3.192 \(\int \log (\sin (x)) \sin ^3(x) \, dx\)

Optimal. Leaf size=40 \[ -\frac{\cos ^3(x)}{9}+\frac{2 \cos (x)}{3}-\frac{2}{3} \tanh ^{-1}(\cos (x))+\frac{1}{3} \cos ^3(x) \log (\sin (x))-\cos (x) \log (\sin (x)) \]

[Out]

(-2*ArcTanh[Cos[x]])/3 + (2*Cos[x])/3 - Cos[x]^3/9 - Cos[x]*Log[Sin[x]] + (Cos[x]^3*Log[Sin[x]])/3

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Rubi [A]  time = 0.0671791, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.875, Rules used = {2633, 2554, 12, 4366, 459, 321, 206} \[ -\frac{\cos ^3(x)}{9}+\frac{2 \cos (x)}{3}-\frac{2}{3} \tanh ^{-1}(\cos (x))+\frac{1}{3} \cos ^3(x) \log (\sin (x))-\cos (x) \log (\sin (x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*ArcTanh[Cos[x]])/3 + (2*Cos[x])/3 - Cos[x]^3/9 - Cos[x]*Log[Sin[x]] + (Cos[x]^3*Log[Sin[x]])/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 4366

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))]^(n_), x_Symbol] :> With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, -Dis
t[d/(b*c), Subst[Int[SubstFor[(1 - d^2*x^2)^((n - 1)/2), Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d]
, x] /; FunctionOfQ[Cos[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && NonsumQ[u] &&
(EqQ[F, Sin] || EqQ[F, sin])

Rule 459

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(b*e*(m + n*(p + 1) + 1)), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m +
n*(p + 1) + 1)), Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0]
&& NeQ[m + n*(p + 1) + 1, 0]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0402819, size = 47, normalized size = 1.18 \[ \frac{1}{36} \left (24 \left (\log \left (\sin \left (\frac{x}{2}\right )\right )-\log \left (\cos \left (\frac{x}{2}\right )\right )\right )+\cos (3 x) (3 \log (\sin (x))-1)-3 \cos (x) (9 \log (\sin (x))-7)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(24*(-Log[Cos[x/2]] + Log[Sin[x/2]]) + Cos[3*x]*(-1 + 3*Log[Sin[x]]) - 3*Cos[x]*(-7 + 9*Log[Sin[x]]))/36

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Maple [C]  time = 0.032, size = 134, normalized size = 3.4 \begin{align*}{\frac{{{\rm e}^{3\,ix}}\ln \left ( 2\,\sin \left ( x \right ) \right ) }{24}}-{\frac{{{\rm e}^{3\,ix}}}{72}}+{\frac{7\,{{\rm e}^{ix}}}{24}}+{\frac{2\,\ln \left ({{\rm e}^{ix}}-1 \right ) }{3}}-{\frac{2\,\ln \left ({{\rm e}^{ix}}+1 \right ) }{3}}-{\frac{3\,{{\rm e}^{ix}}\ln \left ( 2\,\sin \left ( x \right ) \right ) }{8}}-{\frac{3\,{{\rm e}^{-ix}}\ln \left ( 2\,\sin \left ( x \right ) \right ) }{8}}+{\frac{7\,{{\rm e}^{-ix}}}{24}}+{\frac{{{\rm e}^{-3\,ix}}\ln \left ( 2\,\sin \left ( x \right ) \right ) }{24}}-{\frac{{{\rm e}^{-3\,ix}}}{72}}-{\frac{\ln \left ( 2 \right ){{\rm e}^{3\,ix}}}{24}}+{\frac{3\,\ln \left ( 2 \right ){{\rm e}^{ix}}}{8}}+{\frac{3\,\ln \left ( 2 \right ){{\rm e}^{-ix}}}{8}}-{\frac{\ln \left ( 2 \right ){{\rm e}^{-3\,ix}}}{24}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*exp(3*I*x)*ln(2*sin(x))-1/72*exp(3*I*x)+7/24*exp(I*x)+2/3*ln(exp(I*x)-1)-2/3*ln(exp(I*x)+1)-3/8*exp(I*x)*
ln(2*sin(x))-3/8*exp(-I*x)*ln(2*sin(x))+7/24*exp(-I*x)+1/24*exp(-3*I*x)*ln(2*sin(x))-1/72*exp(-3*I*x)-1/24*ln(
2)*exp(3*I*x)+3/8*ln(2)*exp(I*x)+3/8*ln(2)*exp(-I*x)-1/24*ln(2)*exp(-3*I*x)

