3.82 \(\int \frac{A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^2} \, dx\)

Optimal. Leaf size=509 \[ -\frac{d \left (A \left (a^2 d^2+b^2 \left (c^2+2 d^2\right )\right )+a^2 \left (-B c d+2 c^2 C+C d^2\right )-a b B \left (c^2+d^2\right )+b^2 c (c C-B d)\right )}{f \left (a^2+b^2\right ) \left (c^2+d^2\right ) (b c-a d)^2 (c+d \tan (e+f x))}-\frac{x \left (a^2 \left (-A \left (c^2-d^2\right )-2 B c d+c^2 C-C d^2\right )+2 a b \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )-b^2 \left (-A \left (c^2-d^2\right )-2 B c d+c^2 C-C d^2\right )\right )}{\left (a^2+b^2\right )^2 \left (c^2+d^2\right )^2}-\frac{A b^2-a (b B-a C)}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) (c+d \tan (e+f x))}+\frac{b \left (-a^2 b^2 (4 A d+B c)+3 a^3 b B d-2 a^4 C d+a b^3 (2 A c+B d-2 c C)+b^4 (B c-2 A d)\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{f \left (a^2+b^2\right )^2 (b c-a d)^3}+\frac{d \left (b \left (4 A c^2 d^2+2 A d^4-3 B c^3 d-B c d^3+2 c^4 C\right )-a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{f \left (c^2+d^2\right )^2 (b c-a d)^3} \]

[Out]

-(((a^2*(c^2*C - 2*B*c*d - C*d^2 - A*(c^2 - d^2)) - b^2*(c^2*C - 2*B*c*d - C*d^2 - A*(c^2 - d^2)) + 2*a*b*(2*c
*(A - C)*d - B*(c^2 - d^2)))*x)/((a^2 + b^2)^2*(c^2 + d^2)^2)) + (b*(3*a^3*b*B*d - 2*a^4*C*d + b^4*(B*c - 2*A*
d) - a^2*b^2*(B*c + 4*A*d) + a*b^3*(2*A*c - 2*c*C + B*d))*Log[a*Cos[e + f*x] + b*Sin[e + f*x]])/((a^2 + b^2)^2
*(b*c - a*d)^3*f) + (d*(b*(2*c^4*C - 3*B*c^3*d + 4*A*c^2*d^2 - B*c*d^3 + 2*A*d^4) - a*d^2*(2*c*(A - C)*d - B*(
c^2 - d^2)))*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/((b*c - a*d)^3*(c^2 + d^2)^2*f) - (d*(b^2*c*(c*C - B*d) - a
*b*B*(c^2 + d^2) + a^2*(2*c^2*C - B*c*d + C*d^2) + A*(a^2*d^2 + b^2*(c^2 + 2*d^2))))/((a^2 + b^2)*(b*c - a*d)^
2*(c^2 + d^2)*f*(c + d*Tan[e + f*x])) - (A*b^2 - a*(b*B - a*C))/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x]
)*(c + d*Tan[e + f*x]))

________________________________________________________________________________________

Rubi [A]  time = 2.15119, antiderivative size = 508, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {3649, 3651, 3530} \[ -\frac{d \left (a^2 A d^2+a^2 \left (-B c d+2 c^2 C+C d^2\right )-a b B \left (c^2+d^2\right )+A b^2 \left (c^2+2 d^2\right )+b^2 c (c C-B d)\right )}{f \left (a^2+b^2\right ) \left (c^2+d^2\right ) (b c-a d)^2 (c+d \tan (e+f x))}-\frac{x \left (a^2 \left (-A \left (c^2-d^2\right )-2 B c d+c^2 C-C d^2\right )+2 a b \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )-b^2 \left (-A \left (c^2-d^2\right )-2 B c d+c^2 C-C d^2\right )\right )}{\left (a^2+b^2\right )^2 \left (c^2+d^2\right )^2}-\frac{A b^2-a (b B-a C)}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) (c+d \tan (e+f x))}+\frac{b \left (-a^2 b^2 (4 A d+B c)+3 a^3 b B d-2 a^4 C d+a b^3 (2 A c+B d-2 c C)+b^4 (B c-2 A d)\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{f \left (a^2+b^2\right )^2 (b c-a d)^3}+\frac{d \left (b \left (4 A c^2 d^2+2 A d^4-3 B c^3 d-B c d^3+2 c^4 C\right )-a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{f \left (c^2+d^2\right )^2 (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^2),x]

