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 = 0.09, antiderivative size = 79, 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} \[ \frac {1}{6} a^6 c^3 x^6+\frac {2}{5} a^5 c^3 x^5-\frac {1}{4} a^4 c^3 x^4-\frac {4}{3} a^3 c^3 x^3-\frac {1}{2} a^2 c^3 x^2+2 a c^3 x+c^3 \log (x) \]
Antiderivative was successfully verified.
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Rule 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} \, 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*}
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Mathematica [A] time = 0.03, 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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fricas [A] time = 1.36, size = 69, normalized size = 0.87 \[ \frac {1}{6} \, a^{6} c^{3} x^{6} + \frac {2}{5} \, a^{5} c^{3} x^{5} - \frac {1}{4} \, a^{4} c^{3} x^{4} - \frac {4}{3} \, a^{3} c^{3} x^{3} - \frac {1}{2} \, a^{2} c^{3} x^{2} + 2 \, a c^{3} x + c^{3} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 70, normalized size = 0.89 \[ \frac {1}{6} \, a^{6} c^{3} x^{6} + \frac {2}{5} \, a^{5} c^{3} x^{5} - \frac {1}{4} \, a^{4} c^{3} x^{4} - \frac {4}{3} \, a^{3} c^{3} x^{3} - \frac {1}{2} \, a^{2} c^{3} x^{2} + 2 \, a c^{3} x + c^{3} \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 70, normalized size = 0.89 \[ 2 a \,c^{3} x -\frac {a^{2} c^{3} x^{2}}{2}-\frac {4 a^{3} c^{3} x^{3}}{3}-\frac {a^{4} c^{3} x^{4}}{4}+\frac {2 a^{5} c^{3} x^{5}}{5}+\frac {a^{6} c^{3} x^{6}}{6}+c^{3} \ln \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 69, normalized size = 0.87 \[ \frac {1}{6} \, a^{6} c^{3} x^{6} + \frac {2}{5} \, a^{5} c^{3} x^{5} - \frac {1}{4} \, a^{4} c^{3} x^{4} - \frac {4}{3} \, a^{3} c^{3} x^{3} - \frac {1}{2} \, a^{2} c^{3} x^{2} + 2 \, a c^{3} x + c^{3} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 69, normalized size = 0.87 \[ c^3\,\ln \relax (x)-\frac {a^2\,c^3\,x^2}{2}-\frac {4\,a^3\,c^3\,x^3}{3}-\frac {a^4\,c^3\,x^4}{4}+\frac {2\,a^5\,c^3\,x^5}{5}+\frac {a^6\,c^3\,x^6}{6}+2\,a\,c^3\,x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 76, normalized size = 0.96 \[ \frac {a^{6} c^{3} x^{6}}{6} + \frac {2 a^{5} c^{3} x^{5}}{5} - \frac {a^{4} c^{3} x^{4}}{4} - \frac {4 a^{3} c^{3} x^{3}}{3} - \frac {a^{2} c^{3} x^{2}}{2} + 2 a c^{3} x + c^{3} \log {\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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