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])

_______________________________________________________________________________________

Rubi [A]  time = 0.0729363, 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 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{3} \cos (x) \left (2-\cos ^2(x)\right )}{2 \sqrt{\cos ^6(x)-3 \cos ^4(x)+3 \cos ^2(x)}}\right )}{2 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[Sin[x]/Sqrt[1 - Sin[x]^6],x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 9.94088, size = 82, normalized size = 2.1 \[ \frac{\sqrt{3} \sqrt{\cos ^{4}{\left (x \right )} - 3 \cos ^{2}{\left (x \right )} + 3} \cos{\left (x \right )} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (- 3 \cos ^{2}{\left (x \right )} + 6\right )}{6 \sqrt{\cos ^{4}{\left (x \right )} - 3 \cos ^{2}{\left (x \right )} + 3}} \right )}}{6 \sqrt{\cos ^{6}{\left (x \right )} - 3 \cos ^{4}{\left (x \right )} + 3 \cos ^{2}{\left (x \right )}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(sin(x)/(1-sin(x)**6)**(1/2),x)

[Out]

sqrt(3)*sqrt(cos(x)**4 - 3*cos(x)**2 + 3)*cos(x)*atanh(sqrt(3)*(-3*cos(x)**2 + 6
)/(6*sqrt(cos(x)**4 - 3*cos(x)**2 + 3)))/(6*sqrt(cos(x)**6 - 3*cos(x)**4 + 3*cos
(x)**2))

_______________________________________________________________________________________

Mathematica [C]  time = 28.0736, size = 5825, normalized size = 149.36 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

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

[Out]

Result too large to show

_______________________________________________________________________________________

Maple [B]  time = 0.369, size = 67, normalized size = 1.7 \[ -{\frac{\cos \left ( x \right ) \sqrt{3}}{6}\sqrt{3-3\, \left ( \cos \left ( x \right ) \right ) ^{2}+ \left ( \cos \left ( x \right ) \right ) ^{4}}{\it Artanh} \left ({\frac{ \left ( \left ( \cos \left ( x \right ) \right ) ^{2}-2 \right ) \sqrt{3}}{2}{\frac{1}{\sqrt{3-3\, \left ( \cos \left ( x \right ) \right ) ^{2}+ \left ( \cos \left ( x \right ) \right ) ^{4}}}}} \right ){\frac{1}{\sqrt{3\, \left ( \cos \left ( x \right ) \right ) ^{2}-3\, \left ( \cos \left ( x \right ) \right ) ^{4}+ \left ( \cos \left ( x \right ) \right ) ^{6}}}}} \]

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))

_______________________________________________________________________________________

Maxima [A]  time = 1.73364, size = 28, normalized size = 0.72 \[ \frac{1}{6} \, \sqrt{3} \operatorname{arsinh}\left (-\sqrt{3} + \frac{2 \, \sqrt{3}}{\cos \left (x\right )^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(x)/sqrt(-sin(x)^6 + 1),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.279069, size = 86, normalized size = 2.21 \[ \frac{1}{12} \, \sqrt{3} \log \left (\frac{7 \, \sqrt{3} \cos \left (x\right )^{5} - 24 \, \sqrt{3} \cos \left (x\right )^{3} - 12 \, \sqrt{\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2}}{\left (\cos \left (x\right )^{2} - 2\right )} + 24 \, \sqrt{3} \cos \left (x\right )}{\cos \left (x\right )^{5}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(x)/sqrt(-sin(x)^6 + 1),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sin \left (x\right )}{\sqrt{-\sin \left (x\right )^{6} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(x)/sqrt(-sin(x)^6 + 1),x, algorithm="giac")

[Out]

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