3.195 \(\int x^2 \cot ^2(a+i \log (x)) \, dx\)

Optimal. Leaf size=64 \[ -\frac {2 e^{2 i a} x^3}{-x^2+e^{2 i a}}-6 e^{2 i a} x+6 e^{3 i a} \tanh ^{-1}\left (e^{-i a} x\right )-\frac {x^3}{3} \]

[Out]

-6*exp(2*I*a)*x-1/3*x^3-2*exp(2*I*a)*x^3/(exp(2*I*a)-x^2)+6*exp(3*I*a)*arctanh(x/exp(I*a))

________________________________________________________________________________________

Rubi [F]  time = 0.05, 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 = {} \[ \int x^2 \cot ^2(a+i \log (x)) \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^2*Cot[a + I*Log[x]]^2,x]

[Out]

Defer[Int][x^2*Cot[a + I*Log[x]]^2, x]

Rubi steps

\begin {align*} \int x^2 \cot ^2(a+i \log (x)) \, dx &=\int x^2 \cot ^2(a+i \log (x)) \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.13, size = 100, normalized size = 1.56 \[ \frac {2 x (\cos (3 a)+i \sin (3 a))}{\left (x^2-1\right ) \cos (a)-i \left (x^2+1\right ) \sin (a)}-4 i x \sin (2 a)-4 x \cos (2 a)+6 \cos (3 a) \tanh ^{-1}(x (\cos (a)-i \sin (a)))+6 i \sin (3 a) \tanh ^{-1}(x (\cos (a)-i \sin (a)))-\frac {x^3}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Cot[a + I*Log[x]]^2,x]

[Out]

-1/3*x^3 - 4*x*Cos[2*a] + 6*ArcTanh[x*(Cos[a] - I*Sin[a])]*Cos[3*a] - (4*I)*x*Sin[2*a] + (2*x*(Cos[3*a] + I*Si
n[3*a]))/((-1 + x^2)*Cos[a] - I*(1 + x^2)*Sin[a]) + (6*I)*ArcTanh[x*(Cos[a] - I*Sin[a])]*Sin[3*a]

________________________________________________________________________________________

fricas [B]  time = 0.65, size = 102, normalized size = 1.59 \[ -\frac {x^{5} + 11 \, x^{3} e^{\left (2 i \, a\right )} - 9 \, {\left (x^{2} - e^{\left (2 i \, a\right )}\right )} e^{\left (3 i \, a\right )} \log \left ({\left (x e^{\left (2 i \, a\right )} + e^{\left (3 i \, a\right )}\right )} e^{\left (-2 i \, a\right )}\right ) + 9 \, {\left (x^{2} - e^{\left (2 i \, a\right )}\right )} e^{\left (3 i \, a\right )} \log \left ({\left (x e^{\left (2 i \, a\right )} - e^{\left (3 i \, a\right )}\right )} e^{\left (-2 i \, a\right )}\right ) - 18 \, x e^{\left (4 i \, a\right )}}{3 \, {\left (x^{2} - e^{\left (2 i \, a\right )}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(x^5 + 11*x^3*e^(2*I*a) - 9*(x^2 - e^(2*I*a))*e^(3*I*a)*log((x*e^(2*I*a) + e^(3*I*a))*e^(-2*I*a)) + 9*(x^
2 - e^(2*I*a))*e^(3*I*a)*log((x*e^(2*I*a) - e^(3*I*a))*e^(-2*I*a)) - 18*x*e^(4*I*a))/(x^2 - e^(2*I*a))

________________________________________________________________________________________

giac [A]  time = 0.67, size = 83, normalized size = 1.30 \[ -\frac {x^{5}}{3 \, {\left (x^{2} - e^{\left (2 i \, a\right )}\right )}} - \frac {11 \, x^{3} e^{\left (2 i \, a\right )}}{3 \, {\left (x^{2} - e^{\left (2 i \, a\right )}\right )}} - \frac {6 \, \arctan \left (\frac {x}{\sqrt {-e^{\left (2 i \, a\right )}}}\right ) e^{\left (4 i \, a\right )}}{\sqrt {-e^{\left (2 i \, a\right )}}} + \frac {10 \, x e^{\left (4 i \, a\right )}}{x^{2} - e^{\left (2 i \, a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*x^5/(x^2 - e^(2*I*a)) - 11/3*x^3*e^(2*I*a)/(x^2 - e^(2*I*a)) - 6*arctan(x/sqrt(-e^(2*I*a)))*e^(4*I*a)/sqr
t(-e^(2*I*a)) + 10*x*e^(4*I*a)/(x^2 - e^(2*I*a))

________________________________________________________________________________________

maple [A]  time = 0.06, size = 48, normalized size = 0.75 \[ -\frac {7 x^{3}}{3}-\frac {2 x^{3}}{\frac {{\mathrm e}^{2 i a}}{x^{2}}-1}-6 \,{\mathrm e}^{2 i a} x +6 \arctanh \left (x \,{\mathrm e}^{-i a}\right ) {\mathrm e}^{3 i a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-7/3*x^3-2*x^3/(exp(2*I*a)/x^2-1)-6*exp(2*I*a)*x+6*arctanh(x*exp(-I*a))*exp(3*I*a)

