\(\int \frac {\tan ^2(x)}{(a+b \cot ^2(x))^{5/2}} \, dx\) [58]

Optimal result

Integrand size = 17, antiderivative size = 141 \[ \int \frac {\tan ^2(x)}{\left (a+b \cot ^2(x)\right )^{5/2}} \, dx=\frac {\arctan \left (\frac {\sqrt {a-b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )}{(a-b)^{5/2}}+\frac {b \tan (x)}{3 a (a-b) \left (a+b \cot ^2(x)\right )^{3/2}}+\frac {(7 a-4 b) b \tan (x)}{3 a^2 (a-b)^2 \sqrt {a+b \cot ^2(x)}}+\frac {(a-4 b) (3 a-2 b) \sqrt {a+b \cot ^2(x)} \tan (x)}{3 a^3 (a-b)^2} \]

[Out]

arctan(cot(x)*(a-b)^(1/2)/(a+b*cot(x)^2)^(1/2))/(a-b)^(5/2)+1/3*b*tan(x)/a/(a-b)/(a+b*cot(x)^2)^(3/2)+1/3*(7*a
-4*b)*b*tan(x)/a^2/(a-b)^2/(a+b*cot(x)^2)^(1/2)+1/3*(a-4*b)*(3*a-2*b)*(a+b*cot(x)^2)^(1/2)*tan(x)/a^3/(a-b)^2

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {3751, 483, 593, 597, 12, 385, 209} \[ \int \frac {\tan ^2(x)}{\left (a+b \cot ^2(x)\right )^{5/2}} \, dx=\frac {(a-4 b) (3 a-2 b) \tan (x) \sqrt {a+b \cot ^2(x)}}{3 a^3 (a-b)^2}+\frac {b (7 a-4 b) \tan (x)}{3 a^2 (a-b)^2 \sqrt {a+b \cot ^2(x)}}+\frac {\arctan \left (\frac {\sqrt {a-b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )}{(a-b)^{5/2}}+\frac {b \tan (x)}{3 a (a-b) \left (a+b \cot ^2(x)\right )^{3/2}} \]

[In]

Int[Tan[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) + (b*Tan[x])/(3*a*(a - b)*(a + b*Cot[x]^2)^(3/
2)) + ((7*a - 4*b)*b*Tan[x])/(3*a^2*(a - b)^2*Sqrt[a + b*Cot[x]^2]) + ((a - 4*b)*(3*a - 2*b)*Sqrt[a + b*Cot[x]
^2]*Tan[x])/(3*a^3*(a - b)^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 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 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 483

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

Rule 593

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_)*((e_) + (f_.)*(x_)^(n_)), x
_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*g*n*(b*c - a*d)*(p +
1))), x] + Dist[1/(a*n*(b*c - a*d)*(p + 1)), Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f)
*(m + 1) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, e, f, g, m, q}, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 597

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]

