3.206 \(\int x^2 \cos ^2(x) \cot ^3(x) \, dx\)
Optimal. Leaf size=106 \[ 2 i x \text {Li}_2\left (e^{2 i x}\right )-\text {Li}_3\left (e^{2 i x}\right )+\frac {2 i x^3}{3}-\frac {3 x^2}{4}-2 x^2 \log \left (1-e^{2 i x}\right )+\frac {1}{2} x^2 \sin ^2(x)-\frac {1}{2} x^2 \cot ^2(x)-\frac {\sin ^2(x)}{4}-x \cot (x)+\log (\sin (x))+\frac {1}{2} x \sin (x) \cos (x) \]
[Out]
-3/4*x^2+2/3*I*x^3-x*cot(x)-1/2*x^2*cot(x)^2-2*x^2*ln(1-exp(2*I*x))+ln(sin(x))+2*I*x*polylog(2,exp(2*I*x))-pol
ylog(3,exp(2*I*x))+1/2*x*cos(x)*sin(x)-1/4*sin(x)^2+1/2*x^2*sin(x)^2
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Rubi [A] time = 0.28, antiderivative size = 106, normalized size of antiderivative =
1.00, number of steps used = 19, number of rules used = 11, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) =
0.917, Rules used = {4408, 3443, 3310, 30, 3717, 2190, 2531, 2282, 6589, 3720, 3475}
\[ 2 i x \text {PolyLog}\left (2,e^{2 i x}\right )-\text {PolyLog}\left (3,e^{2 i x}\right )+\frac {2 i x^3}{3}-\frac {3 x^2}{4}-2 x^2 \log \left (1-e^{2 i x}\right )+\frac {1}{2} x^2 \sin ^2(x)-\frac {1}{2} x^2 \cot ^2(x)-\frac {\sin ^2(x)}{4}-x \cot (x)+\log (\sin (x))+\frac {1}{2} x \sin (x) \cos (x) \]
Antiderivative was successfully verified.
[In]
Int[x^2*Cos[x]^2*Cot[x]^3,x]
[Out]
(-3*x^2)/4 + ((2*I)/3)*x^3 - x*Cot[x] - (x^2*Cot[x]^2)/2 - 2*x^2*Log[1 - E^((2*I)*x)] + Log[Sin[x]] + (2*I)*x*
PolyLog[2, E^((2*I)*x)] - PolyLog[3, E^((2*I)*x)] + (x*Cos[x]*Sin[x])/2 - Sin[x]^2/4 + (x^2*Sin[x]^2)/2
Rule 30
Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]
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 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 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 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 3443
Int[Cos[(a_.) + (b_.)*(x_)^(n_.)]*(x_)^(m_.)*Sin[(a_.) + (b_.)*(x_)^(n_.)]^(p_.), x_Symbol] :> Simp[(x^(m - n
+ 1)*Sin[a + b*x^n]^(p + 1))/(b*n*(p + 1)), x] - Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*Sin[a + b*x^n]^
(p + 1), x], x] /; FreeQ[{a, b, p}, x] && LtQ[0, n, m + 1] && NeQ[p, -1]
Rule 3475
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]
