Optimal. Leaf size=91 \[ \frac{c (1-a x)^4 \sqrt{c-a^2 c x^2}}{4 a \sqrt{1-a^2 x^2}}-\frac{2 c (1-a x)^3 \sqrt{c-a^2 c x^2}}{3 a \sqrt{1-a^2 x^2}} \]
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Rubi [A] time = 0.0882952, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {6143, 6140, 43} \[ \frac{c (1-a x)^4 \sqrt{c-a^2 c x^2}}{4 a \sqrt{1-a^2 x^2}}-\frac{2 c (1-a x)^3 \sqrt{c-a^2 c x^2}}{3 a \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 6143
Rule 6140
Rule 43
Rubi steps
\begin{align*} \int e^{-\tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx &=\frac{\left (c \sqrt{c-a^2 c x^2}\right ) \int e^{-\tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^{3/2} \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\left (c \sqrt{c-a^2 c x^2}\right ) \int (1-a x)^2 (1+a x) \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\left (c \sqrt{c-a^2 c x^2}\right ) \int \left (2 (1-a x)^2-(1-a x)^3\right ) \, dx}{\sqrt{1-a^2 x^2}}\\ &=-\frac{2 c (1-a x)^3 \sqrt{c-a^2 c x^2}}{3 a \sqrt{1-a^2 x^2}}+\frac{c (1-a x)^4 \sqrt{c-a^2 c x^2}}{4 a \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.030942, size = 57, normalized size = 0.63 \[ \frac{c x \left (3 a^3 x^3-4 a^2 x^2-6 a x+12\right ) \sqrt{c-a^2 c x^2}}{12 \sqrt{1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.029, size = 65, normalized size = 0.7 \begin{align*}{\frac{x \left ( 3\,{x}^{3}{a}^{3}-4\,{a}^{2}{x}^{2}-6\,ax+12 \right ) }{12\, \left ( ax+1 \right ) ^{2} \left ( ax-1 \right ) ^{2}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}\sqrt{-{a}^{2}{x}^{2}+1}} \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^{2} c x^{2} + c\right )}^{\frac{3}{2}} \sqrt{-a^{2} x^{2} + 1}}{a x + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.20506, size = 149, normalized size = 1.64 \begin{align*} -\frac{{\left (3 \, a^{3} c x^{4} - 4 \, a^{2} c x^{3} - 6 \, a c x^{2} + 12 \, c x\right )} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{12 \,{\left (a^{2} x^{2} - 1\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{- \left (a x - 1\right ) \left (a x + 1\right )} \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}}}{a x + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}} \sqrt{-a^{2} x^{2} + 1}}{a x + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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