3.248 \(\int \frac{1}{2 \sin (x)+\sin (2 x)} \, dx\)

Optimal. Leaf size=24 \[ \frac{1}{8} \tan ^2\left (\frac{x}{2}\right )+\frac{1}{4} \log \left (\tan \left (\frac{x}{2}\right )\right ) \]

[Out]

Log[Tan[x/2]]/4 + Tan[x/2]^2/8

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Rubi [A]  time = 0.0457707, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{1}{8} \tan ^2\left (\frac{x}{2}\right )+\frac{1}{4} \log \left (\tan \left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Int[(2*Sin[x] + Sin[2*x])^(-1),x]

[Out]

Log[Tan[x/2]]/4 + Tan[x/2]^2/8

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{2 \sin{\left (x \right )} + \sin{\left (2 x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(2*sin(x) + sin(2*x)), x)

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Mathematica [A]  time = 0.0401553, size = 39, normalized size = 1.62 \[ \frac{1-2 \cos ^2\left (\frac{x}{2}\right ) \left (\log \left (\cos \left (\frac{x}{2}\right )\right )-\log \left (\sin \left (\frac{x}{2}\right )\right )\right )}{4 (\cos (x)+1)} \]

Antiderivative was successfully verified.

[In]  Integrate[(2*Sin[x] + Sin[2*x])^(-1),x]

[Out]

(1 - 2*Cos[x/2]^2*(Log[Cos[x/2]] - Log[Sin[x/2]]))/(4*(1 + Cos[x]))

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Maple [A]  time = 0.076, size = 24, normalized size = 1. \[{\frac{1}{4+4\,\cos \left ( x \right ) }}-{\frac{\ln \left ( 1+\cos \left ( x \right ) \right ) }{8}}+{\frac{\ln \left ( \cos \left ( x \right ) -1 \right ) }{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(2*sin(x)+sin(2*x)),x)

[Out]

1/4/(1+cos(x))-1/8*ln(1+cos(x))+1/8*ln(cos(x)-1)

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Maxima [A]  time = 1.35636, size = 297, normalized size = 12.38 \[ \frac{4 \, \cos \left (2 \, x\right ) \cos \left (x\right ) + 8 \, \cos \left (x\right )^{2} -{\left (2 \,{\left (2 \, \cos \left (x\right ) + 1\right )} \cos \left (2 \, x\right ) + \cos \left (2 \, x\right )^{2} + 4 \, \cos \left (x\right )^{2} + \sin \left (2 \, x\right )^{2} + 4 \, \sin \left (2 \, x\right ) \sin \left (x\right ) + 4 \, \sin \left (x\right )^{2} + 4 \, \cos \left (x\right ) + 1\right )} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) +{\left (2 \,{\left (2 \, \cos \left (x\right ) + 1\right )} \cos \left (2 \, x\right ) + \cos \left (2 \, x\right )^{2} + 4 \, \cos \left (x\right )^{2} + \sin \left (2 \, x\right )^{2} + 4 \, \sin \left (2 \, x\right ) \sin \left (x\right ) + 4 \, \sin \left (x\right )^{2} + 4 \, \cos \left (x\right ) + 1\right )} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) + 4 \, \sin \left (2 \, x\right ) \sin \left (x\right ) + 8 \, \sin \left (x\right )^{2} + 4 \, \cos \left (x\right )}{8 \,{\left (2 \,{\left (2 \, \cos \left (x\right ) + 1\right )} \cos \left (2 \, x\right ) + \cos \left (2 \, x\right )^{2} + 4 \, \cos \left (x\right )^{2} + \sin \left (2 \, x\right )^{2} + 4 \, \sin \left (2 \, x\right ) \sin \left (x\right ) + 4 \, \sin \left (x\right )^{2} + 4 \, \cos \left (x\right ) + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sin(2*x) + 2*sin(x)),x, algorithm="maxima")

[Out]

1/8*(4*cos(2*x)*cos(x) + 8*cos(x)^2 - (2*(2*cos(x) + 1)*cos(2*x) + cos(2*x)^2 +
4*cos(x)^2 + sin(2*x)^2 + 4*sin(2*x)*sin(x) + 4*sin(x)^2 + 4*cos(x) + 1)*log(cos
(x)^2 + sin(x)^2 + 2*cos(x) + 1) + (2*(2*cos(x) + 1)*cos(2*x) + cos(2*x)^2 + 4*c
os(x)^2 + sin(2*x)^2 + 4*sin(2*x)*sin(x) + 4*sin(x)^2 + 4*cos(x) + 1)*log(cos(x)
^2 + sin(x)^2 - 2*cos(x) + 1) + 4*sin(2*x)*sin(x) + 8*sin(x)^2 + 4*cos(x))/(2*(2
*cos(x) + 1)*cos(2*x) + cos(2*x)^2 + 4*cos(x)^2 + sin(2*x)^2 + 4*sin(2*x)*sin(x)
+ 4*sin(x)^2 + 4*cos(x) + 1)

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Fricas [A]  time = 0.224978, size = 47, normalized size = 1.96 \[ -\frac{{\left (\cos \left (x\right ) + 1\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) -{\left (\cos \left (x\right ) + 1\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) - 2}{8 \,{\left (\cos \left (x\right ) + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sin(2*x) + 2*sin(x)),x, algorithm="fricas")

[Out]

-1/8*((cos(x) + 1)*log(1/2*cos(x) + 1/2) - (cos(x) + 1)*log(-1/2*cos(x) + 1/2) -
2)/(cos(x) + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{2 \sin{\left (x \right )} + \sin{\left (2 x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(2*sin(x) + sin(2*x)), x)

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GIAC/XCAS [A]  time = 0.202144, size = 38, normalized size = 1.58 \[ -\frac{\cos \left (x\right ) - 1}{8 \,{\left (\cos \left (x\right ) + 1\right )}} + \frac{1}{8} \,{\rm ln}\left (-\frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sin(2*x) + 2*sin(x)),x, algorithm="giac")

[Out]

-1/8*(cos(x) - 1)/(cos(x) + 1) + 1/8*ln(-(cos(x) - 1)/(cos(x) + 1))