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 {1}{x^2 \left (1+x^2\right ) \left (a+b x^2\right )^{5/2}} \, dx,x,\cot (x)\right ) \\ & = \frac {b \tan (x)}{3 a (a-b) \left (a+b \cot ^2(x)\right )^{3/2}}-\frac {\text {Subst}\left (\int \frac {3 a-4 b-4 b x^2}{x^2 \left (1+x^2\right ) \left (a+b x^2\right )^{3/2}} \, dx,x,\cot (x)\right )}{3 a (a-b)} \\ & = \frac {b \tan (x)}{3 a (a-b) \left (a+b \cot ^2(x)\right )^{3/2}}+\frac {(7 a-4 b) b \tan (x)}{3 a^2 (a-b)^2 \sqrt {a+b \cot ^2(x)}}-\frac {\text {Subst}\left (\int \frac {(a-4 b) (3 a-2 b)-2 (7 a-4 b) b x^2}{x^2 \left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\cot (x)\right )}{3 a^2 (a-b)^2} \\ & = \frac {b \tan (x)}{3 a (a-b) \left (a+b \cot ^2(x)\right )^{3/2}}+\frac {(7 a-4 b) b \tan (x)}{3 a^2 (a-b)^2 \sqrt {a+b \cot ^2(x)}}+\frac {(a-4 b) (3 a-2 b) \sqrt {a+b \cot ^2(x)} \tan (x)}{3 a^3 (a-b)^2}+\frac {\text {Subst}\left (\int \frac {3 a^3}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\cot (x)\right )}{3 a^3 (a-b)^2} \\ & = \frac {b \tan (x)}{3 a (a-b) \left (a+b \cot ^2(x)\right )^{3/2}}+\frac {(7 a-4 b) b \tan (x)}{3 a^2 (a-b)^2 \sqrt {a+b \cot ^2(x)}}+\frac {(a-4 b) (3 a-2 b) \sqrt {a+b \cot ^2(x)} \tan (x)}{3 a^3 (a-b)^2}+\frac {\text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\cot (x)\right )}{(a-b)^2} \\ & = \frac {b \tan (x)}{3 a (a-b) \left (a+b \cot ^2(x)\right )^{3/2}}+\frac {(7 a-4 b) b \tan (x)}{3 a^2 (a-b)^2 \sqrt {a+b \cot ^2(x)}}+\frac {(a-4 b) (3 a-2 b) \sqrt {a+b \cot ^2(x)} \tan (x)}{3 a^3 (a-b)^2}+\frac {\text {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 {\arctan \left (\frac {\sqrt {a-b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )}{(a-b)^{5/2}}+\frac {b \tan (x)}{3 a (a-b) \left (a+b \cot ^2(x)\right )^{3/2}}+\frac {(7 a-4 b) b \tan (x)}{3 a^2 (a-b)^2 \sqrt {a+b \cot ^2(x)}}+\frac {(a-4 b) (3 a-2 b) \sqrt {a+b \cot ^2(x)} \tan (x)}{3 a^3 (a-b)^2} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 8.16 (sec) , antiderivative size = 1450, normalized size of antiderivative = 10.28 \[ \int \frac {\tan ^2(x)}{\left (a+b \cot ^2(x)\right )^{5/2}} \, dx=\frac {\sin ^2(x) \left (-\frac {16 b^3 \left (\cot (x)+\cot ^3(x)\right )^2}{a (a-b)^2}+\frac {40 b \csc ^2(x)}{a-b}+\frac {160 b^2 \cot ^2(x) \csc ^2(x)}{3 a (a-b)}+\frac {64 b^3 \cot ^4(x) \csc ^2(x)}{3 a^2 (a-b)}-\frac {40 b^2 \csc ^4(x)}{(a-b)^2}+\frac {92 (a-b) \cos ^2(x) \operatorname {Hypergeometric2F1}\left (2,2,\frac {9}{2},\frac {(a-b) \cos ^2(x)}{a}\right )}{105 a}+\frac {124 (a-b) b \cos ^2(x) \cot ^2(x) \operatorname {Hypergeometric2F1}\left (2,2,\frac {9}{2},\frac {(a-b) \cos ^2(x)}{a}\right )}{35 a^2}+\frac {152 (a-b) b^2 \cos ^2(x) \cot ^4(x) \operatorname {Hypergeometric2F1}\left (2,2,\frac {9}{2},\frac {(a-b) \cos ^2(x)}{a}\right )}{35 a^3}+\frac {176 (a-b) b^3 \cos ^2(x) \cot ^6(x) \operatorname {Hypergeometric2F1}\left (2,2,\frac {9}{2},\frac {(a-b) \cos ^2(x)}{a}\right )}{105 a^4}+\frac {24 (a-b) \cos ^2(x) \, _3F_2\left (2,2,2;1,\frac {9}{2};\frac {(a-b) \cos ^2(x)}{a}\right )}{35 a}+\frac {16 (a-b) b \cos ^2(x) \cot ^2(x) \, _3F_2\left (2,2,2;1,\frac {9}{2};\frac {(a-b) \cos ^2(x)}{a}\right )}{7 a^2}+\frac {88 (a-b) b^2 \cos ^2(x) \cot ^4(x) \, _3F_2\left (2,2,2;1,\frac {9}{2};\frac {(a-b) \cos ^2(x)}{a}\right )}{35 a^3}+\frac {32 (a-b) b^3 \cos ^2(x) \cot ^6(x) \, _3F_2\left (2,2,2;1,\frac {9}{2};\frac {(a-b) \cos ^2(x)}{a}\right )}{35 a^4}+\frac {16 (a-b) \cos ^2(x) \, _4F_3\left (2,2,2,2;1,1,\frac {9}{2};\frac {(a-b) \cos ^2(x)}{a}\right )}{105 a}+\frac {16 (a-b) b \cos ^2(x) \cot ^2(x) \, _4F_3\left (2,2,2,2;1,1,\frac {9}{2};\frac {(a-b) \cos ^2(x)}{a}\right )}{35 a^2}+\frac {16 (a-b) b^2 \cos ^2(x) \cot ^4(x) \, _4F_3\left (2,2,2,2;1,1,\frac {9}{2};\frac {(a-b) \cos ^2(x)}{a}\right )}{35 a^3}+\frac {16 (a-b) b^3 \cos ^2(x) \cot ^6(x) \, _4F_3\left (2,2,2,2;1,1,\frac {9}{2};\frac {(a-b) \cos ^2(x)}{a}\right )}{105 a^4}+\frac {20 a \sec ^2(x)}{3 (a-b)}-\frac {30 a b \csc ^2(x) \sec ^2(x)}{(a-b)^2}-\frac {5 a^2 \sec ^4(x)}{(a-b)^2}+\frac {5 \arcsin \left (\sqrt {\frac {(a-b) \cos ^2(x)}{a}}\right )}{\left (\frac {(a-b) \cos ^2(x)}{a}\right )^{5/2} \sqrt {\frac {\left (a+b \cot ^2(x)\right ) \sin ^2(x)}{a}}}+\frac {30 b \arcsin \left (\sqrt {\frac {(a-b) \cos ^2(x)}{a}}\right ) \cot ^2(x)}{a \left (\frac {(a-b) \cos ^2(x)}{a}\right )^{5/2} \sqrt {\frac {\left (a+b \cot ^2(x)\right ) \sin ^2(x)}{a}}}+\frac {40 b^2 \arcsin \left (\sqrt {\frac {(a-b) \cos ^2(x)}{a}}\right ) \cot ^4(x)}{a^2 \left (\frac {(a-b) \cos ^2(x)}{a}\right )^{5/2} \sqrt {\frac {\left (a+b \cot ^2(x)\right ) \sin ^2(x)}{a}}}+\frac {16 b^3 \arcsin \left (\sqrt {\frac {(a-b) \cos ^2(x)}{a}}\right ) \cot ^6(x)}{a^3 \left (\frac {(a-b) \cos ^2(x)}{a}\right )^{5/2} \sqrt {\frac {\left (a+b \cot ^2(x)\right ) \sin ^2(x)}{a}}}+\frac {5 \arcsin \left (\sqrt {\frac {(a-b) \cos ^2(x)}{a}}\right )}{\sqrt {\frac {(a-b) \cos ^2(x) \left (a+b \cot ^2(x)\right ) \sin ^2(x)}{a^2}}}+\frac {30 b \arcsin \left (\sqrt {\frac {(a-b) \cos ^2(x)}{a}}\right ) \cot ^2(x)}{a \sqrt {\frac {(a-b) \cos ^2(x) \left (a+b \cot ^2(x)\right ) \sin ^2(x)}{a^2}}}+\frac {40 b^2 \arcsin \left (\sqrt {\frac {(a-b) \cos ^2(x)}{a}}\right ) \cot ^4(x)}{a^2 \sqrt {\frac {(a-b) \cos ^2(x) \left (a+b \cot ^2(x)\right ) \sin ^2(x)}{a^2}}}+\frac {16 b^3 \arcsin \left (\sqrt {\frac {(a-b) \cos ^2(x)}{a}}\right ) \cot ^6(x)}{a^3 \sqrt {\frac {(a-b) \cos ^2(x) \left (a+b \cot ^2(x)\right ) \sin ^2(x)}{a^2}}}-\frac {60 b \arcsin \left (\sqrt {\frac {(a-b) \cos ^2(x)}{a}}\right ) \csc ^2(x)}{(a-b) \sqrt {\frac {(a-b) \cos ^2(x) \left (a+b \cot ^2(x)\right ) \sin ^2(x)}{a^2}}}-\frac {80 b^2 \arcsin \left (\sqrt {\frac {(a-b) \cos ^2(x)}{a}}\right ) \cot ^2(x) \csc ^2(x)}{a (a-b) \sqrt {\frac {(a-b) \cos ^2(x) \left (a+b \cot ^2(x)\right ) \sin ^2(x)}{a^2}}}-\frac {32 b^3 \arcsin \left (\sqrt {\frac {(a-b) \cos ^2(x)}{a}}\right ) \cot ^4(x) \csc ^2(x)}{a^2 (a-b) \sqrt {\frac {(a-b) \cos ^2(x) \left (a+b \cot ^2(x)\right ) \sin ^2(x)}{a^2}}}-\frac {10 a \arcsin \left (\sqrt {\frac {(a-b) \cos ^2(x)}{a}}\right ) \sec ^2(x)}{(a-b) \sqrt {\frac {(a-b) \cos ^2(x) \left (a+b \cot ^2(x)\right ) \sin ^2(x)}{a^2}}}\right ) \tan (x)}{a^2 \sqrt {a+b \cot ^2(x)} \left (1+\frac {b \cot ^2(x)}{a}\right )} \]

