3.202 \(\int (e x)^m \cot ^2(a+i \log (x)) \, dx\)

Optimal. Leaf size=77 \[ -2 x (e x)^m \text{Hypergeometric2F1}\left (1,\frac{1}{2} (-m-1),\frac{1-m}{2},\frac{e^{2 i a}}{x^2}\right )+\frac{2 x (e x)^m}{1-\frac{e^{2 i a}}{x^2}}-\frac{x (e x)^m}{m+1} \]

[Out]

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

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Rubi [F]  time = 0.0695071, 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 = {} \[ \int (e x)^m \cot ^2(a+i \log (x)) \, dx \]

Verification is Not applicable to the result.

[In]

Int[(e*x)^m*Cot[a + I*Log[x]]^2,x]

[Out]

Defer[Int][(e*x)^m*Cot[a + I*Log[x]]^2, x]

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*}

Mathematica [B]  time = 0.429795, size = 169, normalized size = 2.19 \[ \frac{x (e x)^m \left (-\frac{x^4 (\cos (a)-i \sin (a))^2 \text{Hypergeometric2F1}\left (2,\frac{m+5}{2},\frac{m+7}{2},x^2 (\cos (2 a)-i \sin (2 a))\right )}{m+5}-\frac{2 x^2 \text{Hypergeometric2F1}\left (2,\frac{m+3}{2},\frac{m+5}{2},x^2 (\cos (2 a)-i \sin (2 a))\right )}{m+3}-\frac{(\cos (2 a)+2 i \sin (a) \cos (a)) \text{Hypergeometric2F1}\left (2,\frac{m+1}{2},\frac{m+3}{2},x^2 (\cos (2 a)-i \sin (2 a))\right )}{m+1}\right )}{(\cos (a)+i \sin (a))^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(e*x)^m*Cot[a + I*Log[x]]^2,x]

[Out]

(x*(e*x)^m*((-2*x^2*Hypergeometric2F1[2, (3 + m)/2, (5 + m)/2, x^2*(Cos[2*a] - I*Sin[2*a])])/(3 + m) - (x^4*Hy
pergeometric2F1[2, (5 + m)/2, (7 + m)/2, x^2*(Cos[2*a] - I*Sin[2*a])]*(Cos[a] - I*Sin[a])^2)/(5 + m) - (Hyperg
eometric2F1[2, (1 + m)/2, (3 + m)/2, x^2*(Cos[2*a] - I*Sin[2*a])]*(Cos[2*a] + (2*I)*Cos[a]*Sin[a]))/(1 + m)))/
(Cos[a] + I*Sin[a])^2

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Maple [F]  time = 0.06, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m} \left ( \cot \left ( a+i\ln \left ( x \right ) \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m*cot(a+I*ln(x))^2,x)

[Out]

int((e*x)^m*cot(a+I*ln(x))^2,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 )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*cot(a+I*log(x))^2,x, algorithm="maxima")

[Out]

integrate((e*x)^m*cot(a + I*log(x))^2, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{2 \, \left (e x\right )^{m} x -{\left (e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - 1\right )}{\rm integral}\left (\frac{\left (e x\right )^{m}{\left (2 \, m - e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + 3\right )}}{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - 1}, x\right )}{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*cot(a+I*log(x))^2,x, algorithm="fricas")

[Out]

-(2*(e*x)^m*x - (e^(2*I*a - 2*log(x)) - 1)*integral((e*x)^m*(2*m - e^(2*I*a - 2*log(x)) + 3)/(e^(2*I*a - 2*log
(x)) - 1), x))/(e^(2*I*a - 2*log(x)) - 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{m} \cot ^{2}{\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))**2,x)

[Out]

Integral((e*x)**m*cot(a + I*log(x))**2, 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 )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*cot(a+I*log(x))^2,x, algorithm="giac")

[Out]

integrate((e*x)^m*cot(a + I*log(x))^2, x)