Optimal. Leaf size=78 \[ -\frac{2 a \sqrt{c-a^2 c x^2}}{x}-\frac{\sqrt{c-a^2 c x^2}}{2 x^2}-\frac{3}{2} a^2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right ) \]
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Rubi [A] time = 0.246493, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6151, 1807, 807, 266, 63, 208} \[ -\frac{2 a \sqrt{c-a^2 c x^2}}{x}-\frac{\sqrt{c-a^2 c x^2}}{2 x^2}-\frac{3}{2} a^2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right ) \]
Antiderivative was successfully verified.
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Rule 6151
Rule 1807
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{2 \tanh ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x^3} \, dx &=c \int \frac{(1+a x)^2}{x^3 \sqrt{c-a^2 c x^2}} \, dx\\ &=-\frac{\sqrt{c-a^2 c x^2}}{2 x^2}-\frac{1}{2} \int \frac{-4 a c-3 a^2 c x}{x^2 \sqrt{c-a^2 c x^2}} \, dx\\ &=-\frac{\sqrt{c-a^2 c x^2}}{2 x^2}-\frac{2 a \sqrt{c-a^2 c x^2}}{x}+\frac{1}{2} \left (3 a^2 c\right ) \int \frac{1}{x \sqrt{c-a^2 c x^2}} \, dx\\ &=-\frac{\sqrt{c-a^2 c x^2}}{2 x^2}-\frac{2 a \sqrt{c-a^2 c x^2}}{x}+\frac{1}{4} \left (3 a^2 c\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c-a^2 c x}} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{c-a^2 c x^2}}{2 x^2}-\frac{2 a \sqrt{c-a^2 c x^2}}{x}-\frac{3}{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 )\\ &=-\frac{\sqrt{c-a^2 c x^2}}{2 x^2}-\frac{2 a \sqrt{c-a^2 c x^2}}{x}-\frac{3}{2} a^2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right )\\ \end{align*}
Mathematica [A] time = 0.101987, size = 79, normalized size = 1.01 \[ -\frac{(4 a x+1) \sqrt{c-a^2 c x^2}}{2 x^2}-\frac{3}{2} a^2 \sqrt{c} \log \left (\sqrt{c} \sqrt{c-a^2 c x^2}+c\right )+\frac{3}{2} a^2 \sqrt{c} \log (x) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.045, size = 239, normalized size = 3.1 \begin{align*} -2\,{\frac{a \left ( -{a}^{2}c{x}^{2}+c \right ) ^{3/2}}{cx}}-2\,{a}^{3}x\sqrt{-{a}^{2}c{x}^{2}+c}-2\,{\frac{{a}^{3}c}{\sqrt{{a}^{2}c}}\arctan \left ({\frac{\sqrt{{a}^{2}c}x}{\sqrt{-{a}^{2}c{x}^{2}+c}}} \right ) }-{\frac{3\,{a}^{2}}{2}\sqrt{c}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{-{a}^{2}c{x}^{2}+c} \right ) } \right ) }+{\frac{3\,{a}^{2}}{2}\sqrt{-{a}^{2}c{x}^{2}+c}}-2\,{a}^{2}\sqrt{-c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) }+2\,{\frac{{a}^{3}c}{\sqrt{{a}^{2}c}}\arctan \left ({\sqrt{{a}^{2}c}x{\frac{1}{\sqrt{-c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) }}}} \right ) }-{\frac{1}{2\,c{x}^{2}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}} \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{\sqrt{-a^{2} c x^{2} + c}{\left (a x + 1\right )}^{2}}{{\left (a^{2} x^{2} - 1\right )} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.58519, size = 339, normalized size = 4.35 \begin{align*} \left [\frac{3 \, a^{2} \sqrt{c} x^{2} \log \left (-\frac{a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{c} - 2 \, c}{x^{2}}\right ) - 2 \, \sqrt{-a^{2} c x^{2} + c}{\left (4 \, a x + 1\right )}}{4 \, x^{2}}, -\frac{3 \, a^{2} \sqrt{-c} x^{2} \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 (4 \, a x + 1\right )}}{2 \, x^{2}}\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{- a^{2} c x^{2} + c}}{a x^{4} - x^{3}}\, dx - \int \frac{a x \sqrt{- a^{2} c x^{2} + c}}{a x^{4} - x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14897, size = 271, normalized size = 3.47 \begin{align*} \frac{3 \, a^{2} c \arctan \left (-\frac{\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}}{\sqrt{-c}}\right )}{\sqrt{-c}} - \frac{{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{3} a^{2} c - 4 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{2} a \sqrt{-c} c{\left | a \right |} +{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )} a^{2} c^{2} + 4 \, a \sqrt{-c} c^{2}{\left | a \right |}}{{\left ({\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{2} - c\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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