Optimal. Leaf size=14 \[ \log (x)-\frac {1}{2} \coth (a+2 \log (x)) \]
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Rubi [A] time = 0.02, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3473, 8} \[ \log (x)-\frac {1}{2} \coth (a+2 \log (x)) \]
Antiderivative was successfully verified.
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Rule 8
Rule 3473
Rubi steps
\begin {align*} \int \frac {\coth ^2(a+2 \log (x))}{x} \, dx &=\operatorname {Subst}\left (\int \coth ^2(a+2 x) \, dx,x,\log (x)\right )\\ &=-\frac {1}{2} \coth (a+2 \log (x))+\operatorname {Subst}(\int 1 \, dx,x,\log (x))\\ &=-\frac {1}{2} \coth (a+2 \log (x))+\log (x)\\ \end {align*}
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Mathematica [C] time = 0.05, size = 28, normalized size = 2.00 \[ -\frac {1}{2} \coth (a+2 \log (x)) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\tanh ^2(a+2 \log (x))\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 28, normalized size = 2.00 \[ \frac {{\left (x^{4} e^{\left (2 \, a\right )} - 1\right )} \log \relax (x) - 1}{x^{4} e^{\left (2 \, a\right )} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 21, normalized size = 1.50 \[ -\frac {1}{x^{4} e^{\left (2 \, a\right )} - 1} + \frac {1}{4} \, \log \left (x^{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 35, normalized size = 2.50 \[ -\frac {\coth \left (a +2 \ln \relax (x )\right )}{2}-\frac {\ln \left (\coth \left (a +2 \ln \relax (x )\right )-1\right )}{4}+\frac {\ln \left (\coth \left (a +2 \ln \relax (x )\right )+1\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 19, normalized size = 1.36 \[ \frac {1}{2} \, a + \frac {1}{e^{\left (-2 \, a - 4 \, \log \relax (x)\right )} - 1} + \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.19, size = 28, normalized size = 2.00 \[ \ln \relax (x)-\frac {{\mathrm {e}}^{2\,a}\,x^4+1}{2\,\left (x^4\,{\mathrm {e}}^{2\,a}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.36, size = 32, normalized size = 2.29 \[ \begin {cases} \tilde {\infty } \log {\relax (x )} & \text {for}\: a = \log {\left (- \frac {1}{x^{2}} \right )} \vee a = \log {\left (\frac {1}{x^{2}} \right )} \\\log {\relax (x )} - \frac {1}{2 \tanh {\left (a + 2 \log {\relax (x )} \right )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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