\(\int \frac {\tan ^2(x)}{(a+b \cot ^2(x))^{3/2}} \, dx\) [53]

Optimal result

Integrand size = 17, antiderivative size = 92 \[ \int \frac {\tan ^2(x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx=\frac {\arctan \left (\frac {\sqrt {a-b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )}{(a-b)^{3/2}}+\frac {b \tan (x)}{a (a-b) \sqrt {a+b \cot ^2(x)}}+\frac {(a-2 b) \sqrt {a+b \cot ^2(x)} \tan (x)}{a^2 (a-b)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 92, 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, 483, 597, 12, 385, 209} \[ \int \frac {\tan ^2(x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx=\frac {(a-2 b) \tan (x) \sqrt {a+b \cot ^2(x)}}{a^2 (a-b)}+\frac {\arctan \left (\frac {\sqrt {a-b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )}{(a-b)^{3/2}}+\frac {b \tan (x)}{a (a-b) \sqrt {a+b \cot ^2(x)}} \]

[In]

Int[Tan[x]^2/(a + b*Cot[x]^2)^(3/2),x]

[Out]

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

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

Mathematica [C] (warning: unable to verify)

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

Time = 6.95 (sec) , antiderivative size = 674, normalized size of antiderivative = 7.33 \[ \int \frac {\tan ^2(x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx=\frac {\sin ^2(x) \left (\frac {12 b \csc ^2(x)}{a-b}+\frac {8 b^2 \cot ^2(x) \csc ^2(x)}{a (a-b)}+\frac {16 (a-b) \cos ^2(x) \operatorname {Hypergeometric2F1}\left (2,2,\frac {7}{2},\frac {(a-b) \cos ^2(x)}{a}\right )}{15 a}+\frac {8 (a-b) b \cos ^2(x) \cot ^2(x) \operatorname {Hypergeometric2F1}\left (2,2,\frac {7}{2},\frac {(a-b) \cos ^2(x)}{a}\right )}{3 a^2}+\frac {8 (a-b) b^2 \cos ^2(x) \cot ^4(x) \operatorname {Hypergeometric2F1}\left (2,2,\frac {7}{2},\frac {(a-b) \cos ^2(x)}{a}\right )}{5 a^3}+\frac {8 (a-b) \cos ^2(x) \, _3F_2\left (2,2,2;1,\frac {7}{2};\frac {(a-b) \cos ^2(x)}{a}\right )}{15 a}+\frac {16 (a-b) b \cos ^2(x) \cot ^2(x) \, _3F_2\left (2,2,2;1,\frac {7}{2};\frac {(a-b) \cos ^2(x)}{a}\right )}{15 a^2}+\frac {8 (a-b) b^2 \cos ^2(x) \cot ^4(x) \, _3F_2\left (2,2,2;1,\frac {7}{2};\frac {(a-b) \cos ^2(x)}{a}\right )}{15 a^3}+\frac {3 a \sec ^2(x)}{a-b}-\frac {3 \arcsin \left (\sqrt {\frac {(a-b) \cos ^2(x)}{a}}\right )}{\left (\frac {(a-b) \cos ^2(x)}{a}\right )^{3/2} \sqrt {\frac {\left (a+b \cot ^2(x)\right ) \sin ^2(x)}{a}}}-\frac {12 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 )^{3/2} \sqrt {\frac {\left (a+b \cot ^2(x)\right ) \sin ^2(x)}{a}}}-\frac {8 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 )^{3/2} \sqrt {\frac {\left (a+b \cot ^2(x)\right ) \sin ^2(x)}{a}}}+\frac {3 \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 {12 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 {8 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}}}\right ) \tan (x)}{a \sqrt {a+b \cot ^2(x)}} \]

[In]

Integrate[Tan[x]^2/(a + b*Cot[x]^2)^(3/2),x]

[Out]

