Optimal. Leaf size=60 \[ e^a \tanh ^{-1}\left (e^a x^2\right )+\frac {3 e^{2 a} x^2}{2 \left (1-e^{2 a} x^4\right )}-\frac {1}{2 x^2 \left (1-e^{2 a} x^4\right )} \]
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Rubi [F] time = 0.05, 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 = {} \[ \int \frac {\coth ^2(a+2 \log (x))}{x^3} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\coth ^2(a+2 \log (x))}{x^3} \, dx &=\int \frac {\coth ^2(a+2 \log (x))}{x^3} \, dx\\ \end {align*}
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Mathematica [C] time = 3.25, size = 155, normalized size = 2.58 \[ \frac {64 \left (e^{3 a} x^6+e^a x^2\right )^2 \, _4F_3\left (\frac {1}{2},2,2,2;1,1,\frac {7}{2};e^{2 a} x^4\right )+15 \left (e^{4 a} x^8-17 e^{2 a} x^4-\frac {27 e^{-2 a}}{x^4}-77\right )-\frac {15 \left (e^{8 a} x^{16}+4 e^{6 a} x^{12}-54 e^{4 a} x^8-52 e^{2 a} x^4-27\right ) \tanh ^{-1}\left (\sqrt {e^{2 a} x^4}\right )}{\left (e^{2 a} x^4\right )^{3/2}}}{480 x^2} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.40, size = 82, normalized size = 1.37 \[ -\frac {3 \, x^{4} e^{\left (2 \, a\right )} - {\left (x^{6} e^{\left (3 \, a\right )} - x^{2} e^{a}\right )} \log \left (x^{2} e^{a} + 1\right ) + {\left (x^{6} e^{\left (3 \, a\right )} - x^{2} e^{a}\right )} \log \left (x^{2} e^{a} - 1\right ) - 1}{2 \, {\left (x^{6} e^{\left (2 \, a\right )} - x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 57, normalized size = 0.95 \[ \frac {1}{2} \, e^{a} \log \left (x^{2} e^{a} + 1\right ) - \frac {1}{2} \, e^{a} \log \left ({\left | x^{2} e^{a} - 1 \right |}\right ) - \frac {3 \, x^{4} e^{\left (2 \, a\right )} - 1}{2 \, {\left (x^{6} e^{\left (2 \, a\right )} - x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 55, normalized size = 0.92 \[ \frac {-\frac {3 \,{\mathrm e}^{2 a} x^{4}}{2}+\frac {1}{2}}{x^{2} \left (-1+{\mathrm e}^{2 a} x^{4}\right )}-\frac {{\mathrm e}^{a} \ln \left ({\mathrm e}^{a} x^{2}-1\right )}{2}+\frac {{\mathrm e}^{a} \ln \left ({\mathrm e}^{a} x^{2}+1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 50, normalized size = 0.83 \[ \frac {1}{2} \, e^{a} \log \left (\frac {1}{x^{2}} + e^{a}\right ) - \frac {1}{2} \, e^{a} \log \left (\frac {1}{x^{2}} - e^{a}\right ) - \frac {1}{2 \, x^{2}} + \frac {e^{\left (2 \, a\right )}}{x^{2} {\left (\frac {1}{x^{4}} - e^{\left (2 \, a\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.23, size = 48, normalized size = 0.80 \[ \mathrm {atanh}\left (x^2\,\sqrt {{\mathrm {e}}^{2\,a}}\right )\,\sqrt {{\mathrm {e}}^{2\,a}}-\frac {\frac {3\,x^4\,{\mathrm {e}}^{2\,a}}{2}-\frac {1}{2}}{x^6\,{\mathrm {e}}^{2\,a}-x^2} \]
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 \[ \int \frac {\coth ^{2}{\left (a + 2 \log {\relax (x )} \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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