[In]

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

[Out]

(Sin[x]^2*((-16*b^3*(Cot[x] + Cot[x]^3)^2)/(a*(a - b)^2) + (40*b*Csc[x]^2)/(a - b) + (160*b^2*Cot[x]^2*Csc[x]^
2)/(3*a*(a - b)) + (64*b^3*Cot[x]^4*Csc[x]^2)/(3*a^2*(a - b)) - (40*b^2*Csc[x]^4)/(a - b)^2 + (92*(a - b)*Cos[
x]^2*Hypergeometric2F1[2, 2, 9/2, ((a - b)*Cos[x]^2)/a])/(105*a) + (124*(a - b)*b*Cos[x]^2*Cot[x]^2*Hypergeome
tric2F1[2, 2, 9/2, ((a - b)*Cos[x]^2)/a])/(35*a^2) + (152*(a - b)*b^2*Cos[x]^2*Cot[x]^4*Hypergeometric2F1[2, 2
, 9/2, ((a - b)*Cos[x]^2)/a])/(35*a^3) + (176*(a - b)*b^3*Cos[x]^2*Cot[x]^6*Hypergeometric2F1[2, 2, 9/2, ((a -
b)*Cos[x]^2)/a])/(105*a^4) + (24*(a - b)*Cos[x]^2*HypergeometricPFQ[{2, 2, 2}, {1, 9/2}, ((a - b)*Cos[x]^2)/a
])/(35*a) + (16*(a - b)*b*Cos[x]^2*Cot[x]^2*HypergeometricPFQ[{2, 2, 2}, {1, 9/2}, ((a - b)*Cos[x]^2)/a])/(7*a
^2) + (88*(a - b)*b^2*Cos[x]^2*Cot[x]^4*HypergeometricPFQ[{2, 2, 2}, {1, 9/2}, ((a - b)*Cos[x]^2)/a])/(35*a^3)
+ (32*(a - b)*b^3*Cos[x]^2*Cot[x]^6*HypergeometricPFQ[{2, 2, 2}, {1, 9/2}, ((a - b)*Cos[x]^2)/a])/(35*a^4) +
(16*(a - b)*Cos[x]^2*HypergeometricPFQ[{2, 2, 2, 2}, {1, 1, 9/2}, ((a - b)*Cos[x]^2)/a])/(105*a) + (16*(a - b)
*b*Cos[x]^2*Cot[x]^2*HypergeometricPFQ[{2, 2, 2, 2}, {1, 1, 9/2}, ((a - b)*Cos[x]^2)/a])/(35*a^2) + (16*(a - b
)*b^2*Cos[x]^2*Cot[x]^4*HypergeometricPFQ[{2, 2, 2, 2}, {1, 1, 9/2}, ((a - b)*Cos[x]^2)/a])/(35*a^3) + (16*(a
- b)*b^3*Cos[x]^2*Cot[x]^6*HypergeometricPFQ[{2, 2, 2, 2}, {1, 1, 9/2}, ((a - b)*Cos[x]^2)/a])/(105*a^4) + (20
*a*Sec[x]^2)/(3*(a - b)) - (30*a*b*Csc[x]^2*Sec[x]^2)/(a - b)^2 - (5*a^2*Sec[x]^4)/(a - b)^2 + (5*ArcSin[Sqrt[
((a - b)*Cos[x]^2)/a]])/((((a - b)*Cos[x]^2)/a)^(5/2)*Sqrt[((a + b*Cot[x]^2)*Sin[x]^2)/a]) + (30*b*ArcSin[Sqrt
[((a - b)*Cos[x]^2)/a]]*Cot[x]^2)/(a*(((a - b)*Cos[x]^2)/a)^(5/2)*Sqrt[((a + b*Cot[x]^2)*Sin[x]^2)/a]) + (40*b
^2*ArcSin[Sqrt[((a - b)*Cos[x]^2)/a]]*Cot[x]^4)/(a^2*(((a - b)*Cos[x]^2)/a)^(5/2)*Sqrt[((a + b*Cot[x]^2)*Sin[x
]^2)/a]) + (16*b^3*ArcSin[Sqrt[((a - b)*Cos[x]^2)/a]]*Cot[x]^6)/(a^3*(((a - b)*Cos[x]^2)/a)^(5/2)*Sqrt[((a + b
*Cot[x]^2)*Sin[x]^2)/a]) + (5*ArcSin[Sqrt[((a - b)*Cos[x]^2)/a]])/Sqrt[((a - b)*Cos[x]^2*(a + b*Cot[x]^2)*Sin[
x]^2)/a^2] + (30*b*ArcSin[Sqrt[((a - b)*Cos[x]^2)/a]]*Cot[x]^2)/(a*Sqrt[((a - b)*Cos[x]^2*(a + b*Cot[x]^2)*Sin
[x]^2)/a^2]) + (40*b^2*ArcSin[Sqrt[((a - b)*Cos[x]^2)/a]]*Cot[x]^4)/(a^2*Sqrt[((a - b)*Cos[x]^2*(a + b*Cot[x]^
2)*Sin[x]^2)/a^2]) + (16*b^3*ArcSin[Sqrt[((a - b)*Cos[x]^2)/a]]*Cot[x]^6)/(a^3*Sqrt[((a - b)*Cos[x]^2*(a + b*C
ot[x]^2)*Sin[x]^2)/a^2]) - (60*b*ArcSin[Sqrt[((a - b)*Cos[x]^2)/a]]*Csc[x]^2)/((a - b)*Sqrt[((a - b)*Cos[x]^2*
(a + b*Cot[x]^2)*Sin[x]^2)/a^2]) - (80*b^2*ArcSin[Sqrt[((a - b)*Cos[x]^2)/a]]*Cot[x]^2*Csc[x]^2)/(a*(a - b)*Sq
rt[((a - b)*Cos[x]^2*(a + b*Cot[x]^2)*Sin[x]^2)/a^2]) - (32*b^3*ArcSin[Sqrt[((a - b)*Cos[x]^2)/a]]*Cot[x]^4*Cs
c[x]^2)/(a^2*(a - b)*Sqrt[((a - b)*Cos[x]^2*(a + b*Cot[x]^2)*Sin[x]^2)/a^2]) - (10*a*ArcSin[Sqrt[((a - b)*Cos[
x]^2)/a]]*Sec[x]^2)/((a - b)*Sqrt[((a - b)*Cos[x]^2*(a + b*Cot[x]^2)*Sin[x]^2)/a^2]))*Tan[x])/(a^2*Sqrt[a + b*
Cot[x]^2]*(1 + (b*Cot[x]^2)/a))

