Optimal. Leaf size=123 \[ \frac{x (a x+1)^2 \sqrt{c-\frac{c}{a^2 x^2}}}{3 a^2}+\frac{x (a x+1) \sqrt{c-\frac{c}{a^2 x^2}}}{3 a^2}+\frac{x \sqrt{c-\frac{c}{a^2 x^2}}}{a^2}-\frac{x \sqrt{c-\frac{c}{a^2 x^2}} \sin ^{-1}(a x)}{a^2 \sqrt{1-a x} \sqrt{a x+1}} \]
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Rubi [A] time = 0.472021, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {6167, 6159, 6129, 80, 50, 41, 216} \[ \frac{x (a x+1)^2 \sqrt{c-\frac{c}{a^2 x^2}}}{3 a^2}+\frac{x (a x+1) \sqrt{c-\frac{c}{a^2 x^2}}}{3 a^2}+\frac{x \sqrt{c-\frac{c}{a^2 x^2}}}{a^2}-\frac{x \sqrt{c-\frac{c}{a^2 x^2}} \sin ^{-1}(a x)}{a^2 \sqrt{1-a x} \sqrt{a x+1}} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6159
Rule 6129
Rule 80
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^2 \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} x^2 \, dx\\ &=-\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int e^{2 \tanh ^{-1}(a x)} 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{x (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}{3 a^2}-\frac{\left (2 \sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{(1+a x)^{3/2}}{\sqrt{1-a x}} \, dx}{3 a \sqrt{1-a x} \sqrt{1+a x}}\\ &=\frac{\sqrt{c-\frac{c}{a^2 x^2}} x (1+a x)}{3 a^2}+\frac{\sqrt{c-\frac{c}{a^2 x^2}} x (1+a x)^2}{3 a^2}-\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{\sqrt{1+a x}}{\sqrt{1-a x}} \, dx}{a \sqrt{1-a x} \sqrt{1+a x}}\\ &=\frac{\sqrt{c-\frac{c}{a^2 x^2}} x}{a^2}+\frac{\sqrt{c-\frac{c}{a^2 x^2}} x (1+a x)}{3 a^2}+\frac{\sqrt{c-\frac{c}{a^2 x^2}} x (1+a x)^2}{3 a^2}-\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{1}{\sqrt{1-a x} \sqrt{1+a x}} \, dx}{a \sqrt{1-a x} \sqrt{1+a x}}\\ &=\frac{\sqrt{c-\frac{c}{a^2 x^2}} x}{a^2}+\frac{\sqrt{c-\frac{c}{a^2 x^2}} x (1+a x)}{3 a^2}+\frac{\sqrt{c-\frac{c}{a^2 x^2}} x (1+a x)^2}{3 a^2}-\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{a \sqrt{1-a x} \sqrt{1+a x}}\\ &=\frac{\sqrt{c-\frac{c}{a^2 x^2}} x}{a^2}+\frac{\sqrt{c-\frac{c}{a^2 x^2}} x (1+a x)}{3 a^2}+\frac{\sqrt{c-\frac{c}{a^2 x^2}} x (1+a x)^2}{3 a^2}-\frac{\sqrt{c-\frac{c}{a^2 x^2}} x \sin ^{-1}(a x)}{a^2 \sqrt{1-a x} \sqrt{1+a x}}\\ \end{align*}
Mathematica [A] time = 0.0721543, size = 84, normalized size = 0.68 \[ \frac{x \sqrt{c-\frac{c}{a^2 x^2}} \left (\sqrt{a^2 x^2-1} \left (a^2 x^2+3 a x+5\right )+3 \log \left (\sqrt{a^2 x^2-1}+a x\right )\right )}{3 a^2 \sqrt{a^2 x^2-1}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.195, size = 173, normalized size = 1.4 \begin{align*}{\frac{x}{3\,{a}^{3}c}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}}} \left ( \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}} \right ) ^{{\frac{3}{2}}}{a}^{3}+3\,\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}}x{a}^{2}c-3\,{c}^{3/2}\ln \left ( x\sqrt{c}+\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}} \right ) +6\,{c}^{3/2}\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 ) +6\,\sqrt{{\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) c}{{a}^{2}}}}ac \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^{2}}{a x - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71404, size = 436, normalized size = 3.54 \begin{align*} \left [\frac{2 \,{\left (a^{3} x^{3} + 3 \, a^{2} x^{2} + 5 \, 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 )}{6 \, a^{3}}, \frac{{\left (a^{3} x^{3} + 3 \, a^{2} x^{2} + 5 \, 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 )}{3 \, a^{3}}\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^{2} \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 = 1.20295, size = 157, normalized size = 1.28 \begin{align*} \frac{1}{6} \,{\left (2 \, \sqrt{a^{2} c x^{2} - c}{\left (x{\left (\frac{x \mathrm{sgn}\left (x\right )}{a^{2}} + \frac{3 \, \mathrm{sgn}\left (x\right )}{a^{3}}\right )} + \frac{5 \, \mathrm{sgn}\left (x\right )}{a^{4}}\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^{3}{\left | a \right |}} + \frac{{\left (3 \, a \sqrt{c} \log \left ({\left | c \right |}\right ) - 10 \, \sqrt{-c}{\left | a \right |}\right )} \mathrm{sgn}\left (x\right )}{a^{4}{\left | a \right |}}\right )}{\left | a \right |} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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