3.55 \(\int \frac {\cot ^2(x)}{(a+b \cot ^2(x))^{5/2}} \, dx\)

Optimal. Leaf size=94 \[ -\frac {(2 a+b) \cot (x)}{3 a (a-b)^2 \sqrt {a+b \cot ^2(x)}}-\frac {\cot (x)}{3 (a-b) \left (a+b \cot ^2(x)\right )^{3/2}}+\frac {\tan ^{-1}\left (\frac {\sqrt {a-b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )}{(a-b)^{5/2}} \]

[Out]

arctan(cot(x)*(a-b)^(1/2)/(a+b*cot(x)^2)^(1/2))/(a-b)^(5/2)-1/3*cot(x)/(a-b)/(a+b*cot(x)^2)^(3/2)-1/3*(2*a+b)*
cot(x)/a/(a-b)^2/(a+b*cot(x)^2)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.12, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {3670, 471, 527, 12, 377, 203} \[ -\frac {(2 a+b) \cot (x)}{3 a (a-b)^2 \sqrt {a+b \cot ^2(x)}}-\frac {\cot (x)}{3 (a-b) \left (a+b \cot ^2(x)\right )^{3/2}}+\frac {\tan ^{-1}\left (\frac {\sqrt {a-b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )}{(a-b)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[Cot[x]^2/(a + b*Cot[x]^2)^(5/2),x]

[Out]

ArcTan[(Sqrt[a - b]*Cot[x])/Sqrt[a + b*Cot[x]^2]]/(a - b)^(5/2) - Cot[x]/(3*(a - b)*(a + b*Cot[x]^2)^(3/2)) -
((2*a + b)*Cot[x])/(3*a*(a - b)^2*Sqrt[a + b*Cot[x]^2])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 203

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

Rule 377

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 471

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

Rule 527

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> -Simp[
((b*e - a*f)*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d
)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)
*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

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 ^2(x)}{\left (a+b \cot ^2(x)\right )^{5/2}} \, dx &=-\operatorname {Subst}\left (\int \frac {x^2}{\left (1+x^2\right ) \left (a+b x^2\right )^{5/2}} \, dx,x,\cot (x)\right )\\ &=-\frac {\cot (x)}{3 (a-b) \left (a+b \cot ^2(x)\right )^{3/2}}+\frac {\operatorname {Subst}\left (\int \frac {1-2 x^2}{\left (1+x^2\right ) \left (a+b x^2\right )^{3/2}} \, dx,x,\cot (x)\right )}{3 (a-b)}\\ &=-\frac {\cot (x)}{3 (a-b) \left (a+b \cot ^2(x)\right )^{3/2}}-\frac {(2 a+b) \cot (x)}{3 a (a-b)^2 \sqrt {a+b \cot ^2(x)}}+\frac {\operatorname {Subst}\left (\int \frac {3 a}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\cot (x)\right )}{3 a (a-b)^2}\\ &=-\frac {\cot (x)}{3 (a-b) \left (a+b \cot ^2(x)\right )^{3/2}}-\frac {(2 a+b) \cot (x)}{3 a (a-b)^2 \sqrt {a+b \cot ^2(x)}}+\frac {\operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\cot (x)\right )}{(a-b)^2}\\ &=-\frac {\cot (x)}{3 (a-b) \left (a+b \cot ^2(x)\right )^{3/2}}-\frac {(2 a+b) \cot (x)}{3 a (a-b)^2 \sqrt {a+b \cot ^2(x)}}+\frac {\operatorname {Subst}\left (\int \frac {1}{1-(-a+b) x^2} \, dx,x,\frac {\cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )}{(a-b)^2}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {a-b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )}{(a-b)^{5/2}}-\frac {\cot (x)}{3 (a-b) \left (a+b \cot ^2(x)\right )^{3/2}}-\frac {(2 a+b) \cot (x)}{3 a (a-b)^2 \sqrt {a+b \cot ^2(x)}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 6.51, size = 200, normalized size = 2.13 \[ \frac {\tan (x) \left (-\frac {35 a \sin ^2(x) \left (5 a+2 b \cot ^2(x)\right ) \left (a \csc ^2(x) \left ((a-4 b) \cot ^2(x)-3 a\right ) \sqrt {\frac {(a-b) \sin ^2(x) \cos ^2(x) \left (a+b \cot ^2(x)\right )}{a^2}}+3 \left (a+b \cot ^2(x)\right )^2 \sin ^{-1}\left (\sqrt {\frac {(a-b) \cos ^2(x)}{a}}\right )\right )}{\sqrt {\frac {(a-b) \sin ^2(x) \cos ^2(x) \left (a+b \cot ^2(x)\right )}{a^2}}}-12 (a-b)^3 \cos ^4(x) \cot ^2(x) \left (a+b \cot ^2(x)\right ) \, _2F_1\left (2,2;\frac {9}{2};\frac {(a-b) \cos ^2(x)}{a}\right )\right )}{315 a^3 (a-b)^2 \left (a+b \cot ^2(x)\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Cot[x]^2/(a + b*Cot[x]^2)^(5/2),x]