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Maxima [B]  time = 1.00537, size = 242, normalized size = 6.05 \begin{align*} -\frac{4 \,{\left (\frac{3 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right )} \log \left (\frac{2 \, \sin \left (x\right )}{{\left (\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right )}{\left (\cos \left (x\right ) + 1\right )}}\right )}{3 \,{\left (\frac{3 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{3 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac{\sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + 1\right )}} + \frac{2 \,{\left (\frac{12 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{3 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + 5\right )}}{9 \,{\left (\frac{3 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{3 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac{\sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + 1\right )}} - \frac{2}{3} \, \log \left (\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right ) + \frac{2}{3} \, \log \left (\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/3*(3*sin(x)^2/(cos(x) + 1)^2 + 1)*log(2*sin(x)/((sin(x)^2/(cos(x) + 1)^2 + 1)*(cos(x) + 1)))/(3*sin(x)^2/(c
os(x) + 1)^2 + 3*sin(x)^4/(cos(x) + 1)^4 + sin(x)^6/(cos(x) + 1)^6 + 1) + 2/9*(12*sin(x)^2/(cos(x) + 1)^2 + 3*
sin(x)^4/(cos(x) + 1)^4 + 5)/(3*sin(x)^2/(cos(x) + 1)^2 + 3*sin(x)^4/(cos(x) + 1)^4 + sin(x)^6/(cos(x) + 1)^6
+ 1) - 2/3*log(sin(x)^2/(cos(x) + 1)^2 + 1) + 2/3*log(sin(x)^2/(cos(x) + 1)^2)

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Fricas [A]  time = 2.24953, size = 169, normalized size = 4.22 \begin{align*} -\frac{1}{9} \, \cos \left (x\right )^{3} + \frac{1}{3} \,{\left (\cos \left (x\right )^{3} - 3 \, \cos \left (x\right )\right )} \log \left (\sin \left (x\right )\right ) + \frac{2}{3} \, \cos \left (x\right ) - \frac{1}{3} \, \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + \frac{1}{3} \, \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 15.1769, size = 445, normalized size = 11.12 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6*log(tan(x/2)**2 + 1)*tan(x/2)**6/(9*tan(x/2)**6 + 27*tan(x/2)**4 + 27*tan(x/2)**2 + 9) - 18*log(tan(x/2)**2
+ 1)*tan(x/2)**4/(9*tan(x/2)**6 + 27*tan(x/2)**4 + 27*tan(x/2)**2 + 9) + 18*log(tan(x/2)**2 + 1)*tan(x/2)**2/
(9*tan(x/2)**6 + 27*tan(x/2)**4 + 27*tan(x/2)**2 + 9) + 6*log(tan(x/2)**2 + 1)/(9*tan(x/2)**6 + 27*tan(x/2)**4
+ 27*tan(x/2)**2 + 9) + 12*log(tan(x/2))*tan(x/2)**6/(9*tan(x/2)**6 + 27*tan(x/2)**4 + 27*tan(x/2)**2 + 9) +
36*log(tan(x/2))*tan(x/2)**4/(9*tan(x/2)**6 + 27*tan(x/2)**4 + 27*tan(x/2)**2 + 9) - 10*tan(x/2)**6/(9*tan(x/2
)**6 + 27*tan(x/2)**4 + 27*tan(x/2)**2 + 9) + 12*log(2)*tan(x/2)**6/(9*tan(x/2)**6 + 27*tan(x/2)**4 + 27*tan(x
/2)**2 + 9) - 24*tan(x/2)**4/(9*tan(x/2)**6 + 27*tan(x/2)**4 + 27*tan(x/2)**2 + 9) + 36*log(2)*tan(x/2)**4/(9*
tan(x/2)**6 + 27*tan(x/2)**4 + 27*tan(x/2)**2 + 9) - 6*tan(x/2)**2/(9*tan(x/2)**6 + 27*tan(x/2)**4 + 27*tan(x/
2)**2 + 9)

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Giac [A]  time = 1.29915, size = 55, normalized size = 1.38 \begin{align*} -\frac{1}{9} \, \cos \left (x\right )^{3} + \frac{1}{3} \,{\left (\cos \left (x\right )^{3} - 3 \, \cos \left (x\right )\right )} \log \left (\sin \left (x\right )\right ) + \frac{2}{3} \, \cos \left (x\right ) - \frac{1}{3} \, \log \left (\cos \left (x\right ) + 1\right ) + \frac{1}{3} \, \log \left (-\cos \left (x\right ) + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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