Optimal. Leaf size=72 \[ \frac{1}{3} a^6 c^3 x^3+a^5 c^3 x^2-a^4 c^3 x+\frac{a^2 c^3}{x}-4 a^3 c^3 \log (x)-\frac{a c^3}{x^2}-\frac{c^3}{3 x^3} \]
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Rubi [A] time = 0.0929093, antiderivative size = 72, 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} \[ \frac{1}{3} a^6 c^3 x^3+a^5 c^3 x^2-a^4 c^3 x+\frac{a^2 c^3}{x}-4 a^3 c^3 \log (x)-\frac{a c^3}{x^2}-\frac{c^3}{3 x^3} \]
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^4} \, dx &=c^3 \int \frac{(1-a x)^2 (1+a x)^4}{x^4} \, dx\\ &=c^3 \int \left (-a^4+\frac{1}{x^4}+\frac{2 a}{x^3}-\frac{a^2}{x^2}-\frac{4 a^3}{x}+2 a^5 x+a^6 x^2\right ) \, dx\\ &=-\frac{c^3}{3 x^3}-\frac{a c^3}{x^2}+\frac{a^2 c^3}{x}-a^4 c^3 x+a^5 c^3 x^2+\frac{1}{3} a^6 c^3 x^3-4 a^3 c^3 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0293488, size = 55, normalized size = 0.76 \[ c^3 \left (\frac{a^6 x^3}{3}+a^5 x^2-a^4 x+\frac{a^2}{x}-4 a^3 \log (x)-\frac{a}{x^2}-\frac{1}{3 x^3}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.031, size = 69, normalized size = 1. \begin{align*} -{\frac{{c}^{3}}{3\,{x}^{3}}}-{\frac{a{c}^{3}}{{x}^{2}}}+{\frac{{a}^{2}{c}^{3}}{x}}-{a}^{4}{c}^{3}x+{a}^{5}{c}^{3}{x}^{2}+{\frac{{a}^{6}{c}^{3}{x}^{3}}{3}}-4\,{a}^{3}{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 = 0.942724, size = 95, normalized size = 1.32 \begin{align*} \frac{1}{3} \, a^{6} c^{3} x^{3} + a^{5} c^{3} x^{2} - a^{4} c^{3} x - 4 \, a^{3} c^{3} \log \left (x\right ) + \frac{3 \, a^{2} c^{3} x^{2} - 3 \, a c^{3} x - c^{3}}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.03493, size = 153, normalized size = 2.12 \begin{align*} \frac{a^{6} c^{3} x^{6} + 3 \, a^{5} c^{3} x^{5} - 3 \, a^{4} c^{3} x^{4} - 12 \, a^{3} c^{3} x^{3} \log \left (x\right ) + 3 \, a^{2} c^{3} x^{2} - 3 \, a c^{3} x - c^{3}}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.409236, size = 70, normalized size = 0.97 \begin{align*} \frac{a^{6} c^{3} x^{3}}{3} + a^{5} c^{3} x^{2} - a^{4} c^{3} x - 4 a^{3} c^{3} \log{\left (x \right )} + \frac{3 a^{2} c^{3} x^{2} - 3 a c^{3} x - c^{3}}{3 x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15008, size = 96, normalized size = 1.33 \begin{align*} \frac{1}{3} \, a^{6} c^{3} x^{3} + a^{5} c^{3} x^{2} - a^{4} c^{3} x - 4 \, a^{3} c^{3} \log \left ({\left | x \right |}\right ) + \frac{3 \, a^{2} c^{3} x^{2} - 3 \, a c^{3} x - c^{3}}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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