3.24 \(\int \frac{\sin (x)}{4-3 \cos ^3(x)} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0903281, antiderivative size = 98, normalized size of antiderivative = 1., 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 = {3223, 200, 31, 634, 617, 204, 628} \[ -\frac{\log \left (3^{2/3} \cos ^2(x)+2^{2/3} \sqrt [3]{3} \cos (x)+2 \sqrt [3]{2}\right )}{12 \sqrt [3]{6}}+\frac{\log \left (2^{2/3}-\sqrt [3]{3} \cos (x)\right )}{6 \sqrt [3]{6}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{6} \cos (x)+1}{\sqrt{3}}\right )}{2 \sqrt [3]{2} 3^{5/6}} \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]/(4 - 3*Cos[x]^3),x]

[Out]

-ArcTan[(1 + 6^(1/3)*Cos[x])/Sqrt[3]]/(2*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 3223

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])

Rule 200

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 31

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

Rule 634

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 617

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 204

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

Rule 628

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]

Rubi steps

\begin{align*} \int \frac{\sin (x)}{4-3 \cos ^3(x)} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{4-3 x^3} \, dx,x,\cos (x)\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{2^{2/3}-\sqrt [3]{3} x} \, dx,x,\cos (x)\right )}{6 \sqrt [3]{2}}-\frac{\operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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*}

Mathematica [A]  time = 0.0983305, size = 79, normalized size = 0.81 \[ \frac{1}{72} \left (6^{2/3} \left (2 \log \left (2-\sqrt [3]{6} \cos (x)\right )-\log \left (6^{2/3} \cos ^2(x)+2 \sqrt [3]{6} \cos (x)+4\right )\right )-6\ 2^{2/3} \sqrt [6]{3} \tan ^{-1}\left (\frac{\sqrt [3]{6} \cos (x)+1}{\sqrt{3}}\right )\right ) \]

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 = 0.017, size = 80, normalized size = 0.8 \begin{align*}{\frac{\sqrt [3]{4}{3}^{{\frac{2}{3}}}}{36}\ln \left ( \cos \left ( x \right ) -{\frac{\sqrt [3]{4}{3}^{{\frac{2}{3}}}}{3}} \right ) }-{\frac{\sqrt [3]{4}{3}^{{\frac{2}{3}}}}{72}\ln \left ( \left ( \cos \left ( x \right ) \right ) ^{2}+{\frac{\sqrt [3]{4}{3}^{{\frac{2}{3}}}\cos \left ( x \right ) }{3}}+{\frac{{4}^{{\frac{2}{3}}}\sqrt [3]{3}}{3}} \right ) }-{\frac{\sqrt [3]{4}\sqrt [6]{3}}{12}\arctan \left ({\frac{\sqrt{3}}{3} \left ({\frac{{4}^{{\frac{2}{3}}}\sqrt [3]{3}\cos \left ( x \right ) }{2}}+1 \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 1.43933, size = 120, normalized size = 1.22 \begin{align*} -\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{align*}

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 = 1.75272, size = 251, normalized size = 2.56 \begin{align*} -\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{align*}

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 = 3.14148, size = 85, normalized size = 0.87 \begin{align*} \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 \sqrt [3]{6} \right )}}{72} - \frac{2^{\frac{2}{3}} \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{align*}

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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError