\(\int x^3 \cos ^2(x) \cot ^2(x) \, dx\) [202]

Optimal result

Integrand size = 12, antiderivative size = 112 \[ \int x^3 \cos ^2(x) \cot ^2(x) \, dx=\frac {3 x^2}{8}-i x^3-\frac {3 x^4}{8}+\frac {3 \cos ^2(x)}{8}-\frac {3}{4} x^2 \cos ^2(x)-x^3 \cot (x)+3 x^2 \log \left (1-e^{2 i x}\right )-3 i x \operatorname {PolyLog}\left (2,e^{2 i x}\right )+\frac {3}{2} \operatorname {PolyLog}\left (3,e^{2 i x}\right )+\frac {3}{4} x \cos (x) \sin (x)-\frac {1}{2} x^3 \cos (x) \sin (x) \]

[Out]

3/8*x^2-I*x^3-3/8*x^4+3/8*cos(x)^2-3/4*x^2*cos(x)^2-x^3*cot(x)+3*x^2*ln(1-exp(2*I*x))-3*I*x*polylog(2,exp(2*I*
x))+3/2*polylog(3,exp(2*I*x))+3/4*x*cos(x)*sin(x)-1/2*x^3*cos(x)*sin(x)

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {4493, 3392, 30, 3391, 3801, 3798, 2221, 2611, 2320, 6724} \[ \int x^3 \cos ^2(x) \cot ^2(x) \, dx=-3 i x \operatorname {PolyLog}\left (2,e^{2 i x}\right )+\frac {3}{2} \operatorname {PolyLog}\left (3,e^{2 i x}\right )-\frac {3 x^4}{8}-i x^3-x^3 \cot (x)-\frac {1}{2} x^3 \sin (x) \cos (x)+\frac {3 x^2}{8}+3 x^2 \log \left (1-e^{2 i x}\right )-\frac {3}{4} x^2 \cos ^2(x)+\frac {3 \cos ^2(x)}{8}+\frac {3}{4} x \sin (x) \cos (x) \]

[In]

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

[Out]

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

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 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 3391

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

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]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps \begin{align*} \text {integral}& = -\int x^3 \cos ^2(x) \, dx+\int x^3 \cot ^2(x) \, dx \\ & = -\frac {3}{4} x^2 \cos ^2(x)-x^3 \cot (x)-\frac {1}{2} x^3 \cos (x) \sin (x)-\frac {\int x^3 \, dx}{2}+\frac {3}{2} \int x \cos ^2(x) \, dx+3 \int x^2 \cot (x) \, dx-\int x^3 \, dx \\ & = -i x^3-\frac {3 x^4}{8}+\frac {3 \cos ^2(x)}{8}-\frac {3}{4} x^2 \cos ^2(x)-x^3 \cot (x)+\frac {3}{4} x \cos (x) \sin (x)-\frac {1}{2} x^3 \cos (x) \sin (x)-6 i \int \frac {e^{2 i x} x^2}{1-e^{2 i x}} \, dx+\frac {3 \int x \, dx}{4} \\ & = \frac {3 x^2}{8}-i x^3-\frac {3 x^4}{8}+\frac {3 \cos ^2(x)}{8}-\frac {3}{4} x^2 \cos ^2(x)-x^3 \cot (x)+3 x^2 \log \left (1-e^{2 i x}\right )+\frac {3}{4} x \cos (x) \sin (x)-\frac {1}{2} x^3 \cos (x) \sin (x)-6 \int x \log \left (1-e^{2 i x}\right ) \, dx \\ & = \frac {3 x^2}{8}-i x^3-\frac {3 x^4}{8}+\frac {3 \cos ^2(x)}{8}-\frac {3}{4} x^2 \cos ^2(x)-x^3 \cot (x)+3 x^2 \log \left (1-e^{2 i x}\right )-3 i x \operatorname {PolyLog}\left (2,e^{2 i x}\right )+\frac {3}{4} x \cos (x) \sin (x)-\frac {1}{2} x^3 \cos (x) \sin (x)+3 i \int \operatorname {PolyLog}\left (2,e^{2 i x}\right ) \, dx \\ & = \frac {3 x^2}{8}-i x^3-\frac {3 x^4}{8}+\frac {3 \cos ^2(x)}{8}-\frac {3}{4} x^2 \cos ^2(x)-x^3 \cot (x)+3 x^2 \log \left (1-e^{2 i x}\right )-3 i x \operatorname {PolyLog}\left (2,e^{2 i x}\right )+\frac {3}{4} x \cos (x) \sin (x)-\frac {1}{2} x^3 \cos (x) \sin (x)+\frac {3}{2} \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{2 i x}\right ) \\ & = \frac {3 x^2}{8}-i x^3-\frac {3 x^4}{8}+\frac {3 \cos ^2(x)}{8}-\frac {3}{4} x^2 \cos ^2(x)-x^3 \cot (x)+3 x^2 \log \left (1-e^{2 i x}\right )-3 i x \operatorname {PolyLog}\left (2,e^{2 i x}\right )+\frac {3}{2} \operatorname {PolyLog}\left (3,e^{2 i x}\right )+\frac {3}{4} x \cos (x) \sin (x)-\frac {1}{2} x^3 \cos (x) \sin (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.80 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.93 \[ \int x^3 \cos ^2(x) \cot ^2(x) \, dx=\frac {1}{16} \left (-2 i \pi ^3+16 i x^3-6 x^4+3 \cos (2 x)-6 x^2 \cos (2 x)-16 x^3 \cot (x)+48 x^2 \log \left (1-e^{-2 i x}\right )+48 i x \operatorname {PolyLog}\left (2,e^{-2 i x}\right )+24 \operatorname {PolyLog}\left (3,e^{-2 i x}\right )+6 x \sin (2 x)-4 x^3 \sin (2 x)\right ) \]

