3.201 \(\int (e x)^m \cot (a+i \log (x)) \, dx\)

Optimal. Leaf size=70 \[ \frac{i (e x)^{m+1}}{e (m+1)}-\frac{2 i (e x)^{m+1} \text{Hypergeometric2F1}\left (1,\frac{1}{2} (-m-1),\frac{1-m}{2},\frac{e^{2 i a}}{x^2}\right )}{e (m+1)} \]

[Out]

(I*(e*x)^(1 + m))/(e*(1 + m)) - ((2*I)*(e*x)^(1 + m)*Hypergeometric2F1[1, (-1 - m)/2, (1 - m)/2, E^((2*I)*a)/x
^2])/(e*(1 + m))

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Rubi [F]  time = 0.040328, 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 = {} \[ \int (e x)^m \cot (a+i \log (x)) \, dx \]

Verification is Not applicable to the result.

[In]

Int[(e*x)^m*Cot[a + I*Log[x]],x]

[Out]

Defer[Int][(e*x)^m*Cot[a + I*Log[x]], x]

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*}

Mathematica [A]  time = 0.241067, size = 103, normalized size = 1.47 \[ i x (e x)^m \left (\frac{x^2 (\cos (a)-i \sin (a))^2 \text{Hypergeometric2F1}\left (1,\frac{m+3}{2},\frac{m+5}{2},x^2 (\cos (2 a)-i \sin (2 a))\right )}{m+3}+\frac{\text{Hypergeometric2F1}\left (1,\frac{m+1}{2},\frac{m+3}{2},x^2 (\cos (2 a)-i \sin (2 a))\right )}{m+1}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(e*x)^m*Cot[a + I*Log[x]],x]

[Out]

I*x*(e*x)^m*(Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, x^2*(Cos[2*a] - I*Sin[2*a])]/(1 + m) + (x^2*Hypergeome
tric2F1[1, (3 + m)/2, (5 + m)/2, x^2*(Cos[2*a] - I*Sin[2*a])]*(Cos[a] - I*Sin[a])^2)/(3 + m))

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Maple [F]  time = 0.158, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m}\cot \left ( a+i\ln \left ( x \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m*cot(a+I*ln(x)),x)

[Out]

int((e*x)^m*cot(a+I*ln(x)),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{m} \cot \left (a + i \, \log \left (x\right )\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*cot(a+I*log(x)),x, algorithm="maxima")

[Out]

integrate((e*x)^m*cot(a + I*log(x)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\left (e x\right )^{m}{\left (i \, e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + i\right )}}{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*cot(a+I*log(x)),x, algorithm="fricas")

[Out]

integral((e*x)^m*(I*e^(2*I*a - 2*log(x)) + I)/(e^(2*I*a - 2*log(x)) - 1), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{m} \cot{\left (a + i \log{\left (x \right )} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**m*cot(a+I*ln(x)),x)

[Out]

Integral((e*x)**m*cot(a + I*log(x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{m} \cot \left (a + i \, \log \left (x\right )\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*cot(a+I*log(x)),x, algorithm="giac")

[Out]

integrate((e*x)^m*cot(a + I*log(x)), x)