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

Optimal. Leaf size=11 \[ \log \left (\sin ^2(x)-3 \sin (x)+2\right ) \]

[Out]

Log[2 - 3*Sin[x] + Sin[x]^2]

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Rubi [A]  time = 0.0673052, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \log \left (\sin ^2(x)-3 \sin (x)+2\right ) \]

Antiderivative was successfully verified.

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

[Out]

Log[2 - 3*Sin[x] + Sin[x]^2]

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Rubi in Sympy [A]  time = 6.96482, size = 12, normalized size = 1.09 \[ \log{\left (\sin ^{2}{\left (x \right )} - 3 \sin{\left (x \right )} + 2 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(sin(x)**2 - 3*sin(x) + 2)

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Mathematica [B]  time = 0.0625948, size = 26, normalized size = 2.36 \[ \log (2-\sin (x))+2 \log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

2*Log[Cos[x/2] - Sin[x/2]] + Log[2 - Sin[x]]

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Maple [A]  time = 0.043, size = 12, normalized size = 1.1 \[ \ln \left ( 2-3\,\sin \left ( x \right ) + \left ( \sin \left ( x \right ) \right ) ^{2} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

ln(2-3*sin(x)+sin(x)^2)

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Maxima [A]  time = 1.3412, size = 15, normalized size = 1.36 \[ \log \left (\sin \left (x\right )^{2} - 3 \, \sin \left (x\right ) + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(sin(x)^2 - 3*sin(x) + 2)

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Fricas [A]  time = 0.228665, size = 20, normalized size = 1.82 \[ \log \left (-\frac{1}{2} \, \sin \left (x\right ) + 1\right ) + \log \left (-\sin \left (x\right ) + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.284128, size = 12, normalized size = 1.09 \[ \log{\left (\sin{\left (x \right )} - 2 \right )} + \log{\left (\sin{\left (x \right )} - 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(sin(x) - 2) + log(sin(x) - 1)

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GIAC/XCAS [A]  time = 0.213584, size = 20, normalized size = 1.82 \[{\rm ln}\left (-\sin \left (x\right ) + 2\right ) +{\rm ln}\left (-\sin \left (x\right ) + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

ln(-sin(x) + 2) + ln(-sin(x) + 1)