3.245 \(\int \frac {1}{1-\cos (x)+\sin (x)} \, dx\)

Optimal. Leaf size=11 \[ -\log \left (\cot \left (\frac {x}{2}\right )+1\right ) \]

[Out]

-ln(1+cot(1/2*x))

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3121, 31} \[ -\log \left (\cot \left (\frac {x}{2}\right )+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 - Cos[x] + Sin[x])^(-1),x]

[Out]

-Log[1 + Cot[x/2]]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 3121

Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^(-1), x_Symbol] :> Module[{f = Free
Factors[Cot[(d + e*x)/2], x]}, -Dist[f/e, Subst[Int[1/(a + c*f*x), x], x, Cot[(d + e*x)/2]/f], x]] /; FreeQ[{a
, b, c, d, e}, x] && EqQ[a + b, 0]

Rubi steps

\begin {align*} \int \frac {1}{1-\cos (x)+\sin (x)} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,\cot \left (\frac {x}{2}\right )\right )\\ &=-\log \left (1+\cot \left (\frac {x}{2}\right )\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [B]  time = 0.02, size = 24, normalized size = 2.18 \[ \log \left (\sin \left (\frac {x}{2}\right )\right )-\log \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - Cos[x] + Sin[x])^(-1),x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.47, size = 17, normalized size = 1.55 \[ \frac {1}{2} \, \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) - \frac {1}{2} \, \log \left (\sin \relax (x) + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-cos(x)+sin(x)),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.94, size = 17, normalized size = 1.55 \[ -\log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) + 1 \right |}\right ) + \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-cos(x)+sin(x)),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.06, size = 16, normalized size = 1.45 \[ -\ln \left (\tan \left (\frac {x}{2}\right )+1\right )+\ln \left (\tan \left (\frac {x}{2}\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [B]  time = 0.59, size = 25, normalized size = 2.27 \[ -\log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1} + 1\right ) + \log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-cos(x)+sin(x)),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.07, size = 11, normalized size = 1.00 \[ -2\,\mathrm {atanh}\left (2\,\mathrm {tan}\left (\frac {x}{2}\right )+1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sin(x) - cos(x) + 1),x)

[Out]

-2*atanh(2*tan(x/2) + 1)

________________________________________________________________________________________

sympy [A]  time = 0.25, size = 14, normalized size = 1.27 \[ - \log {\left (\tan {\left (\frac {x}{2} \right )} + 1 \right )} + \log {\left (\tan {\left (\frac {x}{2} \right )} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(tan(x/2) + 1) + log(tan(x/2))

________________________________________________________________________________________