3.243 \(\int \frac{1}{3-5 \sin (x)} \, dx\)

Optimal. Leaf size=43 \[ \frac{1}{4} \log \left (3 \cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )-\frac{1}{4} \log \left (\cos \left (\frac{x}{2}\right )-3 \sin \left (\frac{x}{2}\right )\right ) \]

[Out]

-Log[Cos[x/2] - 3*Sin[x/2]]/4 + Log[3*Cos[x/2] - Sin[x/2]]/4

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Rubi [A]  time = 0.0339761, 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 \[ \frac{1}{4} \log \left (3 \cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )-\frac{1}{4} \log \left (\cos \left (\frac{x}{2}\right )-3 \sin \left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Int[(3 - 5*Sin[x])^(-1),x]

[Out]

-Log[Cos[x/2] - 3*Sin[x/2]]/4 + Log[3*Cos[x/2] - Sin[x/2]]/4

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Rubi in Sympy [A]  time = 0.913506, size = 20, normalized size = 0.47 \[ - \frac{\log{\left (- 3 \tan{\left (\frac{x}{2} \right )} + 1 \right )}}{4} + \frac{\log{\left (- \tan{\left (\frac{x}{2} \right )} + 3 \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(3-5*sin(x)),x)

[Out]

-log(-3*tan(x/2) + 1)/4 + log(-tan(x/2) + 3)/4

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Mathematica [A]  time = 0.00889649, size = 43, normalized size = 1. \[ \frac{1}{4} \log \left (3 \cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )-\frac{1}{4} \log \left (\cos \left (\frac{x}{2}\right )-3 \sin \left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(3 - 5*Sin[x])^(-1),x]

[Out]

-Log[Cos[x/2] - 3*Sin[x/2]]/4 + Log[3*Cos[x/2] - Sin[x/2]]/4

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Maple [A]  time = 0.023, size = 22, normalized size = 0.5 \[ -{\frac{1}{4}\ln \left ( -1+3\,\tan \left ( x/2 \right ) \right ) }+{\frac{1}{4}\ln \left ( \tan \left ({\frac{x}{2}} \right ) -3 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(3-5*sin(x)),x)

[Out]

-1/4*ln(-1+3*tan(1/2*x))+1/4*ln(tan(1/2*x)-3)

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Maxima [A]  time = 1.33605, size = 41, normalized size = 0.95 \[ -\frac{1}{4} \, \log \left (\frac{3 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - 1\right ) + \frac{1}{4} \, \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} - 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(5*sin(x) - 3),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.231099, size = 36, normalized size = 0.84 \[ \frac{1}{8} \, \log \left (4 \, \cos \left (x\right ) - 3 \, \sin \left (x\right ) + 5\right ) - \frac{1}{8} \, \log \left (-4 \, \cos \left (x\right ) - 3 \, \sin \left (x\right ) + 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(5*sin(x) - 3),x, algorithm="fricas")

[Out]

1/8*log(4*cos(x) - 3*sin(x) + 5) - 1/8*log(-4*cos(x) - 3*sin(x) + 5)

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Sympy [A]  time = 0.283378, size = 20, normalized size = 0.47 \[ \frac{\log{\left (\tan{\left (\frac{x}{2} \right )} - 3 \right )}}{4} - \frac{\log{\left (\tan{\left (\frac{x}{2} \right )} - \frac{1}{3} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(3-5*sin(x)),x)

[Out]

log(tan(x/2) - 3)/4 - log(tan(x/2) - 1/3)/4

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GIAC/XCAS [A]  time = 0.215824, size = 31, normalized size = 0.72 \[ -\frac{1}{4} \,{\rm ln}\left ({\left | 3 \, \tan \left (\frac{1}{2} \, x\right ) - 1 \right |}\right ) + \frac{1}{4} \,{\rm ln}\left ({\left | \tan \left (\frac{1}{2} \, x\right ) - 3 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(5*sin(x) - 3),x, algorithm="giac")

[Out]

-1/4*ln(abs(3*tan(1/2*x) - 1)) + 1/4*ln(abs(tan(1/2*x) - 3))