Optimal. Leaf size=51 \[ -\frac {3}{4 a^3 c^2 (1-a x)}+\frac {1}{4 a^3 c^2 (1-a x)^2}+\frac {\tanh ^{-1}(a x)}{4 a^3 c^2} \]
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Rubi [A] time = 0.11, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {6150, 88, 207} \[ -\frac {3}{4 a^3 c^2 (1-a x)}+\frac {1}{4 a^3 c^2 (1-a x)^2}+\frac {\tanh ^{-1}(a x)}{4 a^3 c^2} \]
Antiderivative was successfully verified.
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Rule 88
Rule 207
Rule 6150
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^2} \, dx &=\frac {\int \frac {x^2}{(1-a x)^3 (1+a x)} \, dx}{c^2}\\ &=\frac {\int \left (-\frac {1}{2 a^2 (-1+a x)^3}-\frac {3}{4 a^2 (-1+a x)^2}-\frac {1}{4 a^2 \left (-1+a^2 x^2\right )}\right ) \, dx}{c^2}\\ &=\frac {1}{4 a^3 c^2 (1-a x)^2}-\frac {3}{4 a^3 c^2 (1-a x)}-\frac {\int \frac {1}{-1+a^2 x^2} \, dx}{4 a^2 c^2}\\ &=\frac {1}{4 a^3 c^2 (1-a x)^2}-\frac {3}{4 a^3 c^2 (1-a x)}+\frac {\tanh ^{-1}(a x)}{4 a^3 c^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 35, normalized size = 0.69 \[ \frac {3 a x+(a x-1)^2 \tanh ^{-1}(a x)-2}{4 a^3 c^2 (a x-1)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 78, normalized size = 1.53 \[ \frac {6 \, a x + {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (a x + 1\right ) - {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (a x - 1\right ) - 4}{8 \, {\left (a^{5} c^{2} x^{2} - 2 \, a^{4} c^{2} x + a^{3} c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 52, normalized size = 1.02 \[ \frac {\log \left ({\left | a x + 1 \right |}\right )}{8 \, a^{3} c^{2}} - \frac {\log \left ({\left | a x - 1 \right |}\right )}{8 \, a^{3} c^{2}} + \frac {3 \, a x - 2}{4 \, {\left (a x - 1\right )}^{2} a^{3} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 60, normalized size = 1.18 \[ \frac {1}{4 c^{2} a^{3} \left (a x -1\right )^{2}}+\frac {3}{4 c^{2} a^{3} \left (a x -1\right )}-\frac {\ln \left (a x -1\right )}{8 c^{2} a^{3}}+\frac {\ln \left (a x +1\right )}{8 c^{2} a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 66, normalized size = 1.29 \[ \frac {3 \, a x - 2}{4 \, {\left (a^{5} c^{2} x^{2} - 2 \, a^{4} c^{2} x + a^{3} c^{2}\right )}} + \frac {\log \left (a x + 1\right )}{8 \, a^{3} c^{2}} - \frac {\log \left (a x - 1\right )}{8 \, a^{3} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 49, normalized size = 0.96 \[ \frac {\frac {3\,x}{4\,a^2}-\frac {1}{2\,a^3}}{a^2\,c^2\,x^2-2\,a\,c^2\,x+c^2}+\frac {\mathrm {atanh}\left (a\,x\right )}{4\,a^3\,c^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.30, size = 61, normalized size = 1.20 \[ - \frac {- 3 a x + 2}{4 a^{5} c^{2} x^{2} - 8 a^{4} c^{2} x + 4 a^{3} c^{2}} - \frac {\frac {\log {\left (x - \frac {1}{a} \right )}}{8} - \frac {\log {\left (x + \frac {1}{a} \right )}}{8}}{a^{3} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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