Optimal. Leaf size=31 \[ \frac {2 a x+1}{6 a^3 c^3 (1-a x) (a x+1)^3} \]
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Rubi [A] time = 0.09, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6150, 81} \[ \frac {2 a x+1}{6 a^3 c^3 (1-a x) (a x+1)^3} \]
Antiderivative was successfully verified.
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Rule 81
Rule 6150
Rubi steps
\begin {align*} \int \frac {e^{-2 \tanh ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^3} \, dx &=\frac {\int \frac {x^2}{(1-a x)^2 (1+a x)^4} \, dx}{c^3}\\ &=\frac {1+2 a x}{6 a^3 c^3 (1-a x) (1+a x)^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 30, normalized size = 0.97 \[ -\frac {2 a x+1}{6 a^3 c^3 (a x-1) (a x+1)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 49, normalized size = 1.58 \[ -\frac {2 \, a x + 1}{6 \, {\left (a^{7} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{3} - 2 \, a^{4} c^{3} x - a^{3} c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 76, normalized size = 2.45 \[ \frac {1}{32 \, a^{3} c^{3} {\left (\frac {2}{a x + 1} - 1\right )}} + \frac {\frac {3 \, a^{3} c^{6}}{a x + 1} + \frac {6 \, a^{3} c^{6}}{{\left (a x + 1\right )}^{2}} - \frac {4 \, a^{3} c^{6}}{{\left (a x + 1\right )}^{3}}}{48 \, a^{6} c^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 54, normalized size = 1.74 \[ \frac {-\frac {1}{16 a^{3} \left (a x -1\right )}-\frac {1}{12 a^{3} \left (a x +1\right )^{3}}+\frac {1}{8 a^{3} \left (a x +1\right )^{2}}+\frac {1}{16 a^{3} \left (a x +1\right )}}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 49, normalized size = 1.58 \[ -\frac {2 \, a x + 1}{6 \, {\left (a^{7} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{3} - 2 \, a^{4} c^{3} x - a^{3} c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 28, normalized size = 0.90 \[ -\frac {2\,a\,x+1}{6\,a^3\,c^3\,\left (a\,x-1\right )\,{\left (a\,x+1\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.36, size = 49, normalized size = 1.58 \[ \frac {- 2 a x - 1}{6 a^{7} c^{3} x^{4} + 12 a^{6} c^{3} x^{3} - 12 a^{4} c^{3} x - 6 a^{3} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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