Optimal. Leaf size=107 \[ -\frac{\sqrt{1-a^2 x^2}}{x \sqrt{c-a^2 c x^2}}+\frac{a \sqrt{1-a^2 x^2} \log (x)}{\sqrt{c-a^2 c x^2}}-\frac{a \sqrt{1-a^2 x^2} \log (1-a x)}{\sqrt{c-a^2 c x^2}} \]
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Rubi [A] time = 0.190474, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {6153, 6150, 44} \[ -\frac{\sqrt{1-a^2 x^2}}{x \sqrt{c-a^2 c x^2}}+\frac{a \sqrt{1-a^2 x^2} \log (x)}{\sqrt{c-a^2 c x^2}}-\frac{a \sqrt{1-a^2 x^2} \log (1-a x)}{\sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 6153
Rule 6150
Rule 44
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)}}{x^2 \sqrt{c-a^2 c x^2}} \, dx &=\frac{\sqrt{1-a^2 x^2} \int \frac{e^{\tanh ^{-1}(a x)}}{x^2 \sqrt{1-a^2 x^2}} \, dx}{\sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-a^2 x^2} \int \frac{1}{x^2 (1-a x)} \, dx}{\sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-a^2 x^2} \int \left (\frac{1}{x^2}+\frac{a}{x}-\frac{a^2}{-1+a x}\right ) \, dx}{\sqrt{c-a^2 c x^2}}\\ &=-\frac{\sqrt{1-a^2 x^2}}{x \sqrt{c-a^2 c x^2}}+\frac{a \sqrt{1-a^2 x^2} \log (x)}{\sqrt{c-a^2 c x^2}}-\frac{a \sqrt{1-a^2 x^2} \log (1-a x)}{\sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0296793, size = 50, normalized size = 0.47 \[ \frac{\sqrt{1-a^2 x^2} \left (a \log (x)-a \log (1-a x)-\frac{1}{x}\right )}{\sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.089, size = 62, normalized size = 0.6 \begin{align*}{\frac{-a\ln \left ( x \right ) x+\ln \left ( ax-1 \right ) xa+1}{c \left ({a}^{2}{x}^{2}-1 \right ) x}\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) }} \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{a x + 1}{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.64452, size = 894, normalized size = 8.36 \begin{align*} \left [\frac{{\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^{3} x^{3} -{\left (4 \, a^{3} - 6 \, a^{2} + 4 \, a - 1\right )} x^{4} - 6 \, a^{2} x^{2} + 4 \, a x - 1\right )} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} + c}{a^{4} x^{6} - 2 \, a^{3} x^{5} + 2 \, a x^{3} - x^{2}}\right ) - 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}{\left (x - 1\right )}}{2 \,{\left (a^{2} c x^{3} - c x\right )}}, -\frac{{\left (a^{3} x^{3} - a x\right )} \sqrt{-c} \arctan \left (-\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}{\left ({\left (2 \, a^{2} - 2 \, a + 1\right )} x^{2} - 2 \, a x + 1\right )} \sqrt{-c}}{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} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}{\left (x - 1\right )}}{a^{2} c x^{3} - c 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{a x + 1}{x^{2} \sqrt{- \left (a x - 1\right ) \left (a x + 1\right )} \sqrt{- c \left (a x - 1\right ) \left (a x + 1\right )}}\, 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{a x + 1}{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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