\(\int \frac {\tan ^2(d+e x)}{(a+b \tan ^2(d+e x)+c \tan ^4(d+e x))^{3/2}} \, dx\) [51]

Optimal result

Integrand size = 35, antiderivative size = 981 \[ \int \frac {\tan ^2(d+e x)}{\left (a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right )^{3/2}} \, dx=-\frac {\arctan \left (\frac {\sqrt {a-b+c} \tan (d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 (a-b+c)^{3/2} e}+\frac {\tan (d+e x) \left (b^2-2 a c-b c+(b-2 c) c \tan ^2(d+e x)\right )}{(a-b+c) \left (b^2-4 a c\right ) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {(b-2 c) \sqrt {c} \tan (d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{(a-b+c) \left (b^2-4 a c\right ) e \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )}+\frac {\sqrt [4]{a} (b-2 c) \sqrt [4]{c} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}}}{(a-b+c) \left (b^2-4 a c\right ) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}+\frac {\sqrt [4]{c} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}}}{2 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {c}\right ) (a-b+c) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {\left (\sqrt {a}-\sqrt {c}\right ) \sqrt [4]{c} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}}}{2 \sqrt [4]{a} \left (b-2 \sqrt {a} \sqrt {c}\right ) (a-b+c) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {\left (\sqrt {a}+\sqrt {c}\right ) \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {a}-\sqrt {c}\right )^2}{4 \sqrt {a} \sqrt {c}},2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}}}{4 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {c}\right ) \sqrt [4]{c} (a-b+c) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \]

[Out]

-1/2*arctan((a-b+c)^(1/2)*tan(e*x+d)/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2))/(a-b+c)^(3/2)/e+tan(e*x+d)*(b^2-
2*a*c-b*c+(b-2*c)*c*tan(e*x+d)^2)/(a-b+c)/(-4*a*c+b^2)/e/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2)-(b-2*c)*c^(1/
2)*(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2)*tan(e*x+d)/(a-b+c)/(-4*a*c+b^2)/e/(a^(1/2)+c^(1/2)*tan(e*x+d)^2)+a^
(1/4)*(b-2*c)*c^(1/4)*(cos(2*arctan(c^(1/4)*tan(e*x+d)/a^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*tan(e*x+d)/a^(1
/4)))*EllipticE(sin(2*arctan(c^(1/4)*tan(e*x+d)/a^(1/4))),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))*((a+b*tan(e*x+d)^2+
c*tan(e*x+d)^4)/(a^(1/2)+c^(1/2)*tan(e*x+d)^2)^2)^(1/2)*(a^(1/2)+c^(1/2)*tan(e*x+d)^2)/(a-b+c)/(-4*a*c+b^2)/e/
(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2)+1/2*c^(1/4)*(cos(2*arctan(c^(1/4)*tan(e*x+d)/a^(1/4)))^2)^(1/2)/cos(2*
arctan(c^(1/4)*tan(e*x+d)/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*tan(e*x+d)/a^(1/4))),1/2*(2-b/a^(1/2)/c^(1/
2))^(1/2))*((a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)/(a^(1/2)+c^(1/2)*tan(e*x+d)^2)^2)^(1/2)*(a^(1/2)+c^(1/2)*tan(e*x
+d)^2)/a^(1/4)/(a-b+c)/e/(a^(1/2)-c^(1/2))/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2)-1/4*(cos(2*arctan(c^(1/4)*t
an(e*x+d)/a^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*tan(e*x+d)/a^(1/4)))*EllipticPi(sin(2*arctan(c^(1/4)*tan(e*x
+d)/a^(1/4))),-1/4*(a^(1/2)-c^(1/2))^2/a^(1/2)/c^(1/2),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))*(a^(1/2)+c^(1/2))*((a+
b*tan(e*x+d)^2+c*tan(e*x+d)^4)/(a^(1/2)+c^(1/2)*tan(e*x+d)^2)^2)^(1/2)*(a^(1/2)+c^(1/2)*tan(e*x+d)^2)/a^(1/4)/
c^(1/4)/(a-b+c)/e/(a^(1/2)-c^(1/2))/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2)-1/2*c^(1/4)*(cos(2*arctan(c^(1/4)*
tan(e*x+d)/a^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*tan(e*x+d)/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*tan(e*x
+d)/a^(1/4))),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))*(a^(1/2)-c^(1/2))*((a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)/(a^(1/2)+c
^(1/2)*tan(e*x+d)^2)^2)^(1/2)*(a^(1/2)+c^(1/2)*tan(e*x+d)^2)/a^(1/4)/(a-b+c)/e/(b-2*a^(1/2)*c^(1/2))/(a+b*tan(
e*x+d)^2+c*tan(e*x+d)^4)^(1/2)

