Optimal. Leaf size=77 \[ \frac{2 x \sqrt{1-\frac{1}{a^2 x^2}}}{(a x+1) \sqrt{c-a^2 c x^2}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \log (a x+1)}{\sqrt{c-a^2 c x^2}} \]
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Rubi [A] time = 0.163186, antiderivative size = 77, 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 = {6192, 6193, 43} \[ \frac{2 x \sqrt{1-\frac{1}{a^2 x^2}}}{(a x+1) \sqrt{c-a^2 c x^2}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \log (a x+1)}{\sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 6192
Rule 6193
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{-3 \coth ^{-1}(a x)}}{\sqrt{c-a^2 c x^2}} \, dx &=\frac{\left (\sqrt{1-\frac{1}{a^2 x^2}} x\right ) \int \frac{e^{-3 \coth ^{-1}(a x)}}{\sqrt{1-\frac{1}{a^2 x^2}} x} \, dx}{\sqrt{c-a^2 c x^2}}\\ &=\frac{\left (a \sqrt{1-\frac{1}{a^2 x^2}} x\right ) \int \frac{-1+a x}{(1+a x)^2} \, dx}{\sqrt{c-a^2 c x^2}}\\ &=\frac{\left (a \sqrt{1-\frac{1}{a^2 x^2}} x\right ) \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-\frac{1}{a^2 x^2}} x}{(1+a x) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x \log (1+a x)}{\sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0338258, size = 52, normalized size = 0.68 \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}} ((a x+1) \log (a x+1)+2)}{(a x+1) \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.139, size = 62, normalized size = 0.8 \begin{align*} -{\frac{ax\ln \left ( ax+1 \right ) +\ln \left ( ax+1 \right ) +2}{ac \left ( ax-1 \right ) ^{2}}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) } \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{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{-a^{2} c x^{2} + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59553, size = 84, normalized size = 1.09 \begin{align*} -\frac{\sqrt{-a^{2} c}{\left ({\left (a x + 1\right )} \log \left (a x + 1\right ) + 2\right )}}{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{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{\sqrt{-a^{2} c x^{2} + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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