Optimal. Leaf size=31 \[ -\frac {1-2 a x}{6 a^3 c^3 (1-a x)^3 (1+a x)} \]
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Rubi [A]
time = 0.06, 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 = {6285, 82}
\begin {gather*} -\frac {1-2 a x}{6 a^3 c^3 (1-a x)^3 (a x+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 82
Rule 6285
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)^4 (1+a x)^2} \, dx}{c^3}\\ &=-\frac {1-2 a x}{6 a^3 c^3 (1-a x)^3 (1+a x)}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 30, normalized size = 0.97 \begin {gather*} \frac {1-2 a x}{6 a^3 c^3 (-1+a x)^3 (1+a x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 54, normalized size = 1.74
method | result | size |
gosper | \(-\frac {2 a x -1}{6 c^{3} \left (a x -1\right )^{2} \left (a^{2} x^{2}-1\right ) a^{3}}\) | \(33\) |
risch | \(\frac {-\frac {x}{3 a^{2}}+\frac {1}{6 a^{3}}}{c^{3} \left (a x -1\right )^{2} \left (a^{2} x^{2}-1\right )}\) | \(35\) |
norman | \(\frac {-\frac {x^{3}}{3 c}-\frac {a \,x^{4}}{2 c}+\frac {a^{3} x^{6}}{6 c}}{\left (a^{2} x^{2}-1\right )^{3} c^{2}}\) | \(45\) |
default | \(\frac {-\frac {1}{16 a^{3} \left (a x +1\right )}-\frac {1}{12 a^{3} \left (a x -1\right )^{3}}-\frac {1}{8 a^{3} \left (a x -1\right )^{2}}+\frac {1}{16 a^{3} \left (a x -1\right )}}{c^{3}}\) | \(54\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 49, normalized size = 1.58 \begin {gather*} -\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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 49, normalized size = 1.58 \begin {gather*} -\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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.20, size = 48, normalized size = 1.55 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 45, normalized size = 1.45 \begin {gather*} -\frac {1}{16 \, {\left (a x + 1\right )} a^{3} c^{3}} + \frac {3 \, a^{2} x^{2} - 12 \, a x + 5}{48 \, {\left (a x - 1\right )}^{3} a^{3} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 28, normalized size = 0.90 \begin {gather*} -\frac {2\,a\,x-1}{6\,a^3\,c^3\,{\left (a\,x-1\right )}^3\,\left (a\,x+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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