Optimal. Leaf size=60 \[ \frac{x}{1-e^{2 a} x^4}-\frac{1}{2} e^{-a/2} \tan ^{-1}\left (e^{a/2} x\right )-\frac{1}{2} e^{-a/2} \tanh ^{-1}\left (e^{a/2} x\right )+x \]
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Rubi [F] time = 0.0098684, 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 = {} \[ \int \coth ^2(a+2 \log (x)) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \coth ^2(a+2 \log (x)) \, dx &=\int \coth ^2(a+2 \log (x)) \, dx\\ \end{align*}
Mathematica [C] time = 1.95104, size = 153, normalized size = 2.55 \[ \frac{16}{585} e^{2 a} x^5 \left (e^{2 a} x^4+1\right )^2 \text{HypergeometricPFQ}\left (\left \{\frac{5}{4},2,2,2\right \},\left \{1,1,\frac{17}{4}\right \},e^{2 a} x^4\right )+\frac{e^{-4 a} \left (5 \left (e^{8 a} x^{16}-248 e^{6 a} x^{12}+102 e^{4 a} x^8+1208 e^{2 a} x^4+729\right ) \, _2F_1\left (\frac{1}{4},1;\frac{5}{4};e^{2 a} x^4\right )+681 e^{6 a} x^{12}-1483 e^{4 a} x^8-6769 e^{2 a} x^4-3645\right )}{640 x^7} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.043, size = 86, normalized size = 1.4 \begin{align*} x-{\frac{x}{{{\rm e}^{2\,a}}{x}^{4}-1}}+{\frac{1}{4}\ln \left ( x\sqrt{{{\rm e}^{a}}}-1 \right ){\frac{1}{\sqrt{{{\rm e}^{a}}}}}}-{\frac{1}{4}\ln \left ( x\sqrt{{{\rm e}^{a}}}+1 \right ){\frac{1}{\sqrt{{{\rm e}^{a}}}}}}-{\frac{1}{4}\ln \left ( x\sqrt{-{{\rm e}^{a}}}+1 \right ){\frac{1}{\sqrt{-{{\rm e}^{a}}}}}}+{\frac{1}{4}\ln \left ( x\sqrt{-{{\rm e}^{a}}}-1 \right ){\frac{1}{\sqrt{-{{\rm e}^{a}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.57043, size = 81, normalized size = 1.35 \begin{align*} -\frac{1}{2} \, \arctan \left (x e^{\left (\frac{1}{2} \, a\right )}\right ) e^{\left (-\frac{1}{2} \, a\right )} + \frac{1}{4} \, e^{\left (-\frac{1}{2} \, a\right )} \log \left (\frac{x e^{a} - e^{\left (\frac{1}{2} \, a\right )}}{x e^{a} + e^{\left (\frac{1}{2} \, a\right )}}\right ) + x - \frac{x}{x^{4} e^{\left (2 \, a\right )} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.64083, size = 246, normalized size = 4.1 \begin{align*} \frac{4 \, x^{5} e^{\left (3 \, a\right )} - 2 \,{\left (x^{4} e^{\left (2 \, a\right )} - 1\right )} \arctan \left (x e^{\left (\frac{1}{2} \, a\right )}\right ) e^{\left (\frac{1}{2} \, a\right )} +{\left (x^{4} e^{\left (2 \, a\right )} - 1\right )} e^{\left (\frac{1}{2} \, a\right )} \log \left (\frac{x^{2} e^{a} - 2 \, x e^{\left (\frac{1}{2} \, a\right )} + 1}{x^{2} e^{a} - 1}\right ) - 8 \, x e^{a}}{4 \,{\left (x^{4} e^{\left (3 \, a\right )} - e^{a}\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 \coth ^{2}{\left (a + 2 \log{\left (x \right )} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.119, size = 89, normalized size = 1.48 \begin{align*} -\frac{1}{2} \, \arctan \left (x e^{\left (\frac{1}{2} \, a\right )}\right ) e^{\left (-\frac{1}{2} \, a\right )} + \frac{1}{4} \, e^{\left (-\frac{1}{2} \, a\right )} \log \left (\frac{{\left | 2 \, x e^{a} - 2 \, e^{\left (\frac{1}{2} \, a\right )} \right |}}{{\left | 2 \, x e^{a} + 2 \, e^{\left (\frac{1}{2} \, a\right )} \right |}}\right ) + x - \frac{x}{x^{4} e^{\left (2 \, a\right )} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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