\(\int (e x)^m \cot ^2(a+i \log (x)) \, dx\) [202]

Optimal result

Integrand size = 17, antiderivative size = 77 \[ \int (e x)^m \cot ^2(a+i \log (x)) \, dx=-\frac {x (e x)^m}{1+m}+\frac {2 x (e x)^m}{1-\frac {e^{2 i a}}{x^2}}-2 x (e x)^m \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (-1-m),\frac {1-m}{2},\frac {e^{2 i a}}{x^2}\right ) \]

[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 [A] (verified)

Time = 0.13 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {4592, 511, 474, 470, 371} \[ \int (e x)^m \cot ^2(a+i \log (x)) \, dx=-2 x (e x)^m \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (-m-1),\frac {1-m}{2},\frac {e^{2 i a}}{x^2}\right )+\frac {2 x (e x)^m}{1-\frac {e^{2 i a}}{x^2}}-\frac {x (e x)^m}{m+1} \]

[In]

Int[(e*x)^m*Cot[a + I*Log[x]]^2,x]

[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]

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 474

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^2, x_Symbol] :> Simp[(-(b*c - a*
d)^2)*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*b^2*e*n*(p + 1))), x] + Dist[1/(a*b^2*n*(p + 1)), Int[(e*x)^m*(a +
b*x^n)^(p + 1)*Simp[(b*c - a*d)^2*(m + 1) + b^2*c^2*n*(p + 1) + a*b*d^2*n*(p + 1)*x^n, x], x], x] /; FreeQ[{a
, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1]

Rule 511

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

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 )^2 (e x)^m}{\left (1-\frac {e^{2 i a}}{x^2}\right )^2} \, dx \\ & = -\left (\left (\left (\frac {1}{x}\right )^m (e x)^m\right ) \text {Subst}\left (\int \frac {x^{-2-m} \left (-i-i e^{2 i a} x^2\right )^2}{\left (1-e^{2 i a} x^2\right )^2} \, dx,x,\frac {1}{x}\right )\right ) \\ & = \frac {2 x (e x)^m}{1-\frac {e^{2 i a}}{x^2}}+\frac {1}{2} \left (e^{-4 i a} \left (\frac {1}{x}\right )^m (e x)^m\right ) \text {Subst}\left (\int \frac {x^{-2-m} \left (2 e^{4 i a} (3+2 m)-2 e^{6 i a} x^2\right )}{1-e^{2 i a} x^2} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {x (e x)^m}{1+m}+\frac {2 x (e x)^m}{1-\frac {e^{2 i a}}{x^2}}+\left (2 (1+m) \left (\frac {1}{x}\right )^m (e x)^m\right ) \text {Subst}\left (\int \frac {x^{-2-m}}{1-e^{2 i a} x^2} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {x (e x)^m}{1+m}+\frac {2 x (e x)^m}{1-\frac {e^{2 i a}}{x^2}}-2 x (e x)^m \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (-1-m),\frac {1-m}{2},\frac {e^{2 i a}}{x^2}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.09 \[ \int (e x)^m \cot ^2(a+i \log (x)) \, dx=\frac {x (e x)^m \left (-1+4 \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},x^2 (\cos (2 a)-i \sin (2 a))\right )-4 \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{2},\frac {3+m}{2},x^2 (\cos (2 a)-i \sin (2 a))\right )\right )}{1+m} \]

[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)

Maple [F]

\[\int \left (e x \right )^{m} \cot \left (a +i \ln \left (x \right )\right )^{2}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int (e x)^m \cot ^2(a+i \log (x)) \, dx=\int { \left (e x\right )^{m} \cot \left (a + i \, \log \left (x\right )\right )^{2} \,d x } \]

[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)

Sympy [F]

\[ \int (e x)^m \cot ^2(a+i \log (x)) \, dx=\int \left (e x\right )^{m} \cot ^{2}{\left (a + i \log {\left (x \right )} \right )}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int (e x)^m \cot ^2(a+i \log (x)) \, dx=\int { \left (e x\right )^{m} \cot \left (a + i \, \log \left (x\right )\right )^{2} \,d x } \]

[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)

Giac [F]

\[ \int (e x)^m \cot ^2(a+i \log (x)) \, dx=\int { \left (e x\right )^{m} \cot \left (a + i \, \log \left (x\right )\right )^{2} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int (e x)^m \cot ^2(a+i \log (x)) \, dx=\int {\mathrm {cot}\left (a+\ln \left (x\right )\,1{}\mathrm {i}\right )}^2\,{\left (e\,x\right )}^m \,d x \]

[In]

int(cot(a + log(x)*1i)^2*(e*x)^m,x)

[Out]

int(cot(a + log(x)*1i)^2*(e*x)^m, x)