\(\int \cos (x) \log (\sin (x)) \sin ^2(x) \, dx\) [357]

Optimal result

Integrand size = 10, antiderivative size = 20 \[ \int \cos (x) \log (\sin (x)) \sin ^2(x) \, dx=-\frac {1}{9} \sin ^3(x)+\frac {1}{3} \log (\sin (x)) \sin ^3(x) \]

[Out]

-1/9*sin(x)^3+1/3*ln(sin(x))*sin(x)^3

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2644, 30, 2634, 12} \[ \int \cos (x) \log (\sin (x)) \sin ^2(x) \, dx=\frac {1}{3} \sin ^3(x) \log (\sin (x))-\frac {\sin ^3(x)}{9} \]

[In]

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

[Out]

-1/9*Sin[x]^3 + (Log[Sin[x]]*Sin[x]^3)/3

Rule 12

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -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 2644

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(a*f), Subst[Int[
x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \log (\sin (x)) \sin ^3(x)-\int \frac {1}{3} \cos (x) \sin ^2(x) \, dx \\ & = \frac {1}{3} \log (\sin (x)) \sin ^3(x)-\frac {1}{3} \int \cos (x) \sin ^2(x) \, dx \\ & = \frac {1}{3} \log (\sin (x)) \sin ^3(x)-\frac {1}{3} \text {Subst}\left (\int x^2 \, dx,x,\sin (x)\right ) \\ & = -\frac {1}{9} \sin ^3(x)+\frac {1}{3} \log (\sin (x)) \sin ^3(x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75 \[ \int \cos (x) \log (\sin (x)) \sin ^2(x) \, dx=\frac {1}{9} (-1+3 \log (\sin (x))) \sin ^3(x) \]

[In]

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

[Out]

((-1 + 3*Log[Sin[x]])*Sin[x]^3)/9

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85

method	result	size
derivativedivides	\(-\frac {\left (\sin ^{3}\left (x \right )\right )}{9}+\frac {\ln \left (\sin \left (x \right )\right ) \left (\sin ^{3}\left (x \right )\right )}{3}\)	\(17\)
default	\(-\frac {\left (\sin ^{3}\left (x \right )\right )}{9}+\frac {\ln \left (\sin \left (x \right )\right ) \left (\sin ^{3}\left (x \right )\right )}{3}\)	\(17\)
parallelrisch	\(\frac {\left (3 \ln \left (\sin \left (x \right )\right )-1\right ) \left (-\sin \left (3 x \right )+3 \sin \left (x \right )\right )}{36}\)	\(21\)
risch	\(\text {Expression too large to display}\)	\(577\)

[In]

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

[Out]

-1/9*sin(x)^3+1/3*ln(sin(x))*sin(x)^3

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \cos (x) \log (\sin (x)) \sin ^2(x) \, dx=-\frac {1}{3} \, {\left (\cos \left (x\right )^{2} - 1\right )} \log \left (\sin \left (x\right )\right ) \sin \left (x\right ) + \frac {1}{9} \, {\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right ) \]

[In]

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

[Out]

-1/3*(cos(x)^2 - 1)*log(sin(x))*sin(x) + 1/9*(cos(x)^2 - 1)*sin(x)

Sympy [A] (verification not implemented)

Time = 0.64 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \cos (x) \log (\sin (x)) \sin ^2(x) \, dx=\frac {\log {\left (\sin {\left (x \right )} \right )} \sin ^{3}{\left (x \right )}}{3} - \frac {\sin ^{3}{\left (x \right )}}{9} \]

[In]

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

[Out]

log(sin(x))*sin(x)**3/3 - sin(x)**3/9

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \cos (x) \log (\sin (x)) \sin ^2(x) \, dx=\frac {1}{3} \, \log \left (\sin \left (x\right )\right ) \sin \left (x\right )^{3} - \frac {1}{9} \, \sin \left (x\right )^{3} \]

[In]

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

[Out]

1/3*log(sin(x))*sin(x)^3 - 1/9*sin(x)^3

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \cos (x) \log (\sin (x)) \sin ^2(x) \, dx=\frac {1}{3} \, \log \left (\sin \left (x\right )\right ) \sin \left (x\right )^{3} - \frac {1}{9} \, \sin \left (x\right )^{3} \]

[In]

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

[Out]

1/3*log(sin(x))*sin(x)^3 - 1/9*sin(x)^3

Mupad [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.55 \[ \int \cos (x) \log (\sin (x)) \sin ^2(x) \, dx=\frac {{\sin \left (x\right )}^3\,\left (\ln \left (\sin \left (x\right )\right )-\frac {1}{3}\right )}{3} \]

[In]

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

[Out]

(sin(x)^3*(log(sin(x)) - 1/3))/3