[In]

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

[Out]

((-2*I)*Pi^3 + (16*I)*x^3 - 6*x^4 + 3*Cos[2*x] - 6*x^2*Cos[2*x] - 16*x^3*Cot[x] + 48*x^2*Log[1 - E^((-2*I)*x)]
+ (48*I)*x*PolyLog[2, E^((-2*I)*x)] + 24*PolyLog[3, E^((-2*I)*x)] + 6*x*Sin[2*x] - 4*x^3*Sin[2*x])/16

Maple [A] (verified)

Time = 3.78 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.34

method	result	size
risch	\(-\frac {3 x^{4}}{8}+\frac {i \left (4 x^{3}+6 i x^{2}-6 x -3 i\right ) {\mathrm e}^{2 i x}}{32}-\frac {i \left (4 x^{3}-6 i x^{2}-6 x +3 i\right ) {\mathrm e}^{-2 i x}}{32}-\frac {2 i x^{3}}{{\mathrm e}^{2 i x}-1}-2 i x^{3}+3 x^{2} \ln \left ({\mathrm e}^{i x}+1\right )-6 i x \operatorname {polylog}\left (2, -{\mathrm e}^{i x}\right )+6 \operatorname {polylog}\left (3, -{\mathrm e}^{i x}\right )+3 x^{2} \ln \left (1-{\mathrm e}^{i x}\right )-6 i x \operatorname {polylog}\left (2, {\mathrm e}^{i x}\right )+6 \operatorname {polylog}\left (3, {\mathrm e}^{i x}\right )\)	\(150\)

[In]

int(x^3*cos(x)^2*cot(x)^2,x,method=_RETURNVERBOSE)

[Out]

-3/8*x^4+1/32*I*(6*I*x^2+4*x^3-3*I-6*x)*exp(2*I*x)-1/32*I*(-6*I*x^2+4*x^3+3*I-6*x)*exp(-2*I*x)-2*I*x^3/(exp(2*
I*x)-1)-2*I*x^3+3*x^2*ln(exp(I*x)+1)-6*I*x*polylog(2,-exp(I*x))+6*polylog(3,-exp(I*x))+3*x^2*ln(1-exp(I*x))-6*
I*x*polylog(2,exp(I*x))+6*polylog(3,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. 244 vs. \(2 (84) = 168\).

