Integrand size = 35, antiderivative size = 436 \[ \int \frac {\tan ^2(d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \, dx=-\frac {\arctan \left (\frac {\sqrt {a-b+c} \tan (d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 \sqrt {a-b+c} e}+\frac {\sqrt [4]{a} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}}}{2 \left (\sqrt {a}-\sqrt {c}\right ) \sqrt [4]{c} e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {\left (\sqrt {a}+\sqrt {c}\right ) \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {a}-\sqrt {c}\right )^2}{4 \sqrt {a} \sqrt {c}},2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}}}{4 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {c}\right ) \sqrt [4]{c} e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \]
-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))/e/(a-b+c)^(1/2)+1/2*a^(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)/c^(1/4)/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)*tan(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)/e/(a^(1/2)-c^(1/2))/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2)
Result contains complex when optimal does not.
Time = 4.11 (sec) , antiderivative size = 311, normalized size of antiderivative = 0.71 \[ \int \frac {\tan ^2(d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \, dx=-\frac {i \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \tan (d+e x)\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-\operatorname {EllipticPi}\left (\frac {b+\sqrt {b^2-4 a c}}{2 c},i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \tan (d+e x)\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c \tan ^2(d+e x)}{b+\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c \tan ^2(d+e x)}{b-\sqrt {b^2-4 a c}}}}{\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \]
((-I)*(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])] - 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[2]*Sqrt[c/(b + Sq rt[b^2 - 4*a*c])]*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])
Time = 0.58 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.03, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {3042, 4183, 1656, 27, 1416, 2220}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\tan ^2(d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\tan (d+e x)^2}{\sqrt {a+b \tan (d+e x)^2+c \tan (d+e x)^4}}dx\) |
| \(\Big \downarrow \) 4183 |
| \(\displaystyle \frac {\int \frac {\tan ^2(d+e x)}{\left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 1656 |
| \(\displaystyle \frac {\frac {\sqrt {a} \int \frac {1}{\sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}d\tan (d+e x)}{\sqrt {a}-\sqrt {c}}-\frac {\sqrt {a} \int \frac {\sqrt {c} \tan ^2(d+e x)+\sqrt {a}}{\sqrt {a} \left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}d\tan (d+e x)}{\sqrt {a}-\sqrt {c}}}{e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\sqrt {a} \int \frac {1}{\sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}d\tan (d+e x)}{\sqrt {a}-\sqrt {c}}-\frac {\int \frac {\sqrt {c} \tan ^2(d+e x)+\sqrt {a}}{\left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}d\tan (d+e x)}{\sqrt {a}-\sqrt {c}}}{e}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{c} \left (\sqrt {a}-\sqrt {c}\right ) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {\int \frac {\sqrt {c} \tan ^2(d+e x)+\sqrt {a}}{\left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}d\tan (d+e x)}{\sqrt {a}-\sqrt {c}}}{e}\) |
| \(\Big \downarrow \) 2220 |
| \(\displaystyle \frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{c} \left (\sqrt {a}-\sqrt {c}\right ) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {\frac {\left (\sqrt {a}-\sqrt {c}\right ) \arctan \left (\frac {\sqrt {a-b+c} \tan (d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 \sqrt {a-b+c}}+\frac {\left (\sqrt {a}+\sqrt {c}\right ) \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {a}-\sqrt {c}\right )^2}{4 \sqrt {a} \sqrt {c}},2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{4 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}}{\sqrt {a}-\sqrt {c}}}{e}\) |
((a^(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*(Sqrt[a ] - Sqrt[c])*c^(1/4)*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) - (((S qrt[a] - Sqrt[c])*ArcTan[(Sqrt[a - b + c]*Tan[d + e*x])/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]])/(2*Sqrt[a - b + c]) + ((Sqrt[a] + Sqrt[c])*E llipticPi[-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]*Ta n[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sq rt[c]*Tan[d + e*x]^2)^2])/(4*a^(1/4)*c^(1/4)*Sqrt[a + b*Tan[d + e*x]^2 + c *Tan[d + e*x]^4]))/(Sqrt[a] - Sqrt[c]))/e