(Sin[x]^2*((12*b*Csc[x]^2)/(a - b) + (8*b^2*Cot[x]^2*Csc[x]^2)/(a*(a - b)) + (16*(a - b)*Cos[x]^2*Hypergeometr
ic2F1[2, 2, 7/2, ((a - b)*Cos[x]^2)/a])/(15*a) + (8*(a - b)*b*Cos[x]^2*Cot[x]^2*Hypergeometric2F1[2, 2, 7/2, (
(a - b)*Cos[x]^2)/a])/(3*a^2) + (8*(a - b)*b^2*Cos[x]^2*Cot[x]^4*Hypergeometric2F1[2, 2, 7/2, ((a - b)*Cos[x]^
2)/a])/(5*a^3) + (8*(a - b)*Cos[x]^2*HypergeometricPFQ[{2, 2, 2}, {1, 7/2}, ((a - b)*Cos[x]^2)/a])/(15*a) + (1
6*(a - b)*b*Cos[x]^2*Cot[x]^2*HypergeometricPFQ[{2, 2, 2}, {1, 7/2}, ((a - b)*Cos[x]^2)/a])/(15*a^2) + (8*(a -
b)*b^2*Cos[x]^2*Cot[x]^4*HypergeometricPFQ[{2, 2, 2}, {1, 7/2}, ((a - b)*Cos[x]^2)/a])/(15*a^3) + (3*a*Sec[x]
^2)/(a - b) - (3*ArcSin[Sqrt[((a - b)*Cos[x]^2)/a]])/((((a - b)*Cos[x]^2)/a)^(3/2)*Sqrt[((a + b*Cot[x]^2)*Sin[
x]^2)/a]) - (12*b*ArcSin[Sqrt[((a - b)*Cos[x]^2)/a]]*Cot[x]^2)/(a*(((a - b)*Cos[x]^2)/a)^(3/2)*Sqrt[((a + b*Co
t[x]^2)*Sin[x]^2)/a]) - (8*b^2*ArcSin[Sqrt[((a - b)*Cos[x]^2)/a]]*Cot[x]^4)/(a^2*(((a - b)*Cos[x]^2)/a)^(3/2)*
Sqrt[((a + b*Cot[x]^2)*Sin[x]^2)/a]) + (3*ArcSin[Sqrt[((a - b)*Cos[x]^2)/a]])/Sqrt[((a - b)*Cos[x]^2*(a + b*Co
t[x]^2)*Sin[x]^2)/a^2] + (12*b*ArcSin[Sqrt[((a - b)*Cos[x]^2)/a]]*Cot[x]^2)/(a*Sqrt[((a - b)*Cos[x]^2*(a + b*C
ot[x]^2)*Sin[x]^2)/a^2]) + (8*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]))*Tan[x])/(a*Sqrt[a + b*Cot[x]^2])

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(630\) vs. \(2(82)=164\).

Time = 2.93 (sec) , antiderivative size = 631, normalized size of antiderivative = 6.86

