Optimal. Leaf size=38 \[ \frac{c^3}{2 a^3 x^2}+\frac{c^3}{a^2 x}+\frac{c^3 \log (x)}{a}+c^3 x \]
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Rubi [A] time = 0.127107, antiderivative size = 38, 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^{4 \coth ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^3 \, dx &=\int e^{4 \tanh ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^3 \, dx\\ &=-\frac{c^3 \int \frac{e^{4 \tanh ^{-1}(a x)} (1-a x)^3}{x^3} \, dx}{a^3}\\ &=-\frac{c^3 \int \frac{(1-a x) (1+a x)^2}{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 = 0.132784, size = 40, 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 = 37, 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 = 1.06445, size = 46, normalized size = 1.21 \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 = 1.61701, size = 97, normalized size = 2.55 \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 = 0.348838, size = 37, normalized size = 0.97 \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 [B] time = 1.17856, size = 132, normalized size = 3.47 \begin{align*} -\frac{c^{3} \log \left (\frac{{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2}{\left | a \right |}}\right )}{a} + \frac{c^{3} \log \left ({\left | -\frac{1}{a x - 1} - 1 \right |}\right )}{a} + \frac{{\left (2 \, c^{3} + \frac{c^{3}}{a x - 1} - \frac{2 \, c^{3}}{{\left (a x - 1\right )}^{2}}\right )}{\left (a x - 1\right )}}{2 \, a{\left (\frac{1}{a x - 1} + 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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