Optimal. Leaf size=108 \[ -\frac{\sqrt{c-a^2 c x^2}}{c^2 x}-\frac{2 a \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right )}{c^{3/2}}+\frac{2 a (a x+1)}{3 \left (c-a^2 c x^2\right )^{3/2}}+\frac{a (7 a x+6)}{3 c \sqrt{c-a^2 c x^2}} \]
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Rubi [A] time = 0.337047, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6151, 1805, 807, 266, 63, 208} \[ -\frac{\sqrt{c-a^2 c x^2}}{c^2 x}-\frac{2 a \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right )}{c^{3/2}}+\frac{2 a (a x+1)}{3 \left (c-a^2 c x^2\right )^{3/2}}+\frac{a (7 a x+6)}{3 c \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 6151
Rule 1805
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{2 \tanh ^{-1}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{3/2}} \, dx &=c \int \frac{(1+a x)^2}{x^2 \left (c-a^2 c x^2\right )^{5/2}} \, dx\\ &=\frac{2 a (1+a x)}{3 \left (c-a^2 c x^2\right )^{3/2}}-\frac{1}{3} \int \frac{-3-6 a x-4 a^2 x^2}{x^2 \left (c-a^2 c x^2\right )^{3/2}} \, dx\\ &=\frac{2 a (1+a x)}{3 \left (c-a^2 c x^2\right )^{3/2}}+\frac{a (6+7 a x)}{3 c \sqrt{c-a^2 c x^2}}+\frac{\int \frac{3+6 a x}{x^2 \sqrt{c-a^2 c x^2}} \, dx}{3 c}\\ &=\frac{2 a (1+a x)}{3 \left (c-a^2 c x^2\right )^{3/2}}+\frac{a (6+7 a x)}{3 c \sqrt{c-a^2 c x^2}}-\frac{\sqrt{c-a^2 c x^2}}{c^2 x}+\frac{(2 a) \int \frac{1}{x \sqrt{c-a^2 c x^2}} \, dx}{c}\\ &=\frac{2 a (1+a x)}{3 \left (c-a^2 c x^2\right )^{3/2}}+\frac{a (6+7 a x)}{3 c \sqrt{c-a^2 c x^2}}-\frac{\sqrt{c-a^2 c x^2}}{c^2 x}+\frac{a \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c-a^2 c x}} \, dx,x,x^2\right )}{c}\\ &=\frac{2 a (1+a x)}{3 \left (c-a^2 c x^2\right )^{3/2}}+\frac{a (6+7 a x)}{3 c \sqrt{c-a^2 c x^2}}-\frac{\sqrt{c-a^2 c x^2}}{c^2 x}-\frac{2 \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 c^2}\\ &=\frac{2 a (1+a x)}{3 \left (c-a^2 c x^2\right )^{3/2}}+\frac{a (6+7 a x)}{3 c \sqrt{c-a^2 c x^2}}-\frac{\sqrt{c-a^2 c x^2}}{c^2 x}-\frac{2 a \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right )}{c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.130704, size = 89, normalized size = 0.82 \[ \frac{\left (-10 a^2 x^2+14 a x-3\right ) \sqrt{c-a^2 c x^2}}{3 c^2 x (a x-1)^2}-\frac{2 a \log \left (\sqrt{c} \sqrt{c-a^2 c x^2}+c\right )}{c^{3/2}}+\frac{2 a \log (x)}{c^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.042, size = 178, normalized size = 1.7 \begin{align*} -{\frac{1}{cx}{\frac{1}{\sqrt{-{a}^{2}c{x}^{2}+c}}}}+2\,{\frac{{a}^{2}x}{\sqrt{-{a}^{2}c{x}^{2}+c}c}}+2\,{\frac{a}{\sqrt{-{a}^{2}c{x}^{2}+c}c}}-2\,{\frac{a}{{c}^{3/2}}\ln \left ({\frac{2\,c+2\,\sqrt{c}\sqrt{-{a}^{2}c{x}^{2}+c}}{x}} \right ) }-{\frac{2}{3\,c} \left ( x-{a}^{-1} \right ) ^{-1}{\frac{1}{\sqrt{-c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) }}}}+{\frac{4\,{a}^{2}x}{3\,c}{\frac{1}{\sqrt{-c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-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{{\left (a x + 1\right )}^{2}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}}{\left (a^{2} x^{2} - 1\right )} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.19552, size = 514, normalized size = 4.76 \begin{align*} \left [\frac{3 \,{\left (a^{3} x^{3} - 2 \, a^{2} x^{2} + a x\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 ) - \sqrt{-a^{2} c x^{2} + c}{\left (10 \, a^{2} x^{2} - 14 \, a x + 3\right )}}{3 \,{\left (a^{2} c^{2} x^{3} - 2 \, a c^{2} x^{2} + c^{2} x\right )}}, -\frac{6 \,{\left (a^{3} x^{3} - 2 \, a^{2} x^{2} + a x\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right ) + \sqrt{-a^{2} c x^{2} + c}{\left (10 \, a^{2} x^{2} - 14 \, a x + 3\right )}}{3 \,{\left (a^{2} c^{2} x^{3} - 2 \, a c^{2} x^{2} + c^{2} 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{a x}{- a^{3} c x^{5} \sqrt{- a^{2} c x^{2} + c} + a^{2} c x^{4} \sqrt{- a^{2} c x^{2} + c} + a c x^{3} \sqrt{- a^{2} c x^{2} + c} - c x^{2} \sqrt{- a^{2} c x^{2} + c}}\, dx - \int \frac{1}{- a^{3} c x^{5} \sqrt{- a^{2} c x^{2} + c} + a^{2} c x^{4} \sqrt{- a^{2} c x^{2} + c} + a c x^{3} \sqrt{- a^{2} c x^{2} + c} - c x^{2} \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 [A] time = 1.22582, size = 135, normalized size = 1.25 \begin{align*} -2 \, a^{2} \sqrt{-c} c{\left (\frac{2 \,{\left | a \right |} \arctan \left (-\frac{\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}}{\sqrt{-c}}\right )}{a^{3} c^{3}} - \frac{1}{{\left ({\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{2} - c\right )} a^{2} c^{2}}\right )}{\left | a \right |} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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