}} - \int \frac{a^{4} x^{4} + 3 \, a^{3} x^{3} + 3 \, a^{2} x^{2}}{{\left (i \, a^{2} \sqrt{c} x^{2} - i \, \sqrt{c}\right )}{\left (a x + 1\right )}{\left (a x - 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.05115, size = 910, normalized size = 7.4 \begin{align*} \left [-\frac{3 \,{\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \sqrt{-c} \log \left (\frac{a^{6} c x^{6} - 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} + 4 \, a c x +{\left (a^{5} x^{5} - 4 \, a^{4} x^{4} + 6 \, a^{3} x^{3} - 4 \, a^{2} x^{2}\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1}\right ) - 2 \,{\left (a^{3} x^{3} - 3 \, a^{2} x^{2}\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \,{\left (a^{4} c x^{3} - a^{3} c x^{2} - a^{2} c x + a c\right )}}, \frac{3 \,{\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \sqrt{c} \arctan \left (\frac{{\left (a^{2} x^{2} - 2 \, a x + 2\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{3} c x^{3} - 2 \, a^{2} c x^{2} - a c x + 2 \, c}\right ) +{\left (a^{3} x^{3} - 3 \, a^{2} x^{2}\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{4} c x^{3} - a^{3} c x^{2} - a^{2} c x + a c}\right ] \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{\left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}} \sqrt{- c \left (-1 + \frac{1}{a x}\right ) \left (1 + \frac{1}{a x}\right )}}\, dx \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{{\left (a x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 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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