Optimal. Leaf size=39 \[ \frac {c^2}{3 a^4 x^3}+\frac {c^2}{a^3 x^2}+\frac {2 c^2 \log (x)}{a}+c^2 x \]
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Rubi [A] time = 0.15, antiderivative size = 39, 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, 75} \[ \frac {c^2}{a^3 x^2}+\frac {c^2}{3 a^4 x^3}+\frac {2 c^2 \log (x)}{a}+c^2 x \]
Antiderivative was successfully verified.
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Rule 75
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 )^2 \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx\\ &=-\frac {c^2 \int \frac {e^{2 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^2}{x^4} \, dx}{a^4}\\ &=-\frac {c^2 \int \frac {(1-a x) (1+a x)^3}{x^4} \, dx}{a^4}\\ &=-\frac {c^2 \int \left (-a^4+\frac {1}{x^4}+\frac {2 a}{x^3}-\frac {2 a^3}{x}\right ) \, dx}{a^4}\\ &=\frac {c^2}{3 a^4 x^3}+\frac {c^2}{a^3 x^2}+c^2 x+\frac {2 c^2 \log (x)}{a}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 39, normalized size = 1.00 \[ \frac {c^2}{3 a^4 x^3}+\frac {c^2}{a^3 x^2}+\frac {2 c^2 \log (x)}{a}+c^2 x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 43, normalized size = 1.10 \[ \frac {3 \, a^{4} c^{2} x^{4} + 6 \, a^{3} c^{2} x^{3} \log \relax (x) + 3 \, a c^{2} x + c^{2}}{3 \, a^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 36, normalized size = 0.92 \[ c^{2} x + \frac {2 \, c^{2} \log \left ({\left | x \right |}\right )}{a} + \frac {3 \, a c^{2} x + c^{2}}{3 \, a^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 38, normalized size = 0.97 \[ \frac {c^{2}}{3 a^{4} x^{3}}+\frac {c^{2}}{x^{2} a^{3}}+c^{2} x +\frac {2 c^{2} \ln \relax (x )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 35, normalized size = 0.90 \[ c^{2} x + \frac {2 \, c^{2} \log \relax (x)}{a} + \frac {3 \, a c^{2} x + c^{2}}{3 \, a^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 32, normalized size = 0.82 \[ \frac {c^2\,\left (a\,x+a^4\,x^4+2\,a^3\,x^3\,\ln \relax (x)+\frac {1}{3}\right )}{a^4\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 39, normalized size = 1.00 \[ \frac {a^{4} c^{2} x + 2 a^{3} c^{2} \log {\relax (x )} + \frac {3 a c^{2} x + c^{2}}{3 x^{3}}}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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