Optimal. Leaf size=152 \[ \frac{3 \sqrt{c-a^2 c x^2}}{x^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{\sqrt{c-a^2 c x^2}}{2 a x^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 a \log (x) \sqrt{c-a^2 c x^2}}{x \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 a \sqrt{c-a^2 c x^2} \log (a x+1)}{x \sqrt{1-\frac{1}{a^2 x^2}}} \]
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Rubi [A] time = 0.23877, antiderivative size = 152, 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} \[ \frac{3 \sqrt{c-a^2 c x^2}}{x^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{\sqrt{c-a^2 c x^2}}{2 a x^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 a \log (x) \sqrt{c-a^2 c x^2}}{x \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 a \sqrt{c-a^2 c x^2} \log (a x+1)}{x \sqrt{1-\frac{1}{a^2 x^2}}} \]
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^3} \, 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^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^3 (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^3}-\frac{3 a}{x^2}+\frac{4 a^2}{x}-\frac{4 a^3}{1+a x}\right ) \, dx}{a \sqrt{1-\frac{1}{a^2 x^2}} x}\\ &=-\frac{\sqrt{c-a^2 c x^2}}{2 a \sqrt{1-\frac{1}{a^2 x^2}} x^3}+\frac{3 \sqrt{c-a^2 c x^2}}{\sqrt{1-\frac{1}{a^2 x^2}} x^2}+\frac{4 a \sqrt{c-a^2 c x^2} \log (x)}{\sqrt{1-\frac{1}{a^2 x^2}} x}-\frac{4 a \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 = 0.0352233, size = 68, normalized size = 0.45 \[ \frac{\sqrt{c-a^2 c x^2} \left (4 a^2 \log (x)-4 a^2 \log (a x+1)+\frac{3 a}{x}-\frac{1}{2 x^2}\right )}{a x \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.131, size = 77, normalized size = 0.5 \begin{align*}{\frac{ \left ( 8\,{a}^{2}\ln \left ( x \right ){x}^{2}-8\,\ln \left ( ax+1 \right ){a}^{2}{x}^{2}+6\,ax-1 \right ) \left ( ax+1 \right ) }{2\,{x}^{2} \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^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.639, size = 197, normalized size = 1.3 \begin{align*} \frac{8 \, a^{3} \sqrt{-c} x^{2} \log \left (\frac{2 \, a^{3} c x^{2} + 2 \, a^{2} c x + \sqrt{-a^{2} c}{\left (2 \, a x + 1\right )} \sqrt{-c} + a c}{a x^{2} + x}\right ) + \sqrt{-a^{2} c}{\left (6 \, a x - 1\right )}}{2 \, a x^{2}} \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^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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