[Out]

-(((a^2*(c^2*C - 2*B*c*d - C*d^2 - A*(c^2 - d^2)) - b^2*(c^2*C - 2*B*c*d - C*d^2 - A*(c^2 - d^2)) + 2*a*b*(2*c
*(A - C)*d - B*(c^2 - d^2)))*x)/((a^2 + b^2)^2*(c^2 + d^2)^2)) + (b*(3*a^3*b*B*d - 2*a^4*C*d + b^4*(B*c - 2*A*
d) - a^2*b^2*(B*c + 4*A*d) + a*b^3*(2*A*c - 2*c*C + B*d))*Log[a*Cos[e + f*x] + b*Sin[e + f*x]])/((a^2 + b^2)^2
*(b*c - a*d)^3*f) + (d*(b*(2*c^4*C - 3*B*c^3*d + 4*A*c^2*d^2 - B*c*d^3 + 2*A*d^4) - a*d^2*(2*c*(A - C)*d - B*(
c^2 - d^2)))*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/((b*c - a*d)^3*(c^2 + d^2)^2*f) - (d*(a^2*A*d^2 + b^2*c*(c*
C - B*d) - a*b*B*(c^2 + d^2) + A*b^2*(c^2 + 2*d^2) + a^2*(2*c^2*C - B*c*d + C*d^2)))/((a^2 + b^2)*(b*c - a*d)^
2*(c^2 + d^2)*f*(c + d*Tan[e + f*x])) - (A*b^2 - a*(b*B - a*C))/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x]
)*(c + d*Tan[e + f*x]))

Rule 3649

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*t
an[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[((A*b^2 - a*(b*B - a*C))*(a + b*T
an[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2)), x] + Dist[1/((m + 1)*(
b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1)
- b^2*d*(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - a*B - b*C)*Tan[e
+ f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C,
n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] &&  !(ILtQ[n, -1] && ( !I
ntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rule 3651

Int[((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2)/(((a_) + (b_.)*tan[(e_.) + (f_.)
*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[((a*(A*c - c*C + B*d) + b*(B*c - A*d + C*d
))*x)/((a^2 + b^2)*(c^2 + d^2)), x] + (Dist[(A*b^2 - a*b*B + a^2*C)/((b*c - a*d)*(a^2 + b^2)), Int[(b - a*Tan[
e + f*x])/(a + b*Tan[e + f*x]), x], x] - Dist[(c^2*C - B*c*d + A*d^2)/((b*c - a*d)*(c^2 + d^2)), Int[(d - c*Ta
n[e + f*x])/(c + d*Tan[e + f*x]), x], x]) /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ
[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]

Rule 3530

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c*Log[Re
moveContent[a*Cos[e + f*x] + b*Sin[e + f*x], x]])/(b*f), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d,
0] && NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]

