\(\int x^2 \cos ^2(x) \cot ^2(x) \, dx\) [203]

Optimal result

Integrand size = 12, antiderivative size = 83 \[ \int x^2 \cos ^2(x) \cot ^2(x) \, dx=\frac {x}{4}-i x^2-\frac {x^3}{2}-\frac {1}{2} x \cos ^2(x)-x^2 \cot (x)+2 x \log \left (1-e^{2 i x}\right )-i \operatorname {PolyLog}\left (2,e^{2 i x}\right )+\frac {1}{4} \cos (x) \sin (x)-\frac {1}{2} x^2 \cos (x) \sin (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {4493, 3392, 30, 2715, 8, 3801, 3798, 2221, 2317, 2438} \[ \int x^2 \cos ^2(x) \cot ^2(x) \, dx=-i \operatorname {PolyLog}\left (2,e^{2 i x}\right )-\frac {x^3}{2}-i x^2-x^2 \cot (x)-\frac {1}{2} x^2 \sin (x) \cos (x)+\frac {x}{4}+2 x \log \left (1-e^{2 i x}\right )-\frac {1}{2} x \cos ^2(x)+\frac {1}{4} \sin (x) \cos (x) \]

[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 8

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

Rule 30

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

Rule 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2317

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 2438

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 2715

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] && Integ
erQ[2*n]

Rule 3392

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 3798

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 3801

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 4493

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*} \text {integral}& = -\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 \text {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 \operatorname {PolyLog}\left (2,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] (verified)

Time = 0.14 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.87 \[ \int x^2 \cos ^2(x) \cot ^2(x) \, dx=\frac {1}{8} \left (-8 i x^2-4 x^3-2 x \cos (2 x)-8 x^2 \cot (x)+16 x \log \left (1-e^{2 i x}\right )-8 i \operatorname {PolyLog}\left (2,e^{2 i x}\right )+\sin (2 x)-2 x^2 \sin (2 x)\right ) \]

[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

Maple [A] (verified)

Time = 2.60 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.35

method	result	size
risch	\(-\frac {x^{3}}{2}+\frac {i \left (2 x^{2}+2 i x -1\right ) {\mathrm e}^{2 i x}}{16}-\frac {i \left (2 x^{2}-2 i x -1\right ) {\mathrm e}^{-2 i x}}{16}-\frac {2 i x^{2}}{{\mathrm e}^{2 i x}-1}+2 x \ln \left ({\mathrm e}^{i x}+1\right )+2 x \ln \left (1-{\mathrm e}^{i x}\right )-2 i x^{2}-2 i \operatorname {polylog}\left (2, -{\mathrm e}^{i x}\right )-2 i \operatorname {polylog}\left (2, {\mathrm e}^{i x}\right )\)	\(112\)

[In]

int(x^2*cos(x)^2*cot(x)^2,x,method=_RETURNVERBOSE)

[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(e
xp(I*x)+1)+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))

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 162 vs. \(2 (62) = 124\).

Time = 0.26 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.95 \[ \int x^2 \cos ^2(x) \cot ^2(x) \, dx=\frac {{\left (2 \, x^{2} - 1\right )} \cos \left (x\right )^{3} + 4 \, x \log \left (\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) \sin \left (x\right ) + 4 \, x \log \left (\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) \sin \left (x\right ) + 4 \, x \log \left (-\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) \sin \left (x\right ) + 4 \, x \log \left (-\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) \sin \left (x\right ) - {\left (6 \, x^{2} - 1\right )} \cos \left (x\right ) - {\left (2 \, x^{3} + 2 \, x \cos \left (x\right )^{2} - x\right )} \sin \left (x\right ) - 4 i \, {\rm Li}_2\left (\cos \left (x\right ) + i \, \sin \left (x\right )\right ) \sin \left (x\right ) + 4 i \, {\rm Li}_2\left (\cos \left (x\right ) - i \, \sin \left (x\right )\right ) \sin \left (x\right ) + 4 i \, {\rm Li}_2\left (-\cos \left (x\right ) + i \, \sin \left (x\right )\right ) \sin \left (x\right ) - 4 i \, {\rm Li}_2\left (-\cos \left (x\right ) - i \, \sin \left (x\right )\right ) \sin \left (x\right )}{4 \, \sin \left (x\right )} \]

[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)

Sympy [F]

\[ \int x^2 \cos ^2(x) \cot ^2(x) \, dx=\int x^{2} \cos ^{2}{\left (x \right )} \cot ^{2}{\left (x \right )}\, dx \]

[In]

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

[Out]

Integral(x**2*cos(x)**2*cot(x)**2, x)

Maxima [F(-2)]

Exception generated. \[ \int x^2 \cos ^2(x) \cot ^2(x) \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

Giac [F]

\[ \int x^2 \cos ^2(x) \cot ^2(x) \, dx=\int { x^{2} \cos \left (x\right )^{2} \cot \left (x\right )^{2} \,d x } \]

[In]

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

[Out]

integrate(x^2*cos(x)^2*cot(x)^2, x)

Mupad [F(-1)]

Timed out. \[ \int x^2 \cos ^2(x) \cot ^2(x) \, dx=\int x^2\,{\cos \left (x\right )}^2\,{\mathrm {cot}\left (x\right )}^2 \,d x \]

[In]

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

[Out]

int(x^2*cos(x)^2*cot(x)^2, x)