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]

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

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Rubi [A]  time = 0.07, antiderivative size = 9, normalized size of antiderivative = 1.00, 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 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]

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 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])

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*}

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Mathematica [C]  time = 2.22, size = 162, normalized size = 18.00 \[ -\frac {i \sin (x) \cos ^2(x) \sqrt {1-\frac {2 i \tan ^2(x)}{\sqrt {3}-3 i}} \sqrt {1+\frac {2 i \tan ^2(x)}{\sqrt {3}+3 i}} \Pi \left (\frac {3}{2}+\frac {i \sqrt {3}}{2};i \sinh ^{-1}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {3}}} \tan (x)\right )|\frac {3 i-\sqrt {3}}{3 i+\sqrt {3}}\right )}{\sqrt {2} \sqrt {-\frac {i}{\sqrt {3}-3 i}} \sqrt {1-\cos ^6(x)}} \]

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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fricas [B]  time = 1.11, size = 29, normalized size = 3.22 \[ \frac {1}{6} \, \arctan \left (\frac {2 \, \sqrt {-\cos \relax (x)^{6} + 1} \cos \relax (x)^{3}}{2 \, \cos \relax (x)^{6} - 1}\right ) \]

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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giac [A]  time = 0.15, size = 7, normalized size = 0.78 \[ -\frac {1}{3} \, \arcsin \left (\cos \relax (x)^{3}\right ) \]

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)

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maple [F]  time = 1.26, size = 0, normalized size = 0.00 \[ \int \frac {\left (\cos ^{2}\relax (x )\right ) \sin \relax (x )}{\sqrt {1-\left (\cos ^{6}\relax (x )\right )}}\, dx \]

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 = 0.60, size = 18, normalized size = 2.00 \[ \frac {1}{3} \, \arctan \left (\frac {\sqrt {-\cos \relax (x)^{6} + 1}}{\cos \relax (x)^{3}}\right ) \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.11 \[ \int \frac {{\cos \relax (x)}^2\,\sin \relax (x)}{\sqrt {1-{\cos \relax (x)}^6}} \,d x \]

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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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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