3.207 \(\int x \cos ^2(x) \cot ^3(x) \, dx\)

Optimal. Leaf size=73 \[ i \text {Li}_2\left (e^{2 i x}\right )+i x^2-\frac {3 x}{4}-2 x \log \left (1-e^{2 i x}\right )+\frac {1}{2} x \sin ^2(x)-\frac {1}{2} x \cot ^2(x)-\frac {\cot (x)}{2}+\frac {1}{4} \sin (x) \cos (x) \]

[Out]

-3/4*x+I*x^2-1/2*cot(x)-1/2*x*cot(x)^2-2*x*ln(1-exp(2*I*x))+I*polylog(2,exp(2*I*x))+1/4*cos(x)*sin(x)+1/2*x*si
n(x)^2

________________________________________________________________________________________

Rubi [A]  time = 0.16, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {4408, 3443, 2635, 8, 3717, 2190, 2279, 2391, 3720, 3473} \[ i \text {PolyLog}\left (2,e^{2 i x}\right )+i x^2-\frac {3 x}{4}-2 x \log \left (1-e^{2 i x}\right )+\frac {1}{2} x \sin ^2(x)-\frac {1}{2} x \cot ^2(x)-\frac {\cot (x)}{2}+\frac {1}{4} \sin (x) \cos (x) \]

Antiderivative was successfully verified.

[In]

Int[x*Cos[x]^2*Cot[x]^3,x]

[Out]

(-3*x)/4 + I*x^2 - Cot[x]/2 - (x*Cot[x]^2)/2 - 2*x*Log[1 - E^((2*I)*x)] + I*PolyLog[2, E^((2*I)*x)] + (Cos[x]*
Sin[x])/4 + (x*Sin[x]^2)/2

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 3443

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

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 3717

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Simp[(I*(c + d*x)^(m + 1))/(d*
(m + 1)), x] - Dist[2*I, Int[((c + d*x)^m*E^(2*I*k*Pi)*E^(2*I*(e + f*x)))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))
, x], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 3720

Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(c + d*x)^m*(b*Tan[e
+ f*x])^(n - 1))/(f*(n - 1)), x] + (-Dist[(b*d*m)/(f*(n - 1)), Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1)
, x], x] - Dist[b^2, Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n,
1] && GtQ[m, 0]

Rule 4408

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

Rubi steps

\begin {align*} \int x \cos ^2(x) \cot ^3(x) \, dx &=-\int x \cos ^2(x) \cot (x) \, dx+\int x \cot ^3(x) \, dx\\ &=-\frac {1}{2} x \cot ^2(x)+\frac {1}{2} \int \cot ^2(x) \, dx-2 \int x \cot (x) \, dx+\int x \cos (x) \sin (x) \, dx\\ &=-\frac {\cot (x)}{2}-\frac {1}{2} x \cot ^2(x)+\frac {1}{2} x \sin ^2(x)-2 \left (-\frac {i x^2}{2}-2 i \int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx\right )-\frac {\int 1 \, dx}{2}-\frac {1}{2} \int \sin ^2(x) \, dx\\ &=-\frac {x}{2}-\frac {\cot (x)}{2}-\frac {1}{2} x \cot ^2(x)+\frac {1}{4} \cos (x) \sin (x)+\frac {1}{2} x \sin ^2(x)-\frac {\int 1 \, dx}{4}-2 \left (-\frac {i x^2}{2}+x \log \left (1-e^{2 i x}\right )-\int \log \left (1-e^{2 i x}\right ) \, dx\right )\\ &=-\frac {3 x}{4}-\frac {\cot (x)}{2}-\frac {1}{2} x \cot ^2(x)+\frac {1}{4} \cos (x) \sin (x)+\frac {1}{2} x \sin ^2(x)-2 \left (-\frac {i x^2}{2}+x \log \left (1-e^{2 i x}\right )+\frac {1}{2} i \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i x}\right )\right )\\ &=-\frac {3 x}{4}-\frac {\cot (x)}{2}-\frac {1}{2} x \cot ^2(x)-2 \left (-\frac {i x^2}{2}+x \log \left (1-e^{2 i x}\right )-\frac {1}{2} i \text {Li}_2\left (e^{2 i x}\right )\right )+\frac {1}{4} \cos (x) \sin (x)+\frac {1}{2} x \sin ^2(x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.11, size = 62, normalized size = 0.85 \[ \frac {1}{8} \left (8 i \text {Li}_2\left (e^{2 i x}\right )+8 i x^2-16 x \log \left (1-e^{2 i x}\right )+\sin (2 x)-2 x \cos (2 x)-4 \cot (x)-4 x \csc ^2(x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x*Cos[x]^2*Cot[x]^3,x]

