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

Optimal. Leaf size=77 \[ -2 x (e x)^m \, _2F_1\left (1,\frac {1}{2} (-m-1);\frac {1-m}{2};\frac {e^{2 i a}}{x^2}\right )+\frac {2 x (e x)^m}{1-\frac {e^{2 i a}}{x^2}}-\frac {x (e x)^m}{m+1} \]

[Out]

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

________________________________________________________________________________________

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.

[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 [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.

[In]

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

[Out]

(x*(e*x)^m*(-1 + 4*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, x^2*(Cos[2*a] - I*Sin[2*a])] - 4*Hypergeometric2
F1[2, (1 + m)/2, (3 + m)/2, x^2*(Cos[2*a] - I*Sin[2*a])]))/(1 + m)

________________________________________________________________________________________

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.

[In]

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

[Out]

integral(-(x^4 + 2*x^2*e^(2*I*a) + e^(4*I*a))*e^(m*log(e) + m*log(x))/(x^4 - 2*x^2*e^(2*I*a) + e^(4*I*a)), x)

________________________________________________________________________________________

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.

[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)

________________________________________________________________________________________

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.

[In]

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

[Out]

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

________________________________________________________________________________________

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.

[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)

________________________________________________________________________________________

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.

[In]

int(cot(a + log(x)*1i)^2*(e*x)^m,x)

[Out]

int(cot(a + log(x)*1i)^2*(e*x)^m, x)

________________________________________________________________________________________

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.

[In]

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

[Out]

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

________________________________________________________________________________________