\(\int \frac {\cot ^2(x)}{\sqrt {a+b \cot ^2(x)}} \, dx\) [45]

Optimal result

Integrand size = 17, antiderivative size = 64 \[ \int \frac {\cot ^2(x)}{\sqrt {a+b \cot ^2(x)}} \, dx=\frac {\arctan \left (\frac {\sqrt {a-b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )}{\sqrt {a-b}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )}{\sqrt {b}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {3751, 494, 223, 212, 385, 209} \[ \int \frac {\cot ^2(x)}{\sqrt {a+b \cot ^2(x)}} \, dx=\frac {\arctan \left (\frac {\sqrt {a-b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )}{\sqrt {a-b}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )}{\sqrt {b}} \]

[In]

Int[Cot[x]^2/Sqrt[a + b*Cot[x]^2],x]

[Out]

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

Rule 209

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

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 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 385

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

Rule 494

Int[(((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_)^(n_))^(q_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Dist[e^n/b, Int[
(e*x)^(m - n)*(c + d*x^n)^q, x], x] - Dist[a*(e^n/b), Int[(e*x)^(m - n)*((c + d*x^n)^q/(a + b*x^n)), x], x] /;
FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LeQ[n, m, 2*n - 1] && IntBinomialQ[a, b
, c, d, e, m, n, -1, q, x]

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(158\) vs. \(2(64)=128\).

Time = 0.35 (sec) , antiderivative size = 158, normalized size of antiderivative = 2.47 \[ \int \frac {\cot ^2(x)}{\sqrt {a+b \cot ^2(x)}} \, dx=\frac {\left (-\sqrt {-b} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a-b} \cos (x)}{\sqrt {-a-b+(a-b) \cos (2 x)}}\right )+\sqrt {a-b} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {-b} \cos (x)}{\sqrt {-a-b+(a-b) \cos (2 x)}}\right )\right ) \sqrt {(a+b+(-a+b) \cos (2 x)) \csc ^2(x)} \sin (x)}{\sqrt {a-b} \sqrt {-b} \sqrt {-a-b+(a-b) \cos (2 x)}} \]

[In]

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

[Out]

((-(Sqrt[-b]*ArcTanh[(Sqrt[2]*Sqrt[a - b]*Cos[x])/Sqrt[-a - b + (a - b)*Cos[2*x]]]) + Sqrt[a - b]*ArcTanh[(Sqr
t[2]*Sqrt[-b]*Cos[x])/Sqrt[-a - b + (a - b)*Cos[2*x]]])*Sqrt[(a + b + (-a + b)*Cos[2*x])*Csc[x]^2]*Sin[x])/(Sq
rt[a - b]*Sqrt[-b]*Sqrt[-a - b + (a - b)*Cos[2*x]])

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.25

method	result	size
derivativedivides	\(-\frac {\ln \left (\sqrt {b}\, \cot \left (x \right )+\sqrt {a +b \cot \left (x \right )^{2}}\right )}{\sqrt {b}}+\frac {\sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (x \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (x \right )^{2}}}\right )}{b^{2} \left (a -b \right )}\)	\(80\)
default	\(-\frac {\ln \left (\sqrt {b}\, \cot \left (x \right )+\sqrt {a +b \cot \left (x \right )^{2}}\right )}{\sqrt {b}}+\frac {\sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (x \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (x \right )^{2}}}\right )}{b^{2} \left (a -b \right )}\)	\(80\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (52) = 104\).

Time = 0.32 (sec) , antiderivative size = 588, normalized size of antiderivative = 9.19 \[ \int \frac {\cot ^2(x)}{\sqrt {a+b \cot ^2(x)}} \, dx=\left [-\frac {\sqrt {-a + b} b \log \left (-{\left (a - b\right )} \cos \left (2 \, x\right ) + \sqrt {-a + b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) + b\right ) - {\left (a - b\right )} \sqrt {b} \log \left (\frac {{\left (a - 2 \, b\right )} \cos \left (2 \, x\right ) + 2 \, \sqrt {b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) - a - 2 \, b}{\cos \left (2 \, x\right ) - 1}\right )}{2 \, {\left (a b - b^{2}\right )}}, \frac {2 \, {\left (a - b\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right )}{b \cos \left (2 \, x\right ) + b}\right ) - \sqrt {-a + b} b \log \left (-{\left (a - b\right )} \cos \left (2 \, x\right ) + \sqrt {-a + b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) + b\right )}{2 \, {\left (a b - b^{2}\right )}}, \frac {2 \, \sqrt {a - b} 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}} \sin \left (2 \, x\right )}{{\left (a - b\right )} \cos \left (2 \, x\right ) + a - b}\right ) + {\left (a - b\right )} \sqrt {b} \log \left (\frac {{\left (a - 2 \, b\right )} \cos \left (2 \, x\right ) + 2 \, \sqrt {b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) - a - 2 \, b}{\cos \left (2 \, x\right ) - 1}\right )}{2 \, {\left (a b - b^{2}\right )}}, \frac {\sqrt {a - b} 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}} \sin \left (2 \, x\right )}{{\left (a - b\right )} \cos \left (2 \, x\right ) + a - b}\right ) + {\left (a - b\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right )}{b \cos \left (2 \, x\right ) + b}\right )}{a b - b^{2}}\right ] \]