Time = 0.28 (sec) , antiderivative size = 244, normalized size of antiderivative = 2.18 \[ \int x^3 \cos ^2(x) \cot ^2(x) \, dx=\frac {4 \, {\left (2 \, x^{3} - 3 \, x\right )} \cos \left (x\right )^{3} + 24 \, x^{2} \log \left (\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) \sin \left (x\right ) + 24 \, x^{2} \log \left (\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) \sin \left (x\right ) + 24 \, x^{2} \log \left (-\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) \sin \left (x\right ) + 24 \, x^{2} \log \left (-\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) \sin \left (x\right ) - 48 i \, x {\rm Li}_2\left (\cos \left (x\right ) + i \, \sin \left (x\right )\right ) \sin \left (x\right ) + 48 i \, x {\rm Li}_2\left (\cos \left (x\right ) - i \, \sin \left (x\right )\right ) \sin \left (x\right ) + 48 i \, x {\rm Li}_2\left (-\cos \left (x\right ) + i \, \sin \left (x\right )\right ) \sin \left (x\right ) - 48 i \, x {\rm Li}_2\left (-\cos \left (x\right ) - i \, \sin \left (x\right )\right ) \sin \left (x\right ) - 12 \, {\left (2 \, x^{3} - x\right )} \cos \left (x\right ) - 3 \, {\left (2 \, x^{4} + 2 \, {\left (2 \, x^{2} - 1\right )} \cos \left (x\right )^{2} - 2 \, x^{2} + 1\right )} \sin \left (x\right ) + 48 \, {\rm polylog}\left (3, \cos \left (x\right ) + i \, \sin \left (x\right )\right ) \sin \left (x\right ) + 48 \, {\rm polylog}\left (3, \cos \left (x\right ) - i \, \sin \left (x\right )\right ) \sin \left (x\right ) + 48 \, {\rm polylog}\left (3, -\cos \left (x\right ) + i \, \sin \left (x\right )\right ) \sin \left (x\right ) + 48 \, {\rm polylog}\left (3, -\cos \left (x\right ) - i \, \sin \left (x\right )\right ) \sin \left (x\right )}{16 \, \sin \left (x\right )} \]

[In]

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

[Out]

1/16*(4*(2*x^3 - 3*x)*cos(x)^3 + 24*x^2*log(cos(x) + I*sin(x) + 1)*sin(x) + 24*x^2*log(cos(x) - I*sin(x) + 1)*
sin(x) + 24*x^2*log(-cos(x) + I*sin(x) + 1)*sin(x) + 24*x^2*log(-cos(x) - I*sin(x) + 1)*sin(x) - 48*I*x*dilog(
cos(x) + I*sin(x))*sin(x) + 48*I*x*dilog(cos(x) - I*sin(x))*sin(x) + 48*I*x*dilog(-cos(x) + I*sin(x))*sin(x) -
48*I*x*dilog(-cos(x) - I*sin(x))*sin(x) - 12*(2*x^3 - x)*cos(x) - 3*(2*x^4 + 2*(2*x^2 - 1)*cos(x)^2 - 2*x^2 +
1)*sin(x) + 48*polylog(3, cos(x) + I*sin(x))*sin(x) + 48*polylog(3, cos(x) - I*sin(x))*sin(x) + 48*polylog(3,
-cos(x) + I*sin(x))*sin(x) + 48*polylog(3, -cos(x) - I*sin(x))*sin(x))/sin(x)

Sympy [F]

\[ \int x^3 \cos ^2(x) \cot ^2(x) \, dx=\int x^{3} \cos ^{2}{\left (x \right )} \cot ^{2}{\left (x \right )}\, dx \]

[In]

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

[Out]

Integral(x**3*cos(x)**2*cot(x)**2, x)

Maxima [F(-2)]

Exception generated. \[ \int x^3 \cos ^2(x) \cot ^2(x) \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

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

Giac [F]

\[ \int x^3 \cos ^2(x) \cot ^2(x) \, dx=\int { x^{3} \cos \left (x\right )^{2} \cot \left (x\right )^{2} \,d x } \]

[In]

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

[Out]

integrate(x^3*cos(x)^2*cot(x)^2, x)

Mupad [F(-1)]

Timed out. \[ \int x^3 \cos ^2(x) \cot ^2(x) \, dx=\int x^3\,{\cos \left (x\right )}^2\,{\mathrm {cot}\left (x\right )}^2 \,d x \]

[In]

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

[Out]

int(x^3*cos(x)^2*cot(x)^2, x)