Rubi [A] (verified)

Time = 0.94 (sec) , antiderivative size = 981, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {3781, 1329, 1192, 1211, 1117, 1209, 1230, 1720} \[ \int \frac {\tan ^2(d+e x)}{\left (a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right )^{3/2}} \, dx=-\frac {\arctan \left (\frac {\sqrt {a-b+c} \tan (d+e x)}{\sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}\right )}{2 (a-b+c)^{3/2} e}-\frac {(b-2 c) \sqrt {c} \tan (d+e x) \sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}{(a-b+c) \left (b^2-4 a c\right ) e \left (\sqrt {c} \tan ^2(d+e x)+\sqrt {a}\right )}+\frac {\sqrt [4]{a} (b-2 c) \sqrt [4]{c} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (\sqrt {c} \tan ^2(d+e x)+\sqrt {a}\right ) \sqrt {\frac {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}{\left (\sqrt {c} \tan ^2(d+e x)+\sqrt {a}\right )^2}}}{(a-b+c) \left (b^2-4 a c\right ) e \sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}+\frac {\sqrt [4]{c} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (\sqrt {c} \tan ^2(d+e x)+\sqrt {a}\right ) \sqrt {\frac {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}{\left (\sqrt {c} \tan ^2(d+e x)+\sqrt {a}\right )^2}}}{2 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {c}\right ) (a-b+c) e \sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}-\frac {\left (\sqrt {a}-\sqrt {c}\right ) \sqrt [4]{c} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (\sqrt {c} \tan ^2(d+e x)+\sqrt {a}\right ) \sqrt {\frac {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}{\left (\sqrt {c} \tan ^2(d+e x)+\sqrt {a}\right )^2}}}{2 \sqrt [4]{a} \left (b-2 \sqrt {a} \sqrt {c}\right ) (a-b+c) e \sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}-\frac {\left (\sqrt {a}+\sqrt {c}\right ) \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {a}-\sqrt {c}\right )^2}{4 \sqrt {a} \sqrt {c}},2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (\sqrt {c} \tan ^2(d+e x)+\sqrt {a}\right ) \sqrt {\frac {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}{\left (\sqrt {c} \tan ^2(d+e x)+\sqrt {a}\right )^2}}}{4 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {c}\right ) \sqrt [4]{c} (a-b+c) e \sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}+\frac {\tan (d+e x) \left (b^2-c b+(b-2 c) c \tan ^2(d+e x)-2 a c\right )}{(a-b+c) \left (b^2-4 a c\right ) e \sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}} \]

[In]

Int[Tan[d + e*x]^2/(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)^(3/2),x]

[Out]

