Optimal. Leaf size=39 \[ a^4 \left (-c^2\right ) x-2 a^3 c^2 \log (x)-\frac {a c^2}{x^2}-\frac {c^2}{3 x^3} \]
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Rubi [A] time = 0.08, 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, 75} \[ a^4 \left (-c^2\right ) x-2 a^3 c^2 \log (x)-\frac {a c^2}{x^2}-\frac {c^2}{3 x^3} \]
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^4} \, dx &=c^2 \int \frac {(1-a x) (1+a x)^3}{x^4} \, dx\\ &=c^2 \int \left (-a^4+\frac {1}{x^4}+\frac {2 a}{x^3}-\frac {2 a^3}{x}\right ) \, dx\\ &=-\frac {c^2}{3 x^3}-\frac {a c^2}{x^2}-a^4 c^2 x-2 a^3 c^2 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 39, normalized size = 1.00 \[ a^4 \left (-c^2\right ) x-2 a^3 c^2 \log (x)-\frac {a c^2}{x^2}-\frac {c^2}{3 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 40, normalized size = 1.03 \[ -\frac {3 \, a^{4} c^{2} x^{4} + 6 \, a^{3} c^{2} x^{3} \log \relax (x) + 3 \, a c^{2} x + c^{2}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.45, size = 37, normalized size = 0.95 \[ -a^{4} c^{2} x - 2 \, a^{3} c^{2} \log \left ({\left | x \right |}\right ) - \frac {3 \, a c^{2} x + c^{2}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 38, normalized size = 0.97 \[ -\frac {c^{2}}{3 x^{3}}-\frac {a \,c^{2}}{x^{2}}-a^{4} c^{2} x -2 a^{3} c^{2} \ln \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 36, normalized size = 0.92 \[ -a^{4} c^{2} x - 2 \, a^{3} c^{2} \log \relax (x) - \frac {3 \, a c^{2} x + c^{2}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 32, normalized size = 0.82 \[ -\frac {c^2\,\left (3\,a\,x+3\,a^4\,x^4+6\,a^3\,x^3\,\ln \relax (x)+1\right )}{3\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.19, size = 37, normalized size = 0.95 \[ - a^{4} c^{2} x - 2 a^{3} c^{2} \log {\relax (x )} - \frac {3 a c^{2} x + c^{2}}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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