[In]

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

[Out]

[-1/2*(sqrt(-a + b)*b*log(-(a - b)*cos(2*x) + sqrt(-a + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin
(2*x) + b) - (a - b)*sqrt(b)*log(((a - 2*b)*cos(2*x) + 2*sqrt(b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1
))*sin(2*x) - a - 2*b)/(cos(2*x) - 1)))/(a*b - b^2), 1/2*(2*(a - b)*sqrt(-b)*arctan(sqrt(-b)*sqrt(((a - b)*cos
(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x)/(b*cos(2*x) + b)) - sqrt(-a + b)*b*log(-(a - b)*cos(2*x) + sqrt(-a + b
)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x) + b))/(a*b - b^2), 1/2*(2*sqrt(a - b)*b*arctan(-sqr
t(a - b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x)/((a - b)*cos(2*x) + a - b)) + (a - b)*sqrt(b
)*log(((a - 2*b)*cos(2*x) + 2*sqrt(b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x) - a - 2*b)/(cos
(2*x) - 1)))/(a*b - b^2), (sqrt(a - b)*b*arctan(-sqrt(a - b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*s
in(2*x)/((a - b)*cos(2*x) + a - b)) + (a - b)*sqrt(-b)*arctan(sqrt(-b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*
x) - 1))*sin(2*x)/(b*cos(2*x) + b)))/(a*b - b^2)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 229 vs. \(2 (52) = 104\).

Time = 0.54 (sec) , antiderivative size = 229, normalized size of antiderivative = 3.58 \[ \int \frac {\cot ^2(x)}{\sqrt {a+b \cot ^2(x)}} \, dx=\frac {{\left (2 \, a \arctan \left (\frac {\sqrt {-a + b} \sqrt {b}}{\sqrt {a b - b^{2}}}\right ) - 2 \, b \arctan \left (\frac {\sqrt {-a + b} \sqrt {b}}{\sqrt {a b - b^{2}}}\right ) + \sqrt {a b - b^{2}} \log \left (-a - 2 \, \sqrt {-a + b} \sqrt {b} + 2 \, b\right )\right )} \mathrm {sgn}\left (\sin \left (x\right )\right )}{2 \, \sqrt {a b - b^{2}} \sqrt {-a + b}} - \frac {\frac {2 \, \sqrt {-a + b} \arctan \left (\frac {{\left (\sqrt {-a + b} \cos \left (x\right ) - \sqrt {-a \cos \left (x\right )^{2} + b \cos \left (x\right )^{2} + a}\right )}^{2} + a - 2 \, b}{2 \, \sqrt {a b - b^{2}}}\right )}{\sqrt {a b - b^{2}}} + \frac {\log \left ({\left (\sqrt {-a + b} \cos \left (x\right ) - \sqrt {-a \cos \left (x\right )^{2} + b \cos \left (x\right )^{2} + a}\right )}^{2}\right )}{\sqrt {-a + b}}}{2 \, \mathrm {sgn}\left (\sin \left (x\right )\right )} \]

[In]

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

[Out]

1/2*(2*a*arctan(sqrt(-a + b)*sqrt(b)/sqrt(a*b - b^2)) - 2*b*arctan(sqrt(-a + b)*sqrt(b)/sqrt(a*b - b^2)) + sqr
t(a*b - b^2)*log(-a - 2*sqrt(-a + b)*sqrt(b) + 2*b))*sgn(sin(x))/(sqrt(a*b - b^2)*sqrt(-a + b)) - 1/2*(2*sqrt(
-a + b)*arctan(1/2*((sqrt(-a + b)*cos(x) - sqrt(-a*cos(x)^2 + b*cos(x)^2 + a))^2 + a - 2*b)/sqrt(a*b - b^2))/s
qrt(a*b - b^2) + log((sqrt(-a + b)*cos(x) - sqrt(-a*cos(x)^2 + b*cos(x)^2 + a))^2)/sqrt(-a + b))/sgn(sin(x))

Mupad [F(-1)]

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

[In]

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

[Out]

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