Integrand size = 12, antiderivative size = 112 \[ \int x^3 \cos ^2(x) \cot ^2(x) \, dx=\frac {3 x^2}{8}-i x^3-\frac {3 x^4}{8}+\frac {3 \cos ^2(x)}{8}-\frac {3}{4} x^2 \cos ^2(x)-x^3 \cot (x)+3 x^2 \log \left (1-e^{2 i x}\right )-3 i x \operatorname {PolyLog}\left (2,e^{2 i x}\right )+\frac {3}{2} \operatorname {PolyLog}\left (3,e^{2 i x}\right )+\frac {3}{4} x \cos (x) \sin (x)-\frac {1}{2} x^3 \cos (x) \sin (x) \]
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*co s(x)*sin(x)-1/2*x^3*cos(x)*sin(x)
Time = 0.80 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.93 \[ \int x^3 \cos ^2(x) \cot ^2(x) \, dx=\frac {1}{16} \left (-2 i \pi ^3+16 i x^3-6 x^4+3 \cos (2 x)-6 x^2 \cos (2 x)-16 x^3 \cot (x)+48 x^2 \log \left (1-e^{-2 i x}\right )+48 i x \operatorname {PolyLog}\left (2,e^{-2 i x}\right )+24 \operatorname {PolyLog}\left (3,e^{-2 i x}\right )+6 x \sin (2 x)-4 x^3 \sin (2 x)\right ) \]
((-2*I)*Pi^3 + (16*I)*x^3 - 6*x^4 + 3*Cos[2*x] - 6*x^2*Cos[2*x] - 16*x^3*C ot[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
Time = 0.75 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.23, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.500, Rules used = {4908, 3042, 3792, 15, 3042, 3791, 15, 4203, 15, 25, 3042, 25, 4200, 25, 2620, 3011, 2720, 7143}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^3 \cos ^2(x) \cot ^2(x) \, dx\) |
| \(\Big \downarrow \) 4908 |
| \(\displaystyle \int x^3 \cot ^2(x)dx-\int x^3 \cos ^2(x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int x^3 \tan \left (x+\frac {\pi }{2}\right )^2dx-\int x^3 \sin \left (x+\frac {\pi }{2}\right )^2dx\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle -\frac {\int x^3dx}{2}+\int x^3 \tan \left (x+\frac {\pi }{2}\right )^2dx+\frac {3}{2} \int x \cos ^2(x)dx-\frac {1}{2} x^3 \sin (x) \cos (x)-\frac {3}{4} x^2 \cos ^2(x)\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \int x^3 \tan \left (x+\frac {\pi }{2}\right )^2dx+\frac {3}{2} \int x \cos ^2(x)dx-\frac {x^4}{8}-\frac {1}{2} x^3 \sin (x) \cos (x)-\frac {3}{4} x^2 \cos ^2(x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int x^3 \tan \left (x+\frac {\pi }{2}\right )^2dx+\frac {3}{2} \int x \sin \left (x+\frac {\pi }{2}\right )^2dx-\frac {x^4}{8}-\frac {1}{2} x^3 \sin (x) \cos (x)-\frac {3}{4} x^2 \cos ^2(x)\) |
| \(\Big \downarrow \) 3791 |
| \(\displaystyle \int x^3 \tan \left (x+\frac {\pi }{2}\right )^2dx+\frac {3}{2} \left (\frac {\int xdx}{2}+\frac {\cos ^2(x)}{4}+\frac {1}{2} x \sin (x) \cos (x)\right )-\frac {x^4}{8}-\frac {1}{2} x^3 \sin (x) \cos (x)-\frac {3}{4} x^2 \cos ^2(x)\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \int x^3 \tan \left (x+\frac {\pi }{2}\right )^2dx-\frac {x^4}{8}-\frac {1}{2} x^3 \sin (x) \cos (x)-\frac {3}{4} x^2 \cos ^2(x)+\frac {3}{2} \left (\frac {x^2}{4}+\frac {\cos ^2(x)}{4}+\frac {1}{2} x \sin (x) \cos (x)\right )\) |
| \(\Big \downarrow \) 4203 |
| \(\displaystyle -\int x^3dx-3 \int -x^2 \cot (x)dx-\frac {x^4}{8}-x^3 \cot (x)-\frac {1}{2} x^3 \sin (x) \cos (x)-\frac {3}{4} x^2 \cos ^2(x)+\frac {3}{2} \left (\frac {x^2}{4}+\frac {\cos ^2(x)}{4}+\frac {1}{2} x \sin (x) \cos (x)\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -3 \int -x^2 \cot (x)dx-\frac {3 x^4}{8}-x^3 \cot (x)-\frac {1}{2} x^3 \sin (x) \cos (x)-\frac {3}{4} x^2 \cos ^2(x)+\frac {3}{2} \left (\frac {x^2}{4}+\frac {\cos ^2(x)}{4}+\frac {1}{2} x \sin (x) \cos (x)\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 3 \int x^2 \cot (x)dx-\frac {3 x^4}{8}-x^3 \cot (x)-\frac {1}{2} x^3 \sin (x) \cos (x)-\frac {3}{4} x^2 \cos ^2(x)+\frac {3}{2} \left (\frac {x^2}{4}+\frac {\cos ^2(x)}{4}+\frac {1}{2} x \sin (x) \cos (x)\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 3 \int -x^2 \tan \left (x+\frac {\pi }{2}\right )dx-\frac {3 x^4}{8}-x^3 \cot (x)-\frac {1}{2} x^3 \sin (x) \cos (x)-\frac {3}{4} x^2 \cos ^2(x)+\frac {3}{2} \left (\frac {x^2}{4}+\frac {\cos ^2(x)}{4}+\frac {1}{2} x \sin (x) \cos (x)\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -3 \int x^2 \tan \left (x+\frac {\pi }{2}\right )dx-\frac {3 x^4}{8}-x^3 \cot (x)-\frac {1}{2} x^3 \sin (x) \cos (x)-\frac {3}{4} x^2 \cos ^2(x)+\frac {3}{2} \left (\frac {x^2}{4}+\frac {\cos ^2(x)}{4}+\frac {1}{2} x \sin (x) \cos (x)\right )\) |
