\(\int \frac {\cot (x)}{\sqrt {a+a \cot ^2(x)}} \, dx\) [16]

Optimal result

Integrand size = 15, antiderivative size = 10 \[ \int \frac {\cot (x)}{\sqrt {a+a \cot ^2(x)}} \, dx=\frac {1}{\sqrt {a \csc ^2(x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3738, 4209, 32} \[ \int \frac {\cot (x)}{\sqrt {a+a \cot ^2(x)}} \, dx=\frac {1}{\sqrt {a \csc ^2(x)}} \]

[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 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 4209

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*} \text {integral}& = \int \frac {\cot (x)}{\sqrt {a \csc ^2(x)}} \, dx \\ & = -\left (\frac {1}{2} a \text {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] (verified)

Time = 0.03 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {\cot (x)}{\sqrt {a+a \cot ^2(x)}} \, dx=\frac {1}{\sqrt {a \csc ^2(x)}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.10

method	result	size
derivativedivides	\(\frac {1}{\sqrt {a +a \cot \left (x \right )^{2}}}\)	\(11\)
default	\(\frac {1}{\sqrt {a +a \cot \left (x \right )^{2}}}\)	\(11\)
risch	\(\frac {{\mathrm e}^{2 i x}}{2 \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right )}-\frac {1}{2 \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}\)	\(67\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (8) = 16\).

Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 2.70 \[ \int \frac {\cot (x)}{\sqrt {a+a \cot ^2(x)}} \, dx=-\frac {\sqrt {2} \sqrt {-\frac {a}{\cos \left (2 \, x\right ) - 1}} {\left (\cos \left (2 \, x\right ) - 1\right )}}{2 \, a} \]

[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

Sympy [A] (verification not implemented)

Time = 0.54 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\cot (x)}{\sqrt {a+a \cot ^2(x)}} \, dx=\frac {1}{\sqrt {a \cot ^{2}{\left (x \right )} + a}} \]

[In]

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

[Out]

1/sqrt(a*cot(x)**2 + a)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {\cot (x)}{\sqrt {a+a \cot ^2(x)}} \, dx=\frac {1}{\sqrt {\frac {a}{\sin \left (x\right )^{2}}}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.10 \[ \int \frac {\cot (x)}{\sqrt {a+a \cot ^2(x)}} \, dx=\frac {\sin \left (x\right )}{\sqrt {a} \mathrm {sgn}\left (\sin \left (x\right )\right )} \]

[In]

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

[Out]

sin(x)/(sqrt(a)*sgn(sin(x)))

Mupad [B] (verification not implemented)

Time = 12.98 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {\cot (x)}{\sqrt {a+a \cot ^2(x)}} \, dx=\frac {\sqrt {{\sin \left (x\right )}^2}}{\sqrt {a}} \]

[In]

int(cot(x)/(a + a*cot(x)^2)^(1/2),x)

[Out]

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