3.3 \(\int \frac{2+\cos (x)+5 \sin (x)}{4 \cos (x)-2 \sin (x)+\cos (x) \sin (x)-2 \sin ^2(x)} \, dx\)

Optimal. Leaf size=19 \[ \log (\sin (x)+\cos (x)+3)-\log (\sin (x)-3 \cos (x)+1) \]

[Out]

-Log[1 - 3*Cos[x] + Sin[x]] + Log[3 + Cos[x] + Sin[x]]

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Rubi [B]  time = 1.01033, antiderivative size = 42, normalized size of antiderivative = 2.21, number of steps used = 25, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226 \[ \log \left (\tan ^2\left (\frac{x}{2}\right )+\tan \left (\frac{x}{2}\right )+2\right )-\log \left (1-2 \tan \left (\frac{x}{2}\right )\right )-\log \left (\tan \left (\frac{x}{2}\right )+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[(2 + Cos[x] + 5*Sin[x])/(4*Cos[x] - 2*Sin[x] + Cos[x]*Sin[x] - 2*Sin[x]^2),x]

[Out]

-Log[1 - 2*Tan[x/2]] - Log[1 + Tan[x/2]] + Log[2 + Tan[x/2] + Tan[x/2]^2]

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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((2+cos(x)+5*sin(x))/(4*cos(x)-2*sin(x)+cos(x)*sin(x)-2*sin(x)**2),x)

[Out]

Timed out

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Mathematica [A]  time = 0.059474, size = 19, normalized size = 1. \[ \log (\sin (x)+\cos (x)+3)-\log (\sin (x)-3 \cos (x)+1) \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + Cos[x] + 5*Sin[x])/(4*Cos[x] - 2*Sin[x] + Cos[x]*Sin[x] - 2*Sin[x]^2),x]

[Out]

-Log[1 - 3*Cos[x] + Sin[x]] + Log[3 + Cos[x] + Sin[x]]

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Maple [A]  time = 0.105, size = 35, normalized size = 1.8 \[ -\ln \left ( 1+\tan \left ({\frac{x}{2}} \right ) \right ) -\ln \left ( 2\,\tan \left ( x/2 \right ) -1 \right ) +\ln \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+\tan \left ({\frac{x}{2}} \right ) +2 \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+cos(x)+5*sin(x))/(4*cos(x)-2*sin(x)+cos(x)*sin(x)-2*sin(x)^2),x)

[Out]

-ln(1+tan(1/2*x))-ln(2*tan(1/2*x)-1)+ln(tan(1/2*x)^2+tan(1/2*x)+2)

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Maxima [A]  time = 1.59504, size = 72, normalized size = 3.79 \[ -\log \left (\frac{2 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - 1\right ) + \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 2\right ) - \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((cos(x) + 5*sin(x) + 2)/(cos(x)*sin(x) - 2*sin(x)^2 + 4*cos(x) - 2*sin(x)),x, algorithm="maxima")

[Out]

-log(2*sin(x)/(cos(x) + 1) - 1) + log(sin(x)/(cos(x) + 1) + sin(x)^2/(cos(x) + 1
)^2 + 2) - log(sin(x)/(cos(x) + 1) + 1)

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Fricas [A]  time = 0.247355, size = 58, normalized size = 3.05 \[ -\frac{1}{2} \, \log \left (2 \, \cos \left (x\right )^{2} - \frac{1}{2} \,{\left (3 \, \cos \left (x\right ) - 1\right )} \sin \left (x\right ) - \frac{3}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + \frac{1}{2} \, \log \left (\frac{1}{2} \,{\left (\cos \left (x\right ) + 3\right )} \sin \left (x\right ) + \frac{3}{2} \, \cos \left (x\right ) + \frac{5}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((cos(x) + 5*sin(x) + 2)/(cos(x)*sin(x) - 2*sin(x)^2 + 4*cos(x) - 2*sin(x)),x, algorithm="fricas")

[Out]

-1/2*log(2*cos(x)^2 - 1/2*(3*cos(x) - 1)*sin(x) - 3/2*cos(x) + 1/2) + 1/2*log(1/
2*(cos(x) + 3)*sin(x) + 3/2*cos(x) + 5/2)

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Sympy [A]  time = 2.92217, size = 32, normalized size = 1.68 \[ - \log{\left (\tan{\left (\frac{x}{2} \right )} - \frac{1}{2} \right )} - \log{\left (\tan{\left (\frac{x}{2} \right )} + 1 \right )} + \log{\left (\tan ^{2}{\left (\frac{x}{2} \right )} + \tan{\left (\frac{x}{2} \right )} + 2 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2+cos(x)+5*sin(x))/(4*cos(x)-2*sin(x)+cos(x)*sin(x)-2*sin(x)**2),x)

[Out]

-log(tan(x/2) - 1/2) - log(tan(x/2) + 1) + log(tan(x/2)**2 + tan(x/2) + 2)

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GIAC/XCAS [A]  time = 0.222127, size = 49, normalized size = 2.58 \[{\rm ln}\left (\tan \left (\frac{1}{2} \, x\right )^{2} + \tan \left (\frac{1}{2} \, x\right ) + 2\right ) -{\rm ln}\left ({\left | 2 \, \tan \left (\frac{1}{2} \, x\right ) - 1 \right |}\right ) -{\rm ln}\left ({\left | \tan \left (\frac{1}{2} \, x\right ) + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((cos(x) + 5*sin(x) + 2)/(cos(x)*sin(x) - 2*sin(x)^2 + 4*cos(x) - 2*sin(x)),x, algorithm="giac")

[Out]

ln(tan(1/2*x)^2 + tan(1/2*x) + 2) - ln(abs(2*tan(1/2*x) - 1)) - ln(abs(tan(1/2*x
) + 1))