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 = 0.01, 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 = {} \[ \int \coth ^2(a+2 \log (x)) \, dx \]
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*}
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Mathematica [C] time = 2.24, size = 153, normalized size = 2.55 \[ \frac {16}{585} e^{2 a} x^5 \left (e^{2 a} x^4+1\right )^2 \, _4F_3\left (\frac {5}{4},2,2,2;1,1,\frac {17}{4};e^{2 a} x^4\right )+\frac {e^{-4 a} \left (5 \left (e^{8 a} x^{16}-248 e^{6 a} x^{12}+102 e^{4 a} x^8+1208 e^{2 a} x^4+729\right ) \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};e^{2 a} x^4\right )+681 e^{6 a} x^{12}-1483 e^{4 a} x^8-6769 e^{2 a} x^4-3645\right )}{640 x^7} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.41, size = 97, normalized size = 1.62 \[ \frac {4 \, x^{5} e^{\left (3 \, a\right )} - 2 \, {\left (x^{4} e^{\left (2 \, a\right )} - 1\right )} \arctan \left (x e^{\left (\frac {1}{2} \, a\right )}\right ) e^{\left (\frac {1}{2} \, a\right )} + {\left (x^{4} e^{\left (2 \, a\right )} - 1\right )} e^{\left (\frac {1}{2} \, a\right )} \log \left (\frac {x^{2} e^{a} - 2 \, x e^{\left (\frac {1}{2} \, a\right )} + 1}{x^{2} e^{a} - 1}\right ) - 8 \, x e^{a}}{4 \, {\left (x^{4} e^{\left (3 \, a\right )} - e^{a}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 66, normalized size = 1.10 \[ -\frac {1}{2} \, \arctan \left (x e^{\left (\frac {1}{2} \, a\right )}\right ) e^{\left (-\frac {1}{2} \, a\right )} + \frac {1}{4} \, e^{\left (-\frac {1}{2} \, a\right )} \log \left (\frac {{\left | 2 \, x e^{a} - 2 \, e^{\left (\frac {1}{2} \, a\right )} \right |}}{{\left | 2 \, x e^{a} + 2 \, e^{\left (\frac {1}{2} \, a\right )} \right |}}\right ) + x - \frac {x}{x^{4} e^{\left (2 \, a\right )} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 86, normalized size = 1.43 \[ x -\frac {x}{-1+{\mathrm e}^{2 a} x^{4}}+\frac {\ln \left (\sqrt {{\mathrm e}^{a}}\, x -1\right )}{4 \sqrt {{\mathrm e}^{a}}}-\frac {\ln \left (\sqrt {{\mathrm e}^{a}}\, x +1\right )}{4 \sqrt {{\mathrm e}^{a}}}-\frac {\ln \left (x \sqrt {-{\mathrm e}^{a}}+1\right )}{4 \sqrt {-{\mathrm e}^{a}}}+\frac {\ln \left (x \sqrt {-{\mathrm e}^{a}}-1\right )}{4 \sqrt {-{\mathrm e}^{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 60, normalized size = 1.00 \[ -\frac {1}{2} \, \arctan \left (x e^{\left (\frac {1}{2} \, a\right )}\right ) e^{\left (-\frac {1}{2} \, a\right )} + \frac {1}{4} \, e^{\left (-\frac {1}{2} \, a\right )} \log \left (\frac {x e^{a} - e^{\left (\frac {1}{2} \, a\right )}}{x e^{a} + e^{\left (\frac {1}{2} \, a\right )}}\right ) + x - \frac {x}{x^{4} e^{\left (2 \, a\right )} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.21, size = 54, normalized size = 0.90 \[ x-\frac {\mathrm {atan}\left (x\,{\left ({\mathrm {e}}^{2\,a}\right )}^{1/4}\right )}{2\,{\left ({\mathrm {e}}^{2\,a}\right )}^{1/4}}-\frac {x}{x^4\,{\mathrm {e}}^{2\,a}-1}+\frac {\mathrm {atan}\left (x\,{\left ({\mathrm {e}}^{2\,a}\right )}^{1/4}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,{\left ({\mathrm {e}}^{2\,a}\right )}^{1/4}} \]
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 \[ \int \coth ^{2}{\left (a + 2 \log {\relax (x )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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