3.12 \(\int \sqrt{-1-\cot ^2(x)} \, dx\)

Optimal. Leaf size=14 \[ \tan ^{-1}\left (\frac{\cot (x)}{\sqrt{-\csc ^2(x)}}\right ) \]

[Out]

ArcTan[Cot[x]/Sqrt[-Csc[x]^2]]

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Rubi [A]  time = 0.0202224, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3657, 4122, 217, 203} \[ \tan ^{-1}\left (\frac{\cot (x)}{\sqrt{-\csc ^2(x)}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[-1 - Cot[x]^2],x]

[Out]

ArcTan[Cot[x]/Sqrt[-Csc[x]^2]]

Rule 3657

Int[(u_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[ActivateTrig[u*(a*sec[e + f*x]^2)^p]
, x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a, b]

Rule 4122

Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(b*ff)
/f, Subst[Int[(b + b*ff^2*x^2)^(p - 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{b, e, f, p}, x] &&  !IntegerQ[p
]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \sqrt{-1-\cot ^2(x)} \, dx &=\int \sqrt{-\csc ^2(x)} \, dx\\ &=\operatorname{Subst}\left (\int \frac{1}{\sqrt{-1-x^2}} \, dx,x,\cot (x)\right )\\ &=\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\cot (x)}{\sqrt{-\csc ^2(x)}}\right )\\ &=\tan ^{-1}\left (\frac{\cot (x)}{\sqrt{-\csc ^2(x)}}\right )\\ \end{align*}

Mathematica [B]  time = 0.0181687, size = 30, normalized size = 2.14 \[ \frac{\csc (x) \left (\log \left (\cos \left (\frac{x}{2}\right )\right )-\log \left (\sin \left (\frac{x}{2}\right )\right )\right )}{\sqrt{-\csc ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-1 - Cot[x]^2],x]

[Out]

(Csc[x]*(Log[Cos[x/2]] - Log[Sin[x/2]]))/Sqrt[-Csc[x]^2]

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Maple [A]  time = 0.014, size = 15, normalized size = 1.1 \begin{align*} \arctan \left ({\cot \left ( x \right ){\frac{1}{\sqrt{-1- \left ( \cot \left ( x \right ) \right ) ^{2}}}}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-1-cot(x)^2)^(1/2),x)

[Out]

arctan(cot(x)/(-1-cot(x)^2)^(1/2))

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Maxima [A]  time = 1.62221, size = 23, normalized size = 1.64 \begin{align*} -\arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right ) + \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1-cot(x)^2)^(1/2),x, algorithm="maxima")

[Out]

-arctan2(sin(x), cos(x) + 1) + arctan2(sin(x), cos(x) - 1)

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Fricas [C]  time = 1.81229, size = 55, normalized size = 3.93 \begin{align*} i \, \log \left (e^{\left (i \, x\right )} + 1\right ) - i \, \log \left (e^{\left (i \, x\right )} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1-cot(x)^2)^(1/2),x, algorithm="fricas")

[Out]

I*log(e^(I*x) + 1) - I*log(e^(I*x) - 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{- \cot ^{2}{\left (x \right )} - 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1-cot(x)**2)**(1/2),x)

[Out]

Integral(sqrt(-cot(x)**2 - 1), x)

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Giac [C]  time = 1.18356, size = 15, normalized size = 1.07 \begin{align*} i \, \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) \right |}\right ) \mathrm{sgn}\left (\sin \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1-cot(x)^2)^(1/2),x, algorithm="giac")

[Out]

I*log(abs(tan(1/2*x)))*sgn(sin(x))