3.1.12 \(\int \sqrt {-1-\cot ^2(x)} \, dx\) [12]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, 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 = {3738, 4207, 223, 209} \begin {gather*} \text {ArcTan}\left (\frac {\cot (x)}{\sqrt {-\csc ^2(x)}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[-1 - Cot[x]^2],x]

[Out]

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

Rule 209

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

Rule 223

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 3738

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 4207

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
]

Rubi steps

\begin {align*} \int \sqrt {-1-\cot ^2(x)} \, dx &=\int \sqrt {-\csc ^2(x)} \, dx\\ &=\text {Subst}\left (\int \frac {1}{\sqrt {-1-x^2}} \, dx,x,\cot (x)\right )\\ &=\text {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] Leaf count is larger than twice the leaf count of optimal. \(30\) vs. \(2(14)=28\).
time = 0.02, size = 30, normalized size = 2.14 \begin {gather*} \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)}} \end {gather*}

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]

________________________________________________________________________________________

Maple [A]
time = 0.13, size = 15, normalized size = 1.07

method	result	size
derivativedivides	\(\arctan \left (\frac {\cot \left (x \right )}{\sqrt {-1-\left (\cot ^{2}\left (x \right )\right )}}\right )\)	\(15\)
default	\(\arctan \left (\frac {\cot \left (x \right )}{\sqrt {-1-\left (\cot ^{2}\left (x \right )\right )}}\right )\)	\(15\)
risch	\(-2 \sqrt {\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}+1\right ) \sin \left (x \right )+2 \sqrt {\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}-1\right ) \sin \left (x \right )\)	\(60\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-1-cot(x)^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.53, size = 17, normalized size = 1.21 \begin {gather*} -\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 {gather*}

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)

________________________________________________________________________________________

Fricas [C] Result contains complex when optimal does not.
time = 2.70, size = 19, normalized size = 1.36 \begin {gather*} i \, \log \left (e^{\left (i \, x\right )} + 1\right ) - i \, \log \left (e^{\left (i \, x\right )} - 1\right ) \end {gather*}

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)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {- \cot ^{2}{\left (x \right )} - 1}\, dx \end {gather*}

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)

________________________________________________________________________________________

Giac [C] Result contains complex when optimal does not.
time = 0.43, size = 11, normalized size = 0.79 \begin {gather*} i \, \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right ) \mathrm {sgn}\left (\sin \left (x\right )\right ) \end {gather*}

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))

________________________________________________________________________________________

Mupad [B]
time = 0.39, size = 14, normalized size = 1.00 \begin {gather*} \mathrm {atan}\left (\frac {\mathrm {cot}\left (x\right )}{\sqrt {-{\mathrm {cot}\left (x\right )}^2-1}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((- cot(x)^2 - 1)^(1/2),x)

[Out]

atan(cot(x)/(- cot(x)^2 - 1)^(1/2))

________________________________________________________________________________________