Optimal. Leaf size=72 \[ 3 a \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{c-a c x}}\right )-\frac{c \sqrt{1-a^2 x^2}}{x \sqrt{c-a c x}} \]
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Rubi [A] time = 0.160621, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {6128, 879, 875, 208} \[ 3 a \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{c-a c x}}\right )-\frac{c \sqrt{1-a^2 x^2}}{x \sqrt{c-a c x}} \]
Antiderivative was successfully verified.
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Rule 6128
Rule 879
Rule 875
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{-\tanh ^{-1}(a x)} \sqrt{c-a c x}}{x^2} \, dx &=\frac{\int \frac{(c-a c x)^{3/2}}{x^2 \sqrt{1-a^2 x^2}} \, dx}{c}\\ &=-\frac{c \sqrt{1-a^2 x^2}}{x \sqrt{c-a c x}}-\frac{1}{2} (3 a) \int \frac{\sqrt{c-a c x}}{x \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{c \sqrt{1-a^2 x^2}}{x \sqrt{c-a c x}}-\left (3 a^3 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{-a^2 c+a^2 c^2 x^2} \, dx,x,\frac{\sqrt{1-a^2 x^2}}{\sqrt{c-a c x}}\right )\\ &=-\frac{c \sqrt{1-a^2 x^2}}{x \sqrt{c-a c x}}+3 a \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{c-a c x}}\right )\\ \end{align*}
Mathematica [A] time = 0.0303993, size = 52, normalized size = 0.72 \[ \frac{\sqrt{1-a x} \left (3 a c \tanh ^{-1}\left (\sqrt{a x+1}\right )-\frac{c \sqrt{a x+1}}{x}\right )}{\sqrt{c-a c x}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.1, size = 79, normalized size = 1.1 \begin{align*}{\frac{1}{ \left ( ax-1 \right ) x}\sqrt{-c \left ( ax-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1} \left ( -3\,{\it Artanh} \left ({\frac{\sqrt{c \left ( ax+1 \right ) }}{\sqrt{c}}} \right ) xac+\sqrt{c \left ( ax+1 \right ) }\sqrt{c} \right ){\frac{1}{\sqrt{c \left ( ax+1 \right ) }}}{\frac{1}{\sqrt{c}}}} \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} x^{2} + 1} \sqrt{-a c x + c}}{{\left (a x + 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 = 1.86449, size = 446, normalized size = 6.19 \begin{align*} \left [\frac{3 \,{\left (a^{2} x^{2} - a x\right )} \sqrt{c} \log \left (-\frac{a^{2} c x^{2} + a c x - 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 2 \, c}{a x^{2} - x}\right ) + 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{2 \,{\left (a x^{2} - x\right )}}, \frac{3 \,{\left (a^{2} x^{2} - a x\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right ) + \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{a x^{2} - 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{\sqrt{- c \left (a x - 1\right )} \sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}{x^{2} \left (a x + 1\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2062, size = 131, normalized size = 1.82 \begin{align*} -\frac{{\left (a c^{2}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{a c x + c}}{\sqrt{-c}}\right )}{\sqrt{-c}} + \frac{\sqrt{a c x + c}}{a c x}\right )} - \frac{3 \, a c^{\frac{5}{2}} \arctan \left (\frac{\sqrt{2} \sqrt{c}}{\sqrt{-c}}\right ) + \sqrt{2} a \sqrt{-c} c^{2}}{\sqrt{-c} \sqrt{c}}\right )}{\left | c \right |}}{c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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