Rubi steps

\begin{align*} \int \frac{A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^2} \, dx &=-\frac{A b^2-a (b B-a C)}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) (c+d \tan (e+f x))}-\frac{\int \frac{2 A b^2 d-a A (b c-a d)-(b B-a C) (b c+a d)+(A b-a B-b C) (b c-a d) \tan (e+f x)+2 \left (A b^2-a (b B-a C)\right ) d \tan ^2(e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^2} \, dx}{\left (a^2+b^2\right ) (b c-a d)}\\ &=-\frac{d \left (a^2 A d^2+b^2 c (c C-B d)-a b B \left (c^2+d^2\right )+A b^2 \left (c^2+2 d^2\right )+a^2 \left (2 c^2 C-B c d+C d^2\right )\right )}{\left (a^2+b^2\right ) (b c-a d)^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))}-\frac{A b^2-a (b B-a C)}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) (c+d \tan (e+f x))}-\frac{\int \frac{-a^3 d^2 (A c-c C+B d)+2 a^2 A b d \left (c^2+d^2\right )-b^3 (B c-2 A d) \left (c^2+d^2\right )+a b^2 \left (c^3 C+2 c C d^2-B d^3-A \left (c^3+2 c d^2\right )\right )-(b c-a d)^2 (b c C-b B d-A (b c+a d)+a (B c+C d)) \tan (e+f x)+b d \left (a^2 A d^2+b^2 c (c C-B d)-a b B \left (c^2+d^2\right )+A b^2 \left (c^2+2 d^2\right )+a^2 \left (2 c^2 C-B c d+C d^2\right )\right ) \tan ^2(e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))} \, dx}{\left (a^2+b^2\right ) (b c-a d)^2 \left (c^2+d^2\right )}\\ &=-\frac{\left (a^2 \left (c^2 C-2 B c d-C d^2-A \left (c^2-d^2\right )\right )-b^2 \left (c^2 C-2 B c d-C d^2-A \left (c^2-d^2\right )\right )+2 a b \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) x}{\left (a^2+b^2\right )^2 \left (c^2+d^2\right )^2}-\frac{d \left (a^2 A d^2+b^2 c (c C-B d)-a b B \left (c^2+d^2\right )+A b^2 \left (c^2+2 d^2\right )+a^2 \left (2 c^2 C-B c d+C d^2\right )\right )}{\left (a^2+b^2\right ) (b c-a d)^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))}-\frac{A b^2-a (b B-a C)}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) (c+d \tan (e+f x))}+\frac{\left (b \left (3 a^3 b B d-2 a^4 C d+b^4 (B c-2 A d)-a^2 b^2 (B c+4 A d)+a b^3 (2 A c-2 c C+B d)\right )\right ) \int \frac{b-a \tan (e+f x)}{a+b \tan (e+f x)} \, dx}{\left (a^2+b^2\right )^2 (b c-a d)^3}+\frac{\left (d \left (b \left (2 c^4 C-3 B c^3 d+4 A c^2 d^2-B c d^3+2 A d^4\right )-a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right )\right ) \int \frac{d-c \tan (e+f x)}{c+d \tan (e+f x)} \, dx}{(b c-a d)^3 \left (c^2+d^2\right )^2}\\ &=-\frac{\left (a^2 \left (c^2 C-2 B c d-C d^2-A \left (c^2-d^2\right )\right )-b^2 \left (c^2 C-2 B c d-C d^2-A \left (c^2-d^2\right )\right )+2 a b \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) x}{\left (a^2+b^2\right )^2 \left (c^2+d^2\right )^2}+\frac{b \left (3 a^3 b B d-2 a^4 C d+b^4 (B c-2 A d)-a^2 b^2 (B c+4 A d)+a b^3 (2 A c-2 c C+B d)\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{\left (a^2+b^2\right )^2 (b c-a d)^3 f}+\frac{d \left (b \left (2 c^4 C-3 B c^3 d+4 A c^2 d^2-B c d^3+2 A d^4\right )-a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{(b c-a d)^3 \left (c^2+d^2\right )^2 f}-\frac{d \left (a^2 A d^2+b^2 c (c C-B d)-a b B \left (c^2+d^2\right )+A b^2 \left (c^2+2 d^2\right )+a^2 \left (2 c^2 C-B c d+C d^2\right )\right )}{\left (a^2+b^2\right ) (b c-a d)^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))}-\frac{A b^2-a (b B-a C)}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) (c+d \tan (e+f x))}\\ \end{align*}

