3.2 \(\int \frac{1+\cos (x)+2 \sin (x)}{3+\cos ^2(x)+2 \sin (x)-2 \cos (x) \sin (x)} \, dx\)

Optimal. Leaf size=19 \[ -\tan ^{-1}\left (\frac{2 \cos (x)-\sin (x)}{\sin (x)+2}\right ) \]

[Out]

-ArcTan[(2*Cos[x] - Sin[x])/(2 + Sin[x])]

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Rubi [A]  time = 5.99988, antiderivative size = 38, normalized size of antiderivative = 2., number of steps used = 43, number of rules used = 12, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444 \[ \cot \left (\frac{x}{2}\right )-\frac{\sin (x)}{1-\cos (x)}-\tan ^{-1}\left (\frac{2 \cos (x)-\sin (x)}{\sin (x)+2}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 + Cos[x] + 2*Sin[x])/(3 + Cos[x]^2 + 2*Sin[x] - 2*Cos[x]*Sin[x]),x]

[Out]

-ArcTan[(2*Cos[x] - Sin[x])/(2 + Sin[x])] + Cot[x/2] - Sin[x]/(1 - Cos[x])

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+cos(x)+2*sin(x))/(3+cos(x)**2+2*sin(x)-2*cos(x)*sin(x)),x)

[Out]

Timed out

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Mathematica [B]  time = 0.0465095, size = 46, normalized size = 2.42 \[ \frac{1}{2} \tan ^{-1}\left (\frac{\cos (x)+1}{-\sin (x)+\cos (x)-1}\right )-\frac{1}{2} \tan ^{-1}\left (\frac{1}{2} \sec ^2\left (\frac{x}{2}\right ) (-\sin (x)+\cos (x)-1)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + Cos[x] + 2*Sin[x])/(3 + Cos[x]^2 + 2*Sin[x] - 2*Cos[x]*Sin[x]),x]

[Out]

ArcTan[(1 + Cos[x])/(-1 + Cos[x] - Sin[x])]/2 - ArcTan[(Sec[x/2]^2*(-1 + Cos[x]
- Sin[x]))/2]/2

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Maple [A]  time = 0.146, size = 13, normalized size = 0.7 \[ \arctan \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+\tan \left ({\frac{x}{2}} \right ) \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+cos(x)+2*sin(x))/(3+cos(x)^2+2*sin(x)-2*cos(x)*sin(x)),x)

[Out]

arctan(tan(1/2*x)^2+tan(1/2*x))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((cos(x) + 2*sin(x) + 1)/(cos(x)^2 - 2*cos(x)*sin(x) + 2*sin(x) + 3),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 0.240399, size = 65, normalized size = 3.42 \[ \frac{1}{2} \, \arctan \left (-\frac{3 \, \cos \left (x\right )^{2} - 2 \,{\left (3 \, \cos \left (x\right ) + 1\right )} \sin \left (x\right ) - 4 \, \cos \left (x\right ) - 3}{2 \,{\left (2 \, \cos \left (x\right )^{2} +{\left (\cos \left (x\right ) - 3\right )} \sin \left (x\right ) + 4 \, \cos \left (x\right ) - 2\right )}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((cos(x) + 2*sin(x) + 1)/(cos(x)^2 - 2*cos(x)*sin(x) + 2*sin(x) + 3),x, algorithm="fricas")

[Out]

1/2*arctan(-1/2*(3*cos(x)^2 - 2*(3*cos(x) + 1)*sin(x) - 4*cos(x) - 3)/(2*cos(x)^
2 + (cos(x) - 3)*sin(x) + 4*cos(x) - 2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{2 \sin{\left (x \right )} + \cos{\left (x \right )} + 1}{- 2 \sin{\left (x \right )} \cos{\left (x \right )} + 2 \sin{\left (x \right )} + \cos ^{2}{\left (x \right )} + 3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+cos(x)+2*sin(x))/(3+cos(x)**2+2*sin(x)-2*cos(x)*sin(x)),x)

[Out]

Integral((2*sin(x) + cos(x) + 1)/(-2*sin(x)*cos(x) + 2*sin(x) + cos(x)**2 + 3),
x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\cos \left (x\right ) + 2 \, \sin \left (x\right ) + 1}{\cos \left (x\right )^{2} - 2 \, \cos \left (x\right ) \sin \left (x\right ) + 2 \, \sin \left (x\right ) + 3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((cos(x) + 2*sin(x) + 1)/(cos(x)^2 - 2*cos(x)*sin(x) + 2*sin(x) + 3),x, algorithm="giac")

[Out]

integrate((cos(x) + 2*sin(x) + 1)/(cos(x)^2 - 2*cos(x)*sin(x) + 2*sin(x) + 3), x
)