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 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 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 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^2 \cos ^2(x) \cot ^3(x) \, dx &=-\int x^2 \cos ^2(x) \cot (x) \, dx+\int x^2 \cot ^3(x) \, dx\\ &=-\frac {1}{2} x^2 \cot ^2(x)-2 \int x^2 \cot (x) \, dx+\int x \cot ^2(x) \, dx+\int x^2 \cos (x) \sin (x) \, dx\\ &=-x \cot (x)-\frac {1}{2} x^2 \cot ^2(x)+\frac {1}{2} x^2 \sin ^2(x)-2 \left (-\frac {i x^3}{3}-2 i \int \frac {e^{2 i x} x^2}{1-e^{2 i x}} \, dx\right )-\int x \, dx+\int \cot (x) \, dx-\int x \sin ^2(x) \, dx\\ &=-\frac {x^2}{2}-x \cot (x)-\frac {1}{2} x^2 \cot ^2(x)+\log (\sin (x))+\frac {1}{2} x \cos (x) \sin (x)-\frac {\sin ^2(x)}{4}+\frac {1}{2} x^2 \sin ^2(x)-\frac {\int x \, dx}{2}-2 \left (-\frac {i x^3}{3}+x^2 \log \left (1-e^{2 i x}\right )-2 \int x \log \left (1-e^{2 i x}\right ) \, dx\right )\\ &=-\frac {3 x^2}{4}-x \cot (x)-\frac {1}{2} x^2 \cot ^2(x)+\log (\sin (x))+\frac {1}{2} x \cos (x) \sin (x)-\frac {\sin ^2(x)}{4}+\frac {1}{2} x^2 \sin ^2(x)-2 \left (-\frac {i x^3}{3}+x^2 \log \left (1-e^{2 i x}\right )-i x \text {Li}_2\left (e^{2 i x}\right )+i \int \text {Li}_2\left (e^{2 i x}\right ) \, dx\right )\\ &=-\frac {3 x^2}{4}-x \cot (x)-\frac {1}{2} x^2 \cot ^2(x)+\log (\sin (x))+\frac {1}{2} x \cos (x) \sin (x)-\frac {\sin ^2(x)}{4}+\frac {1}{2} x^2 \sin ^2(x)-2 \left (-\frac {i x^3}{3}+x^2 \log \left (1-e^{2 i x}\right )-i x \text {Li}_2\left (e^{2 i x}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 i x}\right )\right )\\ &=-\frac {3 x^2}{4}-x \cot (x)-\frac {1}{2} x^2 \cot ^2(x)+\log (\sin (x))-2 \left (-\frac {i x^3}{3}+x^2 \log \left (1-e^{2 i x}\right )-i x \text {Li}_2\left (e^{2 i x}\right )+\frac {1}{2} \text {Li}_3\left (e^{2 i x}\right )\right )+\frac {1}{2} x \cos (x) \sin (x)-\frac {\sin ^2(x)}{4}+\frac {1}{2} x^2 \sin ^2(x)\\ \end {align*}
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Mathematica [A] time = 0.33, size = 108, normalized size = 1.02 \[ -2 i x \text {Li}_2\left (e^{-2 i x}\right )-\text {Li}_3\left (e^{-2 i x}\right )-\frac {2 i x^3}{3}-2 x^2 \log \left (1-e^{-2 i x}\right )-\frac {1}{4} x^2 \cos (2 x)-\frac {1}{2} x^2 \csc ^2(x)+\frac {1}{4} x \sin (2 x)+\frac {1}{8} \cos (2 x)-x \cot (x)+\log (\sin (x))+\frac {i \pi ^3}{12} \]
Antiderivative was successfully verified.
[In]
Integrate[x^2*Cos[x]^2*Cot[x]^3,x]
[Out]
(I/12)*Pi^3 - ((2*I)/3)*x^3 + Cos[2*x]/8 - (x^2*Cos[2*x])/4 - x*Cot[x] - (x^2*Csc[x]^2)/2 - 2*x^2*Log[1 - E^((
-2*I)*x)] + Log[Sin[x]] - (2*I)*x*PolyLog[2, E^((-2*I)*x)] - PolyLog[3, E^((-2*I)*x)] + (x*Sin[2*x])/4