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(933\) vs. \(2(123)=246\).

Time = 1.56 (sec) , antiderivative size = 934, normalized size of antiderivative = 6.62

method	result	size
default	\(\text {Expression too large to display}\)	\(934\)

[In]

int(tan(x)^2/(a+b*cot(x)^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/6*4^(1/2)/(-a+b)^(1/2)/(a-b)^2/a^3*(-3*(-a+b)^(1/2)*a^2*b^2*(1-cos(x))^8*csc(x)^8+14*(-a+b)^(1/2)*a*b^3*(1-c
os(x))^8*csc(x)^8-8*(-a+b)^(1/2)*b^4*(1-cos(x))^8*csc(x)^8-24*(-a+b)^(1/2)*a^3*b*(1-cos(x))^6*csc(x)^6+96*(-a+
b)^(1/2)*a^2*b^2*(1-cos(x))^6*csc(x)^6-104*(-a+b)^(1/2)*a*b^3*(1-cos(x))^6*csc(x)^6+32*(-a+b)^(1/2)*b^4*(1-cos
(x))^6*csc(x)^6+3*ln(4*(a*(1-cos(x))^2*csc(x)^2-b*(1-cos(x))^2*csc(x)^2+(-a+b)^(1/2)*(b*(1-cos(x))^4*csc(x)^4+
4*a*(1-cos(x))^2*csc(x)^2-2*b*(1-cos(x))^2*csc(x)^2+b)^(1/2)-a+b)/((1-cos(x))^2*csc(x)^2+1))*(b*(1-cos(x))^4*c
sc(x)^4+4*a*(1-cos(x))^2*csc(x)^2-2*b*(1-cos(x))^2*csc(x)^2+b)^(3/2)*a^3*(1-cos(x))^2*csc(x)^2-48*(-a+b)^(1/2)
*a^4*(1-cos(x))^4*csc(x)^4+144*(-a+b)^(1/2)*a^3*b*(1-cos(x))^4*csc(x)^4-234*(-a+b)^(1/2)*a^2*b^2*(1-cos(x))^4*
csc(x)^4+180*(-a+b)^(1/2)*a*b^3*(1-cos(x))^4*csc(x)^4-48*(-a+b)^(1/2)*b^4*(1-cos(x))^4*csc(x)^4-3*ln(4*(a*(1-c
os(x))^2*csc(x)^2-b*(1-cos(x))^2*csc(x)^2+(-a+b)^(1/2)*(b*(1-cos(x))^4*csc(x)^4+4*a*(1-cos(x))^2*csc(x)^2-2*b*
(1-cos(x))^2*csc(x)^2+b)^(1/2)-a+b)/((1-cos(x))^2*csc(x)^2+1))*(b*(1-cos(x))^4*csc(x)^4+4*a*(1-cos(x))^2*csc(x
)^2-2*b*(1-cos(x))^2*csc(x)^2+b)^(3/2)*a^3-24*(-a+b)^(1/2)*a^3*b*(1-cos(x))^2*csc(x)^2+96*(-a+b)^(1/2)*a^2*b^2
*(1-cos(x))^2*csc(x)^2-104*(-a+b)^(1/2)*a*b^3*(1-cos(x))^2*csc(x)^2+32*(-a+b)^(1/2)*b^4*(1-cos(x))^2*csc(x)^2-
3*(-a+b)^(1/2)*a^2*b^2+14*(-a+b)^(1/2)*a*b^3-8*(-a+b)^(1/2)*b^4)*(b*(1-cos(x))^4*csc(x)^4+4*a*(1-cos(x))^2*csc
(x)^2-2*b*(1-cos(x))^2*csc(x)^2+b)/((1-cos(x))^2*csc(x)^2-1)/(1-cos(x))^5*sin(x)^5/(1/(1-cos(x))^2*(b*(1-cos(x
))^4*csc(x)^2+4*a*(1-cos(x))^2-2*b*(1-cos(x))^2+b*sin(x)^2))^(5/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 297 vs. \(2 (123) = 246\).

