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 = 0.09, antiderivative size = 91, normalized size of antiderivative = 1.00, 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 43
Rule 6140
Rule 6143
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*}
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Mathematica [A] time = 0.04, 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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fricas [A] time = 0.58, size = 68, normalized size = 0.75 \[ -\frac {{\left (3 \, a^{3} c x^{4} - 4 \, a^{2} c x^{3} - 6 \, a c x^{2} + 12 \, c x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{12 \, {\left (a^{2} x^{2} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \sqrt {-a^{2} x^{2} + 1}}{a x + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 65, normalized size = 0.71 \[ \frac {x \left (3 x^{3} a^{3}-4 a^{2} x^{2}-6 a x +12\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \sqrt {-a^{2} x^{2}+1}}{12 \left (a x +1\right )^{2} \left (a x -1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \sqrt {-a^{2} x^{2} + 1}}{a x + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c-a^2\,c\,x^2\right )}^{3/2}\,\sqrt {1-a^2\,x^2}}{a\,x+1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{a x + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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