Optimal. Leaf size=70 \[ \frac {i (e x)^{1+m}}{e (1+m)}-\frac {2 i (e x)^{1+m} \, _2F_1\left (1,\frac {1}{2} (-1-m);\frac {1-m}{2};\frac {e^{2 i a}}{x^2}\right )}{e (1+m)} \]
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Rubi [A]
time = 0.04, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {4592, 470, 346,
371} \begin {gather*} \frac {i (e x)^{m+1}}{e (m+1)}-\frac {2 i (e x)^{m+1} \, _2F_1\left (1,\frac {1}{2} (-m-1);\frac {1-m}{2};\frac {e^{2 i a}}{x^2}\right )}{e (m+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 346
Rule 371
Rule 470
Rule 4592
Rubi steps
\begin {align*} \int (e x)^m \cot (a+i \log (x)) \, dx &=\int (e x)^m \cot (a+i \log (x)) \, dx\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 103, normalized size = 1.47 \begin {gather*} i x (e x)^m \left (\frac {\, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};x^2 (\cos (2 a)-i \sin (2 a))\right )}{1+m}+\frac {x^2 \, _2F_1\left (1,\frac {3+m}{2};\frac {5+m}{2};x^2 (\cos (2 a)-i \sin (2 a))\right ) (\cos (a)-i \sin (a))^2}{3+m}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \left (e x \right )^{m} \cot \left (a +i \ln \left (x \right )\right )\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \left (e x\right )^{m} \cot {\left (a + i \log {\left (x \right )} \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \mathrm {cot}\left (a+\ln \left (x\right )\,1{}\mathrm {i}\right )\,{\left (e\,x\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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