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fricas [C] time = 0.54, size = 370, normalized size = 3.49 \[ -\frac {2 \, {\left (2 \, x^{2} - 1\right )} \cos \relax (x)^{4} - 3 \, {\left (2 \, x^{2} - 1\right )} \cos \relax (x)^{2} - 2 \, x^{2} - {\left (16 i \, x \cos \relax (x)^{2} - 16 i \, x\right )} {\rm Li}_2\left (\cos \relax (x) + i \, \sin \relax (x)\right ) - {\left (-16 i \, x \cos \relax (x)^{2} + 16 i \, x\right )} {\rm Li}_2\left (\cos \relax (x) - i \, \sin \relax (x)\right ) - {\left (-16 i \, x \cos \relax (x)^{2} + 16 i \, x\right )} {\rm Li}_2\left (-\cos \relax (x) + i \, \sin \relax (x)\right ) - {\left (16 i \, x \cos \relax (x)^{2} - 16 i \, x\right )} {\rm Li}_2\left (-\cos \relax (x) - i \, \sin \relax (x)\right ) + 4 \, {\left ({\left (2 \, x^{2} - 1\right )} \cos \relax (x)^{2} - 2 \, x^{2} + 1\right )} \log \left (\cos \relax (x) + i \, \sin \relax (x) + 1\right ) + 4 \, {\left ({\left (2 \, x^{2} - 1\right )} \cos \relax (x)^{2} - 2 \, x^{2} + 1\right )} \log \left (\cos \relax (x) - i \, \sin \relax (x) + 1\right ) - 4 \, {\left (\cos \relax (x)^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2} i \, \sin \relax (x) + \frac {1}{2}\right ) - 4 \, {\left (\cos \relax (x)^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \relax (x) - \frac {1}{2} i \, \sin \relax (x) + \frac {1}{2}\right ) + 8 \, {\left (x^{2} \cos \relax (x)^{2} - x^{2}\right )} \log \left (-\cos \relax (x) + i \, \sin \relax (x) + 1\right ) + 8 \, {\left (x^{2} \cos \relax (x)^{2} - x^{2}\right )} \log \left (-\cos \relax (x) - i \, \sin \relax (x) + 1\right ) + 16 \, {\left (\cos \relax (x)^{2} - 1\right )} {\rm polylog}\left (3, \cos \relax (x) + i \, \sin \relax (x)\right ) + 16 \, {\left (\cos \relax (x)^{2} - 1\right )} {\rm polylog}\left (3, \cos \relax (x) - i \, \sin \relax (x)\right ) + 16 \, {\left (\cos \relax (x)^{2} - 1\right )} {\rm polylog}\left (3, -\cos \relax (x) + i \, \sin \relax (x)\right ) + 16 \, {\left (\cos \relax (x)^{2} - 1\right )} {\rm polylog}\left (3, -\cos \relax (x) - i \, \sin \relax (x)\right ) - 4 \, {\left (x \cos \relax (x)^{3} + x \cos \relax (x)\right )} \sin \relax (x) - 1}{8 \, {\left (\cos \relax (x)^{2} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*cos(x)^2*cot(x)^3,x, algorithm="fricas")
[Out]
