Optimal. Leaf size=39 \[ -\frac{c^3}{2 a^3 x^2}+\frac{c^3}{a^2 x}-\frac{c^3 \log (x)}{a}+c^3 x \]
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Rubi [A] time = 0.128253, antiderivative size = 39, 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, 6131, 6129, 75} \[ -\frac{c^3}{2 a^3 x^2}+\frac{c^3}{a^2 x}-\frac{c^3 \log (x)}{a}+c^3 x \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6131
Rule 6129
Rule 75
Rubi steps
\begin{align*} \int e^{2 \coth ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^3 \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^3 \, dx\\ &=\frac{c^3 \int \frac{e^{2 \tanh ^{-1}(a x)} (1-a x)^3}{x^3} \, dx}{a^3}\\ &=\frac{c^3 \int \frac{(1-a x)^2 (1+a x)}{x^3} \, dx}{a^3}\\ &=\frac{c^3 \int \left (a^3+\frac{1}{x^3}-\frac{a}{x^2}-\frac{a^2}{x}\right ) \, dx}{a^3}\\ &=-\frac{c^3}{2 a^3 x^2}+\frac{c^3}{a^2 x}+c^3 x-\frac{c^3 \log (x)}{a}\\ \end{align*}
Mathematica [A] time = 0.120522, size = 41, normalized size = 1.05 \[ -\frac{c^3}{2 a^3 x^2}+\frac{c^3}{a^2 x}-\frac{c^3 \log (a x)}{a}+c^3 x \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.043, size = 38, normalized size = 1. \begin{align*} -{\frac{{c}^{3}}{2\,{x}^{2}{a}^{3}}}+{\frac{{c}^{3}}{{a}^{2}x}}+{c}^{3}x-{\frac{{c}^{3}\ln \left ( x \right ) }{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04247, size = 50, normalized size = 1.28 \begin{align*} c^{3} x - \frac{c^{3} \log \left (x\right )}{a} + \frac{2 \, a c^{3} x - c^{3}}{2 \, a^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46807, size = 97, normalized size = 2.49 \begin{align*} \frac{2 \, a^{3} c^{3} x^{3} - 2 \, a^{2} c^{3} x^{2} \log \left (x\right ) + 2 \, a c^{3} x - c^{3}}{2 \, a^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.338769, size = 37, normalized size = 0.95 \begin{align*} \frac{a^{3} c^{3} x - a^{2} c^{3} \log{\left (x \right )} + \frac{2 a c^{3} x - c^{3}}{2 x^{2}}}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12182, size = 51, normalized size = 1.31 \begin{align*} c^{3} x - \frac{c^{3} \log \left ({\left | x \right |}\right )}{a} + \frac{2 \, a c^{3} x - c^{3}}{2 \, a^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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