3.203 \(\int x^2 \cos ^2(x) \cot ^2(x) \, dx\)

Optimal. Leaf size=83 \[ -i \text{PolyLog}\left (2,e^{2 i x}\right )-\frac{x^3}{2}-i x^2-x^2 \cot (x)-\frac{1}{2} x^2 \sin (x) \cos (x)+\frac{x}{4}+2 x \log \left (1-e^{2 i x}\right )-\frac{1}{2} x \cos ^2(x)+\frac{1}{4} \sin (x) \cos (x) \]

[Out]

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

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Rubi [A]  time = 0.169514, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.833, Rules used = {4408, 3311, 30, 2635, 8, 3720, 3717, 2190, 2279, 2391} \[ -i \text{PolyLog}\left (2,e^{2 i x}\right )-\frac{x^3}{2}-i x^2-x^2 \cot (x)-\frac{1}{2} x^2 \sin (x) \cos (x)+\frac{x}{4}+2 x \log \left (1-e^{2 i x}\right )-\frac{1}{2} x \cos ^2(x)+\frac{1}{4} \sin (x) \cos (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

Rule 3311

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -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 8

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

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 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 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]

Rubi steps

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

Mathematica [A]  time = 0.10048, size = 72, normalized size = 0.87 \[ \frac{1}{8} \left (-8 i \text{PolyLog}\left (2,e^{2 i x}\right )-4 x^3-8 i x^2-2 x^2 \sin (2 x)-8 x^2 \cot (x)+16 x \log \left (1-e^{2 i x}\right )+\sin (2 x)-2 x \cos (2 x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-8*I)*x^2 - 4*x^3 - 2*x*Cos[2*x] - 8*x^2*Cot[x] + 16*x*Log[1 - E^((2*I)*x)] - (8*I)*PolyLog[2, E^((2*I)*x)]
+ Sin[2*x] - 2*x^2*Sin[2*x])/8

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Maple [A]  time = 0.106, size = 112, normalized size = 1.4 \begin{align*} -{\frac{{x}^{3}}{2}}+{\frac{i}{16}} \left ( 2\,ix+2\,{x}^{2}-1 \right ){{\rm e}^{2\,ix}}-{\frac{i}{16}} \left ( -2\,ix+2\,{x}^{2}-1 \right ){{\rm e}^{-2\,ix}}-{\frac{2\,i{x}^{2}}{{{\rm e}^{2\,ix}}-1}}+2\,x\ln \left ( 1-{{\rm e}^{ix}} \right ) +2\,x\ln \left ( 1+{{\rm e}^{ix}} \right ) -2\,i{x}^{2}-2\,i{\it polylog} \left ( 2,{{\rm e}^{ix}} \right ) -2\,i{\it polylog} \left ( 2,-{{\rm e}^{ix}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*x^3+1/16*I*(2*I*x+2*x^2-1)*exp(2*I*x)-1/16*I*(-2*I*x+2*x^2-1)*exp(-2*I*x)-2*I*x^2/(exp(2*I*x)-1)+2*x*ln(1
-exp(I*x))+2*x*ln(1+exp(I*x))-2*I*x^2-2*I*polylog(2,exp(I*x))-2*I*polylog(2,-exp(I*x))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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Fricas [B]  time = 0.558489, size = 551, normalized size = 6.64 \begin{align*} \frac{{\left (2 \, x^{2} - 1\right )} \cos \left (x\right )^{3} + 4 \, x \log \left (\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) \sin \left (x\right ) + 4 \, x \log \left (\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) \sin \left (x\right ) + 4 \, x \log \left (-\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) \sin \left (x\right ) + 4 \, x \log \left (-\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) \sin \left (x\right ) -{\left (6 \, x^{2} - 1\right )} \cos \left (x\right ) -{\left (2 \, x^{3} + 2 \, x \cos \left (x\right )^{2} - x\right )} \sin \left (x\right ) - 4 i \,{\rm Li}_2\left (\cos \left (x\right ) + i \, \sin \left (x\right )\right ) \sin \left (x\right ) + 4 i \,{\rm Li}_2\left (\cos \left (x\right ) - i \, \sin \left (x\right )\right ) \sin \left (x\right ) + 4 i \,{\rm Li}_2\left (-\cos \left (x\right ) + i \, \sin \left (x\right )\right ) \sin \left (x\right ) - 4 i \,{\rm Li}_2\left (-\cos \left (x\right ) - i \, \sin \left (x\right )\right ) \sin \left (x\right )}{4 \, \sin \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*((2*x^2 - 1)*cos(x)^3 + 4*x*log(cos(x) + I*sin(x) + 1)*sin(x) + 4*x*log(cos(x) - I*sin(x) + 1)*sin(x) + 4*
x*log(-cos(x) + I*sin(x) + 1)*sin(x) + 4*x*log(-cos(x) - I*sin(x) + 1)*sin(x) - (6*x^2 - 1)*cos(x) - (2*x^3 +
2*x*cos(x)^2 - x)*sin(x) - 4*I*dilog(cos(x) + I*sin(x))*sin(x) + 4*I*dilog(cos(x) - I*sin(x))*sin(x) + 4*I*dil
og(-cos(x) + I*sin(x))*sin(x) - 4*I*dilog(-cos(x) - I*sin(x))*sin(x))/sin(x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \cos ^{2}{\left (x \right )} \cot ^{2}{\left (x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2*cos(x)**2*cot(x)**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \cos \left (x\right )^{2} \cot \left (x\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2*cos(x)^2*cot(x)^2, x)