-1/2*ArcTan[(Sqrt[a - b + c]*Tan[d + e*x])/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]]/((a - b + c)^(3/2)*e
) + (Tan[d + e*x]*(b^2 - 2*a*c - b*c + (b - 2*c)*c*Tan[d + e*x]^2))/((a - b + c)*(b^2 - 4*a*c)*e*Sqrt[a + b*Ta
n[d + e*x]^2 + c*Tan[d + e*x]^4]) - ((b - 2*c)*Sqrt[c]*Tan[d + e*x]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]
^4])/((a - b + c)*(b^2 - 4*a*c)*e*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)) + (a^(1/4)*(b - 2*c)*c^(1/4)*EllipticE[2
*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[
(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/((a - b + c)*(b^2 - 4*a*c)*e*
Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) + (c^(1/4)*EllipticF[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)],
(2 - b/(Sqrt[a]*Sqrt[c]))/4]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)
/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(2*a^(1/4)*(Sqrt[a] - Sqrt[c])*(a - b + c)*e*Sqrt[a + b*Tan[d + e*x]^2
+ c*Tan[d + e*x]^4]) - ((Sqrt[a] - Sqrt[c])*c^(1/4)*EllipticF[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (2 -
b/(Sqrt[a]*Sqrt[c]))/4]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqr
t[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(2*a^(1/4)*(b - 2*Sqrt[a]*Sqrt[c])*(a - b + c)*e*Sqrt[a + b*Tan[d + e*x]^2
+ c*Tan[d + e*x]^4]) - ((Sqrt[a] + Sqrt[c])*EllipticPi[-1/4*(Sqrt[a] - Sqrt[c])^2/(Sqrt[a]*Sqrt[c]), 2*ArcTan[
(c^(1/4)*Tan[d + e*x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*T
an[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(4*a^(1/4)*(Sqrt[a] - Sqrt[c])*c^(1/4
)*(a - b + c)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])

Rule 1117

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(
a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(
4*c))], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rule 1192

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[x*(a*b*e - d*(b^2 - 2*a
*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Dist[1/(2*a*(p + 1)
*(b^2 - 4*a*c)), Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a +
b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e
^2, 0] && LtQ[p, -1] && IntegerQ[2*p]

Rule 1209

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(
-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 +
q^2*x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c))], x] /; EqQ[e + d*q^2,
0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rule 1211

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 2]}, Dist[(
e + d*q)/q, Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] - Dist[e/q, Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x]
/; NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rule 1230

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[c/a, 2]}, Di
st[(c*d + a*e*q)/(c*d^2 - a*e^2), Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] - Dist[(a*e*(e + d*q))/(c*d^2 - a*e^2)
, Int[(1 + q*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a
*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a]

Rule 1329

Int[(((f_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_))/((d_.) + (e_.)*(x_)^2), x_Symbol] :> Dist[
f^2/(c*d^2 - b*d*e + a*e^2), Int[(f*x)^(m - 2)*(a*e + c*d*x^2)*(a + b*x^2 + c*x^4)^p, x], x] - Dist[d*e*(f^2/(
c*d^2 - b*d*e + a*e^2)), Int[(f*x)^(m - 2)*((a + b*x^2 + c*x^4)^(p + 1)/(d + e*x^2)), x], x] /; FreeQ[{a, b, c
, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && GtQ[m, 0]

Rule 1720

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With[
{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(ArcTan[Rt[-b + c*(d/e) + a*(e/d), 2]*(x/Sqrt[a + b*x^2 + c*x^4])]/(2*d*
e*Rt[-b + c*(d/e) + a*(e/d), 2])), x] + Simp[(B*d + A*e)*(A + B*x^2)*(Sqrt[A^2*((a + b*x^2 + c*x^4)/(a*(A + B*
x^2)^2))]/(4*d*e*A*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticPi[Cancel[-(B*d - A*e)^2/(4*d*e*A*B)], 2*ArcTan[q*x], 1
/2 - b*(A/(4*a*B))], x]] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^
2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 0]