Time = 0.34 (sec) , antiderivative size = 647, normalized size of antiderivative = 4.59 \[ \int \frac {\tan ^2(x)}{\left (a+b \cot ^2(x)\right )^{5/2}} \, dx=\left [-\frac {3 \, {\left (a^{5} \tan \left (x\right )^{4} + 2 \, a^{4} b \tan \left (x\right )^{2} + a^{3} b^{2}\right )} \sqrt {-a + b} \log \left (-\frac {a^{2} \tan \left (x\right )^{4} - 2 \, {\left (3 \, a^{2} - 4 \, a b\right )} \tan \left (x\right )^{2} + a^{2} - 8 \, a b + 8 \, b^{2} + 4 \, {\left (a \tan \left (x\right )^{3} - {\left (a - 2 \, b\right )} \tan \left (x\right )\right )} \sqrt {-a + b} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}}}{\tan \left (x\right )^{4} + 2 \, \tan \left (x\right )^{2} + 1}\right ) - 4 \, {\left (3 \, {\left (a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right )} \tan \left (x\right )^{5} + 3 \, {\left (2 \, a^{4} b - 9 \, a^{3} b^{2} + 11 \, a^{2} b^{3} - 4 \, a b^{4}\right )} \tan \left (x\right )^{3} + {\left (3 \, a^{3} b^{2} - 17 \, a^{2} b^{3} + 22 \, a b^{4} - 8 \, b^{5}\right )} \tan \left (x\right )\right )} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}}}{12 \, {\left (a^{6} b^{2} - 3 \, a^{5} b^{3} + 3 \, a^{4} b^{4} - a^{3} b^{5} + {\left (a^{8} - 3 \, a^{7} b + 3 \, a^{6} b^{2} - a^{5} b^{3}\right )} \tan \left (x\right )^{4} + 2 \, {\left (a^{7} b - 3 \, a^{6} b^{2} + 3 \, a^{5} b^{3} - a^{4} b^{4}\right )} \tan \left (x\right )^{2}\right )}}, \frac {3 \, {\left (a^{5} \tan \left (x\right )^{4} + 2 \, a^{4} b \tan \left (x\right )^{2} + a^{3} b^{2}\right )} \sqrt {a - b} \arctan \left (\frac {2 \, \sqrt {a - b} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}} \tan \left (x\right )}{a \tan \left (x\right )^{2} - a + 2 \, b}\right ) + 2 \, {\left (3 \, {\left (a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right )} \tan \left (x\right )^{5} + 3 \, {\left (2 \, a^{4} b - 9 \, a^{3} b^{2} + 11 \, a^{2} b^{3} - 4 \, a b^{4}\right )} \tan \left (x\right )^{3} + {\left (3 \, a^{3} b^{2} - 17 \, a^{2} b^{3} + 22 \, a b^{4} - 8 \, b^{5}\right )} \tan \left (x\right )\right )} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}}}{6 \, {\left (a^{6} b^{2} - 3 \, a^{5} b^{3} + 3 \, a^{4} b^{4} - a^{3} b^{5} + {\left (a^{8} - 3 \, a^{7} b + 3 \, a^{6} b^{2} - a^{5} b^{3}\right )} \tan \left (x\right )^{4} + 2 \, {\left (a^{7} b - 3 \, a^{6} b^{2} + 3 \, a^{5} b^{3} - a^{4} b^{4}\right )} \tan \left (x\right )^{2}\right )}}\right ] \]

