Optimal. Leaf size=107 \[ -\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-a^2 x^2}}+\frac{3 a x \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-a^2 x^2}}-\frac{4 a x \sqrt{c-\frac{c}{a^2 x^2}} \log (1-a x)}{\sqrt{1-a^2 x^2}} \]
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Rubi [A] time = 0.244818, antiderivative size = 107, 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} \[ -\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-a^2 x^2}}+\frac{3 a x \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-a^2 x^2}}-\frac{4 a x \sqrt{c-\frac{c}{a^2 x^2}} \log (1-a x)}{\sqrt{1-a^2 x^2}} \]
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} \, 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^2} \, 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^2 (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^2}+\frac{3 a}{x}-\frac{4 a^2}{-1+a x}\right ) \, dx}{\sqrt{1-a^2 x^2}}\\ &=-\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-a^2 x^2}}+\frac{3 a \sqrt{c-\frac{c}{a^2 x^2}} x \log (x)}{\sqrt{1-a^2 x^2}}-\frac{4 a \sqrt{c-\frac{c}{a^2 x^2}} x \log (1-a x)}{\sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0328704, size = 49, normalized size = 0.46 \[ \frac{\sqrt{c-\frac{c}{a^2 x^2}} (3 a x \log (x)-4 a x \log (1-a x)-1)}{\sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.148, size = 63, normalized size = 0.6 \begin{align*} -{\frac{3\,a\ln \left ( x \right ) x-4\,\ln \left ( ax-1 \right ) xa-1}{{a}^{2}{x}^{2}-1}\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 = 1.32433, size = 194, normalized size = 1.81 \begin{align*} -\frac{1}{2} \, a^{3}{\left (-\frac{i \, \sqrt{c} \log \left (a x + 1\right )}{a^{3}} - \frac{i \, \sqrt{c} \log \left (a x - 1\right )}{a^{3}}\right )} - \frac{3}{2} \, a^{2}{\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{\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 \, \sqrt{c} \log \left (a x + 1\right ) + \frac{1}{2} i \, \sqrt{c} \log \left (a x - 1\right ) + \frac{i \, \sqrt{c}}{a x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left (a x + 1\right )} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} x^{3} - 2 \, a x^{2} + x}, x\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 \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}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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