[Out]

((-12*(a - b)^3*Cos[x]^4*Cot[x]^2*(a + b*Cot[x]^2)*Hypergeometric2F1[2, 2, 9/2, ((a - b)*Cos[x]^2)/a] - (35*a*
(5*a + 2*b*Cot[x]^2)*Sin[x]^2*(3*ArcSin[Sqrt[((a - b)*Cos[x]^2)/a]]*(a + b*Cot[x]^2)^2 + a*(-3*a + (a - 4*b)*C
ot[x]^2)*Csc[x]^2*Sqrt[((a - b)*Cos[x]^2*(a + b*Cot[x]^2)*Sin[x]^2)/a^2]))/Sqrt[((a - b)*Cos[x]^2*(a + b*Cot[x
]^2)*Sin[x]^2)/a^2])*Tan[x])/(315*a^3*(a - b)^2*(a + b*Cot[x]^2)^(3/2))

________________________________________________________________________________________

fricas [B]  time = 1.35, size = 720, normalized size = 7.66 \[ \left [-\frac {3 \, {\left (a^{3} + 2 \, a^{2} b + a b^{2} + {\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (a^{3} - a b^{2}\right )} \cos \left (2 \, x\right )\right )} \sqrt {-a + b} \log \left (-2 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (2 \, x\right )^{2} + 2 \, {\left ({\left (a - b\right )} \cos \left (2 \, x\right ) - b\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 ) + a^{2} - 2 \, b^{2} + 4 \, {\left (a b - b^{2}\right )} \cos \left (2 \, x\right )\right ) + 4 \, {\left (3 \, a^{3} - a^{2} b - a b^{2} - b^{3} - {\left (3 \, a^{3} - 5 \, a^{2} b + a b^{2} + b^{3}\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}} \sin \left (2 \, x\right )}{12 \, {\left (a^{6} - a^{5} b - 2 \, a^{4} b^{2} + 2 \, a^{3} b^{3} + a^{2} b^{4} - a b^{5} + {\left (a^{6} - 5 \, a^{5} b + 10 \, a^{4} b^{2} - 10 \, a^{3} b^{3} + 5 \, a^{2} b^{4} - a b^{5}\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (a^{6} - 3 \, a^{5} b + 2 \, a^{4} b^{2} + 2 \, a^{3} b^{3} - 3 \, a^{2} b^{4} + a b^{5}\right )} \cos \left (2 \, x\right )\right )}}, \frac {3 \, {\left (a^{3} + 2 \, a^{2} b + a b^{2} + {\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (a^{3} - a b^{2}\right )} \cos \left (2 \, x\right )\right )} \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}} \sin \left (2 \, x\right )}{{\left (a - b\right )} \cos \left (2 \, x\right ) - b}\right ) - 2 \, {\left (3 \, a^{3} - a^{2} b - a b^{2} - b^{3} - {\left (3 \, a^{3} - 5 \, a^{2} b + a b^{2} + b^{3}\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}} \sin \left (2 \, x\right )}{6 \, {\left (a^{6} - a^{5} b - 2 \, a^{4} b^{2} + 2 \, a^{3} b^{3} + a^{2} b^{4} - a b^{5} + {\left (a^{6} - 5 \, a^{5} b + 10 \, a^{4} b^{2} - 10 \, a^{3} b^{3} + 5 \, a^{2} b^{4} - a b^{5}\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (a^{6} - 3 \, a^{5} b + 2 \, a^{4} b^{2} + 2 \, a^{3} b^{3} - 3 \, a^{2} b^{4} + a b^{5}\right )} \cos \left (2 \, x\right )\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^2/(a+b*cot(x)^2)^(5/2),x, algorithm="fricas")

[Out]