________________________________________________________________________________________

maxima [B]  time = 0.36, size = 352, normalized size = 5.50 \[ -\frac {2 \, x^{5} + x^{3} {\left (22 \, \cos \left (2 \, a\right ) + 22 i \, \sin \left (2 \, a\right )\right )} + 18 \, {\left ({\left (-i \, \cos \left (3 \, a\right ) + \sin \left (3 \, a\right )\right )} \arctan \left (\sin \relax (a), x + \cos \relax (a)\right ) + {\left (-i \, \cos \left (3 \, a\right ) + \sin \left (3 \, a\right )\right )} \arctan \left (\sin \relax (a), x - \cos \relax (a)\right )\right )} x^{2} - x {\left (36 \, \cos \left (4 \, a\right ) + 36 i \, \sin \left (4 \, a\right )\right )} + {\left (18 \, {\left (i \, \cos \left (2 \, a\right ) - \sin \left (2 \, a\right )\right )} \cos \left (3 \, a\right ) - {\left (18 \, \cos \left (2 \, a\right ) + 18 i \, \sin \left (2 \, a\right )\right )} \sin \left (3 \, a\right )\right )} \arctan \left (\sin \relax (a), x + \cos \relax (a)\right ) + {\left (18 \, {\left (i \, \cos \left (2 \, a\right ) - \sin \left (2 \, a\right )\right )} \cos \left (3 \, a\right ) - {\left (18 \, \cos \left (2 \, a\right ) + 18 i \, \sin \left (2 \, a\right )\right )} \sin \left (3 \, a\right )\right )} \arctan \left (\sin \relax (a), x - \cos \relax (a)\right ) - {\left (x^{2} {\left (9 \, \cos \left (3 \, a\right ) + 9 i \, \sin \left (3 \, a\right )\right )} - {\left (9 \, \cos \left (2 \, a\right ) + 9 i \, \sin \left (2 \, a\right )\right )} \cos \left (3 \, a\right ) - 9 \, {\left (i \, \cos \left (2 \, a\right ) - \sin \left (2 \, a\right )\right )} \sin \left (3 \, a\right )\right )} \log \left (x^{2} + 2 \, x \cos \relax (a) + \cos \relax (a)^{2} + \sin \relax (a)^{2}\right ) + {\left (x^{2} {\left (9 \, \cos \left (3 \, a\right ) + 9 i \, \sin \left (3 \, a\right )\right )} - {\left (9 \, \cos \left (2 \, a\right ) + 9 i \, \sin \left (2 \, a\right )\right )} \cos \left (3 \, a\right ) + 9 \, {\left (-i \, \cos \left (2 \, a\right ) + \sin \left (2 \, a\right )\right )} \sin \left (3 \, a\right )\right )} \log \left (x^{2} - 2 \, x \cos \relax (a) + \cos \relax (a)^{2} + \sin \relax (a)^{2}\right )}{6 \, x^{2} - 6 \, \cos \left (2 \, a\right ) - 6 i \, \sin \left (2 \, a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*x^5 + x^3*(22*cos(2*a) + 22*I*sin(2*a)) + 18*((-I*cos(3*a) + sin(3*a))*arctan2(sin(a), x + cos(a)) + (-I*c
os(3*a) + sin(3*a))*arctan2(sin(a), x - cos(a)))*x^2 - x*(36*cos(4*a) + 36*I*sin(4*a)) + (18*(I*cos(2*a) - sin
(2*a))*cos(3*a) - (18*cos(2*a) + 18*I*sin(2*a))*sin(3*a))*arctan2(sin(a), x + cos(a)) + (18*(I*cos(2*a) - sin(
2*a))*cos(3*a) - (18*cos(2*a) + 18*I*sin(2*a))*sin(3*a))*arctan2(sin(a), x - cos(a)) - (x^2*(9*cos(3*a) + 9*I*
sin(3*a)) - (9*cos(2*a) + 9*I*sin(2*a))*cos(3*a) - 9*(I*cos(2*a) - sin(2*a))*sin(3*a))*log(x^2 + 2*x*cos(a) +
cos(a)^2 + sin(a)^2) + (x^2*(9*cos(3*a) + 9*I*sin(3*a)) - (9*cos(2*a) + 9*I*sin(2*a))*cos(3*a) + 9*(-I*cos(2*a
) + sin(2*a))*sin(3*a))*log(x^2 - 2*x*cos(a) + cos(a)^2 + sin(a)^2))/(6*x^2 - 6*cos(2*a) - 6*I*sin(2*a))

________________________________________________________________________________________

mupad [B]  time = 2.22, size = 57, normalized size = 0.89 \[ -{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\right )}^{3/2}\,\mathrm {atan}\left (\frac {x\,1{}\mathrm {i}}{\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}}}}\right )\,6{}\mathrm {i}-\frac {x^3}{3}-4\,x\,{\mathrm {e}}^{a\,2{}\mathrm {i}}-\frac {2\,x\,{\mathrm {e}}^{a\,4{}\mathrm {i}}}{{\mathrm {e}}^{a\,2{}\mathrm {i}}-x^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*cot(a + log(x)*1i)^2,x)

[Out]

- exp(a*2i)^(3/2)*atan((x*1i)/exp(a*2i)^(1/2))*6i - x^3/3 - 4*x*exp(a*2i) - (2*x*exp(a*4i))/(exp(a*2i) - x^2)

________________________________________________________________________________________

sympy [A]  time = 0.33, size = 60, normalized size = 0.94 \[ - \frac {x^{3}}{3} - 4 x e^{2 i a} + \frac {2 x e^{4 i a}}{x^{2} - e^{2 i a}} - 3 \left (\log {\left (x - e^{i a} \right )} - \log {\left (x + e^{i a} \right )}\right ) e^{3 i a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x**3/3 - 4*x*exp(2*I*a) + 2*x*exp(4*I*a)/(x**2 - exp(2*I*a)) - 3*(log(x - exp(I*a)) - log(x + exp(I*a)))*exp(
3*I*a)

________________________________________________________________________________________