[Out]

((8*I)*x^2 - 2*x*Cos[2*x] - 4*Cot[x] - 4*x*Csc[x]^2 - 16*x*Log[1 - E^((2*I)*x)] + (8*I)*PolyLog[2, E^((2*I)*x)
] + Sin[2*x])/8

________________________________________________________________________________________

fricas [B]  time = 0.48, size = 203, normalized size = 2.78 \[ -\frac {2 \, x \cos \relax (x)^{4} - 3 \, x \cos \relax (x)^{2} - {\left (4 i \, \cos \relax (x)^{2} - 4 i\right )} {\rm Li}_2\left (\cos \relax (x) + i \, \sin \relax (x)\right ) - {\left (-4 i \, \cos \relax (x)^{2} + 4 i\right )} {\rm Li}_2\left (\cos \relax (x) - i \, \sin \relax (x)\right ) - {\left (-4 i \, \cos \relax (x)^{2} + 4 i\right )} {\rm Li}_2\left (-\cos \relax (x) + i \, \sin \relax (x)\right ) - {\left (4 i \, \cos \relax (x)^{2} - 4 i\right )} {\rm Li}_2\left (-\cos \relax (x) - i \, \sin \relax (x)\right ) + 4 \, {\left (x \cos \relax (x)^{2} - x\right )} \log \left (\cos \relax (x) + i \, \sin \relax (x) + 1\right ) + 4 \, {\left (x \cos \relax (x)^{2} - x\right )} \log \left (\cos \relax (x) - i \, \sin \relax (x) + 1\right ) + 4 \, {\left (x \cos \relax (x)^{2} - x\right )} \log \left (-\cos \relax (x) + i \, \sin \relax (x) + 1\right ) + 4 \, {\left (x \cos \relax (x)^{2} - x\right )} \log \left (-\cos \relax (x) - i \, \sin \relax (x) + 1\right ) - {\left (\cos \relax (x)^{3} + \cos \relax (x)\right )} \sin \relax (x) - x}{4 \, {\left (\cos \relax (x)^{2} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cos(x)^2*cot(x)^3,x, algorithm="fricas")

[Out]

-1/4*(2*x*cos(x)^4 - 3*x*cos(x)^2 - (4*I*cos(x)^2 - 4*I)*dilog(cos(x) + I*sin(x)) - (-4*I*cos(x)^2 + 4*I)*dilo
g(cos(x) - I*sin(x)) - (-4*I*cos(x)^2 + 4*I)*dilog(-cos(x) + I*sin(x)) - (4*I*cos(x)^2 - 4*I)*dilog(-cos(x) -
I*sin(x)) + 4*(x*cos(x)^2 - x)*log(cos(x) + I*sin(x) + 1) + 4*(x*cos(x)^2 - x)*log(cos(x) - I*sin(x) + 1) + 4*
(x*cos(x)^2 - x)*log(-cos(x) + I*sin(x) + 1) + 4*(x*cos(x)^2 - x)*log(-cos(x) - I*sin(x) + 1) - (cos(x)^3 + co
s(x))*sin(x) - x)/(cos(x)^2 - 1)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \cos \relax (x)^{2} \cot \relax (x)^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cos(x)^2*cot(x)^3,x, algorithm="giac")

[Out]

integrate(x*cos(x)^2*cot(x)^3, x)

________________________________________________________________________________________

maple [A]  time = 0.14, size = 109, normalized size = 1.49 \[ i x^{2}-\frac {\left (i+2 x \right ) {\mathrm e}^{2 i x}}{16}-\frac {\left (2 x -i\right ) {\mathrm e}^{-2 i x}}{16}+\frac {2 x \,{\mathrm e}^{2 i x}-i {\mathrm e}^{2 i x}+i}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}-2 x \ln \left (1+{\mathrm e}^{i x}\right )-2 x \ln \left (1-{\mathrm e}^{i x}\right )+2 i \polylog \left (2, -{\mathrm e}^{i x}\right )+2 i \polylog \left (2, {\mathrm e}^{i x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*cos(x)^2*cot(x)^3,x)

[Out]

I*x^2-1/16*(I+2*x)*exp(2*I*x)-1/16*(2*x-I)*exp(-2*I*x)+(2*x*exp(2*I*x)-I*exp(2*I*x)+I)/(exp(2*I*x)-1)^2-2*x*ln
(1+exp(I*x))-2*x*ln(1-exp(I*x))+2*I*polylog(2,-exp(I*x))+2*I*polylog(2,exp(I*x))

