Optimal. Leaf size=40 \[ -\frac {1}{4} a^4 c^2 x^4-\frac {2}{3} a^3 c^2 x^3+2 a c^2 x+c^2 \log (x) \]
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Rubi [A] time = 0.07, antiderivative size = 40, 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, 75} \[ -\frac {1}{4} a^4 c^2 x^4-\frac {2}{3} a^3 c^2 x^3+2 a c^2 x+c^2 \log (x) \]
Antiderivative was successfully verified.
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Rule 75
Rule 6150
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^2}{x} \, dx &=c^2 \int \frac {(1-a x) (1+a x)^3}{x} \, dx\\ &=c^2 \int \left (2 a+\frac {1}{x}-2 a^3 x^2-a^4 x^3\right ) \, dx\\ &=2 a c^2 x-\frac {2}{3} a^3 c^2 x^3-\frac {1}{4} a^4 c^2 x^4+c^2 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 33, normalized size = 0.82 \[ -\frac {1}{12} c^2 \left (3 a^4 x^4+8 a^3 x^3-24 a x-12 \log (x)+3\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 36, normalized size = 0.90 \[ -\frac {1}{4} \, a^{4} c^{2} x^{4} - \frac {2}{3} \, a^{3} c^{2} x^{3} + 2 \, a c^{2} x + c^{2} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 37, normalized size = 0.92 \[ -\frac {1}{4} \, a^{4} c^{2} x^{4} - \frac {2}{3} \, a^{3} c^{2} x^{3} + 2 \, a c^{2} x + c^{2} \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 37, normalized size = 0.92 \[ 2 a \,c^{2} x -\frac {2 a^{3} c^{2} x^{3}}{3}-\frac {a^{4} c^{2} x^{4}}{4}+c^{2} \ln \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 36, normalized size = 0.90 \[ -\frac {1}{4} \, a^{4} c^{2} x^{4} - \frac {2}{3} \, a^{3} c^{2} x^{3} + 2 \, a c^{2} x + c^{2} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 36, normalized size = 0.90 \[ c^2\,\ln \relax (x)-\frac {2\,a^3\,c^2\,x^3}{3}-\frac {a^4\,c^2\,x^4}{4}+2\,a\,c^2\,x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 39, normalized size = 0.98 \[ - \frac {a^{4} c^{2} x^{4}}{4} - \frac {2 a^{3} c^{2} x^{3}}{3} + 2 a c^{2} x + c^{2} \log {\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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