Optimal. Leaf size=77 \[ -2 x (e x)^m \, _2F_1\left (1,\frac {1}{2} (-m-1);\frac {1-m}{2};\frac {e^{2 i a}}{x^2}\right )+\frac {2 x (e x)^m}{1-\frac {e^{2 i a}}{x^2}}-\frac {x (e x)^m}{m+1} \]
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Rubi [F] time = 0.07, 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 = {} \[ \int (e x)^m \cot ^2(a+i \log (x)) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int (e x)^m \cot ^2(a+i \log (x)) \, dx &=\int (e x)^m \cot ^2(a+i \log (x)) \, dx\\ \end {align*}
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Mathematica [A] time = 0.17, size = 84, normalized size = 1.09 \[ \frac {x (e x)^m \left (4 \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};x^2 (\cos (2 a)-i \sin (2 a))\right )-4 \, _2F_1\left (2,\frac {m+1}{2};\frac {m+3}{2};x^2 (\cos (2 a)-i \sin (2 a))\right )-1\right )}{m+1} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (x^{4} + 2 \, x^{2} e^{\left (2 i \, a\right )} + e^{\left (4 i \, a\right )}\right )} e^{\left (m \log \relax (e) + m \log \relax (x)\right )}}{x^{4} - 2 \, x^{2} e^{\left (2 i \, a\right )} + e^{\left (4 i \, a\right )}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e x\right )^{m} \cot \left (a + i \, \log \relax (x)\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \left (e x \right )^{m} \left (\cot ^{2}\left (a +i \ln \relax (x )\right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e x\right )^{m} \cot \left (a + i \, \log \relax (x)\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {cot}\left (a+\ln \relax (x)\,1{}\mathrm {i}\right )}^2\,{\left (e\,x\right )}^m \,d x \]
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 \[ \int \left (e x\right )^{m} \cot ^{2}{\left (a + i \log {\relax (x )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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