3.305 \(\int \frac{1}{4-5 \sin (x)} \, dx\)

Optimal. Leaf size=43 \[ \frac{1}{3} \log \left (2 \cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )-\frac{1}{3} \log \left (\cos \left (\frac{x}{2}\right )-2 \sin \left (\frac{x}{2}\right )\right ) \]

[Out]

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

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Rubi [A]  time = 0.0334494, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ \frac{1}{3} \log \left (2 \cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )-\frac{1}{3} \log \left (\cos \left (\frac{x}{2}\right )-2 \sin \left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Int[(4 - 5*Sin[x])^(-1),x]

[Out]

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

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Rubi in Sympy [A]  time = 0.926556, size = 20, normalized size = 0.47 \[ - \frac{\log{\left (- 2 \tan{\left (\frac{x}{2} \right )} + 1 \right )}}{3} + \frac{\log{\left (- \tan{\left (\frac{x}{2} \right )} + 2 \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(4-5*sin(x)),x)

[Out]

-log(-2*tan(x/2) + 1)/3 + log(-tan(x/2) + 2)/3

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Mathematica [A]  time = 0.0085941, size = 43, normalized size = 1. \[ \frac{1}{3} \log \left (2 \cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )-\frac{1}{3} \log \left (\cos \left (\frac{x}{2}\right )-2 \sin \left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(4 - 5*Sin[x])^(-1),x]

[Out]

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

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Maple [A]  time = 0.021, size = 22, normalized size = 0.5 \[{\frac{1}{3}\ln \left ( \tan \left ({\frac{x}{2}} \right ) -2 \right ) }-{\frac{1}{3}\ln \left ( 2\,\tan \left ( x/2 \right ) -1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(4-5*sin(x)),x)

[Out]

1/3*ln(tan(1/2*x)-2)-1/3*ln(2*tan(1/2*x)-1)

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Maxima [A]  time = 1.35703, size = 41, normalized size = 0.95 \[ -\frac{1}{3} \, \log \left (\frac{2 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - 1\right ) + \frac{1}{3} \, \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(5*sin(x) - 4),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.231989, size = 36, normalized size = 0.84 \[ \frac{1}{6} \, \log \left (\frac{3}{2} \, \cos \left (x\right ) - 2 \, \sin \left (x\right ) + \frac{5}{2}\right ) - \frac{1}{6} \, \log \left (-\frac{3}{2} \, \cos \left (x\right ) - 2 \, \sin \left (x\right ) + \frac{5}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(5*sin(x) - 4),x, algorithm="fricas")

[Out]

1/6*log(3/2*cos(x) - 2*sin(x) + 5/2) - 1/6*log(-3/2*cos(x) - 2*sin(x) + 5/2)

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Sympy [A]  time = 0.287528, size = 20, normalized size = 0.47 \[ \frac{\log{\left (\tan{\left (\frac{x}{2} \right )} - 2 \right )}}{3} - \frac{\log{\left (\tan{\left (\frac{x}{2} \right )} - \frac{1}{2} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(4-5*sin(x)),x)

[Out]

log(tan(x/2) - 2)/3 - log(tan(x/2) - 1/2)/3

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GIAC/XCAS [A]  time = 0.220657, size = 31, normalized size = 0.72 \[ -\frac{1}{3} \,{\rm ln}\left ({\left | 2 \, \tan \left (\frac{1}{2} \, x\right ) - 1 \right |}\right ) + \frac{1}{3} \,{\rm ln}\left ({\left | \tan \left (\frac{1}{2} \, x\right ) - 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(5*sin(x) - 4),x, algorithm="giac")

[Out]

-1/3*ln(abs(2*tan(1/2*x) - 1)) + 1/3*ln(abs(tan(1/2*x) - 2))