Optimal. Leaf size=45 \[ -\frac {c (1-a x)^4 \sqrt {c-a^2 c x^2}}{4 a \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.08, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6143, 6140, 32} \[ -\frac {c (1-a x)^4 \sqrt {c-a^2 c x^2}}{4 a \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 32
Rule 6140
Rule 6143
Rubi steps
\begin {align*} \int e^{-3 \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^{-3 \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)^3 \, dx}{\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 = 58, normalized size = 1.29 \[ \frac {c \left (-\frac {1}{4} a^3 x^4+a^2 x^3-\frac {3 a x^2}{2}+x\right ) \sqrt {c-a^2 c x^2}}{\sqrt {1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.65, size = 67, normalized size = 1.49 \[ \frac {{\left (a^{3} c x^{4} - 4 \, a^{2} c x^{3} + 6 \, a c x^{2} - 4 \, c x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{4 \, {\left (a^{2} x^{2} - 1\right )}} \]
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 \[ \int \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a x + 1\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 64, normalized size = 1.42 \[ \frac {x \left (x^{3} a^{3}-4 a^{2} x^{2}+6 a x -4\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{4 \left (a x -1\right )^{3} \left (a x +1\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 70, normalized size = 1.56 \[ -\frac {{\left (a^{4} c^{\frac {3}{2}} x^{4} - 4 \, a^{3} c^{\frac {3}{2}} x^{3} + 6 \, a^{2} c^{\frac {3}{2}} x^{2} - 4 \, a c^{\frac {3}{2}} x + 4 \, c^{\frac {3}{2}}\right )} {\left (a x + 1\right )} {\left (a x - 1\right )}}{4 \, {\left (a^{3} x^{2} - a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (c-a^2\,c\,x^2\right )}^{3/2}\,{\left (1-a^2\,x^2\right )}^{3/2}}{{\left (a\,x+1\right )}^3} \,d x \]
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 \[ \int \frac {\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{\left (a x + 1\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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