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 = 0.347263, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {6167, 6152, 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 6167
Rule 6152
Rule 1807
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{-2 \coth ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x^3} \, dx &=-\int \frac{e^{-2 \tanh ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x^3} \, dx\\ &=-\left (c \int \frac{(1-a x)^2}{x^3 \sqrt{c-a^2 c x^2}} \, dx\right )\\ &=\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 = 0.113853, size = 76, normalized size = 0.97 \[ \frac{1}{2} \left (\frac{(1-4 a x) \sqrt{c-a^2 c x^2}}{x^2}+3 a^2 \sqrt{c} \log \left (\sqrt{c} \sqrt{c-a^2 c x^2}+c\right )-3 a^2 \sqrt{c} \log (x)\right ) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.056, size = 231, normalized size = 3. \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{-{a}^{2}c \left ( x+{a}^{-1} \right ) ^{2}+2\, \left ( x+{a}^{-1} \right ) ac}+2\,{\frac{{a}^{3}c}{\sqrt{{a}^{2}c}}\arctan \left ({\frac{\sqrt{{a}^{2}c}x}{\sqrt{-{a}^{2}c \left ( x+{a}^{-1} \right ) ^{2}+2\, \left ( x+{a}^{-1} \right ) ac}}} \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 )}}{{\left (a x + 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 = 1.73074, size = 338, normalized size = 4.33 \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{- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x - 1\right )}{x^{3} \left (a x + 1\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13852, size = 270, normalized size = 3.46 \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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