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

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

[Out]

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

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Rubi [A]
time = 0.02, 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 = {2644, 30, 2634, 12} \begin {gather*} \frac {1}{3} \sin ^3(x) \log (\sin (x))-\frac {\sin ^3(x)}{9} \end {gather*}

Antiderivative was successfully verified.

[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*} \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} \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*}

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Mathematica [A]
time = 0.01, size = 15, normalized size = 0.75 \begin {gather*} \frac {1}{9} (-1+3 \log (\sin (x))) \sin ^3(x) \end {gather*}

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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Mathics [A]
time = 2.51, size = 13, normalized size = 0.65 \begin {gather*} \frac {\left (-1+3 \text {Log}\left [\text {Sin}\left [x\right ]\right ]\right ) \text {Sin}\left [x\right ]^3}{9} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[Cos[x]*Log[Sin[x]]*Sin[x]^2,x]')

[Out]

(-1 + 3 Log[Sin[x]]) Sin[x] ^ 3 / 9

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Maple [A]
time = 0.14, size = 17, normalized size = 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	\(-\frac {\left (\sin ^{3}\left (x \right )\right )}{9}+\frac {\ln \left (\sin \left (x \right )\right ) \left (\sin ^{3}\left (x \right )\right )}{3}\)	\(17\)
risch	\(-\frac {{\mathrm e}^{i x} \pi \,\mathrm {csgn}\left (\sin \left (x \right )\right ) \mathrm {csgn}\left (i \sin \left (x \right )\right )^{2}}{16}+\frac {{\mathrm e}^{i x} \pi \,\mathrm {csgn}\left (i {\mathrm e}^{2 i x}-i\right ) \mathrm {csgn}\left (\sin \left (x \right )\right )^{2}}{16}-\frac {{\mathrm e}^{-i x} \pi \,\mathrm {csgn}\left (i {\mathrm e}^{2 i x}-i\right ) \mathrm {csgn}\left (\sin \left (x \right )\right )^{2}}{16}+\frac {{\mathrm e}^{i x} \pi \,\mathrm {csgn}\left (\sin \left (x \right )\right ) \mathrm {csgn}\left (i \sin \left (x \right )\right )}{16}-\frac {{\mathrm e}^{-i x} \pi \,\mathrm {csgn}\left (\sin \left (x \right )\right ) \mathrm {csgn}\left (i \sin \left (x \right )\right )}{16}+\frac {{\mathrm e}^{i x} \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-i x}\right ) \mathrm {csgn}\left (\sin \left (x \right )\right )^{2}}{16}-\frac {{\mathrm e}^{-i x} \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-i x}\right ) \mathrm {csgn}\left (\sin \left (x \right )\right )^{2}}{16}+\frac {{\mathrm e}^{-i x} \pi \,\mathrm {csgn}\left (\sin \left (x \right )\right ) \mathrm {csgn}\left (i \sin \left (x \right )\right )^{2}}{16}-\frac {{\mathrm e}^{-i x} \pi \,\mathrm {csgn}\left (i {\mathrm e}^{2 i x}-i\right ) \mathrm {csgn}\left (i {\mathrm e}^{-i x}\right ) \mathrm {csgn}\left (\sin \left (x \right )\right )}{16}+\frac {{\mathrm e}^{i x} \pi \,\mathrm {csgn}\left (i {\mathrm e}^{2 i x}-i\right ) \mathrm {csgn}\left (i {\mathrm e}^{-i x}\right ) \mathrm {csgn}\left (\sin \left (x \right )\right )}{16}+2 i \left (-\frac {i}{72}+\frac {\pi }{48}-\frac {\pi \,\mathrm {csgn}\left (i {\mathrm e}^{2 i x}-i\right ) \mathrm {csgn}\left (i {\mathrm e}^{-i x}\right ) \mathrm {csgn}\left (\sin \left (x \right )\right )}{48}-\frac {\mathrm {csgn}\left (i \sin \left (x \right )\right ) \mathrm {csgn}\left (\sin \left (x \right )\right ) \pi }{48}-\frac {\mathrm {csgn}\left (\sin \left (x \right )\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{-i x}\right ) \pi }{48}+\frac {\mathrm {csgn}\left (i \sin \left (x \right )\right )^{2} \mathrm {csgn}\left (\sin \left (x \right )\right ) \pi }{48}-\frac {\mathrm {csgn}\left (\sin \left (x \right )\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{2 i x}-i\right ) \pi }{48}-\frac {\mathrm {csgn}\left (i \sin \left (x \right )\right )^{2} \pi }{48}+\frac {\pi \mathrm {csgn}\left (i \sin \left (x \right )\right )^{3}}{48}+\frac {i \ln \left ({\mathrm e}^{2 i x}-1\right )}{24}-\frac {i \ln \left (2\right )}{24}-\frac {\pi \mathrm {csgn}\left (\sin \left (x \right )\right )^{3}}{48}\right ) \sin \left (3 x \right )-\frac {i {\mathrm e}^{-i x} \ln \left (2\right )}{8}+\frac {i {\mathrm e}^{-i x} \ln \left ({\mathrm e}^{2 i x}-1\right )}{8}-\frac {i {\mathrm e}^{i x} \ln \left ({\mathrm e}^{2 i x}-1\right )}{8}+\frac {i {\mathrm e}^{i x} \ln \left (2\right )}{8}-\frac {{\mathrm e}^{-i x} \pi \mathrm {csgn}\left (i \sin \left (x \right )\right )^{2}}{16}+\frac {{\mathrm e}^{-i x} \pi \mathrm {csgn}\left (i \sin \left (x \right )\right )^{3}}{16}+\frac {{\mathrm e}^{i x} \pi \mathrm {csgn}\left (i \sin \left (x \right )\right )^{2}}{16}-\frac {{\mathrm e}^{i x} \pi \mathrm {csgn}\left (i \sin \left (x \right )\right )^{3}}{16}-\frac {i \ln \left ({\mathrm e}^{i x}\right ) \left (-6 i \sin \left (x \right )+2 i \sin \left (3 x \right )\right )}{24}-\frac {{\mathrm e}^{-i x} \pi \mathrm {csgn}\left (\sin \left (x \right )\right )^{3}}{16}+\frac {{\mathrm e}^{i x} \pi \mathrm {csgn}\left (\sin \left (x \right )\right )^{3}}{16}-\frac {i {\mathrm e}^{-i x}}{24}-\frac {{\mathrm e}^{i x} \pi }{16}+\frac {{\mathrm e}^{-i x} \pi }{16}+\frac {i {\mathrm e}^{i x}}{24}\)	\(577\)

Verification of antiderivative is not currently implemented for this CAS.

[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

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Maxima [A]
time = 0.26, size = 16, normalized size = 0.80 \begin {gather*} \frac {1}{3} \, \log \left (\sin \left (x\right )\right ) \sin \left (x\right )^{3} - \frac {1}{9} \, \sin \left (x\right )^{3} \end {gather*}

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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Fricas [A]
time = 0.34, size = 24, normalized size = 1.20 \begin {gather*} -\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 ) \end {gather*}

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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Sympy [A]
time = 0.86, size = 17, normalized size = 0.85 \begin {gather*} \frac {\log {\left (\sin {\left (x \right )} \right )} \sin ^{3}{\left (x \right )}}{3} - \frac {\sin ^{3}{\left (x \right )}}{9} \end {gather*}

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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Giac [A]
time = 0.00, size = 19, normalized size = 0.95 \begin {gather*} -\frac {\sin ^{3}x}{9}+\frac {1}{3} \sin ^{3}x\cdot \ln \left (\sin x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.25, size = 11, normalized size = 0.55 \begin {gather*} \frac {{\sin \left (x\right )}^3\,\left (\ln \left (\sin \left (x\right )\right )-\frac {1}{3}\right )}{3} \end {gather*}

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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