[In]

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

[Out]

[-1/12*(3*(a^5*tan(x)^4 + 2*a^4*b*tan(x)^2 + a^3*b^2)*sqrt(-a + b)*log(-(a^2*tan(x)^4 - 2*(3*a^2 - 4*a*b)*tan(
x)^2 + a^2 - 8*a*b + 8*b^2 + 4*(a*tan(x)^3 - (a - 2*b)*tan(x))*sqrt(-a + b)*sqrt((a*tan(x)^2 + b)/tan(x)^2))/(
tan(x)^4 + 2*tan(x)^2 + 1)) - 4*(3*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*tan(x)^5 + 3*(2*a^4*b - 9*a^3*b^2 + 1
1*a^2*b^3 - 4*a*b^4)*tan(x)^3 + (3*a^3*b^2 - 17*a^2*b^3 + 22*a*b^4 - 8*b^5)*tan(x))*sqrt((a*tan(x)^2 + b)/tan(
x)^2))/(a^6*b^2 - 3*a^5*b^3 + 3*a^4*b^4 - a^3*b^5 + (a^8 - 3*a^7*b + 3*a^6*b^2 - a^5*b^3)*tan(x)^4 + 2*(a^7*b
- 3*a^6*b^2 + 3*a^5*b^3 - a^4*b^4)*tan(x)^2), 1/6*(3*(a^5*tan(x)^4 + 2*a^4*b*tan(x)^2 + a^3*b^2)*sqrt(a - b)*a
rctan(2*sqrt(a - b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)/(a*tan(x)^2 - a + 2*b)) + 2*(3*(a^5 - 3*a^4*b + 3*a
^3*b^2 - a^2*b^3)*tan(x)^5 + 3*(2*a^4*b - 9*a^3*b^2 + 11*a^2*b^3 - 4*a*b^4)*tan(x)^3 + (3*a^3*b^2 - 17*a^2*b^3
+ 22*a*b^4 - 8*b^5)*tan(x))*sqrt((a*tan(x)^2 + b)/tan(x)^2))/(a^6*b^2 - 3*a^5*b^3 + 3*a^4*b^4 - a^3*b^5 + (a^
8 - 3*a^7*b + 3*a^6*b^2 - a^5*b^3)*tan(x)^4 + 2*(a^7*b - 3*a^6*b^2 + 3*a^5*b^3 - a^4*b^4)*tan(x)^2)]

Sympy [F]

\[ \int \frac {\tan ^2(x)}{\left (a+b \cot ^2(x)\right )^{5/2}} \, dx=\int \frac {\tan ^{2}{\left (x \right )}}{\left (a + b \cot ^{2}{\left (x \right )}\right )^{\frac {5}{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\tan ^2(x)}{\left (a+b \cot ^2(x)\right )^{5/2}} \, dx=\int { \frac {\tan \left (x\right )^{2}}{{\left (b \cot \left (x\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 537 vs. \(2 (123) = 246\).

