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

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

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Rubi [A]  time = 0.03, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 12

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

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 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 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*}

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Mathematica [A]  time = 0.01, size = 30, normalized size = 1.00 \[ \frac {\sin ^3(x)}{9}-\sin (x)-\frac {1}{3} \sin ^3(x) \log (\sin (x))+\sin (x) \log (\sin (x)) \]

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos ^3(x) \log (\sin (x)) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Could not integrate

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fricas [A]  time = 0.93, size = 24, normalized size = 0.80 \[ \frac {1}{3} \, {\left (\cos \relax (x)^{2} + 2\right )} \log \left (\sin \relax (x)\right ) \sin \relax (x) - \frac {1}{9} \, {\left (\cos \relax (x)^{2} + 8\right )} \sin \relax (x) \]

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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giac [A]  time = 1.08, size = 26, normalized size = 0.87 \[ -\frac {1}{3} \, \log \left (\sin \relax (x)\right ) \sin \relax (x)^{3} + \frac {1}{9} \, \sin \relax (x)^{3} + \log \left (\sin \relax (x)\right ) \sin \relax (x) - \sin \relax (x) \]

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)

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maple [C]  time = 0.38, size = 197, normalized size = 6.57

method	result	size
default	\(-\frac {i \left (\frac {{\mathrm e}^{3 i x} \ln \left (i \left (-{\mathrm e}^{2 i x}+1\right ) {\mathrm e}^{-i x}\right )}{3}-\frac {{\mathrm e}^{3 i x}}{9}-\frac {11 \,{\mathrm e}^{i x}}{3}+3 \,{\mathrm e}^{i x} \ln \left (i \left (-{\mathrm e}^{2 i x}+1\right ) {\mathrm e}^{-i x}\right )-3 \,{\mathrm e}^{-i x} \ln \left (i \left (-{\mathrm e}^{2 i x}+1\right ) {\mathrm e}^{-i x}\right )+\frac {11 \,{\mathrm e}^{-i x}}{3}-\frac {{\mathrm e}^{-3 i x} \ln \left (i \left (-{\mathrm e}^{2 i x}+1\right ) {\mathrm e}^{-i x}\right )}{3}+\frac {{\mathrm e}^{-3 i x}}{9}-\frac {\ln \relax (2) {\mathrm e}^{3 i x}}{3}-3 \ln \relax (2) {\mathrm e}^{i x}+3 \,{\mathrm e}^{-i x} \ln \relax (2)+\frac {\ln \relax (2) {\mathrm e}^{-3 i x}}{3}\right )}{8}\)	\(197\)
risch	\(\frac {3 i {\mathrm e}^{-i x} \ln \left ({\mathrm e}^{2 i x}-1\right )}{8}+2 i \left (\frac {i \ln \relax (2)}{24}-\frac {i \ln \left ({\mathrm e}^{2 i x}-1\right )}{24}-\frac {\pi }{48}+\frac {i}{72}+\frac {\mathrm {csgn}\left (\sin \relax (x )\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{2 i x}-i\right ) \pi }{48}+\frac {\mathrm {csgn}\left (\sin \relax (x )\right )^{3} \pi }{48}+\frac {\pi \,\mathrm {csgn}\left (\sin \relax (x )\right ) \mathrm {csgn}\left (i \sin \relax (x )\right )}{48}-\frac {\mathrm {csgn}\left (i \sin \relax (x )\right )^{2} \mathrm {csgn}\left (\sin \relax (x )\right ) \pi }{48}+\frac {\mathrm {csgn}\left (i {\mathrm e}^{-i x}\right ) \mathrm {csgn}\left (\sin \relax (x )\right )^{2} \pi }{48}+\frac {\mathrm {csgn}\left (i \sin \relax (x )\right )^{2} \pi }{48}-\frac {\mathrm {csgn}\left (i \sin \relax (x )\right )^{3} \pi }{48}+\frac {\pi \,\mathrm {csgn}\left (i {\mathrm e}^{2 i x}-i\right ) \mathrm {csgn}\left (\sin \relax (x )\right ) \mathrm {csgn}\left (i {\mathrm e}^{-i x}\right )}{48}\right ) \sin \left (3 x \right )+\frac {3 \,{\mathrm e}^{i x} \pi \mathrm {csgn}\left (\sin \relax (x )\right )^{3}}{16}-\frac {3 \,{\mathrm e}^{-i x} \pi \mathrm {csgn}\left (\sin \relax (x )\right )^{3}}{16}+\frac {i \ln \left ({\mathrm e}^{i x}\right ) \left (18 i \sin \relax (x )+2 i \sin \left (3 x \right )\right )}{24}+\frac {3 \,{\mathrm e}^{i x} \pi \mathrm {csgn}\left (i \sin \relax (x )\right )^{2}}{16}+\frac {3 \,{\mathrm e}^{-i x} \pi \mathrm {csgn}\left (i \sin \relax (x )\right )^{3}}{16}-\frac {3 \,{\mathrm e}^{i x} \pi \mathrm {csgn}\left (i \sin \relax (x )\right )^{3}}{16}-\frac {3 \,{\mathrm e}^{-i x} \pi \mathrm {csgn}\left (i \sin \relax (x )\right )^{2}}{16}-\frac {3 i {\mathrm e}^{-i x} \ln \relax (2)}{8}-\frac {3 i {\mathrm e}^{i x} \ln \left ({\mathrm e}^{2 i x}-1\right )}{8}+\frac {3 i {\mathrm e}^{i x} \ln \relax (2)}{8}-\frac {11 i {\mathrm e}^{-i x}}{24}+\frac {3 \,{\mathrm e}^{-i x} \pi }{16}-\frac {3 \,{\mathrm e}^{i x} \pi }{16}+\frac {11 i {\mathrm e}^{i x}}{24}-\frac {3 \,{\mathrm e}^{-i x} \pi \,\mathrm {csgn}\left (i {\mathrm e}^{2 i x}-i\right ) \mathrm {csgn}\left (\sin \relax (x )\right ) \mathrm {csgn}\left (i {\mathrm e}^{-i x}\right )}{16}+\frac {3 \,{\mathrm e}^{i x} \pi \,\mathrm {csgn}\left (i {\mathrm e}^{2 i x}-i\right ) \mathrm {csgn}\left (\sin \relax (x )\right ) \mathrm {csgn}\left (i {\mathrm e}^{-i x}\right )}{16}+\frac {3 \,{\mathrm e}^{i x} \pi \,\mathrm {csgn}\left (\sin \relax (x )\right ) \mathrm {csgn}\left (i \sin \relax (x )\right )}{16}-\frac {3 \,{\mathrm e}^{-i x} \pi \,\mathrm {csgn}\left (\sin \relax (x )\right ) \mathrm {csgn}\left (i \sin \relax (x )\right )}{16}-\frac {3 \,{\mathrm e}^{-i x} \pi \mathrm {csgn}\left (\sin \relax (x )\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{-i x}\right )}{16}+\frac {3 \,{\mathrm e}^{-i x} \pi \,\mathrm {csgn}\left (\sin \relax (x )\right ) \mathrm {csgn}\left (i \sin \relax (x )\right )^{2}}{16}+\frac {3 \,{\mathrm e}^{i x} \pi \mathrm {csgn}\left (\sin \relax (x )\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{-i x}\right )}{16}-\frac {3 \,{\mathrm e}^{i x} \pi \,\mathrm {csgn}\left (\sin \relax (x )\right ) \mathrm {csgn}\left (i \sin \relax (x )\right )^{2}}{16}+\frac {3 \,{\mathrm e}^{i x} \pi \,\mathrm {csgn}\left (i {\mathrm e}^{2 i x}-i\right ) \mathrm {csgn}\left (\sin \relax (x )\right )^{2}}{16}-\frac {3 \,{\mathrm e}^{-i x} \pi \,\mathrm {csgn}\left (i {\mathrm e}^{2 i x}-i\right ) \mathrm {csgn}\left (\sin \relax (x )\right )^{2}}{16}\)	\(577\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*I*(1/3*exp(3*I*x)*ln(I*(-exp(I*x)^2+1)/exp(I*x))-1/9*exp(I*x)^3-11/3*exp(I*x)+3*exp(I*x)*ln(I*(-exp(I*x)^
2+1)/exp(I*x))-3*exp(-I*x)*ln(I*(-exp(I*x)^2+1)/exp(I*x))+11/3/exp(I*x)-1/3*exp(-3*I*x)*ln(I*(-exp(I*x)^2+1)/e
xp(I*x))+1/9/exp(I*x)^3-1/3*ln(2)*exp(I*x)^3-3*ln(2)*exp(I*x)+3*ln(2)/exp(I*x)+1/3*ln(2)/exp(I*x)^3)

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maxima [A]  time = 0.42, size = 25, normalized size = 0.83 \[ \frac {1}{9} \, \sin \relax (x)^{3} - \frac {1}{3} \, {\left (\sin \relax (x)^{3} - 3 \, \sin \relax (x)\right )} \log \left (\sin \relax (x)\right ) - \sin \relax (x) \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \ln \left (\sin \relax (x)\right )\,{\cos \relax (x)}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(log(sin(x))*cos(x)^3, x)

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sympy [A]  time = 5.24, size = 42, normalized size = 1.40 \[ \frac {2 \log {\left (\sin {\relax (x )} \right )} \sin ^{3}{\relax (x )}}{3} + \log {\left (\sin {\relax (x )} \right )} \sin {\relax (x )} \cos ^{2}{\relax (x )} - \frac {8 \sin ^{3}{\relax (x )}}{9} - \sin {\relax (x )} \cos ^{2}{\relax (x )} \]

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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