[-1/12*(3*(a^3 + 2*a^2*b + a*b^2 + (a^3 - 2*a^2*b + a*b^2)*cos(2*x)^2 - 2*(a^3 - a*b^2)*cos(2*x))*sqrt(-a + b)
*log(-2*(a^2 - 2*a*b + b^2)*cos(2*x)^2 + 2*((a - b)*cos(2*x) - b)*sqrt(-a + b)*sqrt(((a - b)*cos(2*x) - a - b)
/(cos(2*x) - 1))*sin(2*x) + a^2 - 2*b^2 + 4*(a*b - b^2)*cos(2*x)) + 4*(3*a^3 - a^2*b - a*b^2 - b^3 - (3*a^3 -
5*a^2*b + a*b^2 + b^3)*cos(2*x))*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x))/(a^6 - a^5*b - 2*a^
4*b^2 + 2*a^3*b^3 + a^2*b^4 - a*b^5 + (a^6 - 5*a^5*b + 10*a^4*b^2 - 10*a^3*b^3 + 5*a^2*b^4 - a*b^5)*cos(2*x)^2
- 2*(a^6 - 3*a^5*b + 2*a^4*b^2 + 2*a^3*b^3 - 3*a^2*b^4 + a*b^5)*cos(2*x)), 1/6*(3*(a^3 + 2*a^2*b + a*b^2 + (a
^3 - 2*a^2*b + a*b^2)*cos(2*x)^2 - 2*(a^3 - a*b^2)*cos(2*x))*sqrt(a - b)*arctan(-sqrt(a - b)*sqrt(((a - b)*cos
(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x)/((a - b)*cos(2*x) - b)) - 2*(3*a^3 - a^2*b - a*b^2 - b^3 - (3*a^3 - 5*
a^2*b + a*b^2 + b^3)*cos(2*x))*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x))/(a^6 - a^5*b - 2*a^4*
b^2 + 2*a^3*b^3 + a^2*b^4 - a*b^5 + (a^6 - 5*a^5*b + 10*a^4*b^2 - 10*a^3*b^3 + 5*a^2*b^4 - a*b^5)*cos(2*x)^2 -
2*(a^6 - 3*a^5*b + 2*a^4*b^2 + 2*a^3*b^3 - 3*a^2*b^4 + a*b^5)*cos(2*x))]

________________________________________________________________________________________

giac [B]  time = 2.40, size = 1025, normalized size = 10.90 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^2/(a+b*cot(x)^2)^(5/2),x, algorithm="giac")

[Out]

1/3*(2*a + b)*sgn(tan(1/2*x))/(a^3*sqrt(b) - 2*a^2*b^(3/2) + a*b^(5/2)) + 1/3*((((2*a^10*b*sgn(tan(1/2*x)) - 1
5*a^9*b^2*sgn(tan(1/2*x)) + 48*a^8*b^3*sgn(tan(1/2*x)) - 84*a^7*b^4*sgn(tan(1/2*x)) + 84*a^6*b^5*sgn(tan(1/2*x
)) - 42*a^5*b^6*sgn(tan(1/2*x)) + 12*a^3*b^8*sgn(tan(1/2*x)) - 6*a^2*b^9*sgn(tan(1/2*x)) + a*b^10*sgn(tan(1/2*
x)))*tan(1/2*x)^2/(a^12 - 10*a^11*b + 45*a^10*b^2 - 120*a^9*b^3 + 210*a^8*b^4 - 252*a^7*b^5 + 210*a^6*b^6 - 12
0*a^5*b^7 + 45*a^4*b^8 - 10*a^3*b^9 + a^2*b^10) + 3*(4*a^11*sgn(tan(1/2*x)) - 34*a^10*b*sgn(tan(1/2*x)) + 127*
a^9*b^2*sgn(tan(1/2*x)) - 272*a^8*b^3*sgn(tan(1/2*x)) + 364*a^7*b^4*sgn(tan(1/2*x)) - 308*a^6*b^5*sgn(tan(1/2*
x)) + 154*a^5*b^6*sgn(tan(1/2*x)) - 32*a^4*b^7*sgn(tan(1/2*x)) - 8*a^3*b^8*sgn(tan(1/2*x)) + 6*a^2*b^9*sgn(tan
(1/2*x)) - a*b^10*sgn(tan(1/2*x)))/(a^12 - 10*a^11*b + 45*a^10*b^2 - 120*a^9*b^3 + 210*a^8*b^4 - 252*a^7*b^5 +
210*a^6*b^6 - 120*a^5*b^7 + 45*a^4*b^8 - 10*a^3*b^9 + a^2*b^10))*tan(1/2*x)^2 - 3*(4*a^11*sgn(tan(1/2*x)) - 3
4*a^10*b*sgn(tan(1/2*x)) + 127*a^9*b^2*sgn(tan(1/2*x)) - 272*a^8*b^3*sgn(tan(1/2*x)) + 364*a^7*b^4*sgn(tan(1/2
*x)) - 308*a^6*b^5*sgn(tan(1/2*x)) + 154*a^5*b^6*sgn(tan(1/2*x)) - 32*a^4*b^7*sgn(tan(1/2*x)) - 8*a^3*b^8*sgn(
tan(1/2*x)) + 6*a^2*b^9*sgn(tan(1/2*x)) - a*b^10*sgn(tan(1/2*x)))/(a^12 - 10*a^11*b + 45*a^10*b^2 - 120*a^9*b^
3 + 210*a^8*b^4 - 252*a^7*b^5 + 210*a^6*b^6 - 120*a^5*b^7 + 45*a^4*b^8 - 10*a^3*b^9 + a^2*b^10))*tan(1/2*x)^2
- (2*a^10*b*sgn(tan(1/2*x)) - 15*a^9*b^2*sgn(tan(1/2*x)) + 48*a^8*b^3*sgn(tan(1/2*x)) - 84*a^7*b^4*sgn(tan(1/2
*x)) + 84*a^6*b^5*sgn(tan(1/2*x)) - 42*a^5*b^6*sgn(tan(1/2*x)) + 12*a^3*b^8*sgn(tan(1/2*x)) - 6*a^2*b^9*sgn(ta
n(1/2*x)) + a*b^10*sgn(tan(1/2*x)))/(a^12 - 10*a^11*b + 45*a^10*b^2 - 120*a^9*b^3 + 210*a^8*b^4 - 252*a^7*b^5
+ 210*a^6*b^6 - 120*a^5*b^7 + 45*a^4*b^8 - 10*a^3*b^9 + a^2*b^10))/(b*tan(1/2*x)^4 + 4*a*tan(1/2*x)^2 - 2*b*ta
n(1/2*x)^2 + b)^(3/2) - 2*arctan(-1/2*(sqrt(b)*tan(1/2*x)^2 - sqrt(b*tan(1/2*x)^4 + 4*a*tan(1/2*x)^2 - 2*b*tan
(1/2*x)^2 + b) + sqrt(b))/sqrt(a - b))/((a^2*sgn(tan(1/2*x)) - 2*a*b*sgn(tan(1/2*x)) + b^2*sgn(tan(1/2*x)))*sq
rt(a - b))

