Optimal. Leaf size=99 \[ -\frac{x (1-a x) \sqrt{c-\frac{c}{a^2 x^2}}}{2 a}-\frac{3 x \sqrt{c-\frac{c}{a^2 x^2}}}{2 a}-\frac{3 x \sqrt{c-\frac{c}{a^2 x^2}} \sin ^{-1}(a x)}{2 a \sqrt{1-a x} \sqrt{a x+1}} \]
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Rubi [A] time = 0.305347, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {6167, 6159, 6129, 50, 41, 216} \[ -\frac{x (1-a x) \sqrt{c-\frac{c}{a^2 x^2}}}{2 a}-\frac{3 x \sqrt{c-\frac{c}{a^2 x^2}}}{2 a}-\frac{3 x \sqrt{c-\frac{c}{a^2 x^2}} \sin ^{-1}(a x)}{2 a \sqrt{1-a x} \sqrt{a x+1}} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6159
Rule 6129
Rule 50
Rule 41
Rule 216
Rubi steps
\begin{align*} \int e^{-2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} x \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} x \, dx\\ &=-\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int e^{-2 \tanh ^{-1}(a x)} \sqrt{1-a x} \sqrt{1+a x} \, dx}{\sqrt{1-a x} \sqrt{1+a x}}\\ &=-\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{(1-a x)^{3/2}}{\sqrt{1+a x}} \, dx}{\sqrt{1-a x} \sqrt{1+a x}}\\ &=-\frac{\sqrt{c-\frac{c}{a^2 x^2}} x (1-a x)}{2 a}-\frac{\left (3 \sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{\sqrt{1-a x}}{\sqrt{1+a x}} \, dx}{2 \sqrt{1-a x} \sqrt{1+a x}}\\ &=-\frac{3 \sqrt{c-\frac{c}{a^2 x^2}} x}{2 a}-\frac{\sqrt{c-\frac{c}{a^2 x^2}} x (1-a x)}{2 a}-\frac{\left (3 \sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{1}{\sqrt{1-a x} \sqrt{1+a x}} \, dx}{2 \sqrt{1-a x} \sqrt{1+a x}}\\ &=-\frac{3 \sqrt{c-\frac{c}{a^2 x^2}} x}{2 a}-\frac{\sqrt{c-\frac{c}{a^2 x^2}} x (1-a x)}{2 a}-\frac{\left (3 \sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{2 \sqrt{1-a x} \sqrt{1+a x}}\\ &=-\frac{3 \sqrt{c-\frac{c}{a^2 x^2}} x}{2 a}-\frac{\sqrt{c-\frac{c}{a^2 x^2}} x (1-a x)}{2 a}-\frac{3 \sqrt{c-\frac{c}{a^2 x^2}} x \sin ^{-1}(a x)}{2 a \sqrt{1-a x} \sqrt{1+a x}}\\ \end{align*}
Mathematica [A] time = 0.0632169, size = 100, normalized size = 1.01 \[ -\frac{x \sqrt{c-\frac{c}{a^2 x^2}} \left (\sqrt{a x+1} \left (a^2 x^2-5 a x+4\right )-6 \sqrt{1-a x} \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )\right )}{2 a \sqrt{1-a x} \sqrt{1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.172, size = 147, normalized size = 1.5 \begin{align*} -{\frac{x}{2\,{a}^{2}}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}}} \left ( -x\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}}{a}^{2}+\sqrt{c}\ln \left ( x\sqrt{c}+\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}} \right ) -4\,\sqrt{c}\ln \left ({\frac{1}{\sqrt{c}} \left ( \sqrt{c}\sqrt{{\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) c}{{a}^{2}}}}+cx \right ) } \right ) +4\,\sqrt{{\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) c}{{a}^{2}}}}a \right ){\frac{1}{\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{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{{\left (a x - 1\right )} \sqrt{c - \frac{c}{a^{2} x^{2}}} x}{a x + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72474, size = 404, normalized size = 4.08 \begin{align*} \left [\frac{2 \,{\left (a^{2} x^{2} - 4 \, a x\right )} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} + 3 \, \sqrt{c} \log \left (2 \, a^{2} c x^{2} + 2 \, a^{2} \sqrt{c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right )}{4 \, a^{2}}, \frac{{\left (a^{2} x^{2} - 4 \, a x\right )} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 3 \, \sqrt{-c} \arctan \left (\frac{a^{2} \sqrt{-c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right )}{2 \, a^{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{x \sqrt{- c \left (-1 + \frac{1}{a x}\right ) \left (1 + \frac{1}{a x}\right )} \left (a x - 1\right )}{a x + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17384, size = 143, normalized size = 1.44 \begin{align*} \frac{1}{4} \,{\left (2 \, \sqrt{a^{2} c x^{2} - c}{\left (\frac{x \mathrm{sgn}\left (x\right )}{a^{2}} - \frac{4 \, \mathrm{sgn}\left (x\right )}{a^{3}}\right )} - \frac{6 \, \sqrt{c} \log \left ({\left | -\sqrt{a^{2} c} x + \sqrt{a^{2} c x^{2} - c} \right |}\right ) \mathrm{sgn}\left (x\right )}{a^{2}{\left | a \right |}} + \frac{{\left (3 \, a \sqrt{c} \log \left ({\left | c \right |}\right ) + 8 \, \sqrt{-c}{\left | a \right |}\right )} \mathrm{sgn}\left (x\right )}{a^{3}{\left | a \right |}}\right )}{\left | a \right |} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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