\(\int \frac {\cot ^2(x)}{\sqrt {a+a \cot ^2(x)}} \, dx\) [15]

Optimal result

Integrand size = 17, antiderivative size = 31 \[ \int \frac {\cot ^2(x)}{\sqrt {a+a \cot ^2(x)}} \, dx=\frac {\cot (x)}{\sqrt {a \csc ^2(x)}}-\frac {\text {arctanh}(\cos (x)) \csc (x)}{\sqrt {a \csc ^2(x)}} \]

[Out]

cot(x)/(a*csc(x)^2)^(1/2)-arctanh(cos(x))*csc(x)/(a*csc(x)^2)^(1/2)

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {3738, 4210, 2672, 327, 212} \[ \int \frac {\cot ^2(x)}{\sqrt {a+a \cot ^2(x)}} \, dx=\frac {\cot (x)}{\sqrt {a \csc ^2(x)}}-\frac {\csc (x) \text {arctanh}(\cos (x))}{\sqrt {a \csc ^2(x)}} \]

[In]

Int[Cot[x]^2/Sqrt[a + a*Cot[x]^2],x]

[Out]

Cot[x]/Sqrt[a*Csc[x]^2] - (ArcTanh[Cos[x]]*Csc[x])/Sqrt[a*Csc[x]^2]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 2672

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> With[{ff = FreeFactors[S
in[e + f*x], x]}, Dist[ff/f, Subst[Int[(ff*x)^(m + n)/(a^2 - ff^2*x^2)^((n + 1)/2), x], x, a*(Sin[e + f*x]/ff)
], x]] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2]

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 4210

Int[(u_.)*((b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sec[e + f*x], x]}, Di
st[(b*ff^n)^IntPart[p]*((b*Sec[e + f*x]^n)^FracPart[p]/(Sec[e + f*x]/ff)^(n*FracPart[p])), Int[ActivateTrig[u]
*(Sec[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])

Rubi steps \begin{align*} \text {integral}& = \int \frac {\cot ^2(x)}{\sqrt {a \csc ^2(x)}} \, dx \\ & = \frac {\csc (x) \int \cos (x) \cot (x) \, dx}{\sqrt {a \csc ^2(x)}} \\ & = -\frac {\csc (x) \text {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,\cos (x)\right )}{\sqrt {a \csc ^2(x)}} \\ & = \frac {\cot (x)}{\sqrt {a \csc ^2(x)}}-\frac {\csc (x) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (x)\right )}{\sqrt {a \csc ^2(x)}} \\ & = \frac {\cot (x)}{\sqrt {a \csc ^2(x)}}-\frac {\text {arctanh}(\cos (x)) \csc (x)}{\sqrt {a \csc ^2(x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {\cot ^2(x)}{\sqrt {a+a \cot ^2(x)}} \, dx=\frac {\csc (x) \left (\cos (x)-\log \left (\cos \left (\frac {x}{2}\right )\right )+\log \left (\sin \left (\frac {x}{2}\right )\right )\right )}{\sqrt {a \csc ^2(x)}} \]

[In]

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

[Out]

(Csc[x]*(Cos[x] - Log[Cos[x/2]] + Log[Sin[x/2]]))/Sqrt[a*Csc[x]^2]

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.23

method	result	size
derivativedivides	\(-\frac {\ln \left (\sqrt {a}\, \cot \left (x \right )+\sqrt {a +a \cot \left (x \right )^{2}}\right )}{\sqrt {a}}+\frac {\cot \left (x \right )}{\sqrt {a +a \cot \left (x \right )^{2}}}\)	\(38\)
default	\(-\frac {\ln \left (\sqrt {a}\, \cot \left (x \right )+\sqrt {a +a \cot \left (x \right )^{2}}\right )}{\sqrt {a}}+\frac {\cot \left (x \right )}{\sqrt {a +a \cot \left (x \right )^{2}}}\)	\(38\)
risch	\(\frac {i {\mathrm e}^{2 i x}}{2 \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right )}+\frac {i}{2 \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}-\frac {i {\mathrm e}^{i x} \ln \left ({\mathrm e}^{i x}+1\right )}{\sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right )}+\frac {i {\mathrm e}^{i x} \ln \left ({\mathrm e}^{i x}-1\right )}{\sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right )}\)	\(157\)

[In]

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

[Out]

-ln(a^(1/2)*cot(x)+(a+a*cot(x)^2)^(1/2))/a^(1/2)+cot(x)/(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. 77 vs. \(2 (27) = 54\).

Time = 0.28 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.48 \[ \int \frac {\cot ^2(x)}{\sqrt {a+a \cot ^2(x)}} \, dx=\frac {\sqrt {2} \sqrt {-\frac {a}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) + \sqrt {a} \log \left (\frac {2 \, \sqrt {2} \sqrt {a} \sqrt {-\frac {a}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) - a \cos \left (2 \, x\right ) - 3 \, a}{\cos \left (2 \, x\right ) - 1}\right )}{2 \, a} \]

[In]

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

[Out]

1/2*(sqrt(2)*sqrt(-a/(cos(2*x) - 1))*sin(2*x) + sqrt(a)*log((2*sqrt(2)*sqrt(a)*sqrt(-a/(cos(2*x) - 1))*sin(2*x
) - a*cos(2*x) - 3*a)/(cos(2*x) - 1)))/a

Sympy [F]

\[ \int \frac {\cot ^2(x)}{\sqrt {a+a \cot ^2(x)}} \, dx=\int \frac {\cot ^{2}{\left (x \right )}}{\sqrt {a \left (\cot ^{2}{\left (x \right )} + 1\right )}}\, dx \]

[In]

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

[Out]

Integral(cot(x)**2/sqrt(a*(cot(x)**2 + 1)), x)

Maxima [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {\cot ^2(x)}{\sqrt {a+a \cot ^2(x)}} \, dx=-\frac {\sqrt {-a} {\left (\arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right ) - \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right )\right )}}{a} \]

[In]

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

[Out]

-sqrt(-a)*(arctan2(sin(x), cos(x) + 1) - arctan2(sin(x), cos(x) - 1))/a

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.58 \[ \int \frac {\cot ^2(x)}{\sqrt {a+a \cot ^2(x)}} \, dx=\frac {1}{2} \, \sqrt {a} {\left (\frac {2 \, \cos \left (x\right )}{a \mathrm {sgn}\left (\sin \left (x\right )\right )} - \frac {\log \left (\cos \left (x\right ) + 1\right )}{a \mathrm {sgn}\left (\sin \left (x\right )\right )} + \frac {\log \left (-\cos \left (x\right ) + 1\right )}{a \mathrm {sgn}\left (\sin \left (x\right )\right )}\right )} \]

[In]

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

[Out]

1/2*sqrt(a)*(2*cos(x)/(a*sgn(sin(x))) - log(cos(x) + 1)/(a*sgn(sin(x))) + log(-cos(x) + 1)/(a*sgn(sin(x))))

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot ^2(x)}{\sqrt {a+a \cot ^2(x)}} \, dx=\int \frac {{\mathrm {cot}\left (x\right )}^2}{\sqrt {a\,{\mathrm {cot}\left (x\right )}^2+a}} \,d x \]

[In]

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

[Out]

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