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

Optimal. Leaf size=12 \[ -\frac{\cot (x)}{\sqrt{\csc ^2(x)}} \]

[Out]

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

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Rubi [A]  time = 0.0205382, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3657, 4122, 191} \[ -\frac{\cot (x)}{\sqrt{\csc ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[1 + Cot[x]^2],x]

[Out]

-(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 191

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]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{1+\cot ^2(x)}} \, dx &=\int \frac{1}{\sqrt{\csc ^2(x)}} \, dx\\ &=-\operatorname{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]  time = 0.0092309, size = 12, normalized size = 1. \[ -\frac{\cot (x)}{\sqrt{\csc ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[1 + Cot[x]^2],x]

[Out]

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

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Maple [A]  time = 0.017, size = 13, normalized size = 1.1 \begin{align*} -{\cot \left ( x \right ){\frac{1}{\sqrt{1+ \left ( \cot \left ( x \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.49327, size = 14, normalized size = 1.17 \begin{align*} -\frac{1}{\sqrt{\tan \left (x\right )^{2} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/sqrt(tan(x)^2 + 1)

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Fricas [A]  time = 1.744, size = 63, normalized size = 5.25 \begin{align*} -\frac{1}{2} \, \sqrt{2} \sqrt{-\frac{1}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 0.723615, size = 14, normalized size = 1.17 \begin{align*} - \frac{\cot{\left (x \right )}}{\sqrt{\cot ^{2}{\left (x \right )} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cot(x)/sqrt(cot(x)**2 + 1)

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Giac [B]  time = 1.18212, size = 38, normalized size = 3.17 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))