Rule 3781

Int[tan[(d_.) + (e_.)*(x_)]^(m_.)*((a_.) + (b_.)*((f_.)*tan[(d_.) + (e_.)*(x_)])^(n_.) + (c_.)*((f_.)*tan[(d_.
) + (e_.)*(x_)])^(n2_.))^(p_), x_Symbol] :> Dist[f/e, Subst[Int[(x/f)^m*((a + b*x^n + c*x^(2*n))^p/(f^2 + x^2)
), x], x, f*Tan[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^2}{\left (1+x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}} \, dx,x,\tan (d+e x)\right )}{e} \\ & = \frac {\text {Subst}\left (\int \frac {a+c x^2}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx,x,\tan (d+e x)\right )}{(a-b+c) e}-\frac {\text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx,x,\tan (d+e x)\right )}{(a-b+c) e} \\ & = \frac {\tan (d+e x) \left (b^2-2 a c-b c+(b-2 c) c \tan ^2(d+e x)\right )}{(a-b+c) \left (b^2-4 a c\right ) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {\sqrt {a} \text {Subst}\left (\int \frac {1+\frac {\sqrt {c} x^2}{\sqrt {a}}}{\left (1+x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx,x,\tan (d+e x)\right )}{\left (\sqrt {a}-\sqrt {c}\right ) (a-b+c) e}+\frac {\sqrt {c} \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2+c x^4}} \, dx,x,\tan (d+e x)\right )}{\left (\sqrt {a}-\sqrt {c}\right ) (a-b+c) e}-\frac {\text {Subst}\left (\int \frac {a (2 a-b) c+a (b-2 c) c x^2}{\sqrt {a+b x^2+c x^4}} \, dx,x,\tan (d+e x)\right )}{a (a-b+c) \left (b^2-4 a c\right ) e} \\ & = -\frac {\arctan \left (\frac {\sqrt {a-b+c} \tan (d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 (a-b+c)^{3/2} e}+\frac {\tan (d+e x) \left (b^2-2 a c-b c+(b-2 c) c \tan ^2(d+e x)\right )}{(a-b+c) \left (b^2-4 a c\right ) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}+\frac {\sqrt [4]{c} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}}}{2 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {c}\right ) (a-b+c) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {\left (\sqrt {a}+\sqrt {c}\right ) \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {a}-\sqrt {c}\right )^2}{4 \sqrt {a} \sqrt {c}},2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}}}{4 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {c}\right ) \sqrt [4]{c} (a-b+c) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {\left (\left (\sqrt {a}-\sqrt {c}\right ) \sqrt {c}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2+c x^4}} \, dx,x,\tan (d+e x)\right )}{\left (b-2 \sqrt {a} \sqrt {c}\right ) (a-b+c) e}+\frac {\left (\sqrt {a} (b-2 c) \sqrt {c}\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+b x^2+c x^4}} \, dx,x,\tan (d+e x)\right )}{(a-b+c) \left (b^2-4 a c\right ) e} \\ & = -\frac {\arctan \left (\frac {\sqrt {a-b+c} \tan (d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 (a-b+c)^{3/2} e}+\frac {\tan (d+e x) \left (b^2-2 a c-b c+(b-2 c) c \tan ^2(d+e x)\right )}{(a-b+c) \left (b^2-4 a c\right ) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {(b-2 c) \sqrt {c} \tan (d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{(a-b+c) \left (b^2-4 a c\right ) e \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )}+\frac {\sqrt [4]{a} (b-2 c) \sqrt [4]{c} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}}}{(a-b+c) \left (b^2-4 a c\right ) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}+\frac {\sqrt [4]{c} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}}}{2 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {c}\right ) (a-b+c) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {\left (\sqrt {a}-\sqrt {c}\right ) \sqrt [4]{c} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}}}{2 \sqrt [4]{a} \left (b-2 \sqrt {a} \sqrt {c}\right ) (a-b+c) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {\left (\sqrt {a}+\sqrt {c}\right ) \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {a}-\sqrt {c}\right )^2}{4 \sqrt {a} \sqrt {c}},2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}}}{4 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {c}\right ) \sqrt [4]{c} (a-b+c) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 19.77 (sec) , antiderivative size = 831, normalized size of antiderivative = 0.85 \[ \int \frac {\tan ^2(d+e x)}{\left (a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right )^{3/2}} \, dx=\frac {\sqrt {\frac {3 a+b+3 c+4 a \cos (2 (d+e x))-4 c \cos (2 (d+e x))+a \cos (4 (d+e x))-b \cos (4 (d+e x))+c \cos (4 (d+e x))}{3+4 \cos (2 (d+e x))+\cos (4 (d+e x))}} \left (\frac {(b-2 c) \sin (2 (d+e x))}{2 (-a+b-c) \left (b^2-4 a c\right )}+\frac {2 b^2 \sin (2 (d+e x))-4 a c \sin (2 (d+e x))-4 c^2 \sin (2 (d+e x))+b^2 \sin (4 (d+e x))-2 a c \sin (4 (d+e x))-2 b c \sin (4 (d+e x))+2 c^2 \sin (4 (d+e x))}{(a-b+c) \left (-b^2+4 a c\right ) (-3 a-b-3 c-4 a \cos (2 (d+e x))+4 c \cos (2 (d+e x))-a \cos (4 (d+e x))+b \cos (4 (d+e x))-c \cos (4 (d+e x)))}\right )}{e}+\frac {\frac {i \sqrt {2} \left ((b-2 c) \left (-b+\sqrt {b^2-4 a c}\right ) E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \tan (d+e x)\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+\left (b^2-b \sqrt {b^2-4 a c}+2 c \left (-2 a+\sqrt {b^2-4 a c}\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \tan (d+e x)\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-2 \left (b^2-4 a c\right ) \operatorname {EllipticPi}\left (\frac {b+\sqrt {b^2-4 a c}}{2 c},i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \tan (d+e x)\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c \tan ^2(d+e x)}{b+\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c \tan ^2(d+e x)}{b-\sqrt {b^2-4 a c}}}}{\sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}}}-\frac {4 (b-2 c) \tan (d+e x) \left (a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right )}{1+\tan ^2(d+e x)}}{4 (a-b+c) \left (-b^2+4 a c\right ) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \]

