Optimal. Leaf size=52 \[ \frac {x}{3 c \sqrt {c-a^2 c x^2}}-\frac {2 (1-a x)}{3 a \left (c-a^2 c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.06, antiderivative size = 52, 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 = {6142, 653, 191} \[ \frac {x}{3 c \sqrt {c-a^2 c x^2}}-\frac {2 (1-a x)}{3 a \left (c-a^2 c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 653
Rule 6142
Rubi steps
\begin {align*} \int \frac {e^{-2 \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx &=c \int \frac {(1-a x)^2}{\left (c-a^2 c x^2\right )^{5/2}} \, dx\\ &=-\frac {2 (1-a x)}{3 a \left (c-a^2 c x^2\right )^{3/2}}+\frac {1}{3} \int \frac {1}{\left (c-a^2 c x^2\right )^{3/2}} \, dx\\ &=-\frac {2 (1-a x)}{3 a \left (c-a^2 c x^2\right )^{3/2}}+\frac {x}{3 c \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 63, normalized size = 1.21 \[ -\frac {\sqrt {1-a x} (a x+2) \sqrt {1-a^2 x^2}}{3 a c (a x+1)^{3/2} \sqrt {c-a^2 c x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.58, size = 47, normalized size = 0.90 \[ -\frac {\sqrt {-a^{2} c x^{2} + c} {\left (a x + 2\right )}}{3 \, {\left (a^{3} c^{2} x^{2} + 2 \, a^{2} c^{2} x + a c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.98, size = 82, normalized size = 1.58 \[ \frac {\frac {2 \, \sqrt {-c} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\relax (a)}{c^{2}} - \frac {3 \, c \sqrt {-c + \frac {2 \, c}{a x + 1}} + {\left (-c + \frac {2 \, c}{a x + 1}\right )}^{\frac {3}{2}}}{c^{3} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\relax (a)}}{6 \, {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 31, normalized size = 0.60 \[ -\frac {\left (a x -1\right )^{2} \left (a x +2\right )}{3 \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 60, normalized size = 1.15 \[ \frac {x}{3 \, \sqrt {-a^{2} c x^{2} + c} c} - \frac {2}{3 \, {\left (\sqrt {-a^{2} c x^{2} + c} a^{2} c x + \sqrt {-a^{2} c x^{2} + c} a c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.98, size = 33, normalized size = 0.63 \[ -\frac {\sqrt {c-a^2\,c\,x^2}\,\left (a\,x+2\right )}{3\,a\,c^2\,{\left (a\,x+1\right )}^2} \]
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 \[ - \int \frac {a x}{- a^{3} c x^{3} \sqrt {- a^{2} c x^{2} + c} - a^{2} c x^{2} \sqrt {- a^{2} c x^{2} + c} + a c x \sqrt {- a^{2} c x^{2} + c} + c \sqrt {- a^{2} c x^{2} + c}}\, dx - \int \left (- \frac {1}{- a^{3} c x^{3} \sqrt {- a^{2} c x^{2} + c} - a^{2} c x^{2} \sqrt {- a^{2} c x^{2} + c} + a c x \sqrt {- a^{2} c x^{2} + c} + c \sqrt {- a^{2} c x^{2} + c}}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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