3.184 \(\int \cos (x) \log (\sin (x)) \sin ^2(x) \, dx\)

Optimal. Leaf size=20 \[ \frac {1}{3} \sin ^3(x) \log (\sin (x))-\frac {\sin ^3(x)}{9} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2564, 30, 2554, 12} \[ \frac {1}{3} \sin ^3(x) \log (\sin (x))-\frac {\sin ^3(x)}{9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-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 30

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

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 2564

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*} \int \cos (x) \log (\sin (x)) \sin ^2(x) \, dx &=\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} \operatorname {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*}

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Mathematica [A]  time = 0.02, size = 15, normalized size = 0.75 \[ \frac {1}{9} \sin ^3(x) (3 \log (\sin (x))-1) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-1 + 3*Log[Sin[x]])*Sin[x]^3)/9

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fricas [A]  time = 0.46, size = 24, normalized size = 1.20 \[ -\frac {1}{3} \, {\left (\cos \relax (x)^{2} - 1\right )} \log \left (\sin \relax (x)\right ) \sin \relax (x) + \frac {1}{9} \, {\left (\cos \relax (x)^{2} - 1\right )} \sin \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.19, size = 16, normalized size = 0.80 \[ \frac {1}{3} \, \log \left (\sin \relax (x)\right ) \sin \relax (x)^{3} - \frac {1}{9} \, \sin \relax (x)^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [A]  time = 0.07, size = 17, normalized size = 0.85 \[ \frac {\ln \left (\sin \relax (x )\right ) \left (\sin ^{3}\relax (x )\right )}{3}-\frac {\left (\sin ^{3}\relax (x )\right )}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.45, size = 16, normalized size = 0.80 \[ \frac {1}{3} \, \log \left (\sin \relax (x)\right ) \sin \relax (x)^{3} - \frac {1}{9} \, \sin \relax (x)^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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mupad [B]  time = 0.43, size = 11, normalized size = 0.55 \[ \frac {{\sin \relax (x)}^3\,\left (\ln \left (\sin \relax (x)\right )-\frac {1}{3}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(sin(x)^3*(log(sin(x)) - 1/3))/3

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sympy [A]  time = 4.93, size = 17, normalized size = 0.85 \[ \frac {\log {\left (\sin {\relax (x )} \right )} \sin ^{3}{\relax (x )}}{3} - \frac {\sin ^{3}{\relax (x )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(sin(x))*sin(x)**3/3 - sin(x)**3/9

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