Optimal. Leaf size=72 \[ \frac {1}{3} a^6 c^3 x^3+a^5 c^3 x^2-a^4 c^3 x-4 a^3 c^3 \log (x)+\frac {a^2 c^3}{x}-\frac {a c^3}{x^2}-\frac {c^3}{3 x^3} \]
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Rubi [A] time = 0.09, antiderivative size = 72, 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} \[ \frac {1}{3} a^6 c^3 x^3+a^5 c^3 x^2-a^4 c^3 x+\frac {a^2 c^3}{x}-4 a^3 c^3 \log (x)-\frac {a c^3}{x^2}-\frac {c^3}{3 x^3} \]
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^4} \, dx &=c^3 \int \frac {(1-a x)^2 (1+a x)^4}{x^4} \, dx\\ &=c^3 \int \left (-a^4+\frac {1}{x^4}+\frac {2 a}{x^3}-\frac {a^2}{x^2}-\frac {4 a^3}{x}+2 a^5 x+a^6 x^2\right ) \, dx\\ &=-\frac {c^3}{3 x^3}-\frac {a c^3}{x^2}+\frac {a^2 c^3}{x}-a^4 c^3 x+a^5 c^3 x^2+\frac {1}{3} a^6 c^3 x^3-4 a^3 c^3 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 55, normalized size = 0.76 \[ c^3 \left (\frac {a^6 x^3}{3}+a^5 x^2-a^4 x-4 a^3 \log (x)+\frac {a^2}{x}-\frac {a}{x^2}-\frac {1}{3 x^3}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 74, normalized size = 1.03 \[ \frac {a^{6} c^{3} x^{6} + 3 \, a^{5} c^{3} x^{5} - 3 \, a^{4} c^{3} x^{4} - 12 \, a^{3} c^{3} x^{3} \log \relax (x) + 3 \, a^{2} c^{3} x^{2} - 3 \, a c^{3} x - c^{3}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 71, normalized size = 0.99 \[ \frac {1}{3} \, a^{6} c^{3} x^{3} + a^{5} c^{3} x^{2} - a^{4} c^{3} x - 4 \, a^{3} c^{3} \log \left ({\left | x \right |}\right ) + \frac {3 \, a^{2} c^{3} x^{2} - 3 \, a c^{3} x - c^{3}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 69, normalized size = 0.96 \[ -\frac {c^{3}}{3 x^{3}}-\frac {a \,c^{3}}{x^{2}}+\frac {a^{2} c^{3}}{x}-a^{4} c^{3} x +a^{5} c^{3} x^{2}+\frac {a^{6} c^{3} x^{3}}{3}-4 a^{3} c^{3} \ln \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 70, normalized size = 0.97 \[ \frac {1}{3} \, a^{6} c^{3} x^{3} + a^{5} c^{3} x^{2} - a^{4} c^{3} x - 4 \, a^{3} c^{3} \log \relax (x) + \frac {3 \, a^{2} c^{3} x^{2} - 3 \, a c^{3} x - c^{3}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 69, normalized size = 0.96 \[ a^5\,c^3\,x^2-a^4\,c^3\,x-\frac {-a^2\,c^3\,x^2+a\,c^3\,x+\frac {c^3}{3}}{x^3}+\frac {a^6\,c^3\,x^3}{3}-4\,a^3\,c^3\,\ln \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.23, size = 70, normalized size = 0.97 \[ \frac {a^{6} c^{3} x^{3}}{3} + a^{5} c^{3} x^{2} - a^{4} c^{3} x - 4 a^{3} c^{3} \log {\relax (x )} + \frac {3 a^{2} c^{3} x^{2} - 3 a c^{3} x - c^{3}}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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