method	result	size
default	\(\frac {\sqrt {4}\, \left (-\sqrt {-a +b}\, a b \left (1-\cos \left (x \right )\right )^{4} \csc \left (x \right )^{4}+2 \sqrt {-a +b}\, b^{2} \left (1-\cos \left (x \right )\right )^{4} \csc \left (x \right )^{4}+\sqrt {b \left (1-\cos \left (x \right )\right )^{4} \csc \left (x \right )^{4}+4 a \left (1-\cos \left (x \right )\right )^{2} \csc \left (x \right )^{2}-2 b \left (1-\cos \left (x \right )\right )^{2} \csc \left (x \right )^{2}+b}\, \ln \left (\frac {4 a \left (1-\cos \left (x \right )\right )^{2} \csc \left (x \right )^{2}-4 b \left (1-\cos \left (x \right )\right )^{2} \csc \left (x \right )^{2}+4 \sqrt {-a +b}\, \sqrt {b \left (1-\cos \left (x \right )\right )^{4} \csc \left (x \right )^{4}+4 a \left (1-\cos \left (x \right )\right )^{2} \csc \left (x \right )^{2}-2 b \left (1-\cos \left (x \right )\right )^{2} \csc \left (x \right )^{2}+b}-4 a +4 b}{\left (1-\cos \left (x \right )\right )^{2} \csc \left (x \right )^{2}+1}\right ) a^{2} \left (1-\cos \left (x \right )\right )^{2} \csc \left (x \right )^{2}-4 \sqrt {-a +b}\, a^{2} \left (1-\cos \left (x \right )\right )^{2} \csc \left (x \right )^{2}+6 \sqrt {-a +b}\, a b \left (1-\cos \left (x \right )\right )^{2} \csc \left (x \right )^{2}-4 \sqrt {-a +b}\, b^{2} \left (1-\cos \left (x \right )\right )^{2} \csc \left (x \right )^{2}-\ln \left (\frac {4 a \left (1-\cos \left (x \right )\right )^{2} \csc \left (x \right )^{2}-4 b \left (1-\cos \left (x \right )\right )^{2} \csc \left (x \right )^{2}+4 \sqrt {-a +b}\, \sqrt {b \left (1-\cos \left (x \right )\right )^{4} \csc \left (x \right )^{4}+4 a \left (1-\cos \left (x \right )\right )^{2} \csc \left (x \right )^{2}-2 b \left (1-\cos \left (x \right )\right )^{2} \csc \left (x \right )^{2}+b}-4 a +4 b}{\left (1-\cos \left (x \right )\right )^{2} \csc \left (x \right )^{2}+1}\right ) \sqrt {b \left (1-\cos \left (x \right )\right )^{4} \csc \left (x \right )^{4}+4 a \left (1-\cos \left (x \right )\right )^{2} \csc \left (x \right )^{2}-2 b \left (1-\cos \left (x \right )\right )^{2} \csc \left (x \right )^{2}+b}\, a^{2}-a \sqrt {-a +b}\, b +2 b^{2} \sqrt {-a +b}\right ) \left (b \left (1-\cos \left (x \right )\right )^{4} \csc \left (x \right )^{4}+4 a \left (1-\cos \left (x \right )\right )^{2} \csc \left (x \right )^{2}-2 b \left (1-\cos \left (x \right )\right )^{2} \csc \left (x \right )^{2}+b \right ) \sin \left (x \right )^{3}}{2 \sqrt {-a +b}\, a^{2} \left (a -b \right ) \left (\left (1-\cos \left (x \right )\right )^{2} \csc \left (x \right )^{2}-1\right ) \left (1-\cos \left (x \right )\right )^{3} \left (\frac {b \left (1-\cos \left (x \right )\right )^{4} \csc \left (x \right )^{2}+4 a \left (1-\cos \left (x \right )\right )^{2}-2 b \left (1-\cos \left (x \right )\right )^{2}+b \sin \left (x \right )^{2}}{\left (1-\cos \left (x \right )\right )^{2}}\right )^{\frac {3}{2}}}\)	\(631\)

[In]

int(tan(x)^2/(a+b*cot(x)^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/2*4^(1/2)/(-a+b)^(1/2)/a^2/(a-b)*(-(-a+b)^(1/2)*a*b*(1-cos(x))^4*csc(x)^4+2*(-a+b)^(1/2)*b^2*(1-cos(x))^4*cs
c(x)^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)^(1/2)*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-co
s(x))^2*csc(x)^2+b)^(1/2)-a+b)/((1-cos(x))^2*csc(x)^2+1))*a^2*(1-cos(x))^2*csc(x)^2-4*(-a+b)^(1/2)*a^2*(1-cos(
x))^2*csc(x)^2+6*(-a+b)^(1/2)*a*b*(1-cos(x))^2*csc(x)^2-4*(-a+b)^(1/2)*b^2*(1-cos(x))^2*csc(x)^2-ln(4*(a*(1-co
s(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)^(1/2)*a^2-a*(-a+b)^(1/2)*b+2*b^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-cos(x))^2*csc(x)^2-1)/(1-cos(x))^3*sin(x)^3/(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))^(3/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 170 vs. \(2 (82) = 164\).

