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

Optimal. Leaf size=39 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{3} \left (\sin ^2(x)+1\right ) \cos (x)}{2 \sqrt{1-\sin ^6(x)}}\right )}{2 \sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0457867, antiderivative size = 50, normalized size of antiderivative = 1.28, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {3216, 1996, 1904, 206} \[ \frac{\tanh ^{-1}\left (\frac{\cos (x) \left (6-3 \cos ^2(x)\right )}{2 \sqrt{3} \sqrt{\cos ^6(x)-3 \cos ^4(x)+3 \cos ^2(x)}}\right )}{2 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]/Sqrt[1 - Sin[x]^6],x]

[Out]

ArcTanh[(Cos[x]*(6 - 3*Cos[x]^2))/(2*Sqrt[3]*Sqrt[3*Cos[x]^2 - 3*Cos[x]^4 + Cos[x]^6])]/(2*Sqrt[3])

Rule 3216

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_.), x_Symbol] :> With[{ff = F
reeFactors[Cos[e + f*x], x]}, -Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*(1 - ff^2*x^2)^(n/2))^p,
x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n/2]

Rule 1996

Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && GeneralizedTrinomialQ[u, x] &&  !Gen
eralizedTrinomialMatchQ[u, x]

Rule 1904

Int[1/Sqrt[(a_.)*(x_)^2 + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(r_.)], x_Symbol] :> Dist[-2/(n - 2), Subst[Int[1/(4*a
- x^2), x], x, (x*(2*a + b*x^(n - 2)))/Sqrt[a*x^2 + b*x^n + c*x^r]], x] /; FreeQ[{a, b, c, n, r}, x] && EqQ[r
, 2*n - 2] && PosQ[n - 2] && NeQ[b^2 - 4*a*c, 0]

Rule 206

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

Rubi steps

\begin{align*} \int \frac{\sin (x)}{\sqrt{1-\sin ^6(x)}} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\left (1-x^2\right )^3}} \, dx,x,\cos (x)\right )\\ &=-\operatorname{Subst}\left (\int \frac{1}{\sqrt{3 x^2-3 x^4+x^6}} \, dx,x,\cos (x)\right )\\ &=\operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{\cos (x) \left (6-3 \cos ^2(x)\right )}{\sqrt{3 \cos ^2(x)-3 \cos ^4(x)+\cos ^6(x)}}\right )\\ &=\frac{\tanh ^{-1}\left (\frac{\cos (x) \left (6-3 \cos ^2(x)\right )}{2 \sqrt{3} \sqrt{3 \cos ^2(x)-3 \cos ^4(x)+\cos ^6(x)}}\right )}{2 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0854961, size = 65, normalized size = 1.67 \[ -\frac{\cos (x) \sqrt{-8 \cos (2 x)+\cos (4 x)+15} \tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} (\cos (2 x)-3)}{\sqrt{-8 \cos (2 x)+\cos (4 x)+15}}\right )}{4 \sqrt{6-6 \sin ^6(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]/Sqrt[1 - Sin[x]^6],x]

[Out]

-(ArcTanh[(Sqrt[3/2]*(-3 + Cos[2*x]))/Sqrt[15 - 8*Cos[2*x] + Cos[4*x]]]*Cos[x]*Sqrt[15 - 8*Cos[2*x] + Cos[4*x]
])/(4*Sqrt[6 - 6*Sin[x]^6])

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Maple [B]  time = 0.242, size = 67, normalized size = 1.7 \begin{align*} -{\frac{\cos \left ( x \right ) \sqrt{3}}{6}\sqrt{3-3\, \left ( \cos \left ( x \right ) \right ) ^{2}+ \left ( \cos \left ( x \right ) \right ) ^{4}}{\it Artanh} \left ({\frac{ \left ( \left ( \cos \left ( x \right ) \right ) ^{2}-2 \right ) \sqrt{3}}{2}{\frac{1}{\sqrt{3-3\, \left ( \cos \left ( x \right ) \right ) ^{2}+ \left ( \cos \left ( x \right ) \right ) ^{4}}}}} \right ){\frac{1}{\sqrt{3\, \left ( \cos \left ( x \right ) \right ) ^{2}-3\, \left ( \cos \left ( x \right ) \right ) ^{4}+ \left ( \cos \left ( x \right ) \right ) ^{6}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6/(3*cos(x)^2-3*cos(x)^4+cos(x)^6)^(1/2)*cos(x)*(3-3*cos(x)^2+cos(x)^4)^(1/2)*3^(1/2)*arctanh(1/2*(cos(x)^2
-2)*3^(1/2)/(3-3*cos(x)^2+cos(x)^4)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (x\right )}{\sqrt{-\sin \left (x\right )^{6} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sin(x)/sqrt(-sin(x)^6 + 1), x)

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Fricas [B]  time = 2.78354, size = 193, normalized size = 4.95 \begin{align*} \frac{1}{12} \, \sqrt{3} \log \left (\frac{7 \, \cos \left (x\right )^{5} - 24 \, \cos \left (x\right )^{3} - 4 \, \sqrt{\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2}}{\left (\sqrt{3} \cos \left (x\right )^{2} - 2 \, \sqrt{3}\right )} + 24 \, \cos \left (x\right )}{\cos \left (x\right )^{5}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*sqrt(3)*log((7*cos(x)^5 - 24*cos(x)^3 - 4*sqrt(cos(x)^6 - 3*cos(x)^4 + 3*cos(x)^2)*(sqrt(3)*cos(x)^2 - 2*
sqrt(3)) + 24*cos(x))/cos(x)^5)

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (x\right )}{\sqrt{-\sin \left (x\right )^{6} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sin(x)/sqrt(-sin(x)^6 + 1), x)