\(\int x \cot ^2(a+i \log (x)) \, dx\) [196]

Optimal result

Integrand size = 13, antiderivative size = 55 \[ \int x \cot ^2(a+i \log (x)) \, dx=-\frac {x^2}{2}-\frac {2 e^{4 i a}}{e^{2 i a}-x^2}-2 e^{2 i a} \log \left (e^{2 i a}-x^2\right ) \]

[Out]

-1/2*x^2-2*exp(4*I*a)/(exp(2*I*a)-x^2)-2*exp(2*I*a)*ln(exp(2*I*a)-x^2)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {4592, 456, 455, 45} \[ \int x \cot ^2(a+i \log (x)) \, dx=-\frac {2 e^{4 i a}}{-x^2+e^{2 i a}}-2 e^{2 i a} \log \left (-x^2+e^{2 i a}\right )-\frac {x^2}{2} \]

[In]

Int[x*Cot[a + I*Log[x]]^2,x]

[Out]

-1/2*x^2 - (2*E^((4*I)*a))/(E^((2*I)*a) - x^2) - 2*E^((2*I)*a)*Log[E^((2*I)*a) - x^2]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 455

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

Rule 456

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

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

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(142\) vs. \(2(55)=110\).

Time = 0.10 (sec) , antiderivative size = 142, normalized size of antiderivative = 2.58 \[ \int x \cot ^2(a+i \log (x)) \, dx=-\frac {x^2}{2}+2 i \arctan \left (\frac {\cot (a)-x^2 \cot (a)}{1+x^2}\right ) \cos (2 a)-\cos (2 a) \log \left (1+x^4-2 x^2 \cos (2 a)\right )-4 \arctan \left (\frac {\cot (a)-x^2 \cot (a)}{1+x^2}\right ) \cos (a) \sin (a)-i \log \left (1+x^4-2 x^2 \cos (2 a)\right ) \sin (2 a)+\frac {2 \cos (3 a)+2 i \sin (3 a)}{\left (-1+x^2\right ) \cos (a)-i \left (1+x^2\right ) \sin (a)} \]

[In]

Integrate[x*Cot[a + I*Log[x]]^2,x]

[Out]

-1/2*x^2 + (2*I)*ArcTan[(Cot[a] - x^2*Cot[a])/(1 + x^2)]*Cos[2*a] - Cos[2*a]*Log[1 + x^4 - 2*x^2*Cos[2*a]] - 4
*ArcTan[(Cot[a] - x^2*Cot[a])/(1 + x^2)]*Cos[a]*Sin[a] - I*Log[1 + x^4 - 2*x^2*Cos[2*a]]*Sin[2*a] + (2*Cos[3*a
] + (2*I)*Sin[3*a])/((-1 + x^2)*Cos[a] - I*(1 + x^2)*Sin[a])

Maple [A] (verified)

Time = 1.03 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.80

method	result	size
risch	\(-\frac {5 x^{2}}{2}-\frac {2 x^{2}}{\frac {{\mathrm e}^{2 i a}}{x^{2}}-1}-2 \,{\mathrm e}^{2 i a} \ln \left ({\mathrm e}^{2 i a}-x^{2}\right )\)	\(44\)

[In]

int(x*cot(a+I*ln(x))^2,x,method=_RETURNVERBOSE)

[Out]

-5/2*x^2-2*x^2/(exp(2*I*a)/x^2-1)-2*exp(2*I*a)*ln(exp(2*I*a)-x^2)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.11 \[ \int x \cot ^2(a+i \log (x)) \, dx=-\frac {x^{4} - x^{2} e^{\left (2 i \, a\right )} + 4 \, {\left (x^{2} e^{\left (2 i \, a\right )} - e^{\left (4 i \, a\right )}\right )} \log \left (x^{2} - e^{\left (2 i \, a\right )}\right ) - 4 \, e^{\left (4 i \, a\right )}}{2 \, {\left (x^{2} - e^{\left (2 i \, a\right )}\right )}} \]

[In]

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

[Out]

-1/2*(x^4 - x^2*e^(2*I*a) + 4*(x^2*e^(2*I*a) - e^(4*I*a))*log(x^2 - e^(2*I*a)) - 4*e^(4*I*a))/(x^2 - e^(2*I*a)
)

Sympy [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.76 \[ \int x \cot ^2(a+i \log (x)) \, dx=- \frac {x^{2}}{2} - 2 e^{2 i a} \log {\left (x^{2} - e^{2 i a} \right )} + \frac {2 e^{4 i a}}{x^{2} - e^{2 i a}} \]

[In]

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

[Out]

-x**2/2 - 2*exp(2*I*a)*log(x**2 - exp(2*I*a)) + 2*exp(4*I*a)/(x**2 - exp(2*I*a))

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 290 vs. \(2 (41) = 82\).

