Optimal. Leaf size=40 \[ -\frac{c^2}{a^3 x^2}+\frac{c^2}{3 a^4 x^3}-\frac{2 c^2 \log (x)}{a}+c^2 x \]
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Rubi [A] time = 0.141085, antiderivative size = 40, normalized size of antiderivative = 1., 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 6167
Rule 6157
Rule 6150
Rule 75
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)^3 (1+a x)}{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*}
Mathematica [A] time = 0.0193561, size = 40, normalized size = 1. \[ -\frac{c^2}{a^3 x^2}+\frac{c^2}{3 a^4 x^3}-\frac{2 c^2 \log (x)}{a}+c^2 x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 39, normalized size = 1. \begin{align*}{\frac{{c}^{2}}{3\,{a}^{4}{x}^{3}}}-{\frac{{c}^{2}}{{x}^{2}{a}^{3}}}+x{c}^{2}-2\,{\frac{{c}^{2}\ln \left ( x \right ) }{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07959, size = 50, normalized size = 1.25 \begin{align*} c^{2} x - \frac{2 \, c^{2} \log \left (x\right )}{a} - \frac{3 \, a c^{2} x - c^{2}}{3 \, a^{4} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.21624, size = 97, normalized size = 2.42 \begin{align*} \frac{3 \, a^{4} c^{2} x^{4} - 6 \, a^{3} c^{2} x^{3} \log \left (x\right ) - 3 \, a c^{2} x + c^{2}}{3 \, a^{4} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.347533, size = 39, normalized size = 0.98 \begin{align*} \frac{a^{4} c^{2} x - 2 a^{3} c^{2} \log{\left (x \right )} - \frac{3 a c^{2} x - c^{2}}{3 x^{3}}}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13297, size = 51, normalized size = 1.27 \begin{align*} 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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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