Integrand size = 33, antiderivative size = 638 \[ \int \frac {\tan ^2(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx=-\frac {\sqrt {2 a-2 c+\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2-(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \arctan \left (\frac {b \left (2 a-2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )+\left (b^2-(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )\right ) \tan (d+e x)}{\sqrt {2} \sqrt {2 a-2 c+\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2-(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \left (a^2+b^2-2 a c+c^2\right )^{3/2} e}+\frac {\sqrt {2 a-2 c-\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2+(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \arctan \left (\frac {b \left (2 a-2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )+\left (b^2-(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )\right ) \tan (d+e x)}{\sqrt {2} \sqrt {2 a-2 c-\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2+(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \left (a^2+b^2-2 a c+c^2\right )^{3/2} e}-\frac {2 \left (a b (a+c)+c \left (2 a^2+b^2-2 a c\right ) \tan (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \]
-1/2*arctan(1/2*(b*(2*a-2*c+(a^2-2*a*c+b^2+c^2)^(1/2))+(b^2-(a-c)*(a-c-(a^ 2-2*a*c+b^2+c^2)^(1/2)))*tan(e*x+d))*2^(1/2)/(2*a-2*c+(a^2-2*a*c+b^2+c^2)^ (1/2))^(1/2)/(a^2-b^2-2*a*c+c^2-(a-c)*(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2)/(a+ b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2))*(2*a-2*c+(a^2-2*a*c+b^2+c^2)^(1/2))^(1 /2)*(a^2-b^2-2*a*c+c^2-(a-c)*(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2)/(a^2-2*a*c+b ^2+c^2)^(3/2)/e*2^(1/2)+1/2*arctan(1/2*(b*(2*a-2*c-(a^2-2*a*c+b^2+c^2)^(1/ 2))+(b^2-(a-c)*(a-c+(a^2-2*a*c+b^2+c^2)^(1/2)))*tan(e*x+d))*2^(1/2)/(2*a-2 *c-(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2)/(a^2-b^2-2*a*c+c^2+(a-c)*(a^2-2*a*c+b^ 2+c^2)^(1/2))^(1/2)/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2))*(2*a-2*c-(a^2-2 *a*c+b^2+c^2)^(1/2))^(1/2)*(a^2-b^2-2*a*c+c^2+(a-c)*(a^2-2*a*c+b^2+c^2)^(1 /2))^(1/2)/(a^2-2*a*c+b^2+c^2)^(3/2)/e*2^(1/2)-2*(a*b*(a+c)+c*(2*a^2-2*a*c +b^2)*tan(e*x+d))/(b^2+(a-c)^2)/(-4*a*c+b^2)/e/(a+b*tan(e*x+d)+c*tan(e*x+d )^2)^(1/2)
Result contains complex when optimal does not.
Time = 5.30 (sec) , antiderivative size = 328, normalized size of antiderivative = 0.51 \[ \int \frac {\tan ^2(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx=\frac {\frac {\left (-b^2 (b+i c)-4 i a^2 c+a \left (i b^2+4 b c+4 i c^2\right )\right ) \text {arctanh}\left (\frac {2 a-i b+(b-2 i c) \tan (d+e x)}{2 \sqrt {a-i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {a-i b-c}}+\frac {i \left (4 a^2 c+b^2 (i b+c)-a \left (b^2+4 i b c+4 c^2\right )\right ) \text {arctanh}\left (\frac {2 a+i b+(b+2 i c) \tan (d+e x)}{2 \sqrt {a+i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {a+i b-c}}-\frac {4 \left (a b (a+c)+c \left (2 a^2+b^2-2 a c\right ) \tan (d+e x)\right )}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}}{2 \left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e} \]
(((-(b^2*(b + I*c)) - (4*I)*a^2*c + a*(I*b^2 + 4*b*c + (4*I)*c^2))*ArcTanh [(2*a - I*b + (b - (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[a + b* Tan[d + e*x] + c*Tan[d + e*x]^2])])/Sqrt[a - I*b - c] + (I*(4*a^2*c + b^2* (I*b + c) - a*(b^2 + (4*I)*b*c + 4*c^2))*ArcTanh[(2*a + I*b + (b + (2*I)*c )*Tan[d + e*x])/(2*Sqrt[a + I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e *x]^2])])/Sqrt[a + I*b - c] - (4*(a*b*(a + c) + c*(2*a^2 + b^2 - 2*a*c)*Ta n[d + e*x]))/Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(2*(b^2 + (a - c )^2)*(b^2 - 4*a*c)*e)
Time = 1.26 (sec) , antiderivative size = 719, normalized size of antiderivative = 1.13, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {3042, 4183, 2137, 27, 1369, 1363, 218}
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.