Mathematica [A]  time = 8.90489, size = 984, normalized size = 1.93 \[ -\frac{A b^2-a (b B-a C)}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) (c+d \tan (e+f x))}-\frac{-\frac{d^2 \left (2 A d b^2-a A (b c-a d)-(b B-a C) (b c+a d)\right )-c \left ((A b-C b-a B) d (b c-a d)-2 c \left (A b^2-a (b B-a C)\right ) d\right )}{(a d-b c) \left (c^2+d^2\right ) f (c+d \tan (e+f x))}-\frac{-\frac{\left (c^2+d^2\right ) \left (-2 C d a^4+3 b B d a^3-b^2 (B c+4 A d) a^2+b^3 (2 A c-2 C c+B d) a+b^4 (B c-2 A d)\right ) \log (a+b \tan (e+f x)) b^2}{\left (a^2+b^2\right ) (b c-a d)}+\frac{(b c-a d)^2 \left (-B c^2 a^2+B d^2 a^2+2 A c d a^2-2 c C d a^2+2 A b c^2 a-2 A b d^2 a+2 b C d^2 a-2 b c^2 C a+4 b B c d a+b^2 B c^2-b^2 B d^2-2 A b^2 c d+2 b^2 c C d-\frac{\sqrt{-b^2} \left (\left (C c^2-2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a^2+2 b \left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) a-b^2 \left (C c^2-2 B d c-C d^2-A \left (c^2-d^2\right )\right )\right )}{b}\right ) \log \left (\sqrt{-b^2}-b \tan (e+f x)\right ) b}{2 \left (a^2+b^2\right ) \left (c^2+d^2\right )}+\frac{(b c-a d)^2 \left (-B c^2 a^2+B d^2 a^2+2 A c d a^2-2 c C d a^2+2 A b c^2 a-2 A b d^2 a+2 b C d^2 a-2 b c^2 C a+4 b B c d a+b^2 B c^2-b^2 B d^2-2 A b^2 c d+2 b^2 c C d+\frac{\sqrt{-b^2} \left (\left (C c^2-2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a^2+2 b \left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) a-b^2 \left (C c^2-2 B d c-C d^2-A \left (c^2-d^2\right )\right )\right )}{b}\right ) \log \left (b \tan (e+f x)+\sqrt{-b^2}\right ) b}{2 \left (a^2+b^2\right ) \left (c^2+d^2\right )}-\frac{\left (a^2+b^2\right ) d \left (b \left (2 C c^4-3 B d c^3+4 A d^2 c^2-B d^3 c+2 A d^4\right )-a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \log (c+d \tan (e+f x)) b}{(b c-a d) \left (c^2+d^2\right )}}{b (a d-b c) \left (c^2+d^2\right ) f}}{\left (a^2+b^2\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^2),x]

[Out]

-((A*b^2 - a*(b*B - a*C))/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x]))) - (-(((b*(b*c
- a*d)^2*(2*a*A*b*c^2 - a^2*B*c^2 + b^2*B*c^2 - 2*a*b*c^2*C + 2*a^2*A*c*d - 2*A*b^2*c*d + 4*a*b*B*c*d - 2*a^2
*c*C*d + 2*b^2*c*C*d - 2*a*A*b*d^2 + a^2*B*d^2 - b^2*B*d^2 + 2*a*b*C*d^2 - (Sqrt[-b^2]*(a^2*(c^2*C - 2*B*c*d -
C*d^2 - A*(c^2 - d^2)) - b^2*(c^2*C - 2*B*c*d - C*d^2 - A*(c^2 - d^2)) + 2*a*b*(2*c*(A - C)*d - B*(c^2 - d^2)
)))/b)*Log[Sqrt[-b^2] - b*Tan[e + f*x]])/(2*(a^2 + b^2)*(c^2 + d^2)) - (b^2*(c^2 + d^2)*(3*a^3*b*B*d - 2*a^4*C
*d + b^4*(B*c - 2*A*d) - a^2*b^2*(B*c + 4*A*d) + a*b^3*(2*A*c - 2*c*C + B*d))*Log[a + b*Tan[e + f*x]])/((a^2 +
b^2)*(b*c - a*d)) + (b*(b*c - a*d)^2*(2*a*A*b*c^2 - a^2*B*c^2 + b^2*B*c^2 - 2*a*b*c^2*C + 2*a^2*A*c*d - 2*A*b
^2*c*d + 4*a*b*B*c*d - 2*a^2*c*C*d + 2*b^2*c*C*d - 2*a*A*b*d^2 + a^2*B*d^2 - b^2*B*d^2 + 2*a*b*C*d^2 + (Sqrt[-
b^2]*(a^2*(c^2*C - 2*B*c*d - C*d^2 - A*(c^2 - d^2)) - b^2*(c^2*C - 2*B*c*d - C*d^2 - A*(c^2 - d^2)) + 2*a*b*(2
*c*(A - C)*d - B*(c^2 - d^2))))/b)*Log[Sqrt[-b^2] + b*Tan[e + f*x]])/(2*(a^2 + b^2)*(c^2 + d^2)) - (b*(a^2 + b
^2)*d*(b*(2*c^4*C - 3*B*c^3*d + 4*A*c^2*d^2 - B*c*d^3 + 2*A*d^4) - a*d^2*(2*c*(A - C)*d - B*(c^2 - d^2)))*Log[
c + d*Tan[e + f*x]])/((b*c - a*d)*(c^2 + d^2)))/(b*(-(b*c) + a*d)*(c^2 + d^2)*f)) - (-(c*(-2*c*(A*b^2 - a*(b*B
- a*C))*d + (A*b - a*B - b*C)*d*(b*c - a*d))) + d^2*(2*A*b^2*d - a*A*(b*c - a*d) - (b*B - a*C)*(b*c + a*d)))/
((-(b*c) + a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x])))/((a^2 + b^2)*(b*c - a*d))

