\(\int \frac {\cos ^2(x) \sin (x)}{\sqrt {1-\cos ^6(x)}} \, dx\) [657]

Optimal result

Integrand size = 19, antiderivative size = 9 \[ \int \frac {\cos ^2(x) \sin (x)}{\sqrt {1-\cos ^6(x)}} \, dx=-\frac {1}{3} \arcsin \left (\cos ^3(x)\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {4420, 281, 222} \[ \int \frac {\cos ^2(x) \sin (x)}{\sqrt {1-\cos ^6(x)}} \, dx=-\frac {1}{3} \arcsin \left (\cos ^3(x)\right ) \]

[In]

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

[Out]

-1/3*ArcSin[Cos[x]^3]

Rule 222

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 281

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 4420

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 2.64 (sec) , antiderivative size = 162, normalized size of antiderivative = 18.00 \[ \int \frac {\cos ^2(x) \sin (x)}{\sqrt {1-\cos ^6(x)}} \, dx=-\frac {i \cos ^2(x) \operatorname {EllipticPi}\left (\frac {3}{2}+\frac {i \sqrt {3}}{2},i \text {arcsinh}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {3}}} \tan (x)\right ),\frac {3 i-\sqrt {3}}{3 i+\sqrt {3}}\right ) \sin (x) \sqrt {1-\frac {2 i \tan ^2(x)}{-3 i+\sqrt {3}}} \sqrt {1+\frac {2 i \tan ^2(x)}{3 i+\sqrt {3}}}}{\sqrt {2} \sqrt {-\frac {i}{-3 i+\sqrt {3}}} \sqrt {1-\cos ^6(x)}} \]

[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])

Maple [F]

\[\int \frac {\cos \left (x \right )^{2} \sin \left (x \right )}{\sqrt {1-\cos \left (x \right )^{6}}}d x\]

[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)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (7) = 14\).

Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 3.22 \[ \int \frac {\cos ^2(x) \sin (x)}{\sqrt {1-\cos ^6(x)}} \, dx=\frac {1}{6} \, \arctan \left (\frac {2 \, \sqrt {-\cos \left (x\right )^{6} + 1} \cos \left (x\right )^{3}}{2 \, \cos \left (x\right )^{6} - 1}\right ) \]

[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))

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^2(x) \sin (x)}{\sqrt {1-\cos ^6(x)}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 18 vs. \(2 (7) = 14\).

Time = 0.30 (sec) , antiderivative size = 18, normalized size of antiderivative = 2.00 \[ \int \frac {\cos ^2(x) \sin (x)}{\sqrt {1-\cos ^6(x)}} \, dx=\frac {1}{3} \, \arctan \left (\frac {\sqrt {-\cos \left (x\right )^{6} + 1}}{\cos \left (x\right )^{3}}\right ) \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \frac {\cos ^2(x) \sin (x)}{\sqrt {1-\cos ^6(x)}} \, dx=-\frac {1}{3} \, \arcsin \left (\cos \left (x\right )^{3}\right ) \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^2(x) \sin (x)}{\sqrt {1-\cos ^6(x)}} \, dx=\int \frac {{\cos \left (x\right )}^2\,\sin \left (x\right )}{\sqrt {1-{\cos \left (x\right )}^6}} \,d x \]

[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)