\(\int \frac {1}{\sqrt {1+\cot ^2(x)}} \, dx\) [10]

Optimal result

Integrand size = 10, antiderivative size = 12 \[ \int \frac {1}{\sqrt {1+\cot ^2(x)}} \, dx=-\frac {\cot (x)}{\sqrt {\csc ^2(x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3738, 4207, 197} \[ \int \frac {1}{\sqrt {1+\cot ^2(x)}} \, dx=-\frac {\cot (x)}{\sqrt {\csc ^2(x)}} \]

[In]

Int[1/Sqrt[1 + Cot[x]^2],x]

[Out]

-(Cot[x]/Sqrt[Csc[x]^2])

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 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*} \text {integral}& = \int \frac {1}{\sqrt {\csc ^2(x)}} \, dx \\ & = -\text {Subst}\left (\int \frac {1}{\left (1+x^2\right )^{3/2}} \, dx,x,\cot (x)\right ) \\ & = -\frac {\cot (x)}{\sqrt {\csc ^2(x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {1+\cot ^2(x)}} \, dx=-\frac {\cot (x)}{\sqrt {\csc ^2(x)}} \]

[In]

Integrate[1/Sqrt[1 + Cot[x]^2],x]

[Out]

-(Cot[x]/Sqrt[Csc[x]^2])

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.08

method	result	size
derivativedivides	\(-\frac {\cot \left (x \right )}{\sqrt {\cot \left (x \right )^{2}+1}}\)	\(13\)
default	\(-\frac {\cot \left (x \right )}{\sqrt {\cot \left (x \right )^{2}+1}}\)	\(13\)
risch	\(-\frac {i {\mathrm e}^{2 i x}}{2 \sqrt {-\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right )}-\frac {i}{2 \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}\)	\(67\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 21 vs. \(2 (10) = 20\).

Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.75 \[ \int \frac {1}{\sqrt {1+\cot ^2(x)}} \, dx=-\frac {1}{2} \, \sqrt {2} \sqrt {-\frac {1}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) \]

[In]

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

[Out]

-1/2*sqrt(2)*sqrt(-1/(cos(2*x) - 1))*sin(2*x)

Sympy [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {1}{\sqrt {1+\cot ^2(x)}} \, dx=- \frac {\cot {\left (x \right )}}{\sqrt {\cot ^{2}{\left (x \right )} + 1}} \]

[In]

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

[Out]

-cot(x)/sqrt(cot(x)**2 + 1)

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\sqrt {1+\cot ^2(x)}} \, dx=-\frac {1}{\sqrt {\tan \left (x\right )^{2} + 1}} \]

[In]

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

[Out]

-1/sqrt(tan(x)^2 + 1)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 28 vs. \(2 (10) = 20\).

Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 2.33 \[ \int \frac {1}{\sqrt {1+\cot ^2(x)}} \, dx=\frac {2}{{\left (\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} - 1\right )} \mathrm {sgn}\left (\sin \left (x\right )\right )} + 2 \, \mathrm {sgn}\left (\sin \left (x\right )\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 13.11 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {1+\cot ^2(x)}} \, dx=-\frac {\sin \left (2\,x\right )}{2\,\sqrt {{\sin \left (x\right )}^2}} \]

[In]

int(1/(cot(x)^2 + 1)^(1/2),x)

[Out]

-sin(2*x)/(2*(sin(x)^2)^(1/2))