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 = 0.10, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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 88
Rule 6150
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.67, size = 75, normalized size = 0.96 \[ \frac {3 \, a^{6} c^{3} x^{6} + 8 \, a^{5} c^{3} x^{5} - 6 \, a^{4} c^{3} x^{4} - 48 \, a^{3} c^{3} x^{3} - 12 \, a^{2} c^{3} x^{2} \log \relax (x) - 24 \, a c^{3} x - 6 \, c^{3}}{12 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.04, size = 70, normalized size = 0.90 \[ \frac {1}{4} \, a^{6} c^{3} x^{4} + \frac {2}{3} \, a^{5} c^{3} x^{3} - \frac {1}{2} \, a^{4} c^{3} x^{2} - 4 \, a^{3} c^{3} x - a^{2} c^{3} \log \left ({\left | x \right |}\right ) - \frac {4 \, a c^{3} x + c^{3}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 71, normalized size = 0.91 \[ -\frac {c^{3}}{2 x^{2}}-\frac {2 a \,c^{3}}{x}-4 a^{3} c^{3} x -\frac {a^{4} c^{3} x^{2}}{2}+\frac {2 a^{5} c^{3} x^{3}}{3}+\frac {a^{6} c^{3} x^{4}}{4}-a^{2} c^{3} \ln \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 69, normalized size = 0.88 \[ \frac {1}{4} \, a^{6} c^{3} x^{4} + \frac {2}{3} \, a^{5} c^{3} x^{3} - \frac {1}{2} \, a^{4} c^{3} x^{2} - 4 \, a^{3} c^{3} x - a^{2} c^{3} \log \relax (x) - \frac {4 \, a c^{3} x + c^{3}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 71, normalized size = 0.91 \[ \frac {2\,a^5\,c^3\,x^3}{3}-4\,a^3\,c^3\,x-\frac {a^4\,c^3\,x^2}{2}-\frac {\frac {c^3}{2}+2\,a\,c^3\,x}{x^2}+\frac {a^6\,c^3\,x^4}{4}-a^2\,c^3\,\ln \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.20, size = 75, normalized size = 0.96 \[ \frac {a^{6} c^{3} x^{4}}{4} + \frac {2 a^{5} c^{3} x^{3}}{3} - \frac {a^{4} c^{3} x^{2}}{2} - 4 a^{3} c^{3} x - a^{2} c^{3} \log {\relax (x )} + \frac {- 4 a c^{3} x - c^{3}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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