Optimal. Leaf size=41 \[ -\frac {1}{3} a^4 c^2 x^3-a^3 c^2 x^2+2 a c^2 \log (x)-\frac {c^2}{x} \]
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Rubi [A] time = 0.08, antiderivative size = 41, 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} \[ -\frac {1}{3} a^4 c^2 x^3-a^3 c^2 x^2+2 a c^2 \log (x)-\frac {c^2}{x} \]
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^2} \, dx &=c^2 \int \frac {(1-a x) (1+a x)^3}{x^2} \, dx\\ &=c^2 \int \left (\frac {1}{x^2}+\frac {2 a}{x}-2 a^3 x-a^4 x^2\right ) \, dx\\ &=-\frac {c^2}{x}-a^3 c^2 x^2-\frac {1}{3} a^4 c^2 x^3+2 a c^2 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 41, normalized size = 1.00 \[ -\frac {1}{3} a^4 c^2 x^3-a^3 c^2 x^2+2 a c^2 \log (x)-\frac {c^2}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 41, normalized size = 1.00 \[ -\frac {a^{4} c^{2} x^{4} + 3 \, a^{3} c^{2} x^{3} - 6 \, a c^{2} x \log \relax (x) + 3 \, c^{2}}{3 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 40, normalized size = 0.98 \[ -\frac {1}{3} \, a^{4} c^{2} x^{3} - a^{3} c^{2} x^{2} + 2 \, a c^{2} \log \left ({\left | x \right |}\right ) - \frac {c^{2}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 40, normalized size = 0.98 \[ -\frac {c^{2}}{x}-a^{3} c^{2} x^{2}-\frac {a^{4} c^{2} x^{3}}{3}+2 a \,c^{2} \ln \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 39, normalized size = 0.95 \[ -\frac {1}{3} \, a^{4} c^{2} x^{3} - a^{3} c^{2} x^{2} + 2 \, a c^{2} \log \relax (x) - \frac {c^{2}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 39, normalized size = 0.95 \[ 2\,a\,c^2\,\ln \relax (x)-\frac {c^2}{x}-a^3\,c^2\,x^2-\frac {a^4\,c^2\,x^3}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 36, normalized size = 0.88 \[ - \frac {a^{4} c^{2} x^{3}}{3} - a^{3} c^{2} x^{2} + 2 a c^{2} \log {\relax (x )} - \frac {c^{2}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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