3.1.24 \(\int \frac {\sin (x)}{4-3 \cos ^3(x)} \, dx\) [24]

Optimal. Leaf size=98 \[ -\frac {\text {ArcTan}\left (\frac {1+\sqrt [3]{6} \cos (x)}{\sqrt {3}}\right )}{2 \sqrt [3]{2} 3^{5/6}}+\frac {\log \left (2^{2/3}-\sqrt [3]{3} \cos (x)\right )}{6 \sqrt [3]{6}}-\frac {\log \left (2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{3} \cos (x)+3^{2/3} \cos ^2(x)\right )}{12 \sqrt [3]{6}} \]

[Out]

-1/12*arctan(1/3*(1+6^(1/3)*cos(x))*3^(1/2))*2^(2/3)*3^(1/6)+1/36*ln(2^(2/3)-3^(1/3)*cos(x))*6^(2/3)-1/72*ln(2
*2^(1/3)+2^(2/3)*3^(1/3)*cos(x)+3^(2/3)*cos(x)^2)*6^(2/3)

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Rubi [A]
time = 0.06, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {3302, 206, 31, 648, 631, 210, 642} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt [3]{6} \cos (x)+1}{\sqrt {3}}\right )}{2 \sqrt [3]{2} 3^{5/6}}-\frac {\log \left (3^{2/3} \cos ^2(x)+2^{2/3} \sqrt [3]{3} \cos (x)+2 \sqrt [3]{2}\right )}{12 \sqrt [3]{6}}+\frac {\log \left (2^{2/3}-\sqrt [3]{3} \cos (x)\right )}{6 \sqrt [3]{6}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[x]/(4 - 3*Cos[x]^3),x]

[Out]

-1/2*ArcTan[(1 + 6^(1/3)*Cos[x])/Sqrt[3]]/(2^(1/3)*3^(5/6)) + Log[2^(2/3) - 3^(1/3)*Cos[x]]/(6*6^(1/3)) - Log[
2*2^(1/3) + 2^(2/3)*3^(1/3)*Cos[x] + 3^(2/3)*Cos[x]^2]/(12*6^(1/3))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 206

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

Rule 210

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

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 3302

Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With
[{ff = FreeFactors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*(c*ff*x)^n)^p, x]
, x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[(m - 1)/2] && (EqQ[n, 4] || GtQ[m, 0
] || IGtQ[p, 0] || IntegersQ[m, p])

Rubi steps

\begin {align*} \int \frac {\sin (x)}{4-3 \cos ^3(x)} \, dx &=-\text {Subst}\left (\int \frac {1}{4-3 x^3} \, dx,x,\cos (x)\right )\\ &=-\frac {\text {Subst}\left (\int \frac {1}{2^{2/3}-\sqrt [3]{3} x} \, dx,x,\cos (x)\right )}{6 \sqrt [3]{2}}-\frac {\text {Subst}\left (\int \frac {2\ 2^{2/3}+\sqrt [3]{3} x}{2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{3} x+3^{2/3} x^2} \, dx,x,\cos (x)\right )}{6 \sqrt [3]{2}}\\ &=\frac {\log \left (2^{2/3}-\sqrt [3]{3} \cos (x)\right )}{6 \sqrt [3]{6}}-\frac {\text {Subst}\left (\int \frac {1}{2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{3} x+3^{2/3} x^2} \, dx,x,\cos (x)\right )}{2\ 2^{2/3}}-\frac {\text {Subst}\left (\int \frac {2^{2/3} \sqrt [3]{3}+2\ 3^{2/3} x}{2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{3} x+3^{2/3} x^2} \, dx,x,\cos (x)\right )}{12 \sqrt [3]{6}}\\ &=\frac {\log \left (2^{2/3}-\sqrt [3]{3} \cos (x)\right )}{6 \sqrt [3]{6}}-\frac {\log \left (2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{3} \cos (x)+3^{2/3} \cos ^2(x)\right )}{12 \sqrt [3]{6}}+\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\sqrt [3]{6} \cos (x)\right )}{2 \sqrt [3]{6}}\\ &=-\frac {\tan ^{-1}\left (\frac {1+\sqrt [3]{6} \cos (x)}{\sqrt {3}}\right )}{2 \sqrt [3]{2} 3^{5/6}}+\frac {\log \left (2^{2/3}-\sqrt [3]{3} \cos (x)\right )}{6 \sqrt [3]{6}}-\frac {\log \left (2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{3} \cos (x)+3^{2/3} \cos ^2(x)\right )}{12 \sqrt [3]{6}}\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 79, normalized size = 0.81 \begin {gather*} \frac {1}{72} \left (-6 2^{2/3} \sqrt [6]{3} \text {ArcTan}\left (\frac {1+\sqrt [3]{6} \cos (x)}{\sqrt {3}}\right )+6^{2/3} \left (2 \log \left (2-\sqrt [3]{6} \cos (x)\right )-\log \left (4+2 \sqrt [3]{6} \cos (x)+6^{2/3} \cos ^2(x)\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]/(4 - 3*Cos[x]^3),x]

