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

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Rubi [A]  time = 0.02, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 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]

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
]

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*}

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Mathematica [A]  time = 0.01, size = 12, normalized size = 1.00 \[ -\frac {\cot (x)}{\sqrt {\csc ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[1 + Cot[x]^2],x]

[Out]

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

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fricas [B]  time = 0.43, size = 21, normalized size = 1.75 \[ -\frac {1}{2} \, \sqrt {2} \sqrt {-\frac {1}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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maple [A]  time = 0.18, size = 13, normalized size = 1.08 \[ -\frac {\cot \relax (x )}{\sqrt {1+\cot ^{2}\relax (x )}} \]

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 = 0.83, size = 10, normalized size = 0.83 \[ -\frac {1}{\sqrt {\tan \relax (x)^{2} + 1}} \]

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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mupad [B]  time = 0.39, size = 12, normalized size = 1.00 \[ -\frac {\sin \left (2\,x\right )}{2\,\sqrt {{\sin \relax (x)}^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cot(x)^2 + 1)^(1/2),x)

[Out]

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

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sympy [A]  time = 0.34, size = 14, normalized size = 1.17 \[ - \frac {\cot {\relax (x )}}{\sqrt {\cot ^{2}{\relax (x )} + 1}} \]

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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