Optimal. Leaf size=78 \[ \frac{1}{4} a^6 c^3 x^4+\frac{2}{3} a^5 c^3 x^3-\frac{1}{2} a^4 c^3 x^2-4 a^3 c^3 x-a^2 c^3 \log (x)-\frac{2 a c^3}{x}-\frac{c^3}{2 x^2} \]
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Rubi [A] time = 0.095062, antiderivative size = 78, 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} \[ \frac{1}{4} a^6 c^3 x^4+\frac{2}{3} a^5 c^3 x^3-\frac{1}{2} a^4 c^3 x^2-4 a^3 c^3 x-a^2 c^3 \log (x)-\frac{2 a c^3}{x}-\frac{c^3}{2 x^2} \]
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^3} \, dx &=c^3 \int \frac{(1-a x)^2 (1+a x)^4}{x^3} \, dx\\ &=c^3 \int \left (-4 a^3+\frac{1}{x^3}+\frac{2 a}{x^2}-\frac{a^2}{x}-a^4 x+2 a^5 x^2+a^6 x^3\right ) \, dx\\ &=-\frac{c^3}{2 x^2}-\frac{2 a c^3}{x}-4 a^3 c^3 x-\frac{1}{2} a^4 c^3 x^2+\frac{2}{3} a^5 c^3 x^3+\frac{1}{4} a^6 c^3 x^4-a^2 c^3 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0239966, size = 58, normalized size = 0.74 \[ \frac{c^3 \left (3 a^6 x^6+8 a^5 x^5-6 a^4 x^4-48 a^3 x^3-12 a^2 x^2 \log (x)-24 a x-6\right )}{12 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 71, normalized size = 0.9 \begin{align*} -{\frac{{c}^{3}}{2\,{x}^{2}}}-2\,{\frac{a{c}^{3}}{x}}-4\,{a}^{3}{c}^{3}x-{\frac{{a}^{4}{c}^{3}{x}^{2}}{2}}+{\frac{2\,{a}^{5}{c}^{3}{x}^{3}}{3}}+{\frac{{a}^{6}{c}^{3}{x}^{4}}{4}}-{a}^{2}{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 = 0.965557, size = 93, normalized size = 1.19 \begin{align*} \frac{1}{4} \, a^{6} c^{3} x^{4} + \frac{2}{3} \, a^{5} c^{3} x^{3} - \frac{1}{2} \, a^{4} c^{3} x^{2} - 4 \, a^{3} c^{3} x - a^{2} c^{3} \log \left (x\right ) - \frac{4 \, a c^{3} x + c^{3}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01215, size = 162, normalized size = 2.08 \begin{align*} \frac{3 \, a^{6} c^{3} x^{6} + 8 \, a^{5} c^{3} x^{5} - 6 \, a^{4} c^{3} x^{4} - 48 \, a^{3} c^{3} x^{3} - 12 \, a^{2} c^{3} x^{2} \log \left (x\right ) - 24 \, a c^{3} x - 6 \, c^{3}}{12 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.370847, size = 73, normalized size = 0.94 \begin{align*} \frac{a^{6} c^{3} x^{4}}{4} + \frac{2 a^{5} c^{3} x^{3}}{3} - \frac{a^{4} c^{3} x^{2}}{2} - 4 a^{3} c^{3} x - a^{2} c^{3} \log{\left (x \right )} - \frac{4 a c^{3} x + c^{3}}{2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16721, size = 95, normalized size = 1.22 \begin{align*} \frac{1}{4} \, a^{6} c^{3} x^{4} + \frac{2}{3} \, a^{5} c^{3} x^{3} - \frac{1}{2} \, a^{4} c^{3} x^{2} - 4 \, a^{3} c^{3} x - a^{2} c^{3} \log \left ({\left | x \right |}\right ) - \frac{4 \, a c^{3} x + c^{3}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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