________________________________________________________________________________________

Maple [B]  time = 0.144, size = 2012, normalized size = 4. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e))^2/(c+d*tan(f*x+e))^2,x)

[Out]

-1/2/f/(a^2+b^2)^2/(c^2+d^2)^2*ln(1+tan(f*x+e)^2)*B*a^2*d^2-1/2/f/(a^2+b^2)^2/(c^2+d^2)^2*ln(1+tan(f*x+e)^2)*B
*b^2*c^2+1/2/f/(a^2+b^2)^2/(c^2+d^2)^2*ln(1+tan(f*x+e)^2)*B*b^2*d^2+1/f/(a^2+b^2)^2/(c^2+d^2)^2*A*arctan(tan(f
*x+e))*a^2*c^2-1/f/(a^2+b^2)^2/(c^2+d^2)^2*A*arctan(tan(f*x+e))*a^2*d^2-4/f/(a^2+b^2)^2/(c^2+d^2)^2*A*arctan(t
an(f*x+e))*a*b*c*d+4/f/(a^2+b^2)^2/(c^2+d^2)^2*C*arctan(tan(f*x+e))*a*b*c*d-2/f/(a^2+b^2)^2/(c^2+d^2)^2*ln(1+t
an(f*x+e)^2)*B*a*b*c*d-4/f*d^3/(a*d-b*c)^3/(c^2+d^2)^2*ln(c+d*tan(f*x+e))*A*b*c^2-2/f*d^4/(a*d-b*c)^3/(c^2+d^2
)^2*ln(c+d*tan(f*x+e))*C*a*c-2/f*d/(a*d-b*c)^3/(c^2+d^2)^2*ln(c+d*tan(f*x+e))*C*b*c^4-1/f*d^3/(a*d-b*c)^3/(c^2
+d^2)^2*ln(c+d*tan(f*x+e))*B*a*c^2+2/f*d^4/(a*d-b*c)^3/(c^2+d^2)^2*ln(c+d*tan(f*x+e))*A*a*c+3/f*d^2/(a*d-b*c)^
3/(c^2+d^2)^2*ln(c+d*tan(f*x+e))*B*b*c^3+1/f*d^4/(a*d-b*c)^3/(c^2+d^2)^2*ln(c+d*tan(f*x+e))*B*b*c-1/f/(a^2+b^2
)^2/(c^2+d^2)^2*ln(1+tan(f*x+e)^2)*A*a^2*c*d-1/f/(a^2+b^2)^2/(c^2+d^2)^2*ln(1+tan(f*x+e)^2)*A*a*b*c^2+1/f/(a^2
+b^2)^2/(c^2+d^2)^2*ln(1+tan(f*x+e)^2)*A*a*b*d^2+1/f/(a^2+b^2)^2/(c^2+d^2)^2*ln(1+tan(f*x+e)^2)*A*b^2*c*d+1/f/
(a^2+b^2)^2/(c^2+d^2)^2*ln(1+tan(f*x+e)^2)*C*a^2*c*d-1/f/(a^2+b^2)^2/(c^2+d^2)^2*ln(1+tan(f*x+e)^2)*C*b^2*c*d+
2/f/(a^2+b^2)^2/(c^2+d^2)^2*B*arctan(tan(f*x+e))*a^2*c*d+2/f/(a^2+b^2)^2/(c^2+d^2)^2*B*arctan(tan(f*x+e))*a*b*
