Optimal. Leaf size=51 \[ -\frac {c^2}{3 a^4 x^3}-\frac {2 c^2}{a^3 x^2}-\frac {6 c^2}{a^2 x}+\frac {4 c^2 \log (x)}{a}+c^2 x \]
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Rubi [A] time = 0.15, antiderivative size = 51, 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, 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 43
Rule 6150
Rule 6157
Rule 6167
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*}
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Mathematica [A] time = 0.02, size = 51, normalized size = 1.00 \[ -\frac {c^2}{3 a^4 x^3}-\frac {2 c^2}{a^3 x^2}-\frac {6 c^2}{a^2 x}+\frac {4 c^2 \log (x)}{a}+c^2 x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 56, normalized size = 1.10 \[ \frac {3 \, a^{4} c^{2} x^{4} + 12 \, a^{3} c^{2} x^{3} \log \relax (x) - 18 \, a^{2} c^{2} x^{2} - 6 \, a c^{2} x - c^{2}}{3 \, a^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.12, size = 112, normalized size = 2.20 \[ -\frac {4 \, c^{2} \log \left (\frac {{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2} {\left | a \right |}}\right )}{a} + \frac {4 \, c^{2} \log \left ({\left | -\frac {1}{a x - 1} - 1 \right |}\right )}{a} + \frac {{\left (3 \, c^{2} + \frac {34 \, c^{2}}{a x - 1} + \frac {66 \, c^{2}}{{\left (a x - 1\right )}^{2}} + \frac {36 \, c^{2}}{{\left (a x - 1\right )}^{3}}\right )} {\left (a x - 1\right )}}{3 \, a {\left (\frac {1}{a x - 1} + 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 50, normalized size = 0.98 \[ -\frac {c^{2}}{3 a^{4} x^{3}}-\frac {2 c^{2}}{x^{2} a^{3}}-\frac {6 c^{2}}{a^{2} x}+c^{2} x +\frac {4 c^{2} \ln \relax (x )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 46, normalized size = 0.90 \[ c^{2} x + \frac {4 \, c^{2} \log \relax (x)}{a} - \frac {18 \, a^{2} c^{2} x^{2} + 6 \, a c^{2} x + c^{2}}{3 \, a^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 43, normalized size = 0.84 \[ -\frac {c^2\,\left (6\,a\,x+18\,a^2\,x^2-3\,a^4\,x^4-12\,a^3\,x^3\,\ln \relax (x)+1\right )}{3\,a^4\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 53, normalized size = 1.04 \[ \frac {a^{4} c^{2} x + 4 a^{3} c^{2} \log {\relax (x )} + \frac {- 18 a^{2} c^{2} x^{2} - 6 a c^{2} x - c^{2}}{3 x^{3}}}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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