Optimal. Leaf size=76 \[ \frac{1}{5} a^6 c^3 x^5+\frac{1}{2} a^5 c^3 x^4-\frac{1}{3} a^4 c^3 x^3-2 a^3 c^3 x^2-a^2 c^3 x+2 a c^3 \log (x)-\frac{c^3}{x} \]
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Rubi [A] time = 0.0908948, antiderivative size = 76, 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, 88} \[ \frac{1}{5} a^6 c^3 x^5+\frac{1}{2} a^5 c^3 x^4-\frac{1}{3} a^4 c^3 x^3-2 a^3 c^3 x^2-a^2 c^3 x+2 a c^3 \log (x)-\frac{c^3}{x} \]
Antiderivative was successfully verified.
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Rule 6150
Rule 88
Rubi steps
\begin{align*} \int \frac{e^{2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^3}{x^2} \, dx &=c^3 \int \frac{(1-a x)^2 (1+a x)^4}{x^2} \, dx\\ &=c^3 \int \left (-a^2+\frac{1}{x^2}+\frac{2 a}{x}-4 a^3 x-a^4 x^2+2 a^5 x^3+a^6 x^4\right ) \, dx\\ &=-\frac{c^3}{x}-a^2 c^3 x-2 a^3 c^3 x^2-\frac{1}{3} a^4 c^3 x^3+\frac{1}{2} a^5 c^3 x^4+\frac{1}{5} a^6 c^3 x^5+2 a c^3 \log (x)\\ \end{align*}
Mathematica [A] time = 0.025573, size = 59, normalized size = 0.78 \[ c^3 \left (\frac{a^6 x^5}{5}+\frac{a^5 x^4}{2}-\frac{a^4 x^3}{3}-2 a^3 x^2-a^2 x+2 a \log (x)-\frac{1}{x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.031, size = 71, normalized size = 0.9 \begin{align*} -{\frac{{c}^{3}}{x}}-{a}^{2}{c}^{3}x-2\,{a}^{3}{c}^{3}{x}^{2}-{\frac{{a}^{4}{c}^{3}{x}^{3}}{3}}+{\frac{{a}^{5}{c}^{3}{x}^{4}}{2}}+{\frac{{a}^{6}{c}^{3}{x}^{5}}{5}}+2\,a{c}^{3}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.956938, size = 95, normalized size = 1.25 \begin{align*} \frac{1}{5} \, a^{6} c^{3} x^{5} + \frac{1}{2} \, a^{5} c^{3} x^{4} - \frac{1}{3} \, a^{4} c^{3} x^{3} - 2 \, a^{3} c^{3} x^{2} - a^{2} c^{3} x + 2 \, a c^{3} \log \left (x\right ) - \frac{c^{3}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.93189, size = 163, normalized size = 2.14 \begin{align*} \frac{6 \, a^{6} c^{3} x^{6} + 15 \, a^{5} c^{3} x^{5} - 10 \, a^{4} c^{3} x^{4} - 60 \, a^{3} c^{3} x^{3} - 30 \, a^{2} c^{3} x^{2} + 60 \, a c^{3} x \log \left (x\right ) - 30 \, c^{3}}{30 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.343045, size = 70, normalized size = 0.92 \begin{align*} \frac{a^{6} c^{3} x^{5}}{5} + \frac{a^{5} c^{3} x^{4}}{2} - \frac{a^{4} c^{3} x^{3}}{3} - 2 a^{3} c^{3} x^{2} - a^{2} c^{3} x + 2 a c^{3} \log{\left (x \right )} - \frac{c^{3}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14991, size = 96, normalized size = 1.26 \begin{align*} \frac{1}{5} \, a^{6} c^{3} x^{5} + \frac{1}{2} \, a^{5} c^{3} x^{4} - \frac{1}{3} \, a^{4} c^{3} x^{3} - 2 \, a^{3} c^{3} x^{2} - a^{2} c^{3} x + 2 \, a c^{3} \log \left ({\left | x \right |}\right ) - \frac{c^{3}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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