Optimal. Leaf size=39 \[ \frac {1}{a^3 c (1-a x)}+\frac {2 \log (1-a x)}{a^3 c}+\frac {x}{a^2 c} \]
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Rubi [A] time = 0.10, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6150, 43} \[ \frac {x}{a^2 c}+\frac {1}{a^3 c (1-a x)}+\frac {2 \log (1-a x)}{a^3 c} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6150
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)} x^2}{c-a^2 c x^2} \, dx &=\frac {\int \frac {x^2}{(1-a x)^2} \, dx}{c}\\ &=\frac {\int \left (\frac {1}{a^2}+\frac {1}{a^2 (-1+a x)^2}+\frac {2}{a^2 (-1+a x)}\right ) \, dx}{c}\\ &=\frac {x}{a^2 c}+\frac {1}{a^3 c (1-a x)}+\frac {2 \log (1-a x)}{a^3 c}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 28, normalized size = 0.72 \[ \frac {a x+\frac {1}{1-a x}+2 \log (1-a x)}{a^3 c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 42, normalized size = 1.08 \[ \frac {a^{2} x^{2} - a x + 2 \, {\left (a x - 1\right )} \log \left (a x - 1\right ) - 1}{a^{4} c x - a^{3} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.88, size = 39, normalized size = 1.00 \[ \frac {x}{a^{2} c} + \frac {2 \, \log \left ({\left | a x - 1 \right |}\right )}{a^{3} c} - \frac {1}{{\left (a x - 1\right )} a^{3} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 39, normalized size = 1.00 \[ \frac {x}{a^{2} c}-\frac {1}{c \,a^{3} \left (a x -1\right )}+\frac {2 \ln \left (a x -1\right )}{c \,a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 40, normalized size = 1.03 \[ -\frac {1}{a^{4} c x - a^{3} c} + \frac {x}{a^{2} c} + \frac {2 \, \log \left (a x - 1\right )}{a^{3} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 38, normalized size = 0.97 \[ \frac {1}{a^3\,c-a^4\,c\,x}+\frac {x}{a^2\,c}+\frac {2\,\ln \left (a\,x-1\right )}{a^3\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 32, normalized size = 0.82 \[ - \frac {1}{a^{4} c x - a^{3} c} + \frac {x}{a^{2} c} + \frac {2 \log {\left (a x - 1 \right )}}{a^{3} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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