Optimal. Leaf size=148 \[ -\frac{3 a \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-a^2 x^2}}-\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{2 x \sqrt{1-a^2 x^2}}+\frac{4 a^2 x \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-a^2 x^2}}-\frac{4 a^2 x \sqrt{c-\frac{c}{a^2 x^2}} \log (1-a x)}{\sqrt{1-a^2 x^2}} \]
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Rubi [A] time = 0.24951, antiderivative size = 148, 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 = {6160, 6150, 88} \[ -\frac{3 a \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-a^2 x^2}}-\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{2 x \sqrt{1-a^2 x^2}}+\frac{4 a^2 x \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-a^2 x^2}}-\frac{4 a^2 x \sqrt{c-\frac{c}{a^2 x^2}} \log (1-a x)}{\sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 6160
Rule 6150
Rule 88
Rubi steps
\begin{align*} \int \frac{e^{3 \tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}}}{x^2} \, dx &=\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{e^{3 \tanh ^{-1}(a x)} \sqrt{1-a^2 x^2}}{x^3} \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{(1+a x)^2}{x^3 (1-a x)} \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \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}{\sqrt{1-a^2 x^2}}\\ &=-\frac{3 a \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-a^2 x^2}}-\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{2 x \sqrt{1-a^2 x^2}}+\frac{4 a^2 \sqrt{c-\frac{c}{a^2 x^2}} x \log (x)}{\sqrt{1-a^2 x^2}}-\frac{4 a^2 \sqrt{c-\frac{c}{a^2 x^2}} x \log (1-a x)}{\sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0446526, size = 64, normalized size = 0.43 \[ \frac{x \sqrt{c-\frac{c}{a^2 x^2}} \left (4 a^2 \log (x)-4 a^2 \log (1-a x)-\frac{3 a}{x}-\frac{1}{2 x^2}\right )}{\sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.148, size = 78, normalized size = 0.5 \begin{align*} -{\frac{8\,{a}^{2}\ln \left ( x \right ){x}^{2}-8\,\ln \left ( ax-1 \right ){a}^{2}{x}^{2}-6\,ax-1}{2\,x \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}}}\sqrt{-{a}^{2}{x}^{2}+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.32495, size = 215, normalized size = 1.45 \begin{align*} -\frac{1}{2} \, a^{3}{\left (\frac{i \, \sqrt{c} \log \left (a x + 1\right )}{a^{2}} - \frac{i \, \sqrt{c} \log \left (a x - 1\right )}{a^{2}}\right )} - \frac{3}{2} \, a^{2}{\left (-\frac{i \, \sqrt{c} \log \left (a x + 1\right )}{a} - \frac{i \, \sqrt{c} \log \left (a x - 1\right )}{a} + \frac{2 i \, \sqrt{c} \log \left (x\right )}{a}\right )} + \frac{1}{2} i \, a \sqrt{c} \log \left (a x + 1\right ) + \frac{1}{2} i \, a \sqrt{c} \log \left (a x - 1\right ) - i \, a \sqrt{c} \log \left (x\right ) - \frac{3}{2} \,{\left (i \, \sqrt{c} \log \left (a x + 1\right ) - i \, \sqrt{c} \log \left (a x - 1\right ) - \frac{2 i \, \sqrt{c}}{a x}\right )} a + \frac{i \, \sqrt{c}}{2 \, a x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.67676, size = 1023, normalized size = 6.91 \begin{align*} \left [\frac{4 \,{\left (a^{3} x^{3} - a x\right )} \sqrt{-c} \log \left (-\frac{4 \, a^{5} c x^{5} -{\left (2 \, a^{6} - 4 \, a^{5} + 6 \, a^{4} - 4 \, a^{3} + a^{2}\right )} c x^{6} -{\left (4 \, a^{4} + 4 \, a^{3} - 6 \, a^{2} + 4 \, a - 1\right )} c x^{4} + 5 \, a^{2} c x^{2} - 4 \, a c x -{\left (4 \, a^{4} x^{4} - 6 \, a^{3} x^{3} -{\left (4 \, a^{4} - 6 \, a^{3} + 4 \, a^{2} - a\right )} x^{5} + 4 \, a^{2} x^{2} - a x\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} + c}{a^{4} x^{6} - 2 \, a^{3} x^{5} + 2 \, a x^{3} - x^{2}}\right ) - \sqrt{-a^{2} x^{2} + 1}{\left ({\left (6 \, a + 1\right )} x^{2} - 6 \, a x - 1\right )} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \,{\left (a^{2} x^{3} - x\right )}}, \frac{8 \,{\left (a^{3} x^{3} - a x\right )} \sqrt{c} \arctan \left (\frac{{\left (2 \, a^{2} x^{2} -{\left (2 \, a^{3} - 2 \, a^{2} + a\right )} x^{3} - a x\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, a^{3} c x^{3} -{\left (2 \, a^{3} - a^{2}\right )} c x^{4} -{\left (a^{2} - 2 \, a + 1\right )} c x^{2} - 2 \, a c x + c}\right ) - \sqrt{-a^{2} x^{2} + 1}{\left ({\left (6 \, a + 1\right )} x^{2} - 6 \, a x - 1\right )} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \,{\left (a^{2} x^{3} - x\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- c \left (-1 + \frac{1}{a x}\right ) \left (1 + \frac{1}{a x}\right )} \left (a x + 1\right )^{3}}{x^{2} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}}}\, dx \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 (a x + 1\right )}^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{{\left (-a^{2} x^{2} + 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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