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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]