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

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Rubi [A]  time = 0.0482915, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, 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 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]

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]

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*}

Mathematica [A]  time = 0.0111266, size = 10, normalized size = 1. \[ \frac{1}{\sqrt{a \csc ^2(x)}} \]

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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Maple [A]  time = 0.019, size = 11, normalized size = 1.1 \begin{align*}{\frac{1}{\sqrt{a+a \left ( \cot \left ( x \right ) \right ) ^{2}}}} \end{align*}

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 = 0.96249, size = 11, normalized size = 1.1 \begin{align*} \frac{1}{\sqrt{\frac{a}{\sin \left (x\right )^{2}}}} \end{align*}

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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Fricas [B]  time = 1.57984, size = 74, normalized size = 7.4 \begin{align*} -\frac{\sqrt{2} \sqrt{-\frac{a}{\cos \left (2 \, x\right ) - 1}}{\left (\cos \left (2 \, x\right ) - 1\right )}}{2 \, a} \end{align*}

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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Sympy [A]  time = 2.77447, size = 12, normalized size = 1.2 \begin{align*} \frac{1}{\sqrt{a \cot ^{2}{\left (x \right )} + a}} \end{align*}

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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Giac [B]  time = 1.2671, size = 39, normalized size = 3.9 \begin{align*} \frac{\sqrt{a} \sin \left (x\right )}{{\left (a \sin \left (x\right )^{2} -{\left (\sin \left (x\right )^{2} - 1\right )} a\right )} \mathrm{sgn}\left (\sin \left (x\right )\right )} \end{align*}

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