Optimal. Leaf size=85 \[ -\frac{1}{3} x^2 \sqrt{c-a^2 c x^2}-\frac{(5-3 a x) \sqrt{c-a^2 c x^2}}{3 a^2}-\frac{\sqrt{c} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{a^2} \]
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Rubi [A] time = 0.183082, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6152, 1809, 780, 217, 203} \[ -\frac{1}{3} x^2 \sqrt{c-a^2 c x^2}-\frac{(5-3 a x) \sqrt{c-a^2 c x^2}}{3 a^2}-\frac{\sqrt{c} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{a^2} \]
Antiderivative was successfully verified.
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Rule 6152
Rule 1809
Rule 780
Rule 217
Rule 203
Rubi steps
\begin{align*} \int e^{-2 \tanh ^{-1}(a x)} x \sqrt{c-a^2 c x^2} \, dx &=c \int \frac{x (1-a x)^2}{\sqrt{c-a^2 c x^2}} \, dx\\ &=-\frac{1}{3} x^2 \sqrt{c-a^2 c x^2}-\frac{\int \frac{x \left (-5 a^2 c+6 a^3 c x\right )}{\sqrt{c-a^2 c x^2}} \, dx}{3 a^2}\\ &=-\frac{1}{3} x^2 \sqrt{c-a^2 c x^2}-\frac{(5-3 a x) \sqrt{c-a^2 c x^2}}{3 a^2}-\frac{c \int \frac{1}{\sqrt{c-a^2 c x^2}} \, dx}{a}\\ &=-\frac{1}{3} x^2 \sqrt{c-a^2 c x^2}-\frac{(5-3 a x) \sqrt{c-a^2 c x^2}}{3 a^2}-\frac{c \operatorname{Subst}\left (\int \frac{1}{1+a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c-a^2 c x^2}}\right )}{a}\\ &=-\frac{1}{3} x^2 \sqrt{c-a^2 c x^2}-\frac{(5-3 a x) \sqrt{c-a^2 c x^2}}{3 a^2}-\frac{\sqrt{c} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{a^2}\\ \end{align*}
Mathematica [A] time = 0.0813597, size = 80, normalized size = 0.94 \[ \frac{3 \sqrt{c} \tan ^{-1}\left (\frac{a x \sqrt{c-a^2 c x^2}}{\sqrt{c} \left (a^2 x^2-1\right )}\right )-\left (a^2 x^2-3 a x+5\right ) \sqrt{c-a^2 c x^2}}{3 a^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.036, size = 154, normalized size = 1.8 \begin{align*}{\frac{1}{3\,{a}^{2}c} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{x}{a}\sqrt{-{a}^{2}c{x}^{2}+c}}+{\frac{c}{a}\arctan \left ({x\sqrt{{a}^{2}c}{\frac{1}{\sqrt{-{a}^{2}c{x}^{2}+c}}}} \right ){\frac{1}{\sqrt{{a}^{2}c}}}}-2\,{\frac{\sqrt{-c{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,ac \left ( x+{a}^{-1} \right ) }}{{a}^{2}}}-2\,{\frac{c}{a\sqrt{{a}^{2}c}}\arctan \left ({\frac{\sqrt{{a}^{2}c}x}{\sqrt{-c{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,ac \left ( x+{a}^{-1} \right ) }}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.41889, size = 347, normalized size = 4.08 \begin{align*} \left [-\frac{2 \, \sqrt{-a^{2} c x^{2} + c}{\left (a^{2} x^{2} - 3 \, a x + 5\right )} - 3 \, \sqrt{-c} \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right )}{6 \, a^{2}}, -\frac{\sqrt{-a^{2} c x^{2} + c}{\left (a^{2} x^{2} - 3 \, a x + 5\right )} - 3 \, \sqrt{c} \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right )}{3 \, 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{- a^{2} c x^{2} + c}}{a x + 1}\, dx - \int \frac{a x^{2} \sqrt{- a^{2} c x^{2} + c}}{a x + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.30079, size = 242, normalized size = 2.85 \begin{align*} \frac{{\left (24 \, a^{4} c^{\frac{3}{2}} \arctan \left (\frac{\sqrt{-c + \frac{2 \, c}{a x + 1}}}{\sqrt{c}}\right ) \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right ) - \frac{{\left (9 \, a^{4}{\left (c - \frac{2 \, c}{a x + 1}\right )}^{2} c^{2} \sqrt{-c + \frac{2 \, c}{a x + 1}} \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right ) + 3 \, a^{4} c^{4} \sqrt{-c + \frac{2 \, c}{a x + 1}} \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right ) + 8 \, a^{4} c^{3}{\left (-c + \frac{2 \, c}{a x + 1}\right )}^{\frac{3}{2}} \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right )\right )}{\left (a x + 1\right )}^{3}}{c^{3}}\right )}{\left | a \right |}}{12 \, a^{7} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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