Time = 0.22 (sec) , antiderivative size = 290, normalized size of antiderivative = 5.27 \[ \int x \cot ^2(a+i \log (x)) \, dx=-\frac {x^{4} - {\left (4 \, {\left (-i \, \cos \left (2 \, a\right ) + \sin \left (2 \, a\right )\right )} \arctan \left (\sin \left (a\right ), x + \cos \left (a\right )\right ) + 4 \, {\left (i \, \cos \left (2 \, a\right ) - \sin \left (2 \, a\right )\right )} \arctan \left (\sin \left (a\right ), x - \cos \left (a\right )\right ) + \cos \left (2 \, a\right ) + i \, \sin \left (2 \, a\right )\right )} x^{2} - 4 \, {\left (i \, \cos \left (2 \, a\right )^{2} - 2 \, \cos \left (2 \, a\right ) \sin \left (2 \, a\right ) - i \, \sin \left (2 \, a\right )^{2}\right )} \arctan \left (\sin \left (a\right ), x + \cos \left (a\right )\right ) - 4 \, {\left (-i \, \cos \left (2 \, a\right )^{2} + 2 \, \cos \left (2 \, a\right ) \sin \left (2 \, a\right ) + i \, \sin \left (2 \, a\right )^{2}\right )} \arctan \left (\sin \left (a\right ), x - \cos \left (a\right )\right ) + 2 \, {\left (x^{2} {\left (\cos \left (2 \, a\right ) + i \, \sin \left (2 \, a\right )\right )} - \cos \left (2 \, a\right )^{2} - 2 i \, \cos \left (2 \, a\right ) \sin \left (2 \, a\right ) + \sin \left (2 \, a\right )^{2}\right )} \log \left (x^{2} + 2 \, x \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (a\right )^{2}\right ) + 2 \, {\left (x^{2} {\left (\cos \left (2 \, a\right ) + i \, \sin \left (2 \, a\right )\right )} - \cos \left (2 \, a\right )^{2} - 2 i \, \cos \left (2 \, a\right ) \sin \left (2 \, a\right ) + \sin \left (2 \, a\right )^{2}\right )} \log \left (x^{2} - 2 \, x \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (a\right )^{2}\right ) - 4 \, \cos \left (4 \, a\right ) - 4 i \, \sin \left (4 \, a\right )}{2 \, {\left (x^{2} - \cos \left (2 \, a\right ) - i \, \sin \left (2 \, a\right )\right )}} \]

[In]

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

[Out]

-1/2*(x^4 - (4*(-I*cos(2*a) + sin(2*a))*arctan2(sin(a), x + cos(a)) + 4*(I*cos(2*a) - sin(2*a))*arctan2(sin(a)
, x - cos(a)) + cos(2*a) + I*sin(2*a))*x^2 - 4*(I*cos(2*a)^2 - 2*cos(2*a)*sin(2*a) - I*sin(2*a)^2)*arctan2(sin
(a), x + cos(a)) - 4*(-I*cos(2*a)^2 + 2*cos(2*a)*sin(2*a) + I*sin(2*a)^2)*arctan2(sin(a), x - cos(a)) + 2*(x^2
*(cos(2*a) + I*sin(2*a)) - cos(2*a)^2 - 2*I*cos(2*a)*sin(2*a) + sin(2*a)^2)*log(x^2 + 2*x*cos(a) + cos(a)^2 +
sin(a)^2) + 2*(x^2*(cos(2*a) + I*sin(2*a)) - cos(2*a)^2 - 2*I*cos(2*a)*sin(2*a) + sin(2*a)^2)*log(x^2 - 2*x*co
s(a) + cos(a)^2 + sin(a)^2) - 4*cos(4*a) - 4*I*sin(4*a))/(x^2 - cos(2*a) - I*sin(2*a))

Giac [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (41) = 82\).

Time = 0.31 (sec) , antiderivative size = 118, normalized size of antiderivative = 2.15 \[ \int x \cot ^2(a+i \log (x)) \, dx=-\frac {x^{4}}{2 \, {\left (x^{2} - e^{\left (2 i \, a\right )}\right )}} - \frac {2 \, x^{2} e^{\left (2 i \, a\right )} \log \left (-x^{2} + e^{\left (2 i \, a\right )}\right )}{x^{2} - e^{\left (2 i \, a\right )}} + \frac {x^{2} e^{\left (2 i \, a\right )}}{2 \, {\left (x^{2} - e^{\left (2 i \, a\right )}\right )}} + \frac {2 \, e^{\left (4 i \, a\right )} \log \left (-x^{2} + e^{\left (2 i \, a\right )}\right )}{x^{2} - e^{\left (2 i \, a\right )}} + \frac {2 \, e^{\left (4 i \, a\right )}}{x^{2} - e^{\left (2 i \, a\right )}} \]

[In]

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

[Out]

-1/2*x^4/(x^2 - e^(2*I*a)) - 2*x^2*e^(2*I*a)*log(-x^2 + e^(2*I*a))/(x^2 - e^(2*I*a)) + 1/2*x^2*e^(2*I*a)/(x^2
- e^(2*I*a)) + 2*e^(4*I*a)*log(-x^2 + e^(2*I*a))/(x^2 - e^(2*I*a)) + 2*e^(4*I*a)/(x^2 - e^(2*I*a))

Mupad [B] (verification not implemented)

Time = 26.69 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.82 \[ \int x \cot ^2(a+i \log (x)) \, dx=-\frac {2\,{\mathrm {e}}^{a\,4{}\mathrm {i}}}{{\mathrm {e}}^{a\,2{}\mathrm {i}}-x^2}-2\,\ln \left (x^2-{\mathrm {e}}^{a\,2{}\mathrm {i}}\right )\,{\mathrm {e}}^{a\,2{}\mathrm {i}}-\frac {x^2}{2} \]

[In]

int(x*cot(a + log(x)*1i)^2,x)

[Out]

- (2*exp(a*4i))/(exp(a*2i) - x^2) - 2*log(x^2 - exp(a*2i))*exp(a*2i) - x^2/2