3.16 \(\int \frac {\cot (x)}{\sqrt {a+a \cot ^2(x)}} \, dx\)

Optimal. Leaf size=10 \[ \frac {1}{\sqrt {a \csc ^2(x)}} \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3657, 4124, 32} \[ \frac {1}{\sqrt {a \csc ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Int[Cot[x]/Sqrt[a + a*Cot[x]^2],x]

[Out]

1/Sqrt[a*Csc[x]^2]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

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 4124

Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> Dist[b/(2*f), Subst[In
t[(-1 + x)^((m - 1)/2)*(b*x)^(p - 1), x], x, Sec[e + f*x]^2], x] /; FreeQ[{b, e, f, p}, x] &&  !IntegerQ[p] &&
IntegerQ[(m - 1)/2]

Rubi steps

\begin {align*} \int \frac {\cot (x)}{\sqrt {a+a \cot ^2(x)}} \, dx &=\int \frac {\cot (x)}{\sqrt {a \csc ^2(x)}} \, dx\\ &=-\left (\frac {1}{2} a \operatorname {Subst}\left (\int \frac {1}{(a x)^{3/2}} \, dx,x,\csc ^2(x)\right )\right )\\ &=\frac {1}{\sqrt {a \csc ^2(x)}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 10, normalized size = 1.00 \[ \frac {1}{\sqrt {a \csc ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[x]/Sqrt[a + a*Cot[x]^2],x]

[Out]

1/Sqrt[a*Csc[x]^2]

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fricas [B]  time = 0.52, size = 27, normalized size = 2.70 \[ -\frac {\sqrt {2} \sqrt {-\frac {a}{\cos \left (2 \, x\right ) - 1}} {\left (\cos \left (2 \, x\right ) - 1\right )}}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.20, size = 12, normalized size = 1.20 \[ \frac {\sqrt {a \sin \relax (x)^{2}}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(a*sin(x)^2)/a

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maple [A]  time = 0.19, size = 11, normalized size = 1.10 \[ \frac {1}{\sqrt {a +a \left (\cot ^{2}\relax (x )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(a+a*cot(x)^2)^(1/2)

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maxima [A]  time = 0.55, size = 8, normalized size = 0.80 \[ \frac {1}{\sqrt {\frac {a}{\sin \relax (x)^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/sqrt(a/sin(x)^2)

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mupad [B]  time = 0.48, size = 10, normalized size = 1.00 \[ \frac {\sqrt {{\sin \relax (x)}^2}}{\sqrt {a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)/(a + a*cot(x)^2)^(1/2),x)

[Out]

(sin(x)^2)^(1/2)/a^(1/2)

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sympy [A]  time = 1.20, size = 12, normalized size = 1.20 \[ \frac {1}{\sqrt {a \cot ^{2}{\relax (x )} + a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/sqrt(a*cot(x)**2 + a)

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