Optimal. Leaf size=83 \[ \frac{2 \sqrt{1-a^2 x^2}}{a (1-a x) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2} \log (1-a x)}{a \sqrt{c-a^2 c x^2}} \]
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Rubi [A] time = 0.0885614, antiderivative size = 83, 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 = {6143, 6140, 43} \[ \frac{2 \sqrt{1-a^2 x^2}}{a (1-a x) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2} \log (1-a x)}{a \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 6143
Rule 6140
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{3 \tanh ^{-1}(a x)}}{\sqrt{c-a^2 c x^2}} \, dx &=\frac{\sqrt{1-a^2 x^2} \int \frac{e^{3 \tanh ^{-1}(a x)}}{\sqrt{1-a^2 x^2}} \, dx}{\sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-a^2 x^2} \int \frac{1+a x}{(1-a x)^2} \, dx}{\sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-a^2 x^2} \int \left (\frac{2}{(-1+a x)^2}+\frac{1}{-1+a x}\right ) \, dx}{\sqrt{c-a^2 c x^2}}\\ &=\frac{2 \sqrt{1-a^2 x^2}}{a (1-a x) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2} \log (1-a x)}{a \sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0331972, size = 51, normalized size = 0.61 \[ \frac{\sqrt{1-a^2 x^2} \left (\frac{2}{1-a x}+\log (1-a x)\right )}{a \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.087, size = 70, normalized size = 0.8 \begin{align*}{\frac{-\ln \left ( ax-1 \right ) xa+\ln \left ( ax-1 \right ) +2}{c \left ({a}^{2}{x}^{2}-1 \right ) a \left ( ax-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) }} \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 (a x + 1\right )}^{3}}{\sqrt{-a^{2} c x^{2} + c}{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.95509, size = 803, normalized size = 9.67 \begin{align*} \left [\frac{4 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} a x +{\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^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x\right )} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} - 2 \, c}{a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1}\right )}{2 \,{\left (a^{4} c x^{3} - a^{3} c x^{2} - a^{2} c x + a c\right )}}, \frac{2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} a x +{\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c}{\left (a^{2} x^{2} - 2 \, a x + 2\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c}}{a^{4} c x^{4} - 2 \, a^{3} c x^{3} - a^{2} c x^{2} + 2 \, a c x}\right )}{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 (a x - 1\right ) \left (a x + 1\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}}{\sqrt{-a^{2} c x^{2} + c}{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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