[Out]

(-6*2^(2/3)*3^(1/6)*ArcTan[(1 + 6^(1/3)*Cos[x])/Sqrt[3]] + 6^(2/3)*(2*Log[2 - 6^(1/3)*Cos[x]] - Log[4 + 2*6^(1
/3)*Cos[x] + 6^(2/3)*Cos[x]^2]))/72

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Maple [A]
time = 0.13, size = 80, normalized size = 0.82

method	result	size
risch	\(-\frac {i \left (\munderset {\textit {\_R} =\RootOf \left (162 \textit {\_Z}^{3}+i\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i x}+12 i \textit {\_R} \,{\mathrm e}^{i x}+1\right )\right )}{2}\)	\(35\)
derivativedivides	\(\frac {4^{\frac {1}{3}} 3^{\frac {2}{3}} \ln \left (\cos \left (x \right )-\frac {4^{\frac {1}{3}} 3^{\frac {2}{3}}}{3}\right )}{36}-\frac {4^{\frac {1}{3}} 3^{\frac {2}{3}} \ln \left (\cos ^{2}\left (x \right )+\frac {4^{\frac {1}{3}} 3^{\frac {2}{3}} \cos \left (x \right )}{3}+\frac {4^{\frac {2}{3}} 3^{\frac {1}{3}}}{3}\right )}{72}-\frac {4^{\frac {1}{3}} 3^{\frac {1}{6}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {4^{\frac {2}{3}} 3^{\frac {1}{3}} \cos \left (x \right )}{2}+1\right )}{3}\right )}{12}\)	\(80\)
default	\(\frac {4^{\frac {1}{3}} 3^{\frac {2}{3}} \ln \left (\cos \left (x \right )-\frac {4^{\frac {1}{3}} 3^{\frac {2}{3}}}{3}\right )}{36}-\frac {4^{\frac {1}{3}} 3^{\frac {2}{3}} \ln \left (\cos ^{2}\left (x \right )+\frac {4^{\frac {1}{3}} 3^{\frac {2}{3}} \cos \left (x \right )}{3}+\frac {4^{\frac {2}{3}} 3^{\frac {1}{3}}}{3}\right )}{72}-\frac {4^{\frac {1}{3}} 3^{\frac {1}{6}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {4^{\frac {2}{3}} 3^{\frac {1}{3}} \cos \left (x \right )}{2}+1\right )}{3}\right )}{12}\)	\(80\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)/(4-3*cos(x)^3),x,method=_RETURNVERBOSE)

[Out]

1/36*4^(1/3)*3^(2/3)*ln(cos(x)-1/3*4^(1/3)*3^(2/3))-1/72*4^(1/3)*3^(2/3)*ln(cos(x)^2+1/3*4^(1/3)*3^(2/3)*cos(x
)+1/3*4^(2/3)*3^(1/3))-1/12*4^(1/3)*3^(1/6)*arctan(1/3*3^(1/2)*(1/2*4^(2/3)*3^(1/3)*cos(x)+1))

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Maxima [A]
time = 0.47, size = 89, normalized size = 0.91 \begin {gather*} -\frac {1}{72} \cdot 4^{\frac {1}{3}} 3^{\frac {2}{3}} \log \left (3^{\frac {2}{3}} \cos \left (x\right )^{2} + 4^{\frac {1}{3}} 3^{\frac {1}{3}} \cos \left (x\right ) + 4^{\frac {2}{3}}\right ) + \frac {1}{36} \cdot 4^{\frac {1}{3}} 3^{\frac {2}{3}} \log \left (\frac {1}{3} \cdot 3^{\frac {2}{3}} {\left (3^{\frac {1}{3}} \cos \left (x\right ) - 4^{\frac {1}{3}}\right )}\right ) - \frac {1}{12} \cdot 4^{\frac {1}{3}} 3^{\frac {1}{6}} \arctan \left (\frac {1}{12} \cdot 4^{\frac {2}{3}} 3^{\frac {1}{6}} {\left (2 \cdot 3^{\frac {2}{3}} \cos \left (x\right ) + 4^{\frac {1}{3}} 3^{\frac {1}{3}}\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(4-3*cos(x)^3),x, algorithm="maxima")

[Out]

-1/72*4^(1/3)*3^(2/3)*log(3^(2/3)*cos(x)^2 + 4^(1/3)*3^(1/3)*cos(x) + 4^(2/3)) + 1/36*4^(1/3)*3^(2/3)*log(1/3*
3^(2/3)*(3^(1/3)*cos(x) - 4^(1/3))) - 1/12*4^(1/3)*3^(1/6)*arctan(1/12*4^(2/3)*3^(1/6)*(2*3^(2/3)*cos(x) + 4^(
1/3)*3^(1/3)))

