3.657 \(\int \frac{\cos ^2(x) \sin (x)}{\sqrt{1-\cos ^6(x)}} \, dx\)

Optimal. Leaf size=9 \[ -\frac{1}{3} \sin ^{-1}\left (\cos ^3(x)\right ) \]

[Out]

-ArcSin[Cos[x]^3]/3

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Rubi [A]  time = 0.0718241, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {4335, 275, 216} \[ -\frac{1}{3} \sin ^{-1}\left (\cos ^3(x)\right ) \]

Antiderivative was successfully verified.

[In]

Int[(Cos[x]^2*Sin[x])/Sqrt[1 - Cos[x]^6],x]

[Out]

-ArcSin[Cos[x]^3]/3

Rule 4335

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, -Dist[d/(
b*c), Subst[Int[SubstFor[1, 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, True]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Sin] || EqQ[F, sin])

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 216

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

Rubi steps

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

Mathematica [C]  time = 2.25209, size = 162, normalized size = 18. \[ -\frac{i \sin (x) \cos ^2(x) \sqrt{1-\frac{2 i \tan ^2(x)}{\sqrt{3}-3 i}} \sqrt{1+\frac{2 i \tan ^2(x)}{\sqrt{3}+3 i}} \Pi \left (\frac{3}{2}+\frac{i \sqrt{3}}{2};i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{-3 i+\sqrt{3}}} \tan (x)\right )|\frac{3 i-\sqrt{3}}{3 i+\sqrt{3}}\right )}{\sqrt{2} \sqrt{-\frac{i}{\sqrt{3}-3 i}} \sqrt{1-\cos ^6(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[x]^2*Sin[x])/Sqrt[1 - Cos[x]^6],x]

[Out]

((-I)*Cos[x]^2*EllipticPi[3/2 + (I/2)*Sqrt[3], I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[3])]*Tan[x]], (3*I - Sqrt[3]
)/(3*I + Sqrt[3])]*Sin[x]*Sqrt[1 - ((2*I)*Tan[x]^2)/(-3*I + Sqrt[3])]*Sqrt[1 + ((2*I)*Tan[x]^2)/(3*I + Sqrt[3]
)])/(Sqrt[2]*Sqrt[(-I)/(-3*I + Sqrt[3])]*Sqrt[1 - Cos[x]^6])

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Maple [F]  time = 0.518, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \cos \left ( x \right ) \right ) ^{2}\sin \left ( x \right ){\frac{1}{\sqrt{1- \left ( \cos \left ( x \right ) \right ) ^{6}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^2*sin(x)/(1-cos(x)^6)^(1/2),x)

[Out]

int(cos(x)^2*sin(x)/(1-cos(x)^6)^(1/2),x)

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Maxima [B]  time = 1.44395, size = 24, normalized size = 2.67 \begin{align*} \frac{1}{3} \, \arctan \left (\frac{\sqrt{-\cos \left (x\right )^{6} + 1}}{\cos \left (x\right )^{3}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^2*sin(x)/(1-cos(x)^6)^(1/2),x, algorithm="maxima")

[Out]

1/3*arctan(sqrt(-cos(x)^6 + 1)/cos(x)^3)

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Fricas [B]  time = 2.85505, size = 82, normalized size = 9.11 \begin{align*} \frac{1}{6} \, \arctan \left (\frac{2 \, \sqrt{-\cos \left (x\right )^{6} + 1} \cos \left (x\right )^{3}}{2 \, \cos \left (x\right )^{6} - 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^2*sin(x)/(1-cos(x)^6)^(1/2),x, algorithm="fricas")

[Out]

1/6*arctan(2*sqrt(-cos(x)^6 + 1)*cos(x)^3/(2*cos(x)^6 - 1))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)**2*sin(x)/(1-cos(x)**6)**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 1.09013, size = 9, normalized size = 1. \begin{align*} -\frac{1}{3} \, \arcsin \left (\cos \left (x\right )^{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^2*sin(x)/(1-cos(x)^6)^(1/2),x, algorithm="giac")

[Out]

-1/3*arcsin(cos(x)^3)