\(\int (e x)^m \cot (a+i \log (x)) \, dx\) [201]

Optimal result

Integrand size = 15, antiderivative size = 70 \[ \int (e x)^m \cot (a+i \log (x)) \, dx=\frac {i (e x)^{1+m}}{e (1+m)}-\frac {2 i (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (-1-m),\frac {1-m}{2},\frac {e^{2 i a}}{x^2}\right )}{e (1+m)} \]

[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)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {4592, 470, 346, 371} \[ \int (e x)^m \cot (a+i \log (x)) \, dx=\frac {i (e x)^{m+1}}{e (m+1)}-\frac {2 i (e x)^{m+1} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (-m-1),\frac {1-m}{2},\frac {e^{2 i a}}{x^2}\right )}{e (m+1)} \]

[In]

Int[(e*x)^m*Cot[a + I*Log[x]],x]

[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))

Rule 346

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(-c^(-1))*(c*x)^(m + 1)*(1/x)^(m + 1),
Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x], x] /; FreeQ[{a, b, c, m, p}, x] && ILtQ[n, 0] &&  !RationalQ[m
]

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 470

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*(e*x)^(m +
1)*((a + b*x^n)^(p + 1)/(b*e*(m + n*(p + 1) + 1))), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m +
n*(p + 1) + 1)), Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0]
&& NeQ[m + n*(p + 1) + 1, 0]

Rule 4592

Int[Cot[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Int[(e*x)^m*((-I - I*E^(2*I*a*d)
*x^(2*I*b*d))/(1 - E^(2*I*a*d)*x^(2*I*b*d)))^p, x] /; FreeQ[{a, b, d, e, m, p}, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (-i-\frac {i e^{2 i a}}{x^2}\right ) (e x)^m}{1-\frac {e^{2 i a}}{x^2}} \, dx \\ & = \frac {i (e x)^{1+m}}{e (1+m)}-2 i \int \frac {(e x)^m}{1-\frac {e^{2 i a}}{x^2}} \, dx \\ & = \frac {i (e x)^{1+m}}{e (1+m)}+\frac {\left (2 i \left (\frac {1}{x}\right )^{1+m} (e x)^{1+m}\right ) \text {Subst}\left (\int \frac {x^{-2-m}}{1-e^{2 i a} x^2} \, dx,x,\frac {1}{x}\right )}{e} \\ & = \frac {i (e x)^{1+m}}{e (1+m)}-\frac {2 i (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (-1-m),\frac {1-m}{2},\frac {e^{2 i a}}{x^2}\right )}{e (1+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.47 \[ \int (e x)^m \cot (a+i \log (x)) \, dx=i x (e x)^m \left (\frac {\operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},x^2 (\cos (2 a)-i \sin (2 a))\right )}{1+m}+\frac {x^2 \operatorname {Hypergeometric2F1}\left (1,\frac {3+m}{2},\frac {5+m}{2},x^2 (\cos (2 a)-i \sin (2 a))\right ) (\cos (a)-i \sin (a))^2}{3+m}\right ) \]

[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))

Maple [F]

\[\int \left (e x \right )^{m} \cot \left (a +i \ln \left (x \right )\right )d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int (e x)^m \cot (a+i \log (x)) \, dx=\int { \left (e x\right )^{m} \cot \left (a + i \, \log \left (x\right )\right ) \,d x } \]

[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)

Sympy [F]

\[ \int (e x)^m \cot (a+i \log (x)) \, dx=\int \left (e x\right )^{m} \cot {\left (a + i \log {\left (x \right )} \right )}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int (e x)^m \cot (a+i \log (x)) \, dx=\int { \left (e x\right )^{m} \cot \left (a + i \, \log \left (x\right )\right ) \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int (e x)^m \cot (a+i \log (x)) \, dx=\int { \left (e x\right )^{m} \cot \left (a + i \, \log \left (x\right )\right ) \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int (e x)^m \cot (a+i \log (x)) \, dx=\int \mathrm {cot}\left (a+\ln \left (x\right )\,1{}\mathrm {i}\right )\,{\left (e\,x\right )}^m \,d x \]

[In]

int(cot(a + log(x)*1i)*(e*x)^m,x)

[Out]

int(cot(a + log(x)*1i)*(e*x)^m, x)