Optimal. Leaf size=51 \[ -\frac{2 c^2}{a^3 x^2}-\frac{c^2}{3 a^4 x^3}-\frac{6 c^2}{a^2 x}+\frac{4 c^2 \log (x)}{a}+c^2 x \]
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Rubi [A] time = 0.147742, antiderivative size = 51, 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, 43} \[ -\frac{2 c^2}{a^3 x^2}-\frac{c^2}{3 a^4 x^3}-\frac{6 c^2}{a^2 x}+\frac{4 c^2 \log (x)}{a}+c^2 x \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6157
Rule 6150
Rule 43
Rubi steps
\begin{align*} \int e^{4 \coth ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^2 \, dx &=\int e^{4 \tanh ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^2 \, dx\\ &=\frac{c^2 \int \frac{e^{4 \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)^4}{x^4} \, dx}{a^4}\\ &=\frac{c^2 \int \left (a^4+\frac{1}{x^4}+\frac{4 a}{x^3}+\frac{6 a^2}{x^2}+\frac{4 a^3}{x}\right ) \, dx}{a^4}\\ &=-\frac{c^2}{3 a^4 x^3}-\frac{2 c^2}{a^3 x^2}-\frac{6 c^2}{a^2 x}+c^2 x+\frac{4 c^2 \log (x)}{a}\\ \end{align*}
Mathematica [A] time = 0.0205453, size = 51, normalized size = 1. \[ -\frac{2 c^2}{a^3 x^2}-\frac{c^2}{3 a^4 x^3}-\frac{6 c^2}{a^2 x}+\frac{4 c^2 \log (x)}{a}+c^2 x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 50, normalized size = 1. \begin{align*} -{\frac{{c}^{2}}{3\,{a}^{4}{x}^{3}}}-2\,{\frac{{c}^{2}}{{x}^{2}{a}^{3}}}-6\,{\frac{{c}^{2}}{{a}^{2}x}}+x{c}^{2}+4\,{\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 = 1.0086, size = 62, normalized size = 1.22 \begin{align*} c^{2} x + \frac{4 \, c^{2} \log \left (x\right )}{a} - \frac{18 \, a^{2} c^{2} x^{2} + 6 \, 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 = 1.49503, size = 122, normalized size = 2.39 \begin{align*} \frac{3 \, a^{4} c^{2} x^{4} + 12 \, a^{3} c^{2} x^{3} \log \left (x\right ) - 18 \, a^{2} c^{2} x^{2} - 6 \, 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 = 0.397397, size = 51, normalized size = 1. \begin{align*} \frac{a^{4} c^{2} x + 4 a^{3} c^{2} \log{\left (x \right )} - \frac{18 a^{2} c^{2} x^{2} + 6 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 [B] time = 1.169, size = 151, normalized size = 2.96 \begin{align*} -\frac{4 \, c^{2} \log \left (\frac{{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2}{\left | a \right |}}\right )}{a} + \frac{4 \, c^{2} \log \left ({\left | -\frac{1}{a x - 1} - 1 \right |}\right )}{a} + \frac{{\left (3 \, c^{2} + \frac{34 \, c^{2}}{a x - 1} + \frac{66 \, c^{2}}{{\left (a x - 1\right )}^{2}} + \frac{36 \, c^{2}}{{\left (a x - 1\right )}^{3}}\right )}{\left (a x - 1\right )}}{3 \, a{\left (\frac{1}{a x - 1} + 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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