Optimal. Leaf size=68 \[ \frac {x^3}{1-e^{2 a} x^4}+\frac {3}{2} e^{-3 a/2} \tan ^{-1}\left (e^{a/2} x\right )-\frac {3}{2} e^{-3 a/2} \tanh ^{-1}\left (e^{a/2} x\right )+\frac {x^3}{3} \]
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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 = {} \[ \int x^2 \coth ^2(a+2 \log (x)) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int x^2 \coth ^2(a+2 \log (x)) \, dx &=\int x^2 \coth ^2(a+2 \log (x)) \, dx\\ \end {align*}
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Mathematica [C] time = 2.95, size = 154, normalized size = 2.26 \[ \frac {16 e^{2 a} x^7 \left (e^{2 a} x^4+1\right )^2 \, _4F_3\left (\frac {7}{4},2,2,2;1,1,\frac {19}{4};e^{2 a} x^4\right )}{1155}+\frac {e^{-4 a} \left (7 \left (27 e^{8 a} x^{16}-632 e^{6 a} x^{12}-398 e^{4 a} x^8+1976 e^{2 a} x^4+1331\right ) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};e^{2 a} x^4\right )+1481 e^{6 a} x^{12}-4787 e^{4 a} x^8-17825 e^{2 a} x^4-9317\right )}{2688 x^5} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.41, size = 104, normalized size = 1.53 \[ \frac {4 \, x^{7} e^{\left (4 \, a\right )} - 16 \, x^{3} e^{\left (2 \, a\right )} + 18 \, {\left (x^{4} e^{\left (2 \, a\right )} - 1\right )} \arctan \left (x e^{\left (\frac {1}{2} \, a\right )}\right ) e^{\left (\frac {1}{2} \, a\right )} + 9 \, {\left (x^{4} e^{\left (2 \, a\right )} - 1\right )} e^{\left (\frac {1}{2} \, a\right )} \log \left (\frac {x^{2} e^{a} - 2 \, x e^{\left (\frac {1}{2} \, a\right )} + 1}{x^{2} e^{a} - 1}\right )}{12 \, {\left (x^{4} e^{\left (4 \, a\right )} - e^{\left (2 \, a\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 72, normalized size = 1.06 \[ \frac {1}{3} \, x^{3} - \frac {x^{3}}{x^{4} e^{\left (2 \, a\right )} - 1} + \frac {3}{2} \, \arctan \left (x e^{\left (\frac {1}{2} \, a\right )}\right ) e^{\left (-\frac {3}{2} \, a\right )} + \frac {3}{4} \, e^{\left (-\frac {3}{2} \, a\right )} \log \left (\frac {{\left | 2 \, x e^{a} - 2 \, e^{\left (\frac {1}{2} \, a\right )} \right |}}{{\left | 2 \, x e^{a} + 2 \, e^{\left (\frac {1}{2} \, a\right )} \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 100, normalized size = 1.47 \[ \frac {x^{3}}{3}-\frac {x^{3}}{-1+{\mathrm e}^{2 a} x^{4}}+\frac {3 \ln \left (-{\mathrm e}^{2 a} x +\left (-{\mathrm e}^{a}\right )^{\frac {3}{2}}\right )}{4 \left (-{\mathrm e}^{a}\right )^{\frac {3}{2}}}-\frac {3 \ln \left ({\mathrm e}^{2 a} x +\left (-{\mathrm e}^{a}\right )^{\frac {3}{2}}\right )}{4 \left (-{\mathrm e}^{a}\right )^{\frac {3}{2}}}+\frac {3 \ln \left (-\sqrt {{\mathrm e}^{a}}\, x +1\right )}{4 \left ({\mathrm e}^{a}\right )^{\frac {3}{2}}}-\frac {3 \ln \left (\sqrt {{\mathrm e}^{a}}\, x +1\right )}{4 \left ({\mathrm e}^{a}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 66, normalized size = 0.97 \[ \frac {1}{3} \, x^{3} - \frac {x^{3}}{x^{4} e^{\left (2 \, a\right )} - 1} + \frac {3}{2} \, \arctan \left (x e^{\left (\frac {1}{2} \, a\right )}\right ) e^{\left (-\frac {3}{2} \, a\right )} + \frac {3}{4} \, e^{\left (-\frac {3}{2} \, a\right )} \log \left (\frac {x e^{a} - e^{\left (\frac {1}{2} \, a\right )}}{x e^{a} + e^{\left (\frac {1}{2} \, a\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.24, size = 60, normalized size = 0.88 \[ \frac {3\,\mathrm {atan}\left (x\,{\left ({\mathrm {e}}^{2\,a}\right )}^{1/4}\right )}{2\,{\left ({\mathrm {e}}^{2\,a}\right )}^{3/4}}-\frac {x^3}{x^4\,{\mathrm {e}}^{2\,a}-1}+\frac {x^3}{3}+\frac {\mathrm {atan}\left (x\,{\left ({\mathrm {e}}^{2\,a}\right )}^{1/4}\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{2\,{\left ({\mathrm {e}}^{2\,a}\right )}^{3/4}} \]
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 \[ \int x^{2} \coth ^{2}{\left (a + 2 \log {\relax (x )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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