Optimal. Leaf size=52 \[ \frac{2 (a x+1)}{\sqrt{c-a^2 c x^2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right )}{\sqrt{c}} \]
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Rubi [A] time = 0.22813, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {6151, 1805, 266, 63, 208} \[ \frac{2 (a x+1)}{\sqrt{c-a^2 c x^2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right )}{\sqrt{c}} \]
Antiderivative was successfully verified.
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Rule 6151
Rule 1805
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{2 \tanh ^{-1}(a x)}}{x \sqrt{c-a^2 c x^2}} \, dx &=c \int \frac{(1+a x)^2}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx\\ &=\frac{2 (1+a x)}{\sqrt{c-a^2 c x^2}}+\int \frac{1}{x \sqrt{c-a^2 c x^2}} \, dx\\ &=\frac{2 (1+a x)}{\sqrt{c-a^2 c x^2}}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c-a^2 c x}} \, dx,x,x^2\right )\\ &=\frac{2 (1+a x)}{\sqrt{c-a^2 c x^2}}-\frac{\operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2 c}} \, dx,x,\sqrt{c-a^2 c x^2}\right )}{a^2 c}\\ &=\frac{2 (1+a x)}{\sqrt{c-a^2 c x^2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right )}{\sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.11062, size = 66, normalized size = 1.27 \[ \frac{2 \sqrt{c-a^2 c x^2}}{c-a c x}-\frac{\log \left (\sqrt{c} \sqrt{c-a^2 c x^2}+c\right )}{\sqrt{c}}+\frac{\log (x)}{\sqrt{c}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.037, size = 80, normalized size = 1.5 \begin{align*} -{\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{-{a}^{2}c{x}^{2}+c} \right ) } \right ){\frac{1}{\sqrt{c}}}}-2\,{\frac{1}{ac}\sqrt{-c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-1}} \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{{\left (a x + 1\right )}^{2}}{\sqrt{-a^{2} c x^{2} + c}{\left (a^{2} x^{2} - 1\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.77843, size = 325, normalized size = 6.25 \begin{align*} \left [\frac{{\left (a x - 1\right )} \sqrt{c} \log \left (-\frac{a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{c} - 2 \, c}{x^{2}}\right ) - 4 \, \sqrt{-a^{2} c x^{2} + c}}{2 \,{\left (a c x - c\right )}}, -\frac{{\left (a x - 1\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right ) + 2 \, \sqrt{-a^{2} c x^{2} + c}}{a c x - c}\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}{a x^{2} \sqrt{- a^{2} c x^{2} + c} - x \sqrt{- a^{2} c x^{2} + c}}\, dx - \int \frac{1}{a x^{2} \sqrt{- a^{2} c x^{2} + c} - x \sqrt{- a^{2} c x^{2} + c}}\, 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*} \mathit{sage}_{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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