| \(\displaystyle \int \frac {\tan ^2(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\tan (d+e x)^2}{\left (a+b \tan (d+e x)+c \tan (d+e x)^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4183 |
| \(\displaystyle \frac {\int \frac {\tan ^2(d+e x)}{\left (\tan ^2(d+e x)+1\right ) \left (c \tan ^2(d+e x)+b \tan (d+e x)+a\right )^{3/2}}d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 2137 |
| \(\displaystyle \frac {-\frac {2 \int \frac {\left (b^2-4 a c\right ) (a-c-b \tan (d+e x))}{2 \left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}d\tan (d+e x)}{\left ((a-c)^2+b^2\right ) \left (b^2-4 a c\right )}-\frac {2 \left (c \left (2 a^2-2 a c+b^2\right ) \tan (d+e x)+a b (a+c)\right )}{\left ((a-c)^2+b^2\right ) \left (b^2-4 a c\right ) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}}{e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int \frac {a-c-b \tan (d+e x)}{\left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}d\tan (d+e x)}{(a-c)^2+b^2}-\frac {2 \left (c \left (2 a^2-2 a c+b^2\right ) \tan (d+e x)+a b (a+c)\right )}{\left ((a-c)^2+b^2\right ) \left (b^2-4 a c\right ) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}}{e}\) |
| \(\Big \downarrow \) 1369 |
| \(\displaystyle \frac {-\frac {\frac {\int \frac {b^2-\left (2 a-2 c+\sqrt {a^2-2 c a+b^2+c^2}\right ) \tan (d+e x) b-(a-c) \left (a-c-\sqrt {a^2-2 c a+b^2+c^2}\right )}{\left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}d\tan (d+e x)}{2 \sqrt {a^2-2 a c+b^2+c^2}}-\frac {\int \frac {b^2-\left (2 a-2 c-\sqrt {a^2-2 c a+b^2+c^2}\right ) \tan (d+e x) b-(a-c) \left (a-c+\sqrt {a^2-2 c a+b^2+c^2}\right )}{\left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}d\tan (d+e x)}{2 \sqrt {a^2-2 a c+b^2+c^2}}}{(a-c)^2+b^2}-\frac {2 \left (c \left (2 a^2-2 a c+b^2\right ) \tan (d+e x)+a b (a+c)\right )}{\left ((a-c)^2+b^2\right ) \left (b^2-4 a c\right ) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}}{e}\) |
| \(\Big \downarrow \) 1363 |
| \(\displaystyle \frac {-\frac {\frac {b \left (\sqrt {a^2-2 a c+b^2+c^2}+2 a-2 c\right ) \left (b^2-(a-c) \left (-\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )\right ) \int \frac {1}{\frac {b \left (b \left (2 a-2 c+\sqrt {a^2-2 c a+b^2+c^2}\right )+\left (b^2-(a-c) \left (a-c-\sqrt {a^2-2 c a+b^2+c^2}\right )\right ) \tan (d+e x)\right )^2}{c \tan ^2(d+e x)+b \tan (d+e x)+a}-2 b \left (2 a-2 c+\sqrt {a^2-2 c a+b^2+c^2}\right ) \left (b^2-(a-c) \left (a-c-\sqrt {a^2-2 c a+b^2+c^2}\right )\right )}d\left (-\frac {b \left (2 a-2 c+\sqrt {a^2-2 c a+b^2+c^2}\right )+\left (b^2-(a-c) \left (a-c-\sqrt {a^2-2 c a+b^2+c^2}\right )\right ) \tan (d+e x)}{\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{\sqrt {a^2-2 a c+b^2+c^2}}-\frac {b \left (-\sqrt {a^2-2 a c+b^2+c^2}+2 a-2 c\right ) \left (b^2-(a-c) \left (\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )\right ) \int \frac {1}{\frac {b \left (b \left (2 a-2 c-\sqrt {a^2-2 c a+b^2+c^2}\right )+\left (b^2-(a-c) \left (a-c+\sqrt {a^2-2 c a+b^2+c^2}\right )\right ) \tan (d+e x)\right )^2}{c \tan ^2(d+e x)+b \tan (d+e x)+a}-2 b \left (2 a-2 c-\sqrt {a^2-2 c a+b^2+c^2}\right ) \left (b^2-(a-c) \left (a-c+\sqrt {a^2-2 c a+b^2+c^2}\right )\right )}d\left (-\frac {b \left (2 a-2 c-\sqrt {a^2-2 c a+b^2+c^2}\right )+\left (b^2-(a-c) \left (a-c+\sqrt {a^2-2 c a+b^2+c^2}\right )\right ) \tan (d+e x)}{\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{\sqrt {a^2-2 a c+b^2+c^2}}}{(a-c)^2+b^2}-\frac {2 \left (c \left (2 a^2-2 a c+b^2\right ) \tan (d+e x)+a b (a+c)\right )}{\left ((a-c)^2+b^2\right ) \left (b^2-4 a c\right ) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}}{e}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {-\frac {\frac {\sqrt {-\sqrt {a^2-2 a c+b^2+c^2}+2 a-2 c} \left (b^2-(a-c) \left (\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )\right ) \arctan \left (\frac {\left (b^2-(a-c) \left (\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )\right ) \tan (d+e x)+b \left (-\sqrt {a^2-2 a c+b^2+c^2}+2 a-2 c\right )}{\sqrt {2} \sqrt {-\sqrt {a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt {(a-c) \sqrt {a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2-2 a c+b^2+c^2} \sqrt {(a-c) \sqrt {a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2}}-\frac {\sqrt {\sqrt {a^2-2 a c+b^2+c^2}+2 a-2 c} \left (b^2-(a-c) \left (-\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )\right ) \arctan \left (\frac {\left (b^2-(a-c) \left (-\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )\right ) \tan (d+e x)+b \left (\sqrt {a^2-2 a c+b^2+c^2}+2 a-2 c\right )}{\sqrt {2} \sqrt {\sqrt {a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt {-(a-c) \sqrt {a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2-2 a c+b^2+c^2} \sqrt {-(a-c) \sqrt {a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2}}}{(a-c)^2+b^2}-\frac {2 \left (c \left (2 a^2-2 a c+b^2\right ) \tan (d+e x)+a b (a+c)\right )}{\left ((a-c)^2+b^2\right ) \left (b^2-4 a c\right ) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}}{e}\) |
(-((-((Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*(b^2 - (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]))*ArcTan[(b*(2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) + (b^2 - (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2])) *Tan[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sq rt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2] ])) + (Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*(b^2 - (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]))*ArcTan[(b*(2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) + (b^2 - (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2])) *Tan[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sq rt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2] ]))/(b^2 + (a - c)^2)) - (2*(a*b*(a + c) + c*(2*a^2 + b^2 - 2*a*c)*Tan[d + e*x]))/((b^2 + (a - c)^2)*(b^2 - 4*a*c)*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]))/e
3.1.22.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((g_) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f _.)*(x_)^2]), x_Symbol] :> Simp[-2*a*g*h Subst[Int[1/Simp[2*a^2*g*h*c + a *e*x^2, x], x], x, Simp[a*h - g*c*x, x]/Sqrt[d + e*x + f*x^2]], x] /; FreeQ [{a, c, d, e, f, g, h}, x] && EqQ[a*h^2*e + 2*g*h*(c*d - a*f) - g^2*c*e, 0]
