Optimal. Leaf size=112 \[ \frac{\sqrt{c-a^2 c x^2}}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\log (x) \sqrt{c-a^2 c x^2}}{a x \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 \sqrt{c-a^2 c x^2} \log (a x+1)}{a x \sqrt{1-\frac{1}{a^2 x^2}}} \]
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Rubi [A] time = 0.160221, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6192, 6193, 72} \[ \frac{\sqrt{c-a^2 c x^2}}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\log (x) \sqrt{c-a^2 c x^2}}{a x \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 \sqrt{c-a^2 c x^2} \log (a x+1)}{a x \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 6192
Rule 6193
Rule 72
Rubi steps
\begin{align*} \int \frac{e^{-3 \coth ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x} \, dx &=\frac{\sqrt{c-a^2 c x^2} \int e^{-3 \coth ^{-1}(a x)} \sqrt{1-\frac{1}{a^2 x^2}} \, dx}{\sqrt{1-\frac{1}{a^2 x^2}} x}\\ &=\frac{\sqrt{c-a^2 c x^2} \int \frac{(-1+a x)^2}{x (1+a x)} \, dx}{a \sqrt{1-\frac{1}{a^2 x^2}} x}\\ &=\frac{\sqrt{c-a^2 c x^2} \int \left (a+\frac{1}{x}-\frac{4 a}{1+a x}\right ) \, dx}{a \sqrt{1-\frac{1}{a^2 x^2}} x}\\ &=\frac{\sqrt{c-a^2 c x^2}}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{c-a^2 c x^2} \log (x)}{a \sqrt{1-\frac{1}{a^2 x^2}} x}-\frac{4 \sqrt{c-a^2 c x^2} \log (1+a x)}{a \sqrt{1-\frac{1}{a^2 x^2}} x}\\ \end{align*}
Mathematica [A] time = 0.0255497, size = 50, normalized size = 0.45 \[ \frac{\sqrt{c-a^2 c x^2} (a x-4 \log (a x+1)+\log (x))}{a x \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.13, size = 57, normalized size = 0.5 \begin{align*}{\frac{ \left ( ax+\ln \left ( x \right ) -4\,\ln \left ( ax+1 \right ) \right ) \left ( ax+1 \right ) }{ \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{\sqrt{-a^{2} c x^{2} + c} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73554, size = 65, normalized size = 0.58 \begin{align*} \frac{\sqrt{-a^{2} c}{\left (a x - 4 \, \log \left (a x + 1\right ) + \log \left (x\right )\right )}}{a} \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{\sqrt{-a^{2} c x^{2} + c} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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