Time = 0.35 (sec) , antiderivative size = 393, normalized size of antiderivative = 4.27 \[ \int \frac {\tan ^2(x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx=\left [\frac {{\left (a^{3} \tan \left (x\right )^{2} + a^{2} b\right )} \sqrt {-a + b} \log \left (-\frac {a^{2} \tan \left (x\right )^{4} - 2 \, {\left (3 \, a^{2} - 4 \, a b\right )} \tan \left (x\right )^{2} + a^{2} - 8 \, a b + 8 \, b^{2} - 4 \, {\left (a \tan \left (x\right )^{3} - {\left (a - 2 \, b\right )} \tan \left (x\right )\right )} \sqrt {-a + b} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}}}{\tan \left (x\right )^{4} + 2 \, \tan \left (x\right )^{2} + 1}\right ) + 4 \, {\left ({\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )} \tan \left (x\right )^{3} + {\left (a^{2} b - 3 \, a b^{2} + 2 \, b^{3}\right )} \tan \left (x\right )\right )} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}}}{4 \, {\left (a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3} + {\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} \tan \left (x\right )^{2}\right )}}, \frac {{\left (a^{3} \tan \left (x\right )^{2} + a^{2} b\right )} \sqrt {a - b} \arctan \left (\frac {2 \, \sqrt {a - b} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}} \tan \left (x\right )}{a \tan \left (x\right )^{2} - a + 2 \, b}\right ) + 2 \, {\left ({\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )} \tan \left (x\right )^{3} + {\left (a^{2} b - 3 \, a b^{2} + 2 \, b^{3}\right )} \tan \left (x\right )\right )} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}}}{2 \, {\left (a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3} + {\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} \tan \left (x\right )^{2}\right )}}\right ] \]

[In]

integrate(tan(x)^2/(a+b*cot(x)^2)^(3/2),x, algorithm="fricas")

[Out]

[1/4*((a^3*tan(x)^2 + a^2*b)*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*((a^3 - 2*a^2*b + a*b^2)*tan(x)^3 + (a^2*b - 3*a*b^2 + 2*b^3)*tan(x))*sqrt((a*tan(x)^2 + b)/tan(x)^2))/
(a^4*b - 2*a^3*b^2 + a^2*b^3 + (a^5 - 2*a^4*b + a^3*b^2)*tan(x)^2), 1/2*((a^3*tan(x)^2 + a^2*b)*sqrt(a - b)*ar
ctan(2*sqrt(a - b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)/(a*tan(x)^2 - a + 2*b)) + 2*((a^3 - 2*a^2*b + a*b^2)
*tan(x)^3 + (a^2*b - 3*a*b^2 + 2*b^3)*tan(x))*sqrt((a*tan(x)^2 + b)/tan(x)^2))/(a^4*b - 2*a^3*b^2 + a^2*b^3 +
(a^5 - 2*a^4*b + a^3*b^2)*tan(x)^2)]

Sympy [F]

\[ \int \frac {\tan ^2(x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx=\int \frac {\tan ^{2}{\left (x \right )}}{\left (a + b \cot ^{2}{\left (x \right )}\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(tan(x)**2/(a+b*cot(x)**2)**(3/2),x)

[Out]

Integral(tan(x)**2/(a + b*cot(x)**2)**(3/2), x)

Maxima [F]

\[ \int \frac {\tan ^2(x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx=\int { \frac {\tan \left (x\right )^{2}}{{\left (b \cot \left (x\right )^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(tan(x)^2/(a+b*cot(x)^2)^(3/2),x, algorithm="maxima")

[Out]

integrate(tan(x)^2/(b*cot(x)^2 + a)^(3/2), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 359 vs. \(2 (82) = 164\).