-1/8*(2*(2*x^2 - 1)*cos(x)^4 - 3*(2*x^2 - 1)*cos(x)^2 - 2*x^2 - (16*I*x*cos(x)^2 - 16*I*x)*dilog(cos(x) + I*si
n(x)) - (-16*I*x*cos(x)^2 + 16*I*x)*dilog(cos(x) - I*sin(x)) - (-16*I*x*cos(x)^2 + 16*I*x)*dilog(-cos(x) + I*s
in(x)) - (16*I*x*cos(x)^2 - 16*I*x)*dilog(-cos(x) - I*sin(x)) + 4*((2*x^2 - 1)*cos(x)^2 - 2*x^2 + 1)*log(cos(x
) + I*sin(x) + 1) + 4*((2*x^2 - 1)*cos(x)^2 - 2*x^2 + 1)*log(cos(x) - I*sin(x) + 1) - 4*(cos(x)^2 - 1)*log(-1/
2*cos(x) + 1/2*I*sin(x) + 1/2) - 4*(cos(x)^2 - 1)*log(-1/2*cos(x) - 1/2*I*sin(x) + 1/2) + 8*(x^2*cos(x)^2 - x^
2)*log(-cos(x) + I*sin(x) + 1) + 8*(x^2*cos(x)^2 - x^2)*log(-cos(x) - I*sin(x) + 1) + 16*(cos(x)^2 - 1)*polylo
g(3, cos(x) + I*sin(x)) + 16*(cos(x)^2 - 1)*polylog(3, cos(x) - I*sin(x)) + 16*(cos(x)^2 - 1)*polylog(3, -cos(
x) + I*sin(x)) + 16*(cos(x)^2 - 1)*polylog(3, -cos(x) - I*sin(x)) - 4*(x*cos(x)^3 + x*cos(x))*sin(x) - 1)/(cos
(x)^2 - 1)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \cos \relax (x)^{2} \cot \relax (x)^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*cos(x)^2*cot(x)^3,x, algorithm="giac")
[Out]
integrate(x^2*cos(x)^2*cot(x)^3, x)
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maple [A] time = 0.16, size = 170, normalized size = 1.60 \[ \frac {2 i x^{3}}{3}-\frac {\left (2 x^{2}+2 i x -1\right ) {\mathrm e}^{2 i x}}{16}-\frac {\left (2 x^{2}-2 i x -1\right ) {\mathrm e}^{-2 i x}}{16}+\frac {2 x \left (x \,{\mathrm e}^{2 i x}-i {\mathrm e}^{2 i x}+i\right )}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}+\ln \left (1+{\mathrm e}^{i x}\right )+\ln \left ({\mathrm e}^{i x}-1\right )-2 \ln \left ({\mathrm e}^{i x}\right )-2 x^{2} \ln \left (1+{\mathrm e}^{i x}\right )+4 i x \polylog \left (2, -{\mathrm e}^{i x}\right )-4 \polylog \left (3, -{\mathrm e}^{i x}\right )-2 x^{2} \ln \left (1-{\mathrm e}^{i x}\right )+4 i x \polylog \left (2, {\mathrm e}^{i x}\right )-4 \polylog \left (3, {\mathrm e}^{i x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^2*cos(x)^2*cot(x)^3,x)
[Out]
2/3*I*x^3-1/16*(2*I*x+2*x^2-1)*exp(2*I*x)-1/16*(-2*I*x+2*x^2-1)*exp(-2*I*x)+2*x*(x*exp(2*I*x)-I*exp(2*I*x)+I)/
(exp(2*I*x)-1)^2+ln(1+exp(I*x))+ln(exp(I*x)-1)-2*ln(exp(I*x))-2*x^2*ln(1+exp(I*x))+4*I*x*polylog(2,-exp(I*x))-
4*polylog(3,-exp(I*x))-2*x^2*ln(1-exp(I*x))+4*I*x*polylog(2,exp(I*x))-4*polylog(3,exp(I*x))