c^2-2/f/(a^2+b^2)^2/(c^2+d^2)^2*B*arctan(tan(f*x+e))*a*b*d^2-2/f/(a^2+b^2)^2/(c^2+d^2)^2*B*arctan(tan(f*x+e))*
b^2*c*d+1/f/(a^2+b^2)^2/(c^2+d^2)^2*ln(1+tan(f*x+e)^2)*C*a*b*c^2-1/f/(a^2+b^2)^2/(c^2+d^2)^2*ln(1+tan(f*x+e)^2
)*C*a*b*d^2+2/f*b/(a^2+b^2)^2/(a*d-b*c)^3*ln(a+b*tan(f*x+e))*a^4*C*d+2/f*b^4/(a^2+b^2)^2/(a*d-b*c)^3*ln(a+b*ta
n(f*x+e))*C*a*c+4/f*b^3/(a^2+b^2)^2/(a*d-b*c)^3*ln(a+b*tan(f*x+e))*A*a^2*d-2/f*b^4/(a^2+b^2)^2/(a*d-b*c)^3*ln(
a+b*tan(f*x+e))*A*a*c-3/f*b^2/(a^2+b^2)^2/(a*d-b*c)^3*ln(a+b*tan(f*x+e))*a^3*B*d+1/f*b^3/(a^2+b^2)^2/(a*d-b*c)
^3*ln(a+b*tan(f*x+e))*B*a^2*c-1/f*b^4/(a^2+b^2)^2/(a*d-b*c)^3*ln(a+b*tan(f*x+e))*B*a*d-1/f*d^3/(a*d-b*c)^2/(c^
2+d^2)/(c+d*tan(f*x+e))*A-1/f*b^3/(a^2+b^2)/(a*d-b*c)^2/(a+b*tan(f*x+e))*A+1/f*d^2/(a*d-b*c)^2/(c^2+d^2)/(c+d*
tan(f*x+e))*B*c-1/f/(a^2+b^2)^2/(c^2+d^2)^2*C*arctan(tan(f*x+e))*b^2*d^2-2/f*d^5/(a*d-b*c)^3/(c^2+d^2)^2*ln(c+
d*tan(f*x+e))*A*b+1/f*d^5/(a*d-b*c)^3/(c^2+d^2)^2*ln(c+d*tan(f*x+e))*B*a-1/f*d/(a*d-b*c)^2/(c^2+d^2)/(c+d*tan(
f*x+e))*c^2*C+2/f*b^5/(a^2+b^2)^2/(a*d-b*c)^3*ln(a+b*tan(f*x+e))*A*d-1/f*b^5/(a^2+b^2)^2/(a*d-b*c)^3*ln(a+b*ta
n(f*x+e))*B*c+1/f*b^2/(a^2+b^2)/(a*d-b*c)^2/(a+b*tan(f*x+e))*B*a+1/2/f/(a^2+b^2)^2/(c^2+d^2)^2*ln(1+tan(f*x+e)
^2)*B*a^2*c^2-1/f/(a^2+b^2)^2/(c^2+d^2)^2*A*arctan(tan(f*x+e))*b^2*c^2+1/f/(a^2+b^2)^2/(c^2+d^2)^2*A*arctan(ta
n(f*x+e))*b^2*d^2-1/f/(a^2+b^2)^2/(c^2+d^2)^2*C*arctan(tan(f*x+e))*a^2*c^2+1/f/(a^2+b^2)^2/(c^2+d^2)^2*C*arcta
n(tan(f*x+e))*a^2*d^2+1/f/(a^2+b^2)^2/(c^2+d^2)^2*C*arctan(tan(f*x+e))*b^2*c^2-1/f*b/(a^2+b^2)/(a*d-b*c)^2/(a+
b*tan(f*x+e))*C*a^2

