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

Optimal. Leaf size=112 \[ -3 i x \text{PolyLog}\left (2,e^{2 i x}\right )+\frac{3}{2} \text{PolyLog}\left (3,e^{2 i x}\right )-\frac{3 x^4}{8}-i x^3+\frac{3 x^2}{8}+3 x^2 \log \left (1-e^{2 i x}\right )-\frac{3}{4} x^2 \cos ^2(x)-x^3 \cot (x)-\frac{1}{2} x^3 \sin (x) \cos (x)+\frac{3 \cos ^2(x)}{8}+\frac{3}{4} x \sin (x) \cos (x) \]

[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

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Rubi [A]  time = 0.185145, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.833, Rules used = {4408, 3311, 30, 3310, 3720, 3717, 2190, 2531, 2282, 6589} \[ -3 i x \text{PolyLog}\left (2,e^{2 i x}\right )+\frac{3}{2} \text{PolyLog}\left (3,e^{2 i x}\right )-\frac{3 x^4}{8}-i x^3+\frac{3 x^2}{8}+3 x^2 \log \left (1-e^{2 i x}\right )-\frac{3}{4} x^2 \cos ^2(x)-x^3 \cot (x)-\frac{1}{2} x^3 \sin (x) \cos (x)+\frac{3 \cos ^2(x)}{8}+\frac{3}{4} x \sin (x) \cos (x) \]

Antiderivative was successfully verified.

[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 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 3310

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 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 2531

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 2282

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 6589

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*} \int x^3 \cos ^2(x) \cot ^2(x) \, dx &=-\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 \text{Li}_2\left (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 \text{Li}_2\left (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 \text{Li}_2\left (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} \operatorname{Subst}\left (\int \frac{\text{Li}_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 \text{Li}_2\left (e^{2 i x}\right )+\frac{3}{2} \text{Li}_3\left (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]  time = 0.175528, size = 104, normalized size = 0.93 \[ \frac{1}{16} \left (48 i x \text{PolyLog}\left (2,e^{-2 i x}\right )+24 \text{PolyLog}\left (3,e^{-2 i x}\right )-6 x^4+16 i x^3+48 x^2 \log \left (1-e^{-2 i x}\right )-4 x^3 \sin (2 x)-6 x^2 \cos (2 x)-16 x^3 \cot (x)+6 x \sin (2 x)+3 \cos (2 x)-2 i \pi ^3\right ) \]

Antiderivative was successfully verified.

[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

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Maple [A]  time = 0.106, size = 150, normalized size = 1.3 \begin{align*} -{\frac{3\,{x}^{4}}{8}}+{\frac{i}{32}} \left ( 6\,i{x}^{2}+4\,{x}^{3}-3\,i-6\,x \right ){{\rm e}^{2\,ix}}-{\frac{i}{32}} \left ( -6\,i{x}^{2}+4\,{x}^{3}+3\,i-6\,x \right ){{\rm e}^{-2\,ix}}-{\frac{2\,i{x}^{3}}{{{\rm e}^{2\,ix}}-1}}-2\,i{x}^{3}+3\,{x}^{2}\ln \left ( 1-{{\rm e}^{ix}} \right ) -6\,ix{\it polylog} \left ( 2,{{\rm e}^{ix}} \right ) +6\,{\it polylog} \left ( 3,{{\rm e}^{ix}} \right ) +3\,{x}^{2}\ln \left ( 1+{{\rm e}^{ix}} \right ) -6\,ix{\it polylog} \left ( 2,-{{\rm e}^{ix}} \right ) +6\,{\it polylog} \left ( 3,-{{\rm e}^{ix}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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(1-exp(I*x))-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))

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

[Out]

Exception raised: RuntimeError

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Fricas [C]  time = 0.594582, size = 851, normalized size = 7.6 \begin{align*} \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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \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**3*cos(x)**2*cot(x)**2,x)

[Out]

Integral(x**3*cos(x)**2*cot(x)**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \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^3*cos(x)^2*cot(x)^2,x, algorithm="giac")

[Out]

integrate(x^3*cos(x)^2*cot(x)^2, x)