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maxima [B] time = 0.75, size = 2855, normalized size = 26.93 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*cos(x)^2*cot(x)^3,x, algorithm="maxima")
[Out]
-(3*(2*x^2 + 2*I*x - 1)*cos(6*x)^2 - (-64*I*x^3 - 24*x^2 + 168*I*x + 12)*cos(4*x)^2 - (-64*I*x^3 + 84*x^2 + 96
*I*x + 6)*cos(2*x)^2 - 3*(2*x^2 + 2*I*x - 1)*sin(6*x)^2 - (64*I*x^3 + 24*x^2 - 168*I*x - 12)*sin(4*x)^2 - (64*
I*x^3 - 84*x^2 - 96*I*x - 6)*sin(2*x)^2 + 6*x^2 - ((192*I*x^2 - 96*I)*cos(4*x)^2 + (192*I*x^2 - 96*I)*cos(2*x)
^2 + (-192*I*x^2 + 96*I)*sin(4*x)^2 + (-192*I*x^2 + 96*I)*sin(2*x)^2 + (-96*I*x^2 + (-96*I*x^2 + 48*I)*cos(4*x
) + (192*I*x^2 - 96*I)*cos(2*x) + 48*(2*x^2 - 1)*sin(4*x) - 96*(2*x^2 - 1)*sin(2*x) + 48*I)*cos(6*x) + (192*I*
x^2 + (-480*I*x^2 + 240*I)*cos(2*x) + 240*(2*x^2 - 1)*sin(2*x) - 96*I)*cos(4*x) + (-96*I*x^2 + 48*I)*cos(2*x)
+ (96*x^2 + 48*(2*x^2 - 1)*cos(4*x) - 96*(2*x^2 - 1)*cos(2*x) + (96*I*x^2 - 48*I)*sin(4*x) + (-192*I*x^2 + 96*
I)*sin(2*x) - 48)*sin(6*x) - (192*x^2 + 192*(2*x^2 - 1)*cos(4*x) - 240*(2*x^2 - 1)*cos(2*x) - (480*I*x^2 - 240
*I)*sin(2*x) - 96)*sin(4*x) + 48*(2*x^2 - 4*(2*x^2 - 1)*cos(2*x) - 1)*sin(2*x))*arctan2(sin(x), cos(x) + 1) -
((48*I*cos(4*x) - 96*I*cos(2*x) - 48*sin(4*x) + 96*sin(2*x) + 48*I)*cos(6*x) + (240*I*cos(2*x) - 240*sin(2*x)
- 96*I)*cos(4*x) - 96*I*cos(4*x)^2 - 96*I*cos(2*x)^2 - (48*cos(4*x) - 96*cos(2*x) + 48*I*sin(4*x) - 96*I*sin(2
*x) + 48)*sin(6*x) + (192*cos(4*x) - 240*cos(2*x) - 240*I*sin(2*x) + 96)*sin(4*x) + 96*I*sin(4*x)^2 + 48*(4*co
s(2*x) - 1)*sin(2*x) + 96*I*sin(2*x)^2 + 48*I*cos(2*x))*arctan2(sin(x), cos(x) - 1) - (-192*I*x^2*cos(4*x)^2 -
192*I*x^2*cos(2*x)^2 + 192*I*x^2*sin(4*x)^2 + 192*I*x^2*sin(2*x)^2 + 96*I*x^2*cos(2*x) + (96*I*x^2*cos(4*x) -
192*I*x^2*cos(2*x) - 96*x^2*sin(4*x) + 192*x^2*sin(2*x) + 96*I*x^2)*cos(6*x) + (480*I*x^2*cos(2*x) - 480*x^2*
sin(2*x) - 192*I*x^2)*cos(4*x) - (96*x^2*cos(4*x) - 192*x^2*cos(2*x) + 96*I*x^2*sin(4*x) - 192*I*x^2*sin(2*x)
+ 96*x^2)*sin(6*x) + 96*(4*x^2*cos(4*x) - 5*x^2*cos(2*x) - 5*I*x^2*sin(2*x) + 2*x^2)*sin(4*x) + 96*(4*x^2*cos(
2*x) - x^2)*sin(2*x))*arctan2(sin(x), -cos(x) + 1) - (32*I*x^3 + 12*x^2 + (32*I*x^3 + 24*x^2 - 72*I*x - 12)*co
s(4*x) + (-64*I*x^3 + 78*x^2 + 90*I*x + 9)*cos(2*x) - (32*x^3 - 24*I*x^2 - 72*x + 12*I)*sin(4*x) + (64*x^3 + 7
8*I*x^2 - 90*x + 9*I)*sin(2*x) - 12*I*x - 6)*cos(6*x) - (-64*I*x^3 - 30*x^2 + (160*I*x^3 - 156*x^2 - 276*I*x -
