\(\int \log (\sin (x)) \sin ^2(x) \, dx\) [191]

Optimal result

Integrand size = 8, antiderivative size = 74 \[ \int \log (\sin (x)) \sin ^2(x) \, dx=\frac {x}{4}+\frac {i x^2}{4}-\frac {1}{2} x \log \left (1-e^{2 i x}\right )+\frac {1}{2} x \log (\sin (x))+\frac {1}{4} i \operatorname {PolyLog}\left (2,e^{2 i x}\right )+\frac {1}{4} \cos (x) \sin (x)-\frac {1}{2} \cos (x) \log (\sin (x)) \sin (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.125, Rules used = {2715, 8, 2634, 12, 6874, 3798, 2221, 2317, 2438} \[ \int \log (\sin (x)) \sin ^2(x) \, dx=\frac {1}{4} i \operatorname {PolyLog}\left (2,e^{2 i x}\right )+\frac {i x^2}{4}+\frac {x}{4}-\frac {1}{2} x \log \left (1-e^{2 i x}\right )+\frac {1}{2} x \log (\sin (x))+\frac {1}{4} \sin (x) \cos (x)-\frac {1}{2} \sin (x) \cos (x) \log (\sin (x)) \]

[In]

Int[Log[Sin[x]]*Sin[x]^2,x]

[Out]

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

Rule 8

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 2634

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[w*(D[u, x]
/u), x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, x]

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 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 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x \log (\sin (x))-\frac {1}{2} \cos (x) \log (\sin (x)) \sin (x)-\int \frac {1}{2} \cot (x) (x-\cos (x) \sin (x)) \, dx \\ & = \frac {1}{2} x \log (\sin (x))-\frac {1}{2} \cos (x) \log (\sin (x)) \sin (x)-\frac {1}{2} \int \cot (x) (x-\cos (x) \sin (x)) \, dx \\ & = \frac {1}{2} x \log (\sin (x))-\frac {1}{2} \cos (x) \log (\sin (x)) \sin (x)-\frac {1}{2} \int \left (-\cos ^2(x)+x \cot (x)\right ) \, dx \\ & = \frac {1}{2} x \log (\sin (x))-\frac {1}{2} \cos (x) \log (\sin (x)) \sin (x)+\frac {1}{2} \int \cos ^2(x) \, dx-\frac {1}{2} \int x \cot (x) \, dx \\ & = \frac {i x^2}{4}+\frac {1}{2} x \log (\sin (x))+\frac {1}{4} \cos (x) \sin (x)-\frac {1}{2} \cos (x) \log (\sin (x)) \sin (x)+i \int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx+\frac {\int 1 \, dx}{4} \\ & = \frac {x}{4}+\frac {i x^2}{4}-\frac {1}{2} x \log \left (1-e^{2 i x}\right )+\frac {1}{2} x \log (\sin (x))+\frac {1}{4} \cos (x) \sin (x)-\frac {1}{2} \cos (x) \log (\sin (x)) \sin (x)+\frac {1}{2} \int \log \left (1-e^{2 i x}\right ) \, dx \\ & = \frac {x}{4}+\frac {i x^2}{4}-\frac {1}{2} x \log \left (1-e^{2 i x}\right )+\frac {1}{2} x \log (\sin (x))+\frac {1}{4} \cos (x) \sin (x)-\frac {1}{2} \cos (x) \log (\sin (x)) \sin (x)-\frac {1}{4} i \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i x}\right ) \\ & = \frac {x}{4}+\frac {i x^2}{4}-\frac {1}{2} x \log \left (1-e^{2 i x}\right )+\frac {1}{2} x \log (\sin (x))+\frac {1}{4} i \text {Li}_2\left (e^{2 i x}\right )+\frac {1}{4} \cos (x) \sin (x)-\frac {1}{2} \cos (x) \log (\sin (x)) \sin (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.80 \[ \int \log (\sin (x)) \sin ^2(x) \, dx=\frac {1}{8} \left (2 x \left (1+i x-2 \log \left (1-e^{2 i x}\right )+2 \log (\sin (x))\right )+2 i \operatorname {PolyLog}\left (2,e^{2 i x}\right )+(1-2 \log (\sin (x))) \sin (2 x)\right ) \]

[In]

Integrate[Log[Sin[x]]*Sin[x]^2,x]

[Out]

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 193 vs. \(2 (54 ) = 108\).

Time = 4.48 (sec) , antiderivative size = 194, normalized size of antiderivative = 2.62

method	result	size
default	\(\frac {i \left (\frac {\ln \left (i \left (1-{\mathrm e}^{2 i x}\right ) {\mathrm e}^{-i x}\right ) {\mathrm e}^{2 i x}}{2}-\frac {{\mathrm e}^{2 i x}}{4}-2 \ln \left ({\mathrm e}^{i x}\right ) \ln \left (i \left (1-{\mathrm e}^{2 i x}\right ) {\mathrm e}^{-i x}\right )-\ln \left ({\mathrm e}^{i x}\right )^{2}+2 \ln \left ({\mathrm e}^{i x}\right ) \ln \left ({\mathrm e}^{i x}+1\right )-2 \operatorname {dilog}\left ({\mathrm e}^{i x}\right )+2 \operatorname {dilog}\left ({\mathrm e}^{i x}+1\right )-\frac {{\mathrm e}^{-2 i x} \ln \left (i \left (1-{\mathrm e}^{2 i x}\right ) {\mathrm e}^{-i x}\right )}{2}+\frac {{\mathrm e}^{-2 i x}}{4}-\ln \left ({\mathrm e}^{i x}\right )-\ln \left (2\right ) \left (\frac {{\mathrm e}^{2 i x}}{2}-2 \ln \left ({\mathrm e}^{i x}\right )-\frac {{\mathrm e}^{-2 i x}}{2}\right )\right )}{4}\)	\(194\)
risch	\(\text {Expression too large to display}\)	\(497\)

