3.4 \(\int \frac{3+7 \cos (x)+2 \sin (x)}{1+4 \cos (x)+3 \cos ^2(x)-5 \sin (x)-\cos (x) \sin (x)} \, dx\)

Optimal. Leaf size=19 \[ \log (\sin (x)+\cos (x)+3)-\log (-2 \sin (x)+\cos (x)+1) \]

[Out]

-Log[1 + Cos[x] - 2*Sin[x]] + Log[3 + Cos[x] + Sin[x]]

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Rubi [A]  time = 3.37346, antiderivative size = 31, normalized size of antiderivative = 1.63, number of steps used = 32, number of rules used = 9, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.257 \[ \log \left (\tan ^2\left (\frac{x}{2}\right )+\tan \left (\frac{x}{2}\right )+2\right )-\log \left (1-2 \tan \left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Int[(3 + 7*Cos[x] + 2*Sin[x])/(1 + 4*Cos[x] + 3*Cos[x]^2 - 5*Sin[x] - Cos[x]*Sin[x]),x]

[Out]

-Log[1 - 2*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((3+7*cos(x)+2*sin(x))/(1+4*cos(x)+3*cos(x)**2-5*sin(x)-cos(x)*sin(x)),x)

[Out]

Timed out

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Mathematica [A]  time = 0.0469354, size = 19, normalized size = 1. \[ \log (\sin (x)+\cos (x)+3)-\log (-2 \sin (x)+\cos (x)+1) \]

Antiderivative was successfully verified.

[In]  Integrate[(3 + 7*Cos[x] + 2*Sin[x])/(1 + 4*Cos[x] + 3*Cos[x]^2 - 5*Sin[x] - Cos[x]*Sin[x]),x]

[Out]

-Log[1 + Cos[x] - 2*Sin[x]] + Log[3 + Cos[x] + Sin[x]]

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Maple [A]  time = 0.149, size = 26, normalized size = 1.4 \[ -\ln \left ( 2\,\tan \left ( x/2 \right ) -1 \right ) +\ln \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+\tan \left ({\frac{x}{2}} \right ) +2 \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3+7*cos(x)+2*sin(x))/(1+4*cos(x)+3*cos(x)^2-5*sin(x)-cos(x)*sin(x)),x)

[Out]

-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 = 1.54425, size = 53, normalized size = 2.79 \[ -\log \left (\frac{2 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - 1\right ) + \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((7*cos(x) + 2*sin(x) + 3)/(3*cos(x)^2 - cos(x)*sin(x) + 4*cos(x) - 5*sin(x) + 1),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)

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Fricas [A]  time = 0.236064, size = 55, normalized size = 2.89 \[ -\frac{1}{2} \, \log \left (-\frac{3}{4} \, \cos \left (x\right )^{2} -{\left (\cos \left (x\right ) + 1\right )} \sin \left (x\right ) + \frac{1}{2} \, \cos \left (x\right ) + \frac{5}{4}\right ) + \frac{1}{2} \, \log \left (\frac{1}{2} \,{\left (\cos \left (x\right ) + 3\right )} \sin \left (x\right ) + \frac{3}{2} \, \cos \left (x\right ) + \frac{5}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((7*cos(x) + 2*sin(x) + 3)/(3*cos(x)^2 - cos(x)*sin(x) + 4*cos(x) - 5*sin(x) + 1),x, algorithm="fricas")

[Out]

-1/2*log(-3/4*cos(x)^2 - (cos(x) + 1)*sin(x) + 1/2*cos(x) + 5/4) + 1/2*log(1/2*(
cos(x) + 3)*sin(x) + 3/2*cos(x) + 5/2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{2 \sin{\left (x \right )} + 7 \cos{\left (x \right )} + 3}{- \sin{\left (x \right )} \cos{\left (x \right )} - 5 \sin{\left (x \right )} + 3 \cos ^{2}{\left (x \right )} + 4 \cos{\left (x \right )} + 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3+7*cos(x)+2*sin(x))/(1+4*cos(x)+3*cos(x)**2-5*sin(x)-cos(x)*sin(x)),x)

[Out]

Integral((2*sin(x) + 7*cos(x) + 3)/(-sin(x)*cos(x) - 5*sin(x) + 3*cos(x)**2 + 4*
cos(x) + 1), x)

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GIAC/XCAS [A]  time = 0.213975, size = 35, normalized size = 1.84 \[{\rm ln}\left (\tan \left (\frac{1}{2} \, x\right )^{2} + \tan \left (\frac{1}{2} \, x\right ) + 2\right ) -{\rm ln}\left ({\left | 2 \, \tan \left (\frac{1}{2} \, x\right ) - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((7*cos(x) + 2*sin(x) + 3)/(3*cos(x)^2 - cos(x)*sin(x) + 4*cos(x) - 5*sin(x) + 1),x, algorithm="giac")

[Out]

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