18)*cos(2*x) - (160*x^3 + 156*I*x^2 - 276*x + 18*I)*sin(2*x) + 30*I*x + 15)*cos(4*x) - (32*I*x^3 + 24*x^2 - 2
4*I*x - 12)*cos(2*x) - (-384*I*x*cos(4*x)^2 - 384*I*x*cos(2*x)^2 + 384*I*x*sin(4*x)^2 + 384*I*x*sin(2*x)^2 + (
192*I*x*cos(4*x) - 384*I*x*cos(2*x) - 192*x*sin(4*x) + 384*x*sin(2*x) + 192*I*x)*cos(6*x) + (960*I*x*cos(2*x)
- 960*x*sin(2*x) - 384*I*x)*cos(4*x) + 192*I*x*cos(2*x) - (192*x*cos(4*x) - 384*x*cos(2*x) + 192*I*x*sin(4*x)
- 384*I*x*sin(2*x) + 192*x)*sin(6*x) + 192*(4*x*cos(4*x) - 5*x*cos(2*x) - 5*I*x*sin(2*x) + 2*x)*sin(4*x) + 192
*(4*x*cos(2*x) - x)*sin(2*x))*dilog(-e^(I*x)) - (-384*I*x*cos(4*x)^2 - 384*I*x*cos(2*x)^2 + 384*I*x*sin(4*x)^2
+ 384*I*x*sin(2*x)^2 + (192*I*x*cos(4*x) - 384*I*x*cos(2*x) - 192*x*sin(4*x) + 384*x*sin(2*x) + 192*I*x)*cos(
6*x) + (960*I*x*cos(2*x) - 960*x*sin(2*x) - 384*I*x)*cos(4*x) + 192*I*x*cos(2*x) - (192*x*cos(4*x) - 384*x*cos
(2*x) + 192*I*x*sin(4*x) - 384*I*x*sin(2*x) + 192*x)*sin(6*x) + 192*(4*x*cos(4*x) - 5*x*cos(2*x) - 5*I*x*sin(2
*x) + 2*x)*sin(4*x) + 192*(4*x*cos(2*x) - x)*sin(2*x))*dilog(e^(I*x)) - (48*(2*x^2 - 1)*cos(4*x)^2 + 48*(2*x^2
- 1)*cos(2*x)^2 - 48*(2*x^2 - 1)*sin(4*x)^2 - 48*(2*x^2 - 1)*sin(2*x)^2 - (48*x^2 + 24*(2*x^2 - 1)*cos(4*x) -
48*(2*x^2 - 1)*cos(2*x) - (-48*I*x^2 + 24*I)*sin(4*x) - (96*I*x^2 - 48*I)*sin(2*x) - 24)*cos(6*x) + (96*x^2 -
120*(2*x^2 - 1)*cos(2*x) + (-240*I*x^2 + 120*I)*sin(2*x) - 48)*cos(4*x) - 24*(2*x^2 - 1)*cos(2*x) + (-48*I*x^
2 + (-48*I*x^2 + 24*I)*cos(4*x) + (96*I*x^2 - 48*I)*cos(2*x) + 24*(2*x^2 - 1)*sin(4*x) - 48*(2*x^2 - 1)*sin(2*
x) + 24*I)*sin(6*x) + (96*I*x^2 + (192*I*x^2 - 96*I)*cos(4*x) + (-240*I*x^2 + 120*I)*cos(2*x) + 120*(2*x^2 - 1
)*sin(2*x) - 48*I)*sin(4*x) + (-48*I*x^2 + (192*I*x^2 - 96*I)*cos(2*x) + 24*I)*sin(2*x))*log(cos(x)^2 + sin(x)
^2 + 2*cos(x) + 1) - (48*(2*x^2 - 1)*cos(4*x)^2 + 48*(2*x^2 - 1)*cos(2*x)^2 - 48*(2*x^2 - 1)*sin(4*x)^2 - 48*(
2*x^2 - 1)*sin(2*x)^2 - (48*x^2 + 24*(2*x^2 - 1)*cos(4*x) - 48*(2*x^2 - 1)*cos(2*x) - (-48*I*x^2 + 24*I)*sin(4
*x) - (96*I*x^2 - 48*I)*sin(2*x) - 24)*cos(6*x) + (96*x^2 - 120*(2*x^2 - 1)*cos(2*x) + (-240*I*x^2 + 120*I)*si
n(2*x) - 48)*cos(4*x) - 24*(2*x^2 - 1)*cos(2*x) + (-48*I*x^2 + (-48*I*x^2 + 24*I)*cos(4*x) + (96*I*x^2 - 48*I)
*cos(2*x) + 24*(2*x^2 - 1)*sin(4*x) - 48*(2*x^2 - 1)*sin(2*x) + 24*I)*sin(6*x) + (96*I*x^2 + (192*I*x^2 - 96*I
)*cos(4*x) + (-240*I*x^2 + 120*I)*cos(2*x) + 120*(2*x^2 - 1)*sin(2*x) - 48*I)*sin(4*x) + (-48*I*x^2 + (192*I*x