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Fricas [A]
time = 0.41, size = 71, normalized size = 0.72 \begin {gather*} -\frac {1}{12} \cdot 6^{\frac {1}{6}} \sqrt {2} \arctan \left (\frac {1}{6} \cdot 6^{\frac {1}{6}} {\left (6^{\frac {2}{3}} \sqrt {2} \cos \left (x\right ) + 6^{\frac {1}{3}} \sqrt {2}\right )}\right ) - \frac {1}{72} \cdot 6^{\frac {2}{3}} \log \left (-3 \, \cos \left (x\right )^{2} - 6^{\frac {2}{3}} \cos \left (x\right ) - 2 \cdot 6^{\frac {1}{3}}\right ) + \frac {1}{36} \cdot 6^{\frac {2}{3}} \log \left (6^{\frac {2}{3}} - 3 \, \cos \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(4-3*cos(x)^3),x, algorithm="fricas")

[Out]

-1/12*6^(1/6)*sqrt(2)*arctan(1/6*6^(1/6)*(6^(2/3)*sqrt(2)*cos(x) + 6^(1/3)*sqrt(2))) - 1/72*6^(2/3)*log(-3*cos
(x)^2 - 6^(2/3)*cos(x) - 2*6^(1/3)) + 1/36*6^(2/3)*log(6^(2/3) - 3*cos(x))

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Sympy [A]
time = 0.54, size = 85, normalized size = 0.87 \begin {gather*} \frac {6^{\frac {2}{3}} \log {\left (\cos {\left (x \right )} - \frac {6^{\frac {2}{3}}}{3} \right )}}{36} - \frac {6^{\frac {2}{3}} \log {\left (36 \cos ^{2}{\left (x \right )} + 12 \cdot 6^{\frac {2}{3}} \cos {\left (x \right )} + 24 \cdot \sqrt [3]{6} \right )}}{72} - \frac {2^{\frac {2}{3}} \cdot \sqrt [6]{3} \operatorname {atan}{\left (\frac {\sqrt [3]{2} \cdot 3^{\frac {5}{6}} \cos {\left (x \right )}}{3} + \frac {\sqrt {3}}{3} \right )}}{12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(4-3*cos(x)**3),x)

[Out]

6**(2/3)*log(cos(x) - 6**(2/3)/3)/36 - 6**(2/3)*log(36*cos(x)**2 + 12*6**(2/3)*cos(x) + 24*6**(1/3))/72 - 2**(
2/3)*3**(1/6)*atan(2**(1/3)*3**(5/6)*cos(x)/3 + sqrt(3)/3)/12

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Giac [A]
time = 0.39, size = 60, normalized size = 0.61 \begin {gather*} -\frac {1}{12} \, \sqrt {3} \left (\frac {4}{3}\right )^{\frac {1}{3}} \arctan \left (\frac {1}{4} \, \sqrt {3} \left (\frac {4}{3}\right )^{\frac {2}{3}} {\left (\left (\frac {4}{3}\right )^{\frac {1}{3}} + 2 \, \cos \left (x\right )\right )}\right ) - \frac {1}{72} \cdot 36^{\frac {1}{3}} \log \left (\cos \left (x\right )^{2} + \left (\frac {4}{3}\right )^{\frac {1}{3}} \cos \left (x\right ) + \left (\frac {4}{3}\right )^{\frac {2}{3}}\right ) + \frac {1}{12} \, \left (\frac {4}{3}\right )^{\frac {1}{3}} \log \left (\left (\frac {4}{3}\right )^{\frac {1}{3}} - \cos \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(4-3*cos(x)^3),x, algorithm="giac")

[Out]

-1/12*sqrt(3)*(4/3)^(1/3)*arctan(1/4*sqrt(3)*(4/3)^(2/3)*((4/3)^(1/3) + 2*cos(x))) - 1/72*36^(1/3)*log(cos(x)^
2 + (4/3)^(1/3)*cos(x) + (4/3)^(2/3)) + 1/12*(4/3)^(1/3)*log((4/3)^(1/3) - cos(x))

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Mupad [B]
time = 0.31, size = 75, normalized size = 0.77 \begin {gather*} \frac {6^{2/3}\,\ln \left (\cos \left (x\right )-\frac {6^{2/3}}{3}\right )}{36}+\frac {6^{2/3}\,\ln \left (\cos \left (x\right )-\frac {6^{2/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{72}-\frac {6^{2/3}\,\ln \left (\cos \left (x\right )+\frac {6^{2/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{72} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-sin(x)/(3*cos(x)^3 - 4),x)

[Out]

(6^(2/3)*log(cos(x) - 6^(2/3)/3))/36 + (6^(2/3)*log(cos(x) - (6^(2/3)*(3^(1/2)*1i - 1))/6)*(3^(1/2)*1i - 1))/7
2 - (6^(2/3)*log(cos(x) + (6^(2/3)*(3^(1/2)*1i + 1))/6)*(3^(1/2)*1i + 1))/72

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