3.3.45 \(\int \frac {1}{1-\cos (x)+\sin (x)} \, dx\) [245]

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

[Out]

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

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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 = {3200, 31} \begin {gather*} -\log \left (\cot \left (\frac {x}{2}\right )+1\right ) \end {gather*}

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 3200

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 &=-\text {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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(24\) vs. \(2(11)=22\).
time = 0.01, size = 24, normalized size = 2.18 \begin {gather*} \log \left (\sin \left (\frac {x}{2}\right )\right )-\log \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right ) \end {gather*}

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]]

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Maple [A]
time = 0.05, size = 16, normalized size = 1.45

method	result	size
default	\(-\ln \left (1+\tan \left (\frac {x}{2}\right )\right )+\ln \left (\tan \left (\frac {x}{2}\right )\right )\)	\(16\)
norman	\(-\ln \left (1+\tan \left (\frac {x}{2}\right )\right )+\ln \left (\tan \left (\frac {x}{2}\right )\right )\)	\(16\)
risch	\(\ln \left ({\mathrm e}^{i x}-1\right )-\ln \left ({\mathrm e}^{i x}+i\right )\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-cos(x)+sin(x)),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (9) = 18\).
time = 1.97, size = 25, normalized size = 2.27 \begin {gather*} -\log \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1\right ) + \log \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right ) \end {gather*}

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))

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Fricas [A]
time = 0.47, size = 17, normalized size = 1.55 \begin {gather*} \frac {1}{2} \, \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) - \frac {1}{2} \, \log \left (\sin \left (x\right ) + 1\right ) \end {gather*}

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)

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Sympy [A]
time = 0.10, size = 14, normalized size = 1.27 \begin {gather*} - \log {\left (\tan {\left (\frac {x}{2} \right )} + 1 \right )} + \log {\left (\tan {\left (\frac {x}{2} \right )} \right )} \end {gather*}

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))

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Giac [A]
time = 0.47, size = 17, normalized size = 1.55 \begin {gather*} -\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 ) \end {gather*}

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)))

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

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)

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