Optimal. Leaf size=138 \[ -\frac {\sqrt {1-a^2 x^2}}{2 a^3 c (a x+1) \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2} \log (1-a x)}{4 a^3 c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {1-a^2 x^2} \log (a x+1)}{4 a^3 c \sqrt {c-a^2 c x^2}} \]
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Rubi [A] time = 0.26, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6153, 6150, 88} \[ -\frac {\sqrt {1-a^2 x^2}}{2 a^3 c (a x+1) \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2} \log (1-a x)}{4 a^3 c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {1-a^2 x^2} \log (a x+1)}{4 a^3 c \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 88
Rule 6150
Rule 6153
Rubi steps
\begin {align*} \int \frac {e^{-\tanh ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{3/2}} \, dx &=\frac {\sqrt {1-a^2 x^2} \int \frac {e^{-\tanh ^{-1}(a x)} x^2}{\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 {x^2}{(1-a x) (1+a x)^2} \, dx}{c \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2} \int \left (-\frac {1}{4 a^2 (-1+a x)}+\frac {1}{2 a^2 (1+a x)^2}-\frac {3}{4 a^2 (1+a x)}\right ) \, dx}{c \sqrt {c-a^2 c x^2}}\\ &=-\frac {\sqrt {1-a^2 x^2}}{2 a^3 c (1+a x) \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2} \log (1-a x)}{4 a^3 c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {1-a^2 x^2} \log (1+a x)}{4 a^3 c \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 72, normalized size = 0.52 \[ -\frac {\sqrt {1-a^2 x^2} ((a x+1) \log (1-a x)+3 (a x+1) \log (a x+1)+2)}{4 a^3 (a c x+c) \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} x^{2}}{a^{5} c^{2} x^{5} + a^{4} c^{2} x^{4} - 2 \, a^{3} c^{2} x^{3} - 2 \, a^{2} c^{2} x^{2} + a c^{2} x + c^{2}}, x\right ) \]
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 \[ \int \frac {\sqrt {-a^{2} x^{2} + 1} x^{2}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} {\left (a x + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 88, normalized size = 0.64 \[ \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (\ln \left (a x -1\right ) x a +3 a x \ln \left (a x +1\right )+\ln \left (a x -1\right )+3 \ln \left (a x +1\right )+2\right )}{4 \left (a^{2} x^{2}-1\right ) c^{2} a^{3} \left (a x +1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 52, normalized size = 0.38 \[ -\frac {\sqrt {c}}{2 \, {\left (a^{4} c^{2} x + a^{3} c^{2}\right )}} - \frac {3 \, \log \left (a x + 1\right )}{4 \, a^{3} c^{\frac {3}{2}}} - \frac {\log \left (a x - 1\right )}{4 \, a^{3} c^{\frac {3}{2}}} \]
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 \[ \int \frac {x^2\,\sqrt {1-a^2\,x^2}}{{\left (c-a^2\,c\,x^2\right )}^{3/2}\,\left (a\,x+1\right )} \,d x \]
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 \[ \int \frac {x^{2} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \left (a x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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