[In]

Integrate[Tan[d + e*x]^2/(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)^(3/2),x]

[Out]

(Sqrt[(3*a + b + 3*c + 4*a*Cos[2*(d + e*x)] - 4*c*Cos[2*(d + e*x)] + a*Cos[4*(d + e*x)] - b*Cos[4*(d + e*x)] +
c*Cos[4*(d + e*x)])/(3 + 4*Cos[2*(d + e*x)] + Cos[4*(d + e*x)])]*(((b - 2*c)*Sin[2*(d + e*x)])/(2*(-a + b - c
)*(b^2 - 4*a*c)) + (2*b^2*Sin[2*(d + e*x)] - 4*a*c*Sin[2*(d + e*x)] - 4*c^2*Sin[2*(d + e*x)] + b^2*Sin[4*(d +
e*x)] - 2*a*c*Sin[4*(d + e*x)] - 2*b*c*Sin[4*(d + e*x)] + 2*c^2*Sin[4*(d + e*x)])/((a - b + c)*(-b^2 + 4*a*c)*
(-3*a - b - 3*c - 4*a*Cos[2*(d + e*x)] + 4*c*Cos[2*(d + e*x)] - a*Cos[4*(d + e*x)] + b*Cos[4*(d + e*x)] - c*Co
s[4*(d + e*x)]))))/e + ((I*Sqrt[2]*((b - 2*c)*(-b + Sqrt[b^2 - 4*a*c])*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(b +
Sqrt[b^2 - 4*a*c])]*Tan[d + e*x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] + (b^2 - b*Sqrt[b^2 - 4*a
*c] + 2*c*(-2*a + Sqrt[b^2 - 4*a*c]))*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Tan[d + e*x]
], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - 2*(b^2 - 4*a*c)*EllipticPi[(b + Sqrt[b^2 - 4*a*c])/(2*c)
, I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Tan[d + e*x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a
*c])])*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*Tan[d + e*x]^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*Tan[d + e*x]^2
)/(b - Sqrt[b^2 - 4*a*c])])/Sqrt[c/(b + Sqrt[b^2 - 4*a*c])] - (4*(b - 2*c)*Tan[d + e*x]*(a + b*Tan[d + e*x]^2
+ c*Tan[d + e*x]^4))/(1 + Tan[d + e*x]^2))/(4*(a - b + c)*(-b^2 + 4*a*c)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d
+ e*x]^4])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3597\) vs. \(2(995)=1990\).