________________________________________________________________________________________

maxima [B]  time = 0.59, size = 1739, normalized size = 23.82 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cos(x)^2*cot(x)^3,x, algorithm="maxima")

[Out]

-((2*x + I)*cos(6*x)^2 - (-32*I*x^2 - 8*x - 4*I)*cos(4*x)^2 - (-32*I*x^2 + 28*x - 16*I)*cos(2*x)^2 - (2*x + I)
*sin(6*x)^2 - (32*I*x^2 + 8*x + 4*I)*sin(4*x)^2 - (32*I*x^2 - 28*x + 16*I)*sin(2*x)^2 - (64*I*x*cos(4*x)^2 + 6
4*I*x*cos(2*x)^2 - 64*I*x*sin(4*x)^2 - 64*I*x*sin(2*x)^2 + (-32*I*x*cos(4*x) + 64*I*x*cos(2*x) + 32*x*sin(4*x)
- 64*x*sin(2*x) - 32*I*x)*cos(6*x) + (-160*I*x*cos(2*x) + 160*x*sin(2*x) + 64*I*x)*cos(4*x) - 32*I*x*cos(2*x)
+ (32*x*cos(4*x) - 64*x*cos(2*x) + 32*I*x*sin(4*x) - 64*I*x*sin(2*x) + 32*x)*sin(6*x) - 32*(4*x*cos(4*x) - 5*
x*cos(2*x) - 5*I*x*sin(2*x) + 2*x)*sin(4*x) - 32*(4*x*cos(2*x) - x)*sin(2*x))*arctan2(sin(x), cos(x) + 1) - (-
64*I*x*cos(4*x)^2 - 64*I*x*cos(2*x)^2 + 64*I*x*sin(4*x)^2 + 64*I*x*sin(2*x)^2 + (32*I*x*cos(4*x) - 64*I*x*cos(
2*x) - 32*x*sin(4*x) + 64*x*sin(2*x) + 32*I*x)*cos(6*x) + (160*I*x*cos(2*x) - 160*x*sin(2*x) - 64*I*x)*cos(4*x
) + 32*I*x*cos(2*x) - (32*x*cos(4*x) - 64*x*cos(2*x) + 32*I*x*sin(4*x) - 64*I*x*sin(2*x) + 32*x)*sin(6*x) + 32
*(4*x*cos(4*x) - 5*x*cos(2*x) - 5*I*x*sin(2*x) + 2*x)*sin(4*x) + 32*(4*x*cos(2*x) - x)*sin(2*x))*arctan2(sin(x
), -cos(x) + 1) - (16*I*x^2 + (16*I*x^2 + 8*x + 4*I)*cos(4*x) + (-32*I*x^2 + 26*x - 17*I)*cos(2*x) - 4*(4*x^2
- 2*I*x + 1)*sin(4*x) + (32*x^2 + 26*I*x + 17)*sin(2*x) + 4*x + 14*I)*cos(6*x) - (-32*I*x^2 + (80*I*x^2 - 52*x
+ 34*I)*cos(2*x) - 2*(40*x^2 + 26*I*x + 17)*sin(2*x) - 10*x - 27*I)*cos(4*x) - (16*I*x^2 + 8*x + 12*I)*cos(2*
x) - ((32*I*cos(4*x) - 64*I*cos(2*x) - 32*sin(4*x) + 64*sin(2*x) + 32*I)*cos(6*x) + (160*I*cos(2*x) - 160*sin(
2*x) - 64*I)*cos(4*x) - 64*I*cos(4*x)^2 - 64*I*cos(2*x)^2 - (32*cos(4*x) - 64*cos(2*x) + 32*I*sin(4*x) - 64*I*
sin(2*x) + 32)*sin(6*x) + (128*cos(4*x) - 160*cos(2*x) - 160*I*sin(2*x) + 64)*sin(4*x) + 64*I*sin(4*x)^2 + 32*
(4*cos(2*x) - 1)*sin(2*x) + 64*I*sin(2*x)^2 + 32*I*cos(2*x))*dilog(-e^(I*x)) - ((32*I*cos(4*x) - 64*I*cos(2*x)
- 32*sin(4*x) + 64*sin(2*x) + 32*I)*cos(6*x) + (160*I*cos(2*x) - 160*sin(2*x) - 64*I)*cos(4*x) - 64*I*cos(4*x
)^2 - 64*I*cos(2*x)^2 - (32*cos(4*x) - 64*cos(2*x) + 32*I*sin(4*x) - 64*I*sin(2*x) + 32)*sin(6*x) + (128*cos(4