^2 - 96*I)*cos(2*x) + 24*I)*sin(2*x))*log(cos(x)^2 + sin(x)^2 - 2*cos(x) + 1) + ((192*cos(4*x) - 384*cos(2*x)
+ 192*I*sin(4*x) - 384*I*sin(2*x) + 192)*cos(6*x) + (960*cos(2*x) + 960*I*sin(2*x) - 384)*cos(4*x) - 384*cos(4
*x)^2 - 384*cos(2*x)^2 - (-192*I*cos(4*x) + 384*I*cos(2*x) + 192*sin(4*x) - 384*sin(2*x) - 192*I)*sin(6*x) - (
768*I*cos(4*x) - 960*I*cos(2*x) + 960*sin(2*x) + 384*I)*sin(4*x) + 384*sin(4*x)^2 - (768*I*cos(2*x) - 192*I)*s
in(2*x) + 384*sin(2*x)^2 + 192*cos(2*x))*polylog(3, -e^(I*x)) + ((192*cos(4*x) - 384*cos(2*x) + 192*I*sin(4*x)
- 384*I*sin(2*x) + 192)*cos(6*x) + (960*cos(2*x) + 960*I*sin(2*x) - 384)*cos(4*x) - 384*cos(4*x)^2 - 384*cos(
2*x)^2 - (-192*I*cos(4*x) + 384*I*cos(2*x) + 192*sin(4*x) - 384*sin(2*x) - 192*I)*sin(6*x) - (768*I*cos(4*x) -
960*I*cos(2*x) + 960*sin(2*x) + 384*I)*sin(4*x) + 384*sin(4*x)^2 - (768*I*cos(2*x) - 192*I)*sin(2*x) + 384*si
n(2*x)^2 + 192*cos(2*x))*polylog(3, e^(I*x)) + (32*x^3 - 12*I*x^2 - (-12*I*x^2 + 12*x + 6*I)*cos(6*x) + (32*x^
3 - 24*I*x^2 - 72*x + 12*I)*cos(4*x) - (64*x^3 + 78*I*x^2 - 90*x + 9*I)*cos(2*x) - (-32*I*x^3 - 24*x^2 + 72*I*
x + 12)*sin(4*x) - (64*I*x^3 - 78*x^2 - 90*I*x - 9)*sin(2*x) - 12*x + 6*I)*sin(6*x) - (64*x^3 - 30*I*x^2 + (12
8*x^3 - 48*I*x^2 - 336*x + 24*I)*cos(4*x) - (160*x^3 + 156*I*x^2 - 276*x + 18*I)*cos(2*x) + (-160*I*x^3 + 156*
x^2 + 276*I*x + 18)*sin(2*x) - 30*x + 15*I)*sin(4*x) + (32*x^3 - 24*I*x^2 - (128*x^3 + 168*I*x^2 - 192*x + 12*
I)*cos(2*x) - 24*x + 12*I)*sin(2*x) - 6*I*x - 3)/((48*cos(4*x) - 96*cos(2*x) + 48*I*sin(4*x) - 96*I*sin(2*x) +
48)*cos(6*x) + (240*cos(2*x) + 240*I*sin(2*x) - 96)*cos(4*x) - 96*cos(4*x)^2 - 96*cos(2*x)^2 - (-48*I*cos(4*x
) + 96*I*cos(2*x) + 48*sin(4*x) - 96*sin(2*x) - 48*I)*sin(6*x) - (192*I*cos(4*x) - 240*I*cos(2*x) + 240*sin(2*
x) + 96*I)*sin(4*x) + 96*sin(4*x)^2 - (192*I*cos(2*x) - 48*I)*sin(2*x) + 96*sin(2*x)^2 + 48*cos(2*x))
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,{\cos \relax (x)}^2\,{\mathrm {cot}\relax (x)}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^2*cos(x)^2*cot(x)^3,x)
[Out]
int(x^2*cos(x)^2*cot(x)^3, x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \cos ^{2}{\relax (x )} \cot ^{3}{\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**2*cos(x)**2*cot(x)**3,x)
[Out]
Integral(x**2*cos(x)**2*cot(x)**3, x)
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