Optimal. Leaf size=47 \[ \frac {\sqrt {1-a^2 x^2}}{2 a c (1-a x)^2 \sqrt {c-a^2 c x^2}} \]
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Rubi [A] time = 0.08, antiderivative size = 47, 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} \[ \frac {\sqrt {1-a^2 x^2}}{2 a c (1-a x)^2 \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 32
Rule 6140
Rule 6143
Rubi steps
\begin {align*} \int \frac {e^{3 \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx &=\frac {\sqrt {1-a^2 x^2} \int \frac {e^{3 \tanh ^{-1}(a x)}}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2} \int \frac {1}{(1-a x)^3} \, dx}{c \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2}}{2 a c (1-a x)^2 \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 53, normalized size = 1.13 \[ -\frac {\sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}{2 a c^2 (a x-1)^3 (a x+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 72, normalized size = 1.53 \[ \frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} {\left (a x^{2} - 2 \, x\right )}}{2 \, {\left (a^{4} c^{2} x^{4} - 2 \, a^{3} c^{2} x^{3} + 2 \, a c^{2} x - c^{2}\right )}} \]
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 \[ \int \frac {{\left (a x + 1\right )}^{3}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 43, normalized size = 0.91 \[ -\frac {\left (a x -1\right ) \left (a x +1\right )^{3}}{2 a \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}} \]
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 \[ \int \frac {{\left (a x + 1\right )}^{3}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.25, size = 74, normalized size = 1.57 \[ \frac {\sqrt {c-a^2\,c\,x^2}}{2\,a^3\,c^2\,\left (\frac {\sqrt {1-a^2\,x^2}}{a^2}+x^2\,\sqrt {1-a^2\,x^2}-\frac {2\,x\,\sqrt {1-a^2\,x^2}}{a}\right )} \]
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 \[ \int \frac {\left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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