*x) - 160*cos(2*x) - 160*I*sin(2*x) + 64)*sin(4*x) + 64*I*sin(4*x)^2 + 32*(4*cos(2*x) - 1)*sin(2*x) + 64*I*sin
(2*x)^2 + 32*I*cos(2*x))*dilog(e^(I*x)) - (32*x*cos(4*x)^2 + 32*x*cos(2*x)^2 - 32*x*sin(4*x)^2 - 32*x*sin(2*x)
^2 - (16*x*cos(4*x) - 32*x*cos(2*x) + 16*I*x*sin(4*x) - 32*I*x*sin(2*x) + 16*x)*cos(6*x) - 16*(5*x*cos(2*x) +
5*I*x*sin(2*x) - 2*x)*cos(4*x) - 16*x*cos(2*x) + (-16*I*x*cos(4*x) + 32*I*x*cos(2*x) + 16*x*sin(4*x) - 32*x*si
n(2*x) - 16*I*x)*sin(6*x) + (64*I*x*cos(4*x) - 80*I*x*cos(2*x) + 80*x*sin(2*x) + 32*I*x)*sin(4*x) + (64*I*x*co
s(2*x) - 16*I*x)*sin(2*x))*log(cos(x)^2 + sin(x)^2 + 2*cos(x) + 1) - (32*x*cos(4*x)^2 + 32*x*cos(2*x)^2 - 32*x
*sin(4*x)^2 - 32*x*sin(2*x)^2 - (16*x*cos(4*x) - 32*x*cos(2*x) + 16*I*x*sin(4*x) - 32*I*x*sin(2*x) + 16*x)*cos
(6*x) - 16*(5*x*cos(2*x) + 5*I*x*sin(2*x) - 2*x)*cos(4*x) - 16*x*cos(2*x) + (-16*I*x*cos(4*x) + 32*I*x*cos(2*x
) + 16*x*sin(4*x) - 32*x*sin(2*x) - 16*I*x)*sin(6*x) + (64*I*x*cos(4*x) - 80*I*x*cos(2*x) + 80*x*sin(2*x) + 32
*I*x)*sin(4*x) + (64*I*x*cos(2*x) - 16*I*x)*sin(2*x))*log(cos(x)^2 + sin(x)^2 - 2*cos(x) + 1) + (16*x^2 + 2*(2
*I*x - 1)*cos(6*x) + 4*(4*x^2 - 2*I*x + 1)*cos(4*x) - (32*x^2 + 26*I*x + 17)*cos(2*x) - (-16*I*x^2 - 8*x - 4*I
)*sin(4*x) - (32*I*x^2 - 26*x + 17*I)*sin(2*x) - 4*I*x + 14)*sin(6*x) - (32*x^2 + 8*(8*x^2 - 2*I*x + 1)*cos(4*
x) - 2*(40*x^2 + 26*I*x + 17)*cos(2*x) + (-80*I*x^2 + 52*x - 34*I)*sin(2*x) - 10*I*x + 27)*sin(4*x) + 4*(4*x^2
- 2*(8*x^2 + 7*I*x + 4)*cos(2*x) - 2*I*x + 3)*sin(2*x) + 2*x - I)/((16*cos(4*x) - 32*cos(2*x) + 16*I*sin(4*x)
- 32*I*sin(2*x) + 16)*cos(6*x) + (80*cos(2*x) + 80*I*sin(2*x) - 32)*cos(4*x) - 32*cos(4*x)^2 - 32*cos(2*x)^2
- (-16*I*cos(4*x) + 32*I*cos(2*x) + 16*sin(4*x) - 32*sin(2*x) - 16*I)*sin(6*x) - (64*I*cos(4*x) - 80*I*cos(2*x
) + 80*sin(2*x) + 32*I)*sin(4*x) + 32*sin(4*x)^2 - (64*I*cos(2*x) - 16*I)*sin(2*x) + 32*sin(2*x)^2 + 16*cos(2*
x))

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,{\cos \relax (x)}^2\,{\mathrm {cot}\relax (x)}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*cos(x)^2*cot(x)^3,x)

[Out]

int(x*cos(x)^2*cot(x)^3, x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \cos ^{2}{\relax (x )} \cot ^{3}{\relax (x )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cos(x)**2*cot(x)**3,x)

[Out]

Integral(x*cos(x)**2*cot(x)**3, x)

________________________________________________________________________________________