Optimal. Leaf size=87 \[ \frac{(1-a x) \sqrt{c-a^2 c x^2}}{2 a}+\frac{3 \sqrt{c-a^2 c x^2}}{2 a}+\frac{3 \sqrt{c} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{2 a} \]
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Rubi [A] time = 0.0753709, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {6142, 671, 641, 217, 203} \[ \frac{(1-a x) \sqrt{c-a^2 c x^2}}{2 a}+\frac{3 \sqrt{c-a^2 c x^2}}{2 a}+\frac{3 \sqrt{c} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{2 a} \]
Antiderivative was successfully verified.
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Rule 6142
Rule 671
Rule 641
Rule 217
Rule 203
Rubi steps
\begin{align*} \int e^{-2 \tanh ^{-1}(a x)} \sqrt{c-a^2 c x^2} \, dx &=c \int \frac{(1-a x)^2}{\sqrt{c-a^2 c x^2}} \, dx\\ &=\frac{(1-a x) \sqrt{c-a^2 c x^2}}{2 a}+\frac{1}{2} (3 c) \int \frac{1-a x}{\sqrt{c-a^2 c x^2}} \, dx\\ &=\frac{3 \sqrt{c-a^2 c x^2}}{2 a}+\frac{(1-a x) \sqrt{c-a^2 c x^2}}{2 a}+\frac{1}{2} (3 c) \int \frac{1}{\sqrt{c-a^2 c x^2}} \, dx\\ &=\frac{3 \sqrt{c-a^2 c x^2}}{2 a}+\frac{(1-a x) \sqrt{c-a^2 c x^2}}{2 a}+\frac{1}{2} (3 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 )\\ &=\frac{3 \sqrt{c-a^2 c x^2}}{2 a}+\frac{(1-a x) \sqrt{c-a^2 c x^2}}{2 a}+\frac{3 \sqrt{c} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{2 a}\\ \end{align*}
Mathematica [A] time = 0.0506014, size = 99, normalized size = 1.14 \[ \frac{\sqrt{c-a^2 c x^2} \left (\sqrt{a x+1} \left (a^2 x^2-5 a x+4\right )-6 \sqrt{1-a x} \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )\right )}{2 a \sqrt{1-a x} \sqrt{1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.034, size = 126, normalized size = 1.5 \begin{align*} -{\frac{x}{2}\sqrt{-{a}^{2}c{x}^{2}+c}}-{\frac{c}{2}\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\,{\frac{c}{\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 = 2.43303, size = 309, normalized size = 3.55 \begin{align*} \left [-\frac{2 \, \sqrt{-a^{2} c x^{2} + c}{\left (a x - 4\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 )}{4 \, a}, -\frac{\sqrt{-a^{2} c x^{2} + c}{\left (a x - 4\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 )}{2 \, a}\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{- a^{2} c x^{2} + c}}{a x + 1}\, dx - \int \frac{a x \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 [A] time = 1.20922, size = 177, normalized size = 2.03 \begin{align*} -\frac{{\left (12 \, a^{3} 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 (3 \, a^{3} c^{3} \sqrt{-c + \frac{2 \, c}{a x + 1}} \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right ) + 5 \, a^{3} c^{2}{\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 )}^{2}}{c^{2}}\right )}{\left | a \right |}}{4 \, a^{5} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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