Int[((g_.) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + ( f_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[(c*d - a*f)^2 + a*c*e^2, 2]}, Simp [1/(2*q) Int[Simp[(-a)*h*e - g*(c*d - a*f - q) + (h*(c*d - a*f + q) - g*c *e)*x, x]/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] - Simp[1/(2*q) Int[ Simp[(-a)*h*e - g*(c*d - a*f + q) + (h*(c*d - a*f - q) - g*c*e)*x, x]/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, c, d, e, f, g, h}, x] && NeQ[e^2 - 4*d*f, 0] && NegQ[(-a)*c]
Int[(Px_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (f_.)*(x_)^2)^(q_ ), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[P x, x, 2]}, Simp[(a + b*x + c*x^2)^(p + 1)*((d + f*x^2)^(q + 1)/((b^2 - 4*a* c)*(b^2*d*f + (c*d - a*f)^2)*(p + 1)))*((A*c - a*C)*((-b)*(c*d + a*f)) + (A *b - a*B)*(2*c^2*d + b^2*f - c*(2*a*f)) + c*(A*(2*c^2*d + b^2*f - c*(2*a*f) ) - B*(b*c*d + a*b*f) + C*(b^2*d - 2*a*(c*d - a*f)))*x), x] + Simp[1/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)*(p + 1)) Int[(a + b*x + c*x^2)^(p + 1) *(d + f*x^2)^q*Simp[(b*B - 2*A*c - 2*a*C)*((c*d - a*f)^2 - (b*d)*((-b)*f))* (p + 1) + (b^2*(C*d + A*f) - b*(B*c*d + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c* C*d - a*C*f)))*(a*f*(p + 1) - c*d*(p + 2)) - (2*f*((A*c - a*C)*((-b)*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(2*a*f)))*(p + q + 2) - (b^2*(C*d + A*f) - b*(B*c*d + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - a*C*f)))*(b*f *(p + 1)))*x - c*f*(b^2*(C*d + A*f) - b*(B*c*d + a*B*f) + 2*(A*c*(c*d - a*f ) - a*(c*C*d - a*C*f)))*(2*p + 2*q + 5)*x^2, x], x], x]] /; FreeQ[{a, b, c, d, f, q}, x] && PolyQ[Px, x, 2] && LtQ[p, -1] && NeQ[b^2*d*f + (c*d - a*f) ^2, 0] && !( !IntegerQ[p] && ILtQ[q, -1]) && !IGtQ[q, 0]
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]
result has leaf size over 500,000. Avoiding possible recursion issues.
Time = 1.13 (sec) , antiderivative size = 11848772, normalized size of antiderivative = 18571.74
\[\text {output too large to display}\]
Leaf count of result is larger than twice the leaf count of optimal. 19326 vs. \(2 (587) = 1174\).
Time = 3.09 (sec) , antiderivative size = 19326, normalized size of antiderivative = 30.29 \[ \int \frac {\tan ^2(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx=\text {Too large to display} \]
\[ \int \frac {\tan ^2(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx=\int \frac {\tan ^{2}{\left (d + e x \right )}}{\left (a + b \tan {\left (d + e x \right )} + c \tan ^{2}{\left (d + e x \right )}\right )^{\frac {3}{2}}}\, dx \]
Exception generated. \[ \int \frac {\tan ^2(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Timed out. \[ \int \frac {\tan ^2(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\tan ^2(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx=\int \frac {{\mathrm {tan}\left (d+e\,x\right )}^2}{{\left (c\,{\mathrm {tan}\left (d+e\,x\right )}^2+b\,\mathrm {tan}\left (d+e\,x\right )+a\right )}^{3/2}} \,d x \]