Time = 0.33 (sec) , antiderivative size = 359, normalized size of antiderivative = 3.90 \[ \int \frac {\tan ^2(x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx=\frac {{\left (a^{3} \log \left (-a - 2 \, \sqrt {-a + b} \sqrt {b} + 2 \, b\right ) + a^{2} \sqrt {-a + b} \sqrt {b} \log \left (-a - 2 \, \sqrt {-a + b} \sqrt {b} + 2 \, b\right ) - a^{2} b \log \left (-a - 2 \, \sqrt {-a + b} \sqrt {b} + 2 \, b\right ) + 2 \, a^{3} - 4 \, a^{2} b + 2 \, a \sqrt {-a + b} b^{\frac {3}{2}} - 2 \, \sqrt {-a + b} b^{\frac {5}{2}} + 2 \, b^{3}\right )} \mathrm {sgn}\left (\sin \left (x\right )\right )}{2 \, {\left (a^{4} \sqrt {-a + b} - a^{4} \sqrt {b} - 2 \, a^{3} \sqrt {-a + b} b + 2 \, a^{3} b^{\frac {3}{2}} + a^{2} \sqrt {-a + b} b^{2} - a^{2} b^{\frac {5}{2}}\right )}} + \frac {\frac {2 \, \sqrt {-a \cos \left (x\right )^{2} + b \cos \left (x\right )^{2} + a} b^{2} \cos \left (x\right )}{{\left (a^{3} - a^{2} b\right )} {\left (a \cos \left (x\right )^{2} - b \cos \left (x\right )^{2} - a\right )}} - \frac {\log \left ({\left (\sqrt {-a + b} \cos \left (x\right ) - \sqrt {-a \cos \left (x\right )^{2} + b \cos \left (x\right )^{2} + a}\right )}^{2}\right )}{{\left (a - b\right )} \sqrt {-a + b}} - \frac {4 \, \sqrt {-a + b}}{{\left ({\left (\sqrt {-a + b} \cos \left (x\right ) - \sqrt {-a \cos \left (x\right )^{2} + b \cos \left (x\right )^{2} + a}\right )}^{2} - a\right )} a}}{2 \, \mathrm {sgn}\left (\sin \left (x\right )\right )} \]

[In]

integrate(tan(x)^2/(a+b*cot(x)^2)^(3/2),x, algorithm="giac")

[Out]

1/2*(a^3*log(-a - 2*sqrt(-a + b)*sqrt(b) + 2*b) + a^2*sqrt(-a + b)*sqrt(b)*log(-a - 2*sqrt(-a + b)*sqrt(b) + 2
*b) - a^2*b*log(-a - 2*sqrt(-a + b)*sqrt(b) + 2*b) + 2*a^3 - 4*a^2*b + 2*a*sqrt(-a + b)*b^(3/2) - 2*sqrt(-a +
b)*b^(5/2) + 2*b^3)*sgn(sin(x))/(a^4*sqrt(-a + b) - a^4*sqrt(b) - 2*a^3*sqrt(-a + b)*b + 2*a^3*b^(3/2) + a^2*s
qrt(-a + b)*b^2 - a^2*b^(5/2)) + 1/2*(2*sqrt(-a*cos(x)^2 + b*cos(x)^2 + a)*b^2*cos(x)/((a^3 - a^2*b)*(a*cos(x)
^2 - b*cos(x)^2 - a)) - log((sqrt(-a + b)*cos(x) - sqrt(-a*cos(x)^2 + b*cos(x)^2 + a))^2)/((a - b)*sqrt(-a + b
)) - 4*sqrt(-a + b)/(((sqrt(-a + b)*cos(x) - sqrt(-a*cos(x)^2 + b*cos(x)^2 + a))^2 - a)*a))/sgn(sin(x))

Mupad [F(-1)]

Timed out. \[ \int \frac {\tan ^2(x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx=\int \frac {{\mathrm {tan}\left (x\right )}^2}{{\left (b\,{\mathrm {cot}\left (x\right )}^2+a\right )}^{3/2}} \,d x \]

[In]

int(tan(x)^2/(a + b*cot(x)^2)^(3/2),x)

[Out]

int(tan(x)^2/(a + b*cot(x)^2)^(3/2), x)