Optimal. Leaf size=76 \[ -\frac {c^3}{5 a^6 x^5}-\frac {c^3}{2 a^5 x^4}+\frac {c^3}{3 a^4 x^3}+\frac {2 c^3}{a^3 x^2}+\frac {c^3}{a^2 x}+\frac {2 c^3 \log (x)}{a}+c^3 x \]
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Rubi [A] time = 0.16, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6167, 6157, 6150, 88} \[ \frac {2 c^3}{a^3 x^2}+\frac {c^3}{3 a^4 x^3}-\frac {c^3}{2 a^5 x^4}-\frac {c^3}{5 a^6 x^5}+\frac {c^3}{a^2 x}+\frac {2 c^3 \log (x)}{a}+c^3 x \]
Antiderivative was successfully verified.
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Rule 88
Rule 6150
Rule 6157
Rule 6167
Rubi steps
\begin {align*} \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx\\ &=\frac {c^3 \int \frac {e^{2 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^3}{x^6} \, dx}{a^6}\\ &=\frac {c^3 \int \frac {(1-a x)^2 (1+a x)^4}{x^6} \, dx}{a^6}\\ &=\frac {c^3 \int \left (a^6+\frac {1}{x^6}+\frac {2 a}{x^5}-\frac {a^2}{x^4}-\frac {4 a^3}{x^3}-\frac {a^4}{x^2}+\frac {2 a^5}{x}\right ) \, dx}{a^6}\\ &=-\frac {c^3}{5 a^6 x^5}-\frac {c^3}{2 a^5 x^4}+\frac {c^3}{3 a^4 x^3}+\frac {2 c^3}{a^3 x^2}+\frac {c^3}{a^2 x}+c^3 x+\frac {2 c^3 \log (x)}{a}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 76, normalized size = 1.00 \[ -\frac {c^3}{5 a^6 x^5}-\frac {c^3}{2 a^5 x^4}+\frac {c^3}{3 a^4 x^3}+\frac {2 c^3}{a^3 x^2}+\frac {c^3}{a^2 x}+\frac {2 c^3 \log (x)}{a}+c^3 x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 78, normalized size = 1.03 \[ \frac {30 \, a^{6} c^{3} x^{6} + 60 \, a^{5} c^{3} x^{5} \log \relax (x) + 30 \, a^{4} c^{3} x^{4} + 60 \, a^{3} c^{3} x^{3} + 10 \, a^{2} c^{3} x^{2} - 15 \, a c^{3} x - 6 \, c^{3}}{30 \, a^{6} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 71, normalized size = 0.93 \[ c^{3} x + \frac {2 \, c^{3} \log \left ({\left | x \right |}\right )}{a} + \frac {30 \, a^{4} c^{3} x^{4} + 60 \, a^{3} c^{3} x^{3} + 10 \, a^{2} c^{3} x^{2} - 15 \, a c^{3} x - 6 \, c^{3}}{30 \, a^{6} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 71, normalized size = 0.93 \[ -\frac {c^{3}}{5 a^{6} x^{5}}-\frac {c^{3}}{2 a^{5} x^{4}}+\frac {c^{3}}{3 a^{4} x^{3}}+\frac {2 c^{3}}{x^{2} a^{3}}+\frac {c^{3}}{a^{2} x}+c^{3} x +\frac {2 c^{3} \ln \relax (x )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 70, normalized size = 0.92 \[ c^{3} x + \frac {2 \, c^{3} \log \relax (x)}{a} + \frac {30 \, a^{4} c^{3} x^{4} + 60 \, a^{3} c^{3} x^{3} + 10 \, a^{2} c^{3} x^{2} - 15 \, a c^{3} x - 6 \, c^{3}}{30 \, a^{6} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.22, size = 56, normalized size = 0.74 \[ \frac {c^3\,\left (\frac {a^2\,x^2}{3}-\frac {a\,x}{2}+2\,a^3\,x^3+a^4\,x^4+a^6\,x^6+2\,a^5\,x^5\,\ln \relax (x)-\frac {1}{5}\right )}{a^6\,x^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.33, size = 76, normalized size = 1.00 \[ \frac {a^{6} c^{3} x + 2 a^{5} c^{3} \log {\relax (x )} + \frac {30 a^{4} c^{3} x^{4} + 60 a^{3} c^{3} x^{3} + 10 a^{2} c^{3} x^{2} - 15 a c^{3} x - 6 c^{3}}{30 x^{5}}}{a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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