Time = 0.36 (sec) , antiderivative size = 537, normalized size of antiderivative = 3.81 \[ \int \frac {\tan ^2(x)}{\left (a+b \cot ^2(x)\right )^{5/2}} \, dx=\frac {{\left (3 \, a^{4} \sqrt {b} \log \left (-a - 2 \, \sqrt {-a + b} \sqrt {b} + 2 \, b\right ) + 3 \, a^{3} \sqrt {-a + b} b \log \left (-a - 2 \, \sqrt {-a + b} \sqrt {b} + 2 \, b\right ) - 3 \, a^{3} b^{\frac {3}{2}} \log \left (-a - 2 \, \sqrt {-a + b} \sqrt {b} + 2 \, b\right ) + 6 \, a^{4} \sqrt {b} - 18 \, a^{3} b^{\frac {3}{2}} + 16 \, a^{2} \sqrt {-a + b} b^{2} + 2 \, a^{2} b^{\frac {5}{2}} - 26 \, a \sqrt {-a + b} b^{3} + 20 \, a b^{\frac {7}{2}} + 10 \, \sqrt {-a + b} b^{4} - 10 \, b^{\frac {9}{2}}\right )} \mathrm {sgn}\left (\sin \left (x\right )\right )}{6 \, {\left (a^{6} \sqrt {-a + b} \sqrt {b} - a^{6} b - 3 \, a^{5} \sqrt {-a + b} b^{\frac {3}{2}} + 3 \, a^{5} b^{2} + 3 \, a^{4} \sqrt {-a + b} b^{\frac {5}{2}} - 3 \, a^{4} b^{3} - a^{3} \sqrt {-a + b} b^{\frac {7}{2}} + a^{3} b^{4}\right )}} - \frac {\frac {2 \, {\left (\frac {{\left (9 \, a^{5} b^{2} - 23 \, a^{4} b^{3} + 19 \, a^{3} b^{4} - 5 \, a^{2} b^{5}\right )} \cos \left (x\right )^{2}}{a^{8} - 3 \, a^{7} b + 3 \, a^{6} b^{2} - a^{5} b^{3}} - \frac {3 \, {\left (3 \, a^{5} b^{2} - 5 \, a^{4} b^{3} + 2 \, a^{3} b^{4}\right )}}{a^{8} - 3 \, a^{7} b + 3 \, a^{6} b^{2} - a^{5} b^{3}}\right )} \cos \left (x\right )}{{\left (a \cos \left (x\right )^{2} - b \cos \left (x\right )^{2} - a\right )} \sqrt {-a \cos \left (x\right )^{2} + b \cos \left (x\right )^{2} + a}} + \frac {3 \, \log \left ({\left (\sqrt {-a + b} \cos \left (x\right ) - \sqrt {-a \cos \left (x\right )^{2} + b \cos \left (x\right )^{2} + a}\right )}^{2}\right )}{{\left (a^{2} - 2 \, a b + b^{2}\right )} \sqrt {-a + b}} + \frac {12 \, \sqrt {-a + b}}{{\left ({\left (\sqrt {-a + b} \cos \left (x\right ) - \sqrt {-a \cos \left (x\right )^{2} + b \cos \left (x\right )^{2} + a}\right )}^{2} - a\right )} a^{2}}}{6 \, \mathrm {sgn}\left (\sin \left (x\right )\right )} \]

[In]

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

[Out]

1/6*(3*a^4*sqrt(b)*log(-a - 2*sqrt(-a + b)*sqrt(b) + 2*b) + 3*a^3*sqrt(-a + b)*b*log(-a - 2*sqrt(-a + b)*sqrt(
b) + 2*b) - 3*a^3*b^(3/2)*log(-a - 2*sqrt(-a + b)*sqrt(b) + 2*b) + 6*a^4*sqrt(b) - 18*a^3*b^(3/2) + 16*a^2*sqr
t(-a + b)*b^2 + 2*a^2*b^(5/2) - 26*a*sqrt(-a + b)*b^3 + 20*a*b^(7/2) + 10*sqrt(-a + b)*b^4 - 10*b^(9/2))*sgn(s
in(x))/(a^6*sqrt(-a + b)*sqrt(b) - a^6*b - 3*a^5*sqrt(-a + b)*b^(3/2) + 3*a^5*b^2 + 3*a^4*sqrt(-a + b)*b^(5/2)
- 3*a^4*b^3 - a^3*sqrt(-a + b)*b^(7/2) + a^3*b^4) - 1/6*(2*((9*a^5*b^2 - 23*a^4*b^3 + 19*a^3*b^4 - 5*a^2*b^5)
*cos(x)^2/(a^8 - 3*a^7*b + 3*a^6*b^2 - a^5*b^3) - 3*(3*a^5*b^2 - 5*a^4*b^3 + 2*a^3*b^4)/(a^8 - 3*a^7*b + 3*a^6
*b^2 - a^5*b^3))*cos(x)/((a*cos(x)^2 - b*cos(x)^2 - a)*sqrt(-a*cos(x)^2 + b*cos(x)^2 + a)) + 3*log((sqrt(-a +
b)*cos(x) - sqrt(-a*cos(x)^2 + b*cos(x)^2 + a))^2)/((a^2 - 2*a*b + b^2)*sqrt(-a + b)) + 12*sqrt(-a + b)/(((sqr
t(-a + b)*cos(x) - sqrt(-a*cos(x)^2 + b*cos(x)^2 + a))^2 - a)*a^2))/sgn(sin(x))

Mupad [F(-1)]

Timed out. \[ \int \frac {\tan ^2(x)}{\left (a+b \cot ^2(x)\right )^{5/2}} \, dx=\int \frac {{\mathrm {tan}\left (x\right )}^2}{{\left (b\,{\mathrm {cot}\left (x\right )}^2+a\right )}^{5/2}} \,d x \]

[In]

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

[Out]

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