| \(\Big \downarrow \) 4200 |
| \(\displaystyle -3 \left (\frac {i x^3}{3}-2 i \int -\frac {e^{2 i x} x^2}{1-e^{2 i x}}dx\right )-\frac {3 x^4}{8}-x^3 \cot (x)-\frac {1}{2} x^3 \sin (x) \cos (x)-\frac {3}{4} x^2 \cos ^2(x)+\frac {3}{2} \left (\frac {x^2}{4}+\frac {\cos ^2(x)}{4}+\frac {1}{2} x \sin (x) \cos (x)\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -3 \left (2 i \int \frac {e^{2 i x} x^2}{1-e^{2 i x}}dx+\frac {i x^3}{3}\right )-\frac {3 x^4}{8}-x^3 \cot (x)-\frac {1}{2} x^3 \sin (x) \cos (x)-\frac {3}{4} x^2 \cos ^2(x)+\frac {3}{2} \left (\frac {x^2}{4}+\frac {\cos ^2(x)}{4}+\frac {1}{2} x \sin (x) \cos (x)\right )\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -3 \left (2 i \left (\frac {1}{2} i x^2 \log \left (1-e^{2 i x}\right )-i \int x \log \left (1-e^{2 i x}\right )dx\right )+\frac {i x^3}{3}\right )-\frac {3 x^4}{8}-x^3 \cot (x)-\frac {1}{2} x^3 \sin (x) \cos (x)-\frac {3}{4} x^2 \cos ^2(x)+\frac {3}{2} \left (\frac {x^2}{4}+\frac {\cos ^2(x)}{4}+\frac {1}{2} x \sin (x) \cos (x)\right )\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -3 \left (2 i \left (\frac {1}{2} i x^2 \log \left (1-e^{2 i x}\right )-i \left (\frac {1}{2} i x \operatorname {PolyLog}\left (2,e^{2 i x}\right )-\frac {1}{2} i \int \operatorname {PolyLog}\left (2,e^{2 i x}\right )dx\right )\right )+\frac {i x^3}{3}\right )-\frac {3 x^4}{8}-x^3 \cot (x)-\frac {1}{2} x^3 \sin (x) \cos (x)-\frac {3}{4} x^2 \cos ^2(x)+\frac {3}{2} \left (\frac {x^2}{4}+\frac {\cos ^2(x)}{4}+\frac {1}{2} x \sin (x) \cos (x)\right )\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -3 \left (2 i \left (\frac {1}{2} i x^2 \log \left (1-e^{2 i x}\right )-i \left (\frac {1}{2} i x \operatorname {PolyLog}\left (2,e^{2 i x}\right )-\frac {1}{4} \int e^{-2 i x} \operatorname {PolyLog}\left (2,e^{2 i x}\right )de^{2 i x}\right )\right )+\frac {i x^3}{3}\right )-\frac {3 x^4}{8}-x^3 \cot (x)-\frac {1}{2} x^3 \sin (x) \cos (x)-\frac {3}{4} x^2 \cos ^2(x)+\frac {3}{2} \left (\frac {x^2}{4}+\frac {\cos ^2(x)}{4}+\frac {1}{2} x \sin (x) \cos (x)\right )\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle -3 \left (2 i \left (\frac {1}{2} i x^2 \log \left (1-e^{2 i x}\right )-i \left (\frac {1}{2} i x \operatorname {PolyLog}\left (2,e^{2 i x}\right )-\frac {1}{4} \operatorname {PolyLog}\left (3,e^{2 i x}\right )\right )\right )+\frac {i x^3}{3}\right )-\frac {3 x^4}{8}-x^3 \cot (x)-\frac {1}{2} x^3 \sin (x) \cos (x)-\frac {3}{4} x^2 \cos ^2(x)+\frac {3}{2} \left (\frac {x^2}{4}+\frac {\cos ^2(x)}{4}+\frac {1}{2} x \sin (x) \cos (x)\right )\) |
(-3*x^4)/8 - (3*x^2*Cos[x]^2)/4 - x^3*Cot[x] - 3*((I/3)*x^3 + (2*I)*((I/2) *x^2*Log[1 - E^((2*I)*x)] - I*((I/2)*x*PolyLog[2, E^((2*I)*x)] - PolyLog[3 , E^((2*I)*x)]/4))) - (x^3*Cos[x]*Sin[x])/2 + (3*(x^2/4 + Cos[x]^2/4 + (x* Cos[x]*Sin[x])/2))/2
3.3.2.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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] - Si mp[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]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
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] + Simp[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]
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] + (-Simp[b*(c + d*x)*Cos[e + f*x ]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^2*((n - 1)/n) Int[(c + d* x)*(b*Sin[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]
Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 2*((n - 1)/n) Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 *m*((m - 1)/(f^2*n^2)) Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol ] :> Simp[I*((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[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]
Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symb ol] :> Simp[b*(c + d*x)^m*((b*Tan[e + f*x])^(n - 1)/(f*(n - 1))), x] + (-Si mp[b*d*(m/(f*(n - 1))) Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1), x] , x] - Simp[b^2 Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; Free Q[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 0]
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] /; Fr eeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> 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]
Time = 3.78 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.34
| method | result | size |
| risch | \(-\frac {3 x^{4}}{8}+\frac {i \left (4 x^{3}+6 i x^{2}-6 x -3 i\right ) {\mathrm e}^{2 i x}}{32}-\frac {i \left (4 x^{3}-6 i x^{2}-6 x +3 i\right ) {\mathrm e}^{-2 i x}}{32}-\frac {2 i x^{3}}{{\mathrm e}^{2 i x}-1}-2 i x^{3}+3 x^{2} \ln \left ({\mathrm e}^{i x}+1\right )-6 i x \operatorname {polylog}\left (2, -{\mathrm e}^{i x}\right )+6 \operatorname {polylog}\left (3, -{\mathrm e}^{i x}\right )+3 x^{2} \ln \left (1-{\mathrm e}^{i x}\right )-6 i x \operatorname {polylog}\left (2, {\mathrm e}^{i x}\right )+6 \operatorname {polylog}\left (3, {\mathrm e}^{i x}\right )\) | \(150\) |
-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))
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 \[ \int x^3 \cos ^2(x) \cot ^2(x) \, dx=\frac {4 \, {\left (2 \, x^{3} - 3 \, x\right )} \cos \left (x\right )^{3} + 24 \, x^{2} \log \left (\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) \sin \left (x\right ) + 24 \, x^{2} \log \left (\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) \sin \left (x\right ) + 24 \, x^{2} \log \left (-\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) \sin \left (x\right ) + 24 \, x^{2} \log \left (-\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) \sin \left (x\right ) - 48 i \, x {\rm Li}_2\left (\cos \left (x\right ) + i \, \sin \left (x\right )\right ) \sin \left (x\right ) + 48 i \, x {\rm Li}_2\left (\cos \left (x\right ) - i \, \sin \left (x\right )\right ) \sin \left (x\right ) + 48 i \, x {\rm Li}_2\left (-\cos \left (x\right ) + i \, \sin \left (x\right )\right ) \sin \left (x\right ) - 48 i \, x {\rm Li}_2\left (-\cos \left (x\right ) - i \, \sin \left (x\right )\right ) \sin \left (x\right ) - 12 \, {\left (2 \, x^{3} - x\right )} \cos \left (x\right ) - 3 \, {\left (2 \, x^{4} + 2 \, {\left (2 \, x^{2} - 1\right )} \cos \left (x\right )^{2} - 2 \, x^{2} + 1\right )} \sin \left (x\right ) + 48 \, {\rm polylog}\left (3, \cos \left (x\right ) + i \, \sin \left (x\right )\right ) \sin \left (x\right ) + 48 \, {\rm polylog}\left (3, \cos \left (x\right ) - i \, \sin \left (x\right )\right ) \sin \left (x\right ) + 48 \, {\rm polylog}\left (3, -\cos \left (x\right ) + i \, \sin \left (x\right )\right ) \sin \left (x\right ) + 48 \, {\rm polylog}\left (3, -\cos \left (x\right ) - i \, \sin \left (x\right )\right ) \sin \left (x\right )}{16 \, \sin \left (x\right )} \]
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(co s(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)
\[ \int x^3 \cos ^2(x) \cot ^2(x) \, dx=\int x^{3} \cos ^{2}{\left (x \right )} \cot ^{2}{\left (x \right )}\, dx \]
Exception generated. \[ \int x^3 \cos ^2(x) \cot ^2(x) \, dx=\text {Exception raised: RuntimeError} \]
\[ \int x^3 \cos ^2(x) \cot ^2(x) \, dx=\int { x^{3} \cos \left (x\right )^{2} \cot \left (x\right )^{2} \,d x } \]
Timed out. \[ \int x^3 \cos ^2(x) \cot ^2(x) \, dx=\int x^3\,{\cos \left (x\right )}^2\,{\mathrm {cot}\left (x\right )}^2 \,d x \]