________________________________________________________________________________________

Maxima [B]  time = 1.81216, size = 1600, normalized size = 3.14 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e))^2/(c+d*tan(f*x+e))^2,x, algorithm="maxima")

[Out]

1/2*(2*(((A - C)*a^2 + 2*B*a*b - (A - C)*b^2)*c^2 + 2*(B*a^2 - 2*(A - C)*a*b - B*b^2)*c*d - ((A - C)*a^2 + 2*B
*a*b - (A - C)*b^2)*d^2)*(f*x + e)/((a^4 + 2*a^2*b^2 + b^4)*c^4 + 2*(a^4 + 2*a^2*b^2 + b^4)*c^2*d^2 + (a^4 + 2
*a^2*b^2 + b^4)*d^4) - 2*((B*a^2*b^3 - 2*(A - C)*a*b^4 - B*b^5)*c + (2*C*a^4*b - 3*B*a^3*b^2 + 4*A*a^2*b^3 - B
*a*b^4 + 2*A*b^5)*d)*log(b*tan(f*x + e) + a)/((a^4*b^3 + 2*a^2*b^5 + b^7)*c^3 - 3*(a^5*b^2 + 2*a^3*b^4 + a*b^6
)*c^2*d + 3*(a^6*b + 2*a^4*b^3 + a^2*b^5)*c*d^2 - (a^7 + 2*a^5*b^2 + a^3*b^4)*d^3) + 2*(2*C*b*c^4*d - 3*B*b*c^
3*d^2 + (B*a + 4*A*b)*c^2*d^3 - (2*(A - C)*a + B*b)*c*d^4 - (B*a - 2*A*b)*d^5)*log(d*tan(f*x + e) + c)/(b^3*c^
7 - 3*a*b^2*c^6*d + 3*a^2*b*c*d^6 - a^3*d^7 + (3*a^2*b + 2*b^3)*c^5*d^2 - (a^3 + 6*a*b^2)*c^4*d^3 + (6*a^2*b +
b^3)*c^3*d^4 - (2*a^3 + 3*a*b^2)*c^2*d^5) + ((B*a^2 - 2*(A - C)*a*b - B*b^2)*c^2 - 2*((A - C)*a^2 + 2*B*a*b -
(A - C)*b^2)*c*d - (B*a^2 - 2*(A - C)*a*b - B*b^2)*d^2)*log(tan(f*x + e)^2 + 1)/((a^4 + 2*a^2*b^2 + b^4)*c^4
+ 2*(a^4 + 2*a^2*b^2 + b^4)*c^2*d^2 + (a^4 + 2*a^2*b^2 + b^4)*d^4) - 2*((C*a^2*b - B*a*b^2 + A*b^3)*c^3 + (C*a
^3 + C*a*b^2)*c^2*d - (B*a^3 - C*a^2*b + 2*B*a*b^2 - A*b^3)*c*d^2 + (A*a^3 + A*a*b^2)*d^3 + ((2*C*a^2*b - B*a*
b^2 + (A + C)*b^3)*c^2*d - (B*a^2*b + B*b^3)*c*d^2 + ((A + C)*a^2*b - B*a*b^2 + 2*A*b^3)*d^3)*tan(f*x + e))/((
a^3*b^2 + a*b^4)*c^5 - 2*(a^4*b + a^2*b^3)*c^4*d + (a^5 + 2*a^3*b^2 + a*b^4)*c^3*d^2 - 2*(a^4*b + a^2*b^3)*c^2
*d^3 + (a^5 + a^3*b^2)*c*d^4 + ((a^2*b^3 + b^5)*c^4*d - 2*(a^3*b^2 + a*b^4)*c^3*d^2 + (a^4*b + 2*a^2*b^3 + b^5
)*c^2*d^3 - 2*(a^3*b^2 + a*b^4)*c*d^4 + (a^4*b + a^2*b^3)*d^5)*tan(f*x + e)^2 + ((a^2*b^3 + b^5)*c^5 - (a^3*b^
2 + a*b^4)*c^4*d - (a^4*b - b^5)*c^3*d^2 + (a^5 - a*b^4)*c^2*d^3 - (a^4*b + a^2*b^3)*c*d^4 + (a^5 + a^3*b^2)*d
^5)*tan(f*x + e)))/f

________________________________________________________________________________________

Fricas [B]  time = 35.2913, size = 8519, normalized size = 16.74 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e))^2/(c+d*tan(f*x+e))^2,x, algorithm="fricas")

[Out]