________________________________________________________________________________________

maple [B]  time = 0.20, size = 166, normalized size = 1.77 \[ -\frac {\cot \relax (x )}{3 a \left (a +b \left (\cot ^{2}\relax (x )\right )\right )^{\frac {3}{2}}}-\frac {2 \cot \relax (x )}{3 a^{2} \sqrt {a +b \left (\cot ^{2}\relax (x )\right )}}-\frac {b \cot \relax (x )}{\left (a -b \right )^{2} a \sqrt {a +b \left (\cot ^{2}\relax (x )\right )}}-\frac {b \cot \relax (x )}{3 \left (a -b \right ) a \left (a +b \left (\cot ^{2}\relax (x )\right )\right )^{\frac {3}{2}}}-\frac {2 b \cot \relax (x )}{3 \left (a -b \right ) a^{2} \sqrt {a +b \left (\cot ^{2}\relax (x )\right )}}+\frac {\sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {\left (a -b \right ) b^{2} \cot \relax (x )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \left (\cot ^{2}\relax (x )\right )}}\right )}{\left (a -b \right )^{3} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)^2/(a+b*cot(x)^2)^(5/2),x)

[Out]

-1/3*cot(x)/a/(a+b*cot(x)^2)^(3/2)-2/3/a^2*cot(x)/(a+b*cot(x)^2)^(1/2)-b/(a-b)^2*cot(x)/a/(a+b*cot(x)^2)^(1/2)
-1/3*b/(a-b)*cot(x)/a/(a+b*cot(x)^2)^(3/2)-2/3*b/(a-b)/a^2*cot(x)/(a+b*cot(x)^2)^(1/2)+1/(a-b)^3*(b^4*(a-b))^(
1/2)/b^2*arctan((a-b)*b^2/(b^4*(a-b))^(1/2)/(a+b*cot(x)^2)^(1/2)*cot(x))

________________________________________________________________________________________

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)^2/(a+b*cot(x)^2)^(5/2),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(b-a>0)', see `assume?` for mor
e details)Is b-a positive or negative?

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {cot}\relax (x)}^2}{{\left (b\,{\mathrm {cot}\relax (x)}^2+a\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)^2/(a + b*cot(x)^2)^(5/2),x)

[Out]

int(cot(x)^2/(a + b*cot(x)^2)^(5/2), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot ^{2}{\relax (x )}}{\left (a + b \cot ^{2}{\relax (x )}\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)**2/(a+b*cot(x)**2)**(5/2),x)

[Out]

Integral(cot(x)**2/(a + b*cot(x)**2)**(5/2), x)

________________________________________________________________________________________