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 = 0.0802476, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, 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 6150
Rule 75
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*}
Mathematica [A] time = 0.0150206, size = 39, normalized size = 1. \[ a^4 \left (-c^2\right ) x-2 a^3 c^2 \log (x)-\frac{a c^2}{x^2}-\frac{c^2}{3 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.031, size = 38, normalized size = 1. \begin{align*} -{\frac{{c}^{2}}{3\,{x}^{3}}}-{\frac{a{c}^{2}}{{x}^{2}}}-{a}^{4}{c}^{2}x-2\,{a}^{3}{c}^{2}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.948565, size = 49, normalized size = 1.26 \begin{align*} -a^{4} c^{2} x - 2 \, a^{3} c^{2} \log \left (x\right ) - \frac{3 \, a c^{2} x + c^{2}}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67061, size = 90, normalized size = 2.31 \begin{align*} -\frac{3 \, a^{4} c^{2} x^{4} + 6 \, a^{3} c^{2} x^{3} \log \left (x\right ) + 3 \, a c^{2} x + c^{2}}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.35641, size = 37, normalized size = 0.95 \begin{align*} - a^{4} c^{2} x - 2 a^{3} c^{2} \log{\left (x \right )} - \frac{3 a c^{2} x + c^{2}}{3 x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13319, size = 50, normalized size = 1.28 \begin{align*} -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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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