Time = 0.60 (sec) , antiderivative size = 3598, normalized size of antiderivative = 3.67

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(3598\)
default	\(\text {Expression too large to display}\)	\(3598\)

[In]

int(tan(e*x+d)^2/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/e*(-2*c*(1/2/a*b/(4*a*c-b^2)*tan(e*x+d)^3-1/2*(2*a*c-b^2)/a/(4*a*c-b^2)/c*tan(e*x+d))/((tan(e*x+d)^4+b/c*tan
(e*x+d)^2+a/c)*c)^(1/2)+1/4*(1/a-(2*a*c-b^2)/a/(4*a*c-b^2))*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(4-2*(-b
+(-4*a*c+b^2)^(1/2))/a*tan(e*x+d)^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2))/a*tan(e*x+d)^2)^(1/2)/(a+b*tan(e*x+d)^2
+c*tan(e*x+d)^4)^(1/2)*EllipticF(1/2*tan(e*x+d)*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a
*c+b^2)^(1/2))/a/c)^(1/2))-1/2*b/(4*a*c-b^2)*c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(4-2*(-b+(-4*a*c+b^2)
^(1/2))/a*tan(e*x+d)^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2))/a*tan(e*x+d)^2)^(1/2)/(a+b*tan(e*x+d)^2+c*tan(e*x+d)
^4)^(1/2)/(b+(-4*a*c+b^2)^(1/2))*(EllipticF(1/2*tan(e*x+d)*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2
*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))-EllipticE(1/2*tan(e*x+d)*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(
-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2)))+2*c*(1/2*(2*a*c-b^2+b*c)/a/(4*a*c-b^2)/(a-b+c)*tan(e*x+d)^3+1/2*(3*
a*b*c-2*a*c^2-b^3+b^2*c)/a/(4*a*c-b^2)/(a-b+c)/c*tan(e*x+d))/((tan(e*x+d)^4+b/c*tan(e*x+d)^2+a/c)*c)^(1/2)+1/4
*2^(1/2)/(-1/a*b+1/a*(-4*a*c+b^2)^(1/2))^(1/2)*(4+2/a*b*tan(e*x+d)^2-2/a*tan(e*x+d)^2*(-4*a*c+b^2)^(1/2))^(1/2
)*(4+2/a*b*tan(e*x+d)^2+2/a*tan(e*x+d)^2*(-4*a*c+b^2)^(1/2))^(1/2)/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2)*Ell
ipticF(1/2*tan(e*x+d)*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))
/a/(a-b+c)*b-1/4*2^(1/2)/(-1/a*b+1/a*(-4*a*c+b^2)^(1/2))^(1/2)*(4+2/a*b*tan(e*x+d)^2-2/a*tan(e*x+d)^2*(-4*a*c+
b^2)^(1/2))^(1/2)*(4+2/a*b*tan(e*x+d)^2+2/a*tan(e*x+d)^2*(-4*a*c+b^2)^(1/2))^(1/2)/(a+b*tan(e*x+d)^2+c*tan(e*x
+d)^4)^(1/2)*EllipticF(1/2*tan(e*x+d)*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1
/2))/a/c)^(1/2))/a/(a-b+c)*c-3/4*2^(1/2)/(-1/a*b+1/a*(-4*a*c+b^2)^(1/2))^(1/2)*(4+2/a*b*tan(e*x+d)^2-2/a*tan(e
*x+d)^2*(-4*a*c+b^2)^(1/2))^(1/2)*(4+2/a*b*tan(e*x+d)^2+2/a*tan(e*x+d)^2*(-4*a*c+b^2)^(1/2))^(1/2)/(a+b*tan(e*
x+d)^2+c*tan(e*x+d)^4)^(1/2)*EllipticF(1/2*tan(e*x+d)*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b
+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))/(4*a*c-b^2)/(a-b+c)*b*c+1/2*2^(1/2)/(-1/a*b+1/a*(-4*a*c+b^2)^(1/2))^(1/2)*(4+
2/a*b*tan(e*x+d)^2-2/a*tan(e*x+d)^2*(-4*a*c+b^2)^(1/2))^(1/2)*(4+2/a*b*tan(e*x+d)^2+2/a*tan(e*x+d)^2*(-4*a*c+b
^2)^(1/2))^(1/2)/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2)*EllipticF(1/2*tan(e*x+d)*2^(1/2)*((-b+(-4*a*c+b^2)^(1
/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))/(4*a*c-b^2)/(a-b+c)*c^2+1/4*2^(1/2)/(-1/a*b+1/a*(
-4*a*c+b^2)^(1/2))^(1/2)*(4+2/a*b*tan(e*x+d)^2-2/a*tan(e*x+d)^2*(-4*a*c+b^2)^(1/2))^(1/2)*(4+2/a*b*tan(e*x+d)^
2+2/a*tan(e*x+d)^2*(-4*a*c+b^2)^(1/2))^(1/2)/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2)*EllipticF(1/2*tan(e*x+d)*