3.1.42.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
Int[(x_)^2/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]) , x_Symbol] :> With[{q = Rt[c/a, 2]}, Simp[(-a)*((e + d*q)/(c*d^2 - a*e^2)) Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] + Simp[a*d*((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]] /; Fr eeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a] && NeQ[c*d^2 - a*e^2, 0]
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))*(A rcTan[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)*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/(4*d*e*q*Sqrt[a + b*x^2 + c*x^4]))*El lipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*ArcTan[q*x], 1/2 - b/(4*a*q^2)], x]] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] & & EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && PosQ[-b + c*(d/e) + a*(e/d)]
Int[tan[(d_.) + (e_.)*(x_)]^(m_.)*((a_.) + (b_.)*((f_.)*tan[(d_.) + (e_.)*( x_)])^(n_.) + (c_.)*((f_.)*tan[(d_.) + (e_.)*(x_)])^(n2_.))^(p_), x_Symbol] :> Simp[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[n 2, 2*n] && NeQ[b^2 - 4*a*c, 0]
Time = 0.61 (sec) , antiderivative size = 402, normalized size of antiderivative = 0.92
| method | result | size |
| derivativedivides | \(\frac {\frac {\sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) \tan \left (e x +d \right )^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) \tan \left (e x +d \right )^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {\tan \left (e x +d \right ) \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {a +b \tan \left (e x +d \right )^{2}+c \tan \left (e x +d \right )^{4}}}-\frac {\sqrt {2}\, \sqrt {1+\frac {b \tan \left (e x +d \right )^{2}}{2 a}-\frac {\tan \left (e x +d \right )^{2} \sqrt {-4 a c +b^{2}}}{2 a}}\, \sqrt {1+\frac {b \tan \left (e x +d \right )^{2}}{2 a}+\frac {\tan \left (e x +d \right )^{2} \sqrt {-4 a c +b^{2}}}{2 a}}\, \operatorname {EllipticPi}\left (\frac {\tan \left (e x +d \right ) \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, -\frac {2 a}{-b +\sqrt {-4 a c +b^{2}}}, \frac {\sqrt {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}\, \sqrt {2}}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}\right )}{\sqrt {-\frac {b}{a}+\frac {\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {a +b \tan \left (e x +d \right )^{2}+c \tan \left (e x +d \right )^{4}}}}{e}\) | \(402\) |
| default | \(\frac {\frac {\sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) \tan \left (e x +d \right )^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) \tan \left (e x +d \right )^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {\tan \left (e x +d \right ) \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {a +b \tan \left (e x +d \right )^{2}+c \tan \left (e x +d \right )^{4}}}-\frac {\sqrt {2}\, \sqrt {1+\frac {b \tan \left (e x +d \right )^{2}}{2 a}-\frac {\tan \left (e x +d \right )^{2} \sqrt {-4 a c +b^{2}}}{2 a}}\, \sqrt {1+\frac {b \tan \left (e x +d \right )^{2}}{2 a}+\frac {\tan \left (e x +d \right )^{2} \sqrt {-4 a c +b^{2}}}{2 a}}\, \operatorname {EllipticPi}\left (\frac {\tan \left (e x +d \right ) \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, -\frac {2 a}{-b +\sqrt {-4 a c +b^{2}}}, \frac {\sqrt {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}\, \sqrt {2}}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}\right )}{\sqrt {-\frac {b}{a}+\frac {\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {a +b \tan \left (e x +d \right )^{2}+c \tan \left (e x +d \right )^{4}}}}{e}\) | \(402\) |
1/e*(1/4*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))-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)))
\[ \int \frac {\tan ^2(d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \, dx=\int { \frac {\tan \left (e x + d\right )^{2}}{\sqrt {c \tan \left (e x + d\right )^{4} + b \tan \left (e x + d\right )^{2} + a}} \,d x } \]
\[ \int \frac {\tan ^2(d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \, dx=\int \frac {\tan ^{2}{\left (d + e x \right )}}{\sqrt {a + b \tan ^{2}{\left (d + e x \right )} + c \tan ^{4}{\left (d + e x \right )}}}\, dx \]
\[ \int \frac {\tan ^2(d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \, dx=\int { \frac {\tan \left (e x + d\right )^{2}}{\sqrt {c \tan \left (e x + d\right )^{4} + b \tan \left (e x + d\right )^{2} + a}} \,d x } \]
Timed out. \[ \int \frac {\tan ^2(d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\tan ^2(d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \, dx=\int \frac {{\mathrm {tan}\left (d+e\,x\right )}^2}{\sqrt {c\,{\mathrm {tan}\left (d+e\,x\right )}^4+b\,{\mathrm {tan}\left (d+e\,x\right )}^2+a}} \,d x \]