[In]

int(ln(sin(x))*sin(x)^2,x,method=_RETURNVERBOSE)

[Out]

1/4*I*(1/2*ln(I*(-exp(I*x)^2+1)/exp(I*x))*exp(2*I*x)-1/4*exp(I*x)^2-2*ln(exp(I*x))*ln(I*(-exp(I*x)^2+1)/exp(I*
x))-ln(exp(I*x))^2+2*ln(exp(I*x))*ln(exp(I*x)+1)-2*dilog(exp(I*x))+2*dilog(exp(I*x)+1)-1/2*exp(-2*I*x)*ln(I*(-
exp(I*x)^2+1)/exp(I*x))+1/4/exp(I*x)^2-ln(exp(I*x))-ln(2)*(1/2*exp(I*x)^2-2*ln(exp(I*x))-1/2/exp(I*x)^2))

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. 120 vs. \(2 (49) = 98\).

Time = 0.34 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.62 \[ \int \log (\sin (x)) \sin ^2(x) \, dx=-\frac {1}{4} \, x \log \left (\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) - \frac {1}{4} \, x \log \left (\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) - \frac {1}{4} \, x \log \left (-\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) - \frac {1}{4} \, x \log \left (-\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) - \frac {1}{2} \, {\left (\cos \left (x\right ) \sin \left (x\right ) - x\right )} \log \left (\sin \left (x\right )\right ) + \frac {1}{4} \, \cos \left (x\right ) \sin \left (x\right ) + \frac {1}{4} \, x + \frac {1}{4} i \, {\rm Li}_2\left (\cos \left (x\right ) + i \, \sin \left (x\right )\right ) - \frac {1}{4} i \, {\rm Li}_2\left (\cos \left (x\right ) - i \, \sin \left (x\right )\right ) - \frac {1}{4} i \, {\rm Li}_2\left (-\cos \left (x\right ) + i \, \sin \left (x\right )\right ) + \frac {1}{4} i \, {\rm Li}_2\left (-\cos \left (x\right ) - i \, \sin \left (x\right )\right ) \]

[In]

integrate(log(sin(x))*sin(x)^2,x, algorithm="fricas")

[Out]

-1/4*x*log(cos(x) + I*sin(x) + 1) - 1/4*x*log(cos(x) - I*sin(x) + 1) - 1/4*x*log(-cos(x) + I*sin(x) + 1) - 1/4
*x*log(-cos(x) - I*sin(x) + 1) - 1/2*(cos(x)*sin(x) - x)*log(sin(x)) + 1/4*cos(x)*sin(x) + 1/4*x + 1/4*I*dilog
(cos(x) + I*sin(x)) - 1/4*I*dilog(cos(x) - I*sin(x)) - 1/4*I*dilog(-cos(x) + I*sin(x)) + 1/4*I*dilog(-cos(x) -
I*sin(x))

Sympy [F]

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

[In]

integrate(ln(sin(x))*sin(x)**2,x)

[Out]

Integral(log(sin(x))*sin(x)**2, x)

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (49) = 98\).

Time = 0.46 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.41 \[ \int \log (\sin (x)) \sin ^2(x) \, dx=\frac {1}{4} i \, x^{2} - \frac {1}{2} i \, x \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right ) + \frac {1}{2} i \, x \arctan \left (\sin \left (x\right ), -\cos \left (x\right ) + 1\right ) - \frac {1}{4} \, x \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) - \frac {1}{4} \, x \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) + \frac {1}{4} \, {\left (2 \, x - \sin \left (2 \, x\right )\right )} \log \left (\sin \left (x\right )\right ) + \frac {1}{4} \, x + \frac {1}{2} i \, {\rm Li}_2\left (-e^{\left (i \, x\right )}\right ) + \frac {1}{2} i \, {\rm Li}_2\left (e^{\left (i \, x\right )}\right ) + \frac {1}{8} \, \sin \left (2 \, x\right ) \]

[In]

integrate(log(sin(x))*sin(x)^2,x, algorithm="maxima")

[Out]

1/4*I*x^2 - 1/2*I*x*arctan2(sin(x), cos(x) + 1) + 1/2*I*x*arctan2(sin(x), -cos(x) + 1) - 1/4*x*log(cos(x)^2 +
sin(x)^2 + 2*cos(x) + 1) - 1/4*x*log(cos(x)^2 + sin(x)^2 - 2*cos(x) + 1) + 1/4*(2*x - sin(2*x))*log(sin(x)) +
1/4*x + 1/2*I*dilog(-e^(I*x)) + 1/2*I*dilog(e^(I*x)) + 1/8*sin(2*x)

Giac [F]

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

[In]

integrate(log(sin(x))*sin(x)^2,x, algorithm="giac")

[Out]

integrate(log(sin(x))*sin(x)^2, x)

Mupad [F(-1)]

Timed out. \[ \int \log (\sin (x)) \sin ^2(x) \, dx=\int \ln \left (\sin \left (x\right )\right )\,{\sin \left (x\right )}^2 \,d x \]

[In]

int(log(sin(x))*sin(x)^2,x)

[Out]

int(log(sin(x))*sin(x)^2, x)