2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))/a/(4*a*c-b^2)/(a-b+c)
*b^3-1/4*2^(1/2)/(-1/a*b+1/a*(-4*a*c+b^2)^(1/2))^(1/2)*(4+2/a*b*tan(e*x+d)^2-2/a*tan(e*x+d)^2*(-4*a*c+b^2)^(1/
2))^(1/2)*(4+2/a*b*tan(e*x+d)^2+2/a*tan(e*x+d)^2*(-4*a*c+b^2)^(1/2))^(1/2)/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(
1/2)*EllipticF(1/2*tan(e*x+d)*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c
)^(1/2))/a*b^2/(4*a*c-b^2)*c/(a-b+c)+c^2/(a-b+c)/(4*a*c-b^2)*2^(1/2)/(-1/a*b+1/a*(-4*a*c+b^2)^(1/2))^(1/2)*(4+
2/a*b*tan(e*x+d)^2-2/a*tan(e*x+d)^2*(-4*a*c+b^2)^(1/2))^(1/2)*(4+2/a*b*tan(e*x+d)^2+2/a*tan(e*x+d)^2*(-4*a*c+b
^2)^(1/2))^(1/2)/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2)/(b+(-4*a*c+b^2)^(1/2))*EllipticF(1/2*tan(e*x+d)*2^(1/
2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))*a-1/2*c/(a-b+c)/(4*a*c-b^2
)*2^(1/2)/(-1/a*b+1/a*(-4*a*c+b^2)^(1/2))^(1/2)*(4+2/a*b*tan(e*x+d)^2-2/a*tan(e*x+d)^2*(-4*a*c+b^2)^(1/2))^(1/
2)*(4+2/a*b*tan(e*x+d)^2+2/a*tan(e*x+d)^2*(-4*a*c+b^2)^(1/2))^(1/2)/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2)/(b
+(-4*a*c+b^2)^(1/2))*EllipticF(1/2*tan(e*x+d)*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c
+b^2)^(1/2))/a/c)^(1/2))*b^2+1/2*c^2/(a-b+c)/(4*a*c-b^2)*2^(1/2)/(-1/a*b+1/a*(-4*a*c+b^2)^(1/2))^(1/2)*(4+2/a*
b*tan(e*x+d)^2-2/a*tan(e*x+d)^2*(-4*a*c+b^2)^(1/2))^(1/2)*(4+2/a*b*tan(e*x+d)^2+2/a*tan(e*x+d)^2*(-4*a*c+b^2)^
(1/2))^(1/2)/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2)/(b+(-4*a*c+b^2)^(1/2))*EllipticF(1/2*tan(e*x+d)*2^(1/2)*(
(-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))*b-c^2/(a-b+c)/(4*a*c-b^2)*2^(1
/2)/(-1/a*b+1/a*(-4*a*c+b^2)^(1/2))^(1/2)*(4+2/a*b*tan(e*x+d)^2-2/a*tan(e*x+d)^2*(-4*a*c+b^2)^(1/2))^(1/2)*(4+
2/a*b*tan(e*x+d)^2+2/a*tan(e*x+d)^2*(-4*a*c+b^2)^(1/2))^(1/2)/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2)/(b+(-4*a
*c+b^2)^(1/2))*EllipticE(1/2*tan(e*x+d)*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^
(1/2))/a/c)^(1/2))*a+1/2*c/(a-b+c)/(4*a*c-b^2)*2^(1/2)/(-1/a*b+1/a*(-4*a*c+b^2)^(1/2))^(1/2)*(4+2/a*b*tan(e*x+
d)^2-2/a*tan(e*x+d)^2*(-4*a*c+b^2)^(1/2))^(1/2)*(4+2/a*b*tan(e*x+d)^2+2/a*tan(e*x+d)^2*(-4*a*c+b^2)^(1/2))^(1/
2)/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2)/(b+(-4*a*c+b^2)^(1/2))*EllipticE(1/2*tan(e*x+d)*2^(1/2)*((-b+(-4*a*
c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))*b^2-1/2*c^2/(a-b+c)/(4*a*c-b^2)*2^(1/2)/
(-1/a*b+1/a*(-4*a*c+b^2)^(1/2))^(1/2)*(4+2/a*b*tan(e*x+d)^2-2/a*tan(e*x+d)^2*(-4*a*c+b^2)^(1/2))^(1/2)*(4+2/a*
b*tan(e*x+d)^2+2/a*tan(e*x+d)^2*(-4*a*c+b^2)^(1/2))^(1/2)/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2)/(b+(-4*a*c+b
^2)^(1/2))*EllipticE(1/2*tan(e*x+d)*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2
))/a/c)^(1/2))*b-1/(a-b+c)*2^(1/2)/(-1/a*b+1/a*(-4*a*c+b^2)^(1/2))^(1/2)*(1+1/2/a*b*tan(e*x+d)^2-1/2/a*tan(e*x
+d)^2*(-4*a*c+b^2)^(1/2))^(1/2)*(1+1/2/a*b*tan(e*x+d)^2+1/2/a*tan(e*x+d)^2*(-4*a*c+b^2)^(1/2))^(1/2)/(a+b*tan(
e*x+d)^2+c*tan(e*x+d)^4)^(1/2)*EllipticPi(1/2*tan(e*x+d)*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),-2/(-b+(-4*
a*c+b^2)^(1/2))*a,(-1/2*(b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)))