-1/2*(2*(C*a^2*b^4 - B*a*b^5 + A*b^6)*c^6 - 2*(C*a^3*b^3 - B*a^2*b^4 + A*a*b^5)*c^5*d + 4*(C*a^2*b^4 - B*a*b^5
+ A*b^6)*c^4*d^2 + 2*(C*a^5*b + 2*B*a^2*b^4 - (2*A - C)*a*b^5)*c^3*d^3 - 2*(C*a^6 + B*a^5*b + 2*C*a^4*b^2 + 2
*B*a^3*b^3 + 2*B*a*b^5 - A*b^6)*c^2*d^4 + 2*(B*a^6 + A*a^5*b + 2*B*a^4*b^2 + (2*A - C)*a^3*b^3 + 2*B*a^2*b^4)*
c*d^5 - 2*(A*a^6 + 2*A*a^4*b^2 + A*a^2*b^4)*d^6 - 2*(((A - C)*a^3*b^3 + 2*B*a^2*b^4 - (A - C)*a*b^5)*c^6 - (3*
(A - C)*a^4*b^2 + 4*B*a^3*b^3 + (A - C)*a^2*b^4 + 2*B*a*b^5)*c^5*d + (3*(A - C)*a^5*b + 8*(A - C)*a^3*b^3 + 4*
B*a^2*b^4 + (A - C)*a*b^5)*c^4*d^2 - ((A - C)*a^6 - 4*B*a^5*b + 8*(A - C)*a^4*b^2 + 3*(A - C)*a^2*b^4)*c^3*d^3
- (2*B*a^6 - (A - C)*a^5*b + 4*B*a^4*b^2 - 3*(A - C)*a^3*b^3)*c^2*d^4 + ((A - C)*a^6 + 2*B*a^5*b - (A - C)*a^
4*b^2)*c*d^5)*f*x - 2*((C*a^3*b^3 - B*a^2*b^4 + A*a*b^5)*c^5*d + (B*a^3*b^3 - (A - 2*C)*a^2*b^4 + C*b^6)*c^4*d
^2 - (C*a^5*b + B*a^4*b^2 + 4*B*a^2*b^4 - (2*A - C)*a*b^5 + B*b^6)*c^3*d^3 + (B*a^5*b + (A - 2*C)*a^4*b^2 + 4*
B*a^3*b^3 + B*a*b^5 + A*b^6)*c^2*d^4 - (A*a^5*b + (2*A - C)*a^3*b^3 + B*a^2*b^4)*c*d^5 - (C*a^4*b^2 - B*a^3*b^
3 + A*a^2*b^4)*d^6 + (((A - C)*a^2*b^4 + 2*B*a*b^5 - (A - C)*b^6)*c^5*d - (3*(A - C)*a^3*b^3 + 4*B*a^2*b^4 + (
A - C)*a*b^5 + 2*B*b^6)*c^4*d^2 + (3*(A - C)*a^4*b^2 + 8*(A - C)*a^2*b^4 + 4*B*a*b^5 + (A - C)*b^6)*c^3*d^3 -
((A - C)*a^5*b - 4*B*a^4*b^2 + 8*(A - C)*a^3*b^3 + 3*(A - C)*a*b^5)*c^2*d^4 - (2*B*a^5*b - (A - C)*a^4*b^2 + 4
*B*a^3*b^3 - 3*(A - C)*a^2*b^4)*c*d^5 + ((A - C)*a^5*b + 2*B*a^4*b^2 - (A - C)*a^3*b^3)*d^6)*f*x)*tan(f*x + e)
^2 + ((B*a^3*b^3 - 2*(A - C)*a^2*b^4 - B*a*b^5)*c^6 + (2*C*a^5*b - 3*B*a^4*b^2 + 4*A*a^3*b^3 - B*a^2*b^4 + 2*A
*a*b^5)*c^5*d + 2*(B*a^3*b^3 - 2*(A - C)*a^2*b^4 - B*a*b^5)*c^4*d^2 + 2*(2*C*a^5*b - 3*B*a^4*b^2 + 4*A*a^3*b^3
- B*a^2*b^4 + 2*A*a*b^5)*c^3*d^3 + (B*a^3*b^3 - 2*(A - C)*a^2*b^4 - B*a*b^5)*c^2*d^4 + (2*C*a^5*b - 3*B*a^4*b
^2 + 4*A*a^3*b^3 - B*a^2*b^4 + 2*A*a*b^5)*c*d^5 + ((B*a^2*b^4 - 2*(A - C)*a*b^5 - B*b^6)*c^5*d + (2*C*a^4*b^2
- 3*B*a^3*b^3 + 4*A*a^2*b^4 - B*a*b^5 + 2*A*b^6)*c^4*d^2 + 2*(B*a^2*b^4 - 2*(A - C)*a*b^5 - B*b^6)*c^3*d^3 + 2
*(2*C*