Optimal. Leaf size=76 \[ \frac {1}{5} a^6 c^3 x^5+\frac {1}{2} a^5 c^3 x^4-\frac {1}{3} a^4 c^3 x^3-2 a^3 c^3 x^2-a^2 c^3 x+2 a c^3 \log (x)-\frac {c^3}{x} \]
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Rubi [A] time = 0.09, antiderivative size = 76, 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} \[ \frac {1}{5} a^6 c^3 x^5+\frac {1}{2} a^5 c^3 x^4-\frac {1}{3} a^4 c^3 x^3-2 a^3 c^3 x^2-a^2 c^3 x+2 a c^3 \log (x)-\frac {c^3}{x} \]
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^2} \, dx &=c^3 \int \frac {(1-a x)^2 (1+a x)^4}{x^2} \, dx\\ &=c^3 \int \left (-a^2+\frac {1}{x^2}+\frac {2 a}{x}-4 a^3 x-a^4 x^2+2 a^5 x^3+a^6 x^4\right ) \, dx\\ &=-\frac {c^3}{x}-a^2 c^3 x-2 a^3 c^3 x^2-\frac {1}{3} a^4 c^3 x^3+\frac {1}{2} a^5 c^3 x^4+\frac {1}{5} a^6 c^3 x^5+2 a c^3 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 59, normalized size = 0.78 \[ c^3 \left (\frac {a^6 x^5}{5}+\frac {a^5 x^4}{2}-\frac {a^4 x^3}{3}-2 a^3 x^2-a^2 x+2 a \log (x)-\frac {1}{x}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 75, normalized size = 0.99 \[ \frac {6 \, a^{6} c^{3} x^{6} + 15 \, a^{5} c^{3} x^{5} - 10 \, a^{4} c^{3} x^{4} - 60 \, a^{3} c^{3} x^{3} - 30 \, a^{2} c^{3} x^{2} + 60 \, a c^{3} x \log \relax (x) - 30 \, c^{3}}{30 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 71, normalized size = 0.93 \[ \frac {1}{5} \, a^{6} c^{3} x^{5} + \frac {1}{2} \, a^{5} c^{3} x^{4} - \frac {1}{3} \, a^{4} c^{3} x^{3} - 2 \, a^{3} c^{3} x^{2} - a^{2} c^{3} x + 2 \, a c^{3} \log \left ({\left | x \right |}\right ) - \frac {c^{3}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 71, normalized size = 0.93 \[ -\frac {c^{3}}{x}-a^{2} c^{3} x -2 a^{3} c^{3} x^{2}-\frac {a^{4} c^{3} x^{3}}{3}+\frac {a^{5} c^{3} x^{4}}{2}+\frac {a^{6} c^{3} x^{5}}{5}+2 a \,c^{3} \ln \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 70, normalized size = 0.92 \[ \frac {1}{5} \, a^{6} c^{3} x^{5} + \frac {1}{2} \, a^{5} c^{3} x^{4} - \frac {1}{3} \, a^{4} c^{3} x^{3} - 2 \, a^{3} c^{3} x^{2} - a^{2} c^{3} x + 2 \, a c^{3} \log \relax (x) - \frac {c^{3}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 70, normalized size = 0.92 \[ 2\,a\,c^3\,\ln \relax (x)-a^2\,c^3\,x-\frac {c^3}{x}-2\,a^3\,c^3\,x^2-\frac {a^4\,c^3\,x^3}{3}+\frac {a^5\,c^3\,x^4}{2}+\frac {a^6\,c^3\,x^5}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 70, normalized size = 0.92 \[ \frac {a^{6} c^{3} x^{5}}{5} + \frac {a^{5} c^{3} x^{4}}{2} - \frac {a^{4} c^{3} x^{3}}{3} - 2 a^{3} c^{3} x^{2} - a^{2} c^{3} x + 2 a c^{3} \log {\relax (x )} - \frac {c^{3}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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