Optimal. Leaf size=146 \[ \frac{3 \sqrt{c-\frac{c}{a^2 x^2}}}{x \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{2 a x^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 a \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 a \sqrt{c-\frac{c}{a^2 x^2}} \log (a x+1)}{\sqrt{1-\frac{1}{a^2 x^2}}} \]
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Rubi [A] time = 0.278066, antiderivative size = 146, 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 = {6197, 6193, 88} \[ \frac{3 \sqrt{c-\frac{c}{a^2 x^2}}}{x \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{2 a x^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 a \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 a \sqrt{c-\frac{c}{a^2 x^2}} \log (a x+1)}{\sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 6197
Rule 6193
Rule 88
Rubi steps
\begin{align*} \int \frac{e^{-3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}}}{x^2} \, dx &=\frac{\sqrt{c-\frac{c}{a^2 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}}}\\ &=\frac{\sqrt{c-\frac{c}{a^2 x^2}} \int \frac{(-1+a x)^2}{x^3 (1+a x)} \, dx}{a \sqrt{1-\frac{1}{a^2 x^2}}}\\ &=\frac{\sqrt{c-\frac{c}{a^2 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}}}\\ &=-\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{2 a \sqrt{1-\frac{1}{a^2 x^2}} x^2}+\frac{3 \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}} x}+\frac{4 a \sqrt{c-\frac{c}{a^2 x^2}} \log (x)}{\sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 a \sqrt{c-\frac{c}{a^2 x^2}} \log (1+a x)}{\sqrt{1-\frac{1}{a^2 x^2}}}\\ \end{align*}
Mathematica [A] time = 0.0460094, size = 65, normalized size = 0.45 \[ \frac{\sqrt{c-\frac{c}{a^2 x^2}} \left (4 a^2 \log (x)-4 a^2 \log (a x+1)+\frac{3 a}{x}-\frac{1}{2 x^2}\right )}{a \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.229, size = 82, normalized size = 0.6 \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\, \left ( ax-1 \right ) ^{2}x}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}}} \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{c - \frac{c}{a^{2} x^{2}}} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.69383, size = 194, normalized size = 1.33 \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^{2} 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{c - \frac{c}{a^{2} x^{2}}} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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