Optimal. Leaf size=21 \[ \frac {1}{2 x^2}-e^a \tanh ^{-1}\left (e^a x^2\right ) \]
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Rubi [F] time = 0.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\coth (a+2 \log (x))}{x^3} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\coth (a+2 \log (x))}{x^3} \, dx &=\int \frac {\coth (a+2 \log (x))}{x^3} \, dx\\ \end {align*}
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Mathematica [A] time = 0.16, size = 27, normalized size = 1.29 \[ \frac {1}{2 x^2}-(\sinh (a)+\cosh (a)) \tanh ^{-1}\left (\frac {\cosh (a)-\sinh (a)}{x^2}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 38, normalized size = 1.81 \[ -\frac {x^{2} e^{a} \log \left (x^{2} e^{a} + 1\right ) - x^{2} e^{a} \log \left (x^{2} e^{a} - 1\right ) - 1}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 33, normalized size = 1.57 \[ -\frac {1}{2} \, e^{a} \log \left (x^{2} e^{a} + 1\right ) + \frac {1}{2} \, e^{a} \log \left ({\left | x^{2} e^{a} - 1 \right |}\right ) + \frac {1}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 35, normalized size = 1.67 \[ \frac {1}{2 x^{2}}-\frac {{\mathrm e}^{a} \ln \left (-{\mathrm e}^{a} x^{2}-1\right )}{2}+\frac {{\mathrm e}^{a} \ln \left (-{\mathrm e}^{a} x^{2}+1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 30, normalized size = 1.43 \[ -\frac {1}{2} \, e^{a} \log \left (\frac {1}{x^{2}} + e^{a}\right ) + \frac {1}{2} \, e^{a} \log \left (\frac {1}{x^{2}} - e^{a}\right ) + \frac {1}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.21, size = 25, normalized size = 1.19 \[ \frac {1}{2\,x^2}-\mathrm {atanh}\left (x^2\,\sqrt {{\mathrm {e}}^{2\,a}}\right )\,\sqrt {{\mathrm {e}}^{2\,a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\coth {\left (a + 2 \log {\relax (x )} \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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