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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]