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

Optimal. Leaf size=33 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b}} \]

[Out]

arctanh((a+b*cot(x)^2)^(1/2)/(a-b)^(1/2))/(a-b)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {3670, 444, 63, 208} \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b}} \]

Antiderivative was successfully verified.

[In]

Int[Cot[x]/Sqrt[a + b*Cot[x]^2],x]

[Out]

ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]]/Sqrt[a - b]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 3670

Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol]
:> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(c*ff)/f, Subst[Int[(((d*ff*x)/c)^m*(a + b*(ff*x)^n)^p)/(c^
2 + ff^2*x^2), x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ
[n, 2] || EqQ[n, 4] || (IntegerQ[p] && RationalQ[n]))

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 33, normalized size = 1.00 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]]/Sqrt[a - b]

________________________________________________________________________________________

fricas [B]  time = 0.52, size = 127, normalized size = 3.85 \[ \left [\frac {\log \left (-\sqrt {a - b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} {\left (\cos \left (2 \, x\right ) - 1\right )} - {\left (a - b\right )} \cos \left (2 \, x\right ) + a\right )}{2 \, \sqrt {a - b}}, \frac {\sqrt {-a + b} \arctan \left (-\frac {\sqrt {-a + b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}}{a - b}\right )}{a - b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*log(-sqrt(a - b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*(cos(2*x) - 1) - (a - b)*cos(2*x) + a)/s
qrt(a - b), sqrt(-a + b)*arctan(-sqrt(-a + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))/(a - b))/(a - b)
]

________________________________________________________________________________________

giac [B]  time = 0.56, size = 70, normalized size = 2.12 \[ \frac {\log \left ({\left | 2 \, {\left (\sqrt {a - b} \cos \relax (x)^{2} - \sqrt {a \cos \relax (x)^{4} - b \cos \relax (x)^{4} - 2 \, a \cos \relax (x)^{2} + b \cos \relax (x)^{2} + a}\right )} \sqrt {a - b} - 2 \, a + b \right |}\right )}{2 \, \sqrt {a - b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(abs(2*(sqrt(a - b)*cos(x)^2 - sqrt(a*cos(x)^4 - b*cos(x)^4 - 2*a*cos(x)^2 + b*cos(x)^2 + a))*sqrt(a -
b) - 2*a + b))/sqrt(a - b)

________________________________________________________________________________________

maple [A]  time = 0.15, size = 29, normalized size = 0.88 \[ -\frac {\arctan \left (\frac {\sqrt {a +b \left (\cot ^{2}\relax (x )\right )}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(-a+b)^(1/2)*arctan((a+b*cot(x)^2)^(1/2)/(-a+b)^(1/2))

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a-4*b>0)', see `assume?` for
more details)Is 4*a-4*b positive or negative?

________________________________________________________________________________________

mupad [B]  time = 0.96, size = 27, normalized size = 0.82 \[ \frac {\mathrm {atanh}\left (\frac {\sqrt {b\,{\mathrm {cot}\relax (x)}^2+a}}{\sqrt {a-b}}\right )}{\sqrt {a-b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)/(a + b*cot(x)^2)^(1/2),x)

[Out]

atanh((a + b*cot(x)^2)^(1/2)/(a - b)^(1/2))/(a - b)^(1/2)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot {\relax (x )}}{\sqrt {a + b \cot ^{2}{\relax (x )}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________