3.1.44 \(\int \frac {\sin (x)}{\sqrt {1-\sin ^6(x)}} \, dx\) [44]

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

[Out]

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

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Rubi [A]
time = 0.03, 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 = {3295, 2021, 1918, 212} \begin {gather*} \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}} \end {gather*}

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 212

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

Rule 1918

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 2021

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

Rule 3295

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]

Rubi steps

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

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Mathematica [A]
time = 0.06, size = 65, normalized size = 1.67 \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} (-3+\cos (2 x))}{\sqrt {15-8 \cos (2 x)+\cos (4 x)}}\right ) \cos (x) \sqrt {15-8 \cos (2 x)+\cos (4 x)}}{4 \sqrt {6-6 \sin ^6(x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(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]])/Sqrt[6 - 6*Sin[x]^6]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(66\) vs. \(2(29)=58\).
time = 0.26, size = 67, normalized size = 1.72

method	result	size
default	\(-\frac {\cos \left (x \right ) \sqrt {\cos ^{4}\left (x \right )-3 \left (\cos ^{2}\left (x \right )\right )+3}\, \sqrt {3}\, \arctanh \left (\frac {\left (\cos ^{2}\left (x \right )-2\right ) \sqrt {3}}{2 \sqrt {\cos ^{4}\left (x \right )-3 \left (\cos ^{2}\left (x \right )\right )+3}}\right )}{6 \sqrt {3 \left (\cos ^{2}\left (x \right )\right )-3 \left (\cos ^{4}\left (x \right )\right )+\cos ^{6}\left (x \right )}}\)	\(67\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (29) = 58\).
time = 0.48, size = 63, normalized size = 1.62 \begin {gather*} \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 {gather*}

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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 4847 deep

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (29) = 58\).
time = 0.51, size = 67, normalized size = 1.72 \begin {gather*} -\frac {\sqrt {3} \log \left (\cos \left (x\right )^{2} + \sqrt {3} - \sqrt {\cos \left (x\right )^{4} - 3 \, \cos \left (x\right )^{2} + 3}\right ) - \sqrt {3} \log \left (-\cos \left (x\right )^{2} + \sqrt {3} + \sqrt {\cos \left (x\right )^{4} - 3 \, \cos \left (x\right )^{2} + 3}\right )}{6 \, \mathrm {sgn}\left (\cos \left (x\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(sqrt(3)*log(cos(x)^2 + sqrt(3) - sqrt(cos(x)^4 - 3*cos(x)^2 + 3)) - sqrt(3)*log(-cos(x)^2 + sqrt(3) + sq
rt(cos(x)^4 - 3*cos(x)^2 + 3)))/sgn(cos(x))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\sin \left (x\right )}{\sqrt {1-{\sin \left (x\right )}^6}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)/(1 - sin(x)^6)^(1/2),x)

[Out]

int(sin(x)/(1 - sin(x)^6)^(1/2), x)

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