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 = 0.13, antiderivative size = 38, 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, 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 75
Rule 6129
Rule 6131
Rule 6167
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*}
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Mathematica [A] time = 0.15, 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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fricas [A] time = 0.47, size = 43, normalized size = 1.13 \[ \frac {2 \, a^{3} c^{3} x^{3} + 2 \, a^{2} c^{3} x^{2} \log \relax (x) + 2 \, a c^{3} x + c^{3}}{2 \, a^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 98, normalized size = 2.58 \[ -\frac {c^{3} \log \left (\frac {{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2} {\left | a \right |}}\right )}{a} + \frac {c^{3} \log \left ({\left | -\frac {1}{a x - 1} - 1 \right |}\right )}{a} + \frac {{\left (2 \, c^{3} + \frac {c^{3}}{a x - 1} - \frac {2 \, c^{3}}{{\left (a x - 1\right )}^{2}}\right )} {\left (a x - 1\right )}}{2 \, a {\left (\frac {1}{a x - 1} + 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 37, normalized size = 0.97 \[ \frac {c^{3}}{2 x^{2} a^{3}}+\frac {c^{3}}{a^{2} x}+c^{3} x +\frac {c^{3} \ln \relax (x )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 34, normalized size = 0.89 \[ c^{3} x + \frac {c^{3} \log \relax (x)}{a} + \frac {2 \, a c^{3} x + c^{3}}{2 \, a^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.18, size = 31, normalized size = 0.82 \[ \frac {c^3\,\left (a\,x+a^3\,x^3+a^2\,x^2\,\ln \relax (x)+\frac {1}{2}\right )}{a^3\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 37, normalized size = 0.97 \[ \frac {a^{3} c^{3} x + a^{2} c^{3} \log {\relax (x )} + \frac {2 a c^{3} x + c^{3}}{2 x^{2}}}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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