3.81 \(\int \frac{1}{1-\sin (x)} \, dx\)

Optimal. Leaf size=11 \[ \frac{\cos (x)}{1-\sin (x)} \]

[Out]

Cos[x]/(1 - Sin[x])

_______________________________________________________________________________________

Rubi [A]  time = 0.015708, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{\cos (x)}{1-\sin (x)} \]

Antiderivative was successfully verified.

[In]  Int[(1 - Sin[x])^(-1),x]

[Out]

Cos[x]/(1 - Sin[x])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 0.51639, size = 7, normalized size = 0.64 \[ \frac{\cos{\left (x \right )}}{- \sin{\left (x \right )} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(1-sin(x)),x)

[Out]

cos(x)/(-sin(x) + 1)

_______________________________________________________________________________________

Mathematica [B]  time = 0.0104177, size = 25, normalized size = 2.27 \[ \frac{2 \sin \left (\frac{x}{2}\right )}{\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - Sin[x])^(-1),x]

[Out]

(2*Sin[x/2])/(Cos[x/2] - Sin[x/2])

_______________________________________________________________________________________

Maple [A]  time = 0.003, size = 11, normalized size = 1. \[ -2\, \left ( -1+\tan \left ( x/2 \right ) \right ) ^{-1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1-sin(x)),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.35804, size = 20, normalized size = 1.82 \[ -\frac{2}{\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/(sin(x)/(cos(x) + 1) - 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.211194, size = 23, normalized size = 2.09 \[ \frac{\cos \left (x\right ) + \sin \left (x\right ) + 1}{\cos \left (x\right ) - \sin \left (x\right ) + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(cos(x) + sin(x) + 1)/(cos(x) - sin(x) + 1)

_______________________________________________________________________________________

Sympy [A]  time = 0.714072, size = 8, normalized size = 0.73 \[ - \frac{2}{\tan{\left (\frac{x}{2} \right )} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(1-sin(x)),x)

[Out]

-2/(tan(x/2) - 1)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.215994, size = 14, normalized size = 1.27 \[ -\frac{2}{\tan \left (\frac{1}{2} \, x\right ) - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/(tan(1/2*x) - 1)