\(\int \cos ^3(x) \log (\sin (x)) \, dx\) [641]

Optimal result

Integrand size = 8, antiderivative size = 30 \[ \int \cos ^3(x) \log (\sin (x)) \, dx=-\sin (x)+\log (\sin (x)) \sin (x)+\frac {\sin ^3(x)}{9}-\frac {1}{3} \log (\sin (x)) \sin ^3(x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2713, 2634, 12, 4441} \[ \int \cos ^3(x) \log (\sin (x)) \, dx=\frac {\sin ^3(x)}{9}-\sin (x)-\frac {1}{3} \sin ^3(x) \log (\sin (x))+\sin (x) \log (\sin (x)) \]

[In]

Int[Cos[x]^3*Log[Sin[x]],x]

[Out]

-Sin[x] + Log[Sin[x]]*Sin[x] + Sin[x]^3/9 - (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 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 2713

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rule 4441

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, Dist[d/(b
*c), Subst[Int[SubstFor[1, Sin[c*(a + b*x)]/d, u, x], x], x, Sin[c*(a + b*x)]/d], x] /; FunctionOfQ[Sin[c*(a +
b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Cos] || EqQ[F, cos])

Rubi steps \begin{align*} \text {integral}& = \log (\sin (x)) \sin (x)-\frac {1}{3} \log (\sin (x)) \sin ^3(x)-\int \frac {1}{6} \cos (x) (5+\cos (2 x)) \, dx \\ & = \log (\sin (x)) \sin (x)-\frac {1}{3} \log (\sin (x)) \sin ^3(x)-\frac {1}{6} \int \cos (x) (5+\cos (2 x)) \, dx \\ & = \log (\sin (x)) \sin (x)-\frac {1}{3} \log (\sin (x)) \sin ^3(x)-\frac {1}{6} \text {Subst}\left (\int \left (6-2 x^2\right ) \, dx,x,\sin (x)\right ) \\ & = -\sin (x)+\log (\sin (x)) \sin (x)+\frac {\sin ^3(x)}{9}-\frac {1}{3} \log (\sin (x)) \sin ^3(x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \cos ^3(x) \log (\sin (x)) \, dx=-\sin (x)+\log (\sin (x)) \sin (x)+\frac {\sin ^3(x)}{9}-\frac {1}{3} \log (\sin (x)) \sin ^3(x) \]

[In]

Integrate[Cos[x]^3*Log[Sin[x]],x]

[Out]

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

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87

method	result	size
parallelrisch	\(\frac {\left (3 \ln \left (\sin \left (x \right )\right )-1\right ) \sin \left (3 x \right )}{36}+\frac {3 \ln \left (\sin \left (x \right )\right ) \sin \left (x \right )}{4}-\frac {11 \sin \left (x \right )}{12}\)	\(26\)
derivativedivides	\(-\sin \left (x \right )+\ln \left (\sin \left (x \right )\right ) \sin \left (x \right )+\frac {\left (\sin ^{3}\left (x \right )\right )}{9}-\frac {\ln \left (\sin \left (x \right )\right ) \left (\sin ^{3}\left (x \right )\right )}{3}\)	\(27\)
default	\(-\sin \left (x \right )+\ln \left (\sin \left (x \right )\right ) \sin \left (x \right )+\frac {\left (\sin ^{3}\left (x \right )\right )}{9}-\frac {\ln \left (\sin \left (x \right )\right ) \left (\sin ^{3}\left (x \right )\right )}{3}\)	\(27\)
risch	\(\text {Expression too large to display}\)	\(577\)

[In]

int(cos(x)^3*ln(sin(x)),x,method=_RETURNVERBOSE)

[Out]

1/36*(3*ln(sin(x))-1)*sin(3*x)+3/4*ln(sin(x))*sin(x)-11/12*sin(x)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \cos ^3(x) \log (\sin (x)) \, dx=\frac {1}{3} \, {\left (\cos \left (x\right )^{2} + 2\right )} \log \left (\sin \left (x\right )\right ) \sin \left (x\right ) - \frac {1}{9} \, {\left (\cos \left (x\right )^{2} + 8\right )} \sin \left (x\right ) \]

[In]

integrate(cos(x)^3*log(sin(x)),x, algorithm="fricas")

[Out]

1/3*(cos(x)^2 + 2)*log(sin(x))*sin(x) - 1/9*(cos(x)^2 + 8)*sin(x)

Sympy [A] (verification not implemented)

Time = 0.66 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40 \[ \int \cos ^3(x) \log (\sin (x)) \, dx=\frac {2 \log {\left (\sin {\left (x \right )} \right )} \sin ^{3}{\left (x \right )}}{3} + \log {\left (\sin {\left (x \right )} \right )} \sin {\left (x \right )} \cos ^{2}{\left (x \right )} - \frac {8 \sin ^{3}{\left (x \right )}}{9} - \sin {\left (x \right )} \cos ^{2}{\left (x \right )} \]

[In]

integrate(cos(x)**3*ln(sin(x)),x)

[Out]

2*log(sin(x))*sin(x)**3/3 + log(sin(x))*sin(x)*cos(x)**2 - 8*sin(x)**3/9 - sin(x)*cos(x)**2

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83 \[ \int \cos ^3(x) \log (\sin (x)) \, dx=\frac {1}{9} \, \sin \left (x\right )^{3} - \frac {1}{3} \, {\left (\sin \left (x\right )^{3} - 3 \, \sin \left (x\right )\right )} \log \left (\sin \left (x\right )\right ) - \sin \left (x\right ) \]

[In]

integrate(cos(x)^3*log(sin(x)),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \cos ^3(x) \log (\sin (x)) \, dx=-\frac {1}{3} \, \log \left (\sin \left (x\right )\right ) \sin \left (x\right )^{3} + \frac {1}{9} \, \sin \left (x\right )^{3} + \log \left (\sin \left (x\right )\right ) \sin \left (x\right ) - \sin \left (x\right ) \]

[In]

integrate(cos(x)^3*log(sin(x)),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \cos ^3(x) \log (\sin (x)) \, dx=\int \ln \left (\sin \left (x\right )\right )\,{\cos \left (x\right )}^3 \,d x \]

[In]

int(log(sin(x))*cos(x)^3,x)

[Out]

int(log(sin(x))*cos(x)^3, x)