Optimal. Leaf size=114 \[ \frac{\sqrt{c-a^2 c x^2}}{a x^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 \log (x) \sqrt{c-a^2 c x^2}}{x \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 \sqrt{c-a^2 c x^2} \log (1-a x)}{x \sqrt{1-\frac{1}{a^2 x^2}}} \]
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Rubi [A] time = 0.237844, antiderivative size = 114, 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, 88} \[ \frac{\sqrt{c-a^2 c x^2}}{a x^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 \log (x) \sqrt{c-a^2 c x^2}}{x \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 \sqrt{c-a^2 c x^2} \log (1-a x)}{x \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 6192
Rule 6193
Rule 88
Rubi steps
\begin{align*} \int \frac{e^{3 \coth ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x^2} \, dx &=\frac{\sqrt{c-a^2 c x^2} \int \frac{e^{3 \coth ^{-1}(a x)} \sqrt{1-\frac{1}{a^2 x^2}}}{x} \, 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^2 (-1+a x)} \, dx}{a \sqrt{1-\frac{1}{a^2 x^2}} x}\\ &=\frac{\sqrt{c-a^2 c x^2} \int \left (-\frac{1}{x^2}-\frac{3 a}{x}+\frac{4 a^2}{-1+a x}\right ) \, dx}{a \sqrt{1-\frac{1}{a^2 x^2}} x}\\ &=\frac{\sqrt{c-a^2 c x^2}}{a \sqrt{1-\frac{1}{a^2 x^2}} x^2}-\frac{3 \sqrt{c-a^2 c x^2} \log (x)}{\sqrt{1-\frac{1}{a^2 x^2}} x}+\frac{4 \sqrt{c-a^2 c x^2} \log (1-a x)}{\sqrt{1-\frac{1}{a^2 x^2}} x}\\ \end{align*}
Mathematica [A] time = 0.0316096, size = 55, normalized size = 0.48 \[ \frac{\sqrt{c-a^2 c x^2} \left (-3 a \log (x)+4 a \log (1-a x)+\frac{1}{x}\right )}{a x \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.184, size = 65, normalized size = 0.6 \begin{align*} -{\frac{ \left ( 3\,a\ln \left ( x \right ) x-4\,\ln \left ( ax-1 \right ) xa-1 \right ) \left ( ax-1 \right ) }{x \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}}{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 = 1.62928, size = 81, normalized size = 0.71 \begin{align*} \frac{\sqrt{-a^{2} c}{\left (4 \, a x \log \left (a x - 1\right ) - 3 \, a x \log \left (x\right ) + 1\right )}}{a x} \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}}{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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