Fricas [F(-1)]

Timed out. \[ \int \frac {\tan ^2(d+e x)}{\left (a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right )^{3/2}} \, dx=\text {Timed out} \]

[In]

integrate(tan(e*x+d)^2/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(3/2),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int \frac {\tan ^2(d+e x)}{\left (a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right )^{3/2}} \, dx=\int \frac {\tan ^{2}{\left (d + e x \right )}}{\left (a + b \tan ^{2}{\left (d + e x \right )} + c \tan ^{4}{\left (d + e x \right )}\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(tan(e*x+d)**2/(a+b*tan(e*x+d)**2+c*tan(e*x+d)**4)**(3/2),x)

[Out]

Integral(tan(d + e*x)**2/(a + b*tan(d + e*x)**2 + c*tan(d + e*x)**4)**(3/2), x)

Maxima [F]

\[ \int \frac {\tan ^2(d+e x)}{\left (a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right )^{3/2}} \, dx=\int { \frac {\tan \left (e x + d\right )^{2}}{{\left (c \tan \left (e x + d\right )^{4} + b \tan \left (e x + d\right )^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(tan(e*x+d)^2/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(3/2),x, algorithm="maxima")

[Out]

integrate(tan(e*x + d)^2/(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)^(3/2), x)

Giac [F(-1)]

Timed out. \[ \int \frac {\tan ^2(d+e x)}{\left (a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right )^{3/2}} \, dx=\text {Timed out} \]

[In]

integrate(tan(e*x+d)^2/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(3/2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {\tan ^2(d+e x)}{\left (a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right )^{3/2}} \, dx=\int \frac {{\mathrm {tan}\left (d+e\,x\right )}^2}{{\left (c\,{\mathrm {tan}\left (d+e\,x\right )}^4+b\,{\mathrm {tan}\left (d+e\,x\right )}^2+a\right )}^{3/2}} \,d x \]

[In]

int(tan(d + e*x)^2/(a + b*tan(d + e*x)^2 + c*tan(d + e*x)^4)^(3/2),x)

[Out]

int(tan(d + e*x)^2/(a + b*tan(d + e*x)^2 + c*tan(d + e*x)^4)^(3/2), x)