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

Optimal. Leaf size=70 \[ \frac {i (e x)^{m+1}}{e (m+1)}-\frac {2 i (e x)^{m+1} \, _2F_1\left (1,\frac {1}{2} (-m-1);\frac {1-m}{2};\frac {e^{2 i a}}{x^2}\right )}{e (m+1)} \]

[Out]

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

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Rubi [F]  time = 0.04, 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 = {} \[ \int (e x)^m \cot (a+i \log (x)) \, dx \]

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*}

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Mathematica [A]  time = 0.26, size = 103, normalized size = 1.47 \[ i x (e x)^m \left (\frac {x^2 (\cos (a)-i \sin (a))^2 \, _2F_1\left (1,\frac {m+3}{2};\frac {m+5}{2};x^2 (\cos (2 a)-i \sin (2 a))\right )}{m+3}+\frac {\, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};x^2 (\cos (2 a)-i \sin (2 a))\right )}{m+1}\right ) \]

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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fricas [F]  time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (i \, x^{2} + i \, e^{\left (2 i \, a\right )}\right )} e^{\left (m \log \relax (e) + m \log \relax (x)\right )}}{x^{2} - e^{\left (2 i \, a\right )}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e x\right )^{m} \cot \left (a + i \, \log \relax (x)\right )\,{d x} \]

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)

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maple [F]  time = 0.12, size = 0, normalized size = 0.00 \[ \int \left (e x \right )^{m} \cot \left (a +i \ln \relax (x )\right )\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e x\right )^{m} \cot \left (a + i \, \log \relax (x)\right )\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \mathrm {cot}\left (a+\ln \relax (x)\,1{}\mathrm {i}\right )\,{\left (e\,x\right )}^m \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(a + log(x)*1i)*(e*x)^m,x)

[Out]

int(cot(a + log(x)*1i)*(e*x)^m, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e x\right )^{m} \cot {\left (a + i \log {\relax (x )} \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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