\(\int \cot (x) \sqrt {a+b \cot ^2(x)} \, dx\) [20]

Optimal result

Integrand size = 15, antiderivative size = 48 \[ \int \cot (x) \sqrt {a+b \cot ^2(x)} \, dx=\sqrt {a-b} \text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )-\sqrt {a+b \cot ^2(x)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3751, 455, 52, 65, 214} \[ \int \cot (x) \sqrt {a+b \cot ^2(x)} \, dx=\sqrt {a-b} \text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )-\sqrt {a+b \cot ^2(x)} \]

[In]

Int[Cot[x]*Sqrt[a + b*Cot[x]^2],x]

[Out]

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

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

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 214

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

Rule 455

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 3751

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00 \[ \int \cot (x) \sqrt {a+b \cot ^2(x)} \, dx=\sqrt {a-b} \text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )-\sqrt {a+b \cot ^2(x)} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.48

method	result	size
derivativedivides	\(-\sqrt {a +b \cot \left (x \right )^{2}}+\frac {b \arctan \left (\frac {\sqrt {a +b \cot \left (x \right )^{2}}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}-\frac {a \arctan \left (\frac {\sqrt {a +b \cot \left (x \right )^{2}}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}\)	\(71\)
default	\(-\sqrt {a +b \cot \left (x \right )^{2}}+\frac {b \arctan \left (\frac {\sqrt {a +b \cot \left (x \right )^{2}}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}-\frac {a \arctan \left (\frac {\sqrt {a +b \cot \left (x \right )^{2}}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}\)	\(71\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (40) = 80\).

Time = 0.34 (sec) , antiderivative size = 248, normalized size of antiderivative = 5.17 \[ \int \cot (x) \sqrt {a+b \cot ^2(x)} \, dx=\left [\frac {1}{4} \, \sqrt {a - b} \log \left (-2 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (2 \, x\right )^{2} - 2 \, a^{2} + b^{2} - 2 \, {\left ({\left (a - b\right )} \cos \left (2 \, x\right )^{2} - {\left (2 \, a - b\right )} \cos \left (2 \, x\right ) + a\right )} \sqrt {a - b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} + 4 \, {\left (a^{2} - a b\right )} \cos \left (2 \, x\right )\right ) - \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}, \frac {1}{2} \, \sqrt {-a + b} \arctan \left (-\frac {\sqrt {-a + b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} {\left (\cos \left (2 \, x\right ) - 1\right )}}{{\left (a - b\right )} \cos \left (2 \, x\right ) - a}\right ) - \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}\right ] \]

[In]

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

[Out]

[1/4*sqrt(a - b)*log(-2*(a^2 - 2*a*b + b^2)*cos(2*x)^2 - 2*a^2 + b^2 - 2*((a - b)*cos(2*x)^2 - (2*a - b)*cos(2
*x) + a)*sqrt(a - b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)) + 4*(a^2 - a*b)*cos(2*x)) - sqrt(((a - b)
*cos(2*x) - a - b)/(cos(2*x) - 1)), 1/2*sqrt(-a + b)*arctan(-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)) - sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \cot (x) \sqrt {a+b \cot ^2(x)} \, dx=\text {Exception raised: ValueError} \]

[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 detail

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (40) = 80\).

Time = 0.34 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.98 \[ \int \cot (x) \sqrt {a+b \cot ^2(x)} \, dx=-\frac {1}{2} \, {\left (\sqrt {a - b} \log \left ({\left (\sqrt {a - b} \sin \left (x\right ) - \sqrt {a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b}\right )}^{2}\right ) - \frac {4 \, \sqrt {a - b} b}{{\left (\sqrt {a - b} \sin \left (x\right ) - \sqrt {a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b}\right )}^{2} - b}\right )} \mathrm {sgn}\left (\sin \left (x\right )\right ) \]

[In]

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

[Out]

-1/2*(sqrt(a - b)*log((sqrt(a - b)*sin(x) - sqrt(a*sin(x)^2 - b*sin(x)^2 + b))^2) - 4*sqrt(a - b)*b/((sqrt(a -
b)*sin(x) - sqrt(a*sin(x)^2 - b*sin(x)^2 + b))^2 - b))*sgn(sin(x))

Mupad [B] (verification not implemented)

Time = 13.96 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.10 \[ \int \cot (x) \sqrt {a+b \cot ^2(x)} \, dx=-\sqrt {b\,{\mathrm {cot}\left (x\right )}^2+a}-2\,\mathrm {atan}\left (\frac {2\,\sqrt {b\,{\mathrm {cot}\left (x\right )}^2+a}\,\sqrt {\frac {b}{4}-\frac {a}{4}}}{a-b}\right )\,\sqrt {\frac {b}{4}-\frac {a}{4}} \]

[In]

int(cot(x)*(a + b*cot(x)^2)^(1/2),x)

[Out]

- (a + b*cot(x)^2)^(1/2) - 2*atan((2*(a + b*cot(x)^2)^(1/2)*(b/4 - a/4)^(1/2))/(a - b))*(b/4 - a/4)^(1/2)