3.641 \(\int \cos ^3(x) \log (\sin (x)) \, dx\)

Optimal. Leaf size=30 \[ \frac{\sin ^3(x)}{9}-\sin (x)-\frac{1}{3} \sin ^3(x) \log (\sin (x))+\sin (x) \log (\sin (x)) \]

[Out]

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

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Rubi [A]  time = 0.0348104, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2633, 2554, 12, 4356} \[ \frac{\sin ^3(x)}{9}-\sin (x)-\frac{1}{3} \sin ^3(x) \log (\sin (x))+\sin (x) \log (\sin (x)) \]

Antiderivative was successfully verified.

[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 2633

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 2554

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 12

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

Rule 4356

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*} \int \cos ^3(x) \log (\sin (x)) \, dx &=\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} \operatorname{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]  time = 0.0077247, size = 30, normalized size = 1. \[ \frac{\sin ^3(x)}{9}-\sin (x)-\frac{1}{3} \sin ^3(x) \log (\sin (x))+\sin (x) \log (\sin (x)) \]

Antiderivative was successfully verified.

[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

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Maple [C]  time = 0.035, size = 126, normalized size = 4.2 \begin{align*} -{\frac{i}{24}}{{\rm e}^{3\,ix}}\ln \left ( 2\,\sin \left ( x \right ) \right ) +{\frac{i}{72}}{{\rm e}^{3\,ix}}+{\frac{11\,i}{24}}{{\rm e}^{ix}}-{\frac{3\,i}{8}}{{\rm e}^{ix}}\ln \left ( 2\,\sin \left ( x \right ) \right ) +{\frac{3\,i}{8}}{{\rm e}^{-ix}}\ln \left ( 2\,\sin \left ( x \right ) \right ) -{\frac{11\,i}{24}}{{\rm e}^{-ix}}+{\frac{i}{24}}{{\rm e}^{-3\,ix}}\ln \left ( 2\,\sin \left ( x \right ) \right ) -{\frac{i}{72}}{{\rm e}^{-3\,ix}}+{\frac{i}{24}}\ln \left ( 2 \right ){{\rm e}^{3\,ix}}+{\frac{3\,i}{8}}\ln \left ( 2 \right ){{\rm e}^{ix}}-{\frac{i}{24}}\ln \left ( 2 \right ){{\rm e}^{-3\,ix}}-{\frac{3\,i}{8}}\ln \left ( 2 \right ){{\rm e}^{-ix}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*I*exp(3*I*x)*ln(2*sin(x))+1/72*I*exp(3*I*x)+11/24*I*exp(I*x)-3/8*I*exp(I*x)*ln(2*sin(x))+3/8*I*exp(-I*x)
*ln(2*sin(x))-11/24*I*exp(-I*x)+1/24*I*exp(-3*I*x)*ln(2*sin(x))-1/72*I*exp(-3*I*x)+1/24*I*ln(2)*exp(3*I*x)+3/8
*I*ln(2)*exp(I*x)-1/24*I*ln(2)*exp(-3*I*x)-3/8*I*ln(2)*exp(-I*x)

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Maxima [A]  time = 0.958534, size = 34, normalized size = 1.13 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 2.61872, size = 90, normalized size = 3. \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 5.09148, size = 42, normalized size = 1.4 \begin{align*} \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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Giac [A]  time = 1.07313, size = 35, normalized size = 1.17 \begin{align*} -\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)