Optimal. Leaf size=112 \[ \frac{7 a c \sqrt{1-a^2 x^2}}{4 x \sqrt{c-a c x}}-\frac{c \sqrt{1-a^2 x^2}}{2 x^2 \sqrt{c-a c x}}-\frac{7}{4} a^2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{c-a c x}}\right ) \]
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Rubi [A] time = 0.195998, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {6128, 879, 873, 875, 208} \[ \frac{7 a c \sqrt{1-a^2 x^2}}{4 x \sqrt{c-a c x}}-\frac{c \sqrt{1-a^2 x^2}}{2 x^2 \sqrt{c-a c x}}-\frac{7}{4} a^2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{c-a c x}}\right ) \]
Antiderivative was successfully verified.
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Rule 6128
Rule 879
Rule 873
Rule 875
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{-\tanh ^{-1}(a x)} \sqrt{c-a c x}}{x^3} \, dx &=\frac{\int \frac{(c-a c x)^{3/2}}{x^3 \sqrt{1-a^2 x^2}} \, dx}{c}\\ &=-\frac{c \sqrt{1-a^2 x^2}}{2 x^2 \sqrt{c-a c x}}-\frac{1}{4} (7 a) \int \frac{\sqrt{c-a c x}}{x^2 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{c \sqrt{1-a^2 x^2}}{2 x^2 \sqrt{c-a c x}}+\frac{7 a c \sqrt{1-a^2 x^2}}{4 x \sqrt{c-a c x}}+\frac{1}{8} \left (7 a^2\right ) \int \frac{\sqrt{c-a c x}}{x \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{c \sqrt{1-a^2 x^2}}{2 x^2 \sqrt{c-a c x}}+\frac{7 a c \sqrt{1-a^2 x^2}}{4 x \sqrt{c-a c x}}+\frac{1}{4} \left (7 a^4 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}}{2 x^2 \sqrt{c-a c x}}+\frac{7 a c \sqrt{1-a^2 x^2}}{4 x \sqrt{c-a c x}}-\frac{7}{4} a^2 \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 = 0.032561, size = 64, normalized size = 0.57 \[ -\frac{c \sqrt{1-a x} \left (7 a^2 x^2 \tanh ^{-1}\left (\sqrt{a x+1}\right )+(2-7 a x) \sqrt{a x+1}\right )}{4 x^2 \sqrt{c-a c x}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.102, size = 101, normalized size = 0.9 \begin{align*}{\frac{1}{ \left ( 4\,ax-4 \right ){x}^{2}}\sqrt{-c \left ( ax-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1} \left ( 7\,c{\it Artanh} \left ({\frac{\sqrt{c \left ( ax+1 \right ) }}{\sqrt{c}}} \right ){x}^{2}{a}^{2}-7\,xa\sqrt{c \left ( ax+1 \right ) }\sqrt{c}+2\,\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^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.92857, size = 501, normalized size = 4.47 \begin{align*} \left [\frac{7 \,{\left (a^{3} x^{3} - a^{2} x^{2}\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}{\left (7 \, a x - 2\right )}}{8 \,{\left (a x^{3} - x^{2}\right )}}, -\frac{7 \,{\left (a^{3} x^{3} - a^{2} x^{2}\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}{\left (7 \, a x - 2\right )}}{4 \,{\left (a x^{3} - x^{2}\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{\sqrt{- c \left (a x - 1\right )} \sqrt{- \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 [A] time = 1.31209, size = 159, normalized size = 1.42 \begin{align*} \frac{{\left (a^{2} c^{3}{\left (\frac{7 \, \arctan \left (\frac{\sqrt{a c x + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c} + \frac{7 \,{\left (a c x + c\right )}^{\frac{3}{2}} - 9 \, \sqrt{a c x + c} c}{a^{2} c^{3} x^{2}}\right )} - \frac{7 \, a^{2} c^{2} \arctan \left (\frac{\sqrt{2} \sqrt{c}}{\sqrt{-c}}\right ) + 5 \, \sqrt{2} a^{2} \sqrt{-c} c^{\frac{3}{2}}}{\sqrt{-c}}\right )}{\left | c \right |}}{4 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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