Optimal. Leaf size=79 \[ \frac{1}{6} a^6 c^3 x^6+\frac{2}{5} a^5 c^3 x^5-\frac{1}{4} a^4 c^3 x^4-\frac{4}{3} a^3 c^3 x^3-\frac{1}{2} a^2 c^3 x^2+2 a c^3 x+c^3 \log (x) \]
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Rubi [A] time = 0.0856803, antiderivative size = 79, 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} \[ \frac{1}{6} a^6 c^3 x^6+\frac{2}{5} a^5 c^3 x^5-\frac{1}{4} a^4 c^3 x^4-\frac{4}{3} a^3 c^3 x^3-\frac{1}{2} a^2 c^3 x^2+2 a c^3 x+c^3 \log (x) \]
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} \, dx &=c^3 \int \frac{(1-a x)^2 (1+a x)^4}{x} \, dx\\ &=c^3 \int \left (2 a+\frac{1}{x}-a^2 x-4 a^3 x^2-a^4 x^3+2 a^5 x^4+a^6 x^5\right ) \, dx\\ &=2 a c^3 x-\frac{1}{2} a^2 c^3 x^2-\frac{4}{3} a^3 c^3 x^3-\frac{1}{4} a^4 c^3 x^4+\frac{2}{5} a^5 c^3 x^5+\frac{1}{6} a^6 c^3 x^6+c^3 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0290283, size = 51, normalized size = 0.65 \[ c^3 \left (\frac{1}{60} a x \left (10 a^5 x^5+24 a^4 x^4-15 a^3 x^3-80 a^2 x^2-30 a x+120\right )+\log (x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 70, normalized size = 0.9 \begin{align*} 2\,a{c}^{3}x-{\frac{{a}^{2}{c}^{3}{x}^{2}}{2}}-{\frac{4\,{a}^{3}{c}^{3}{x}^{3}}{3}}-{\frac{{a}^{4}{c}^{3}{x}^{4}}{4}}+{\frac{2\,{a}^{5}{c}^{3}{x}^{5}}{5}}+{\frac{{a}^{6}{c}^{3}{x}^{6}}{6}}+{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 = 0.961181, size = 93, normalized size = 1.18 \begin{align*} \frac{1}{6} \, a^{6} c^{3} x^{6} + \frac{2}{5} \, a^{5} c^{3} x^{5} - \frac{1}{4} \, a^{4} c^{3} x^{4} - \frac{4}{3} \, a^{3} c^{3} x^{3} - \frac{1}{2} \, a^{2} c^{3} x^{2} + 2 \, a c^{3} x + c^{3} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.9625, size = 154, normalized size = 1.95 \begin{align*} \frac{1}{6} \, a^{6} c^{3} x^{6} + \frac{2}{5} \, a^{5} c^{3} x^{5} - \frac{1}{4} \, a^{4} c^{3} x^{4} - \frac{4}{3} \, a^{3} c^{3} x^{3} - \frac{1}{2} \, a^{2} c^{3} x^{2} + 2 \, a c^{3} x + c^{3} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.32037, size = 76, normalized size = 0.96 \begin{align*} \frac{a^{6} c^{3} x^{6}}{6} + \frac{2 a^{5} c^{3} x^{5}}{5} - \frac{a^{4} c^{3} x^{4}}{4} - \frac{4 a^{3} c^{3} x^{3}}{3} - \frac{a^{2} c^{3} x^{2}}{2} + 2 a c^{3} x + c^{3} \log{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18015, size = 95, normalized size = 1.2 \begin{align*} \frac{1}{6} \, a^{6} c^{3} x^{6} + \frac{2}{5} \, a^{5} c^{3} x^{5} - \frac{1}{4} \, a^{4} c^{3} x^{4} - \frac{4}{3} \, a^{3} c^{3} x^{3} - \frac{1}{2} \, a^{2} c^{3} x^{2} + 2 \, a c^{3} x + c^{3} \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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