Optimal. Leaf size=53 \[ \frac {1}{a^4 c (1-a x)}+\frac {3 \log (1-a x)}{a^4 c}+\frac {2 x}{a^3 c}+\frac {x^2}{2 a^2 c} \]
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Rubi [A] time = 0.10, antiderivative size = 53, 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} \[ \frac {x^2}{2 a^2 c}+\frac {2 x}{a^3 c}+\frac {1}{a^4 c (1-a x)}+\frac {3 \log (1-a x)}{a^4 c} \]
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^3}{c-a^2 c x^2} \, dx &=\frac {\int \frac {x^3}{(1-a x)^2} \, dx}{c}\\ &=\frac {\int \left (\frac {2}{a^3}+\frac {x}{a^2}+\frac {1}{a^3 (-1+a x)^2}+\frac {3}{a^3 (-1+a x)}\right ) \, dx}{c}\\ &=\frac {2 x}{a^3 c}+\frac {x^2}{2 a^2 c}+\frac {1}{a^4 c (1-a x)}+\frac {3 \log (1-a x)}{a^4 c}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 41, normalized size = 0.77 \[ \frac {a^2 x^2+4 a x+\frac {2}{1-a x}+6 \log (1-a x)}{2 a^4 c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 51, normalized size = 0.96 \[ \frac {a^{3} x^{3} + 3 \, a^{2} x^{2} - 4 \, a x + 6 \, {\left (a x - 1\right )} \log \left (a x - 1\right ) - 2}{2 \, {\left (a^{5} c x - a^{4} c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 53, normalized size = 1.00 \[ \frac {3 \, \log \left ({\left | a x - 1 \right |}\right )}{a^{4} c} + \frac {a^{2} c x^{2} + 4 \, a c x}{2 \, a^{4} c^{2}} - \frac {1}{{\left (a x - 1\right )} a^{4} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 51, normalized size = 0.96 \[ \frac {x^{2}}{2 a^{2} c}+\frac {2 x}{a^{3} c}-\frac {1}{c \,a^{4} \left (a x -1\right )}+\frac {3 \ln \left (a x -1\right )}{c \,a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 49, normalized size = 0.92 \[ -\frac {1}{a^{5} c x - a^{4} c} + \frac {a x^{2} + 4 \, x}{2 \, a^{3} c} + \frac {3 \, \log \left (a x - 1\right )}{a^{4} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.89, size = 54, normalized size = 1.02 \[ \frac {1}{a\,\left (a^3\,c-a^4\,c\,x\right )}+\frac {2\,x}{a^3\,c}+\frac {x^2}{2\,a^2\,c}+\frac {3\,\ln \left (a\,x-1\right )}{a^4\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 44, normalized size = 0.83 \[ - \frac {1}{a^{5} c x - a^{4} c} + \frac {x^{2}}{2 a^{2} c} + \frac {2 x}{a^{3} c} + \frac {3 \log {\left (a x - 1 \right )}}{a^{4} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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