Optimal. Leaf size=60 \[ \frac{3 e^{2 a} x^2}{2 \left (1-e^{2 a} x^4\right )}-\frac{1}{2 x^2 \left (1-e^{2 a} x^4\right )}+e^a \tanh ^{-1}\left (e^a x^2\right ) \]
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Rubi [F] time = 0.0460271, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., 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*}
Mathematica [C] time = 2.86299, size = 155, normalized size = 2.58 \[ \frac{64 \left (e^{3 a} x^6+e^a x^2\right )^2 \text{HypergeometricPFQ}\left (\left \{\frac{1}{2},2,2,2\right \},\left \{1,1,\frac{7}{2}\right \},e^{2 a} x^4\right )+15 \left (e^{4 a} x^8-17 e^{2 a} x^4-\frac{27 e^{-2 a}}{x^4}-77\right )-\frac{15 \left (e^{8 a} x^{16}+4 e^{6 a} x^{12}-54 e^{4 a} x^8-52 e^{2 a} x^4-27\right ) \tanh ^{-1}\left (\sqrt{e^{2 a} x^4}\right )}{\left (e^{2 a} x^4\right )^{3/2}}}{480 x^2} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.023, size = 55, normalized size = 0.9 \begin{align*}{\frac{1}{{x}^{2} \left ({{\rm e}^{2\,a}}{x}^{4}-1 \right ) } \left ( -{\frac{3\,{{\rm e}^{2\,a}}{x}^{4}}{2}}+{\frac{1}{2}} \right ) }+{\frac{{{\rm e}^{a}}\ln \left ({{\rm e}^{a}}{x}^{2}+1 \right ) }{2}}-{\frac{{{\rm e}^{a}}\ln \left ({{\rm e}^{a}}{x}^{2}-1 \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07843, size = 68, normalized size = 1.13 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.58052, size = 178, normalized size = 2.97 \begin{align*} -\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\coth ^{2}{\left (a + 2 \log{\left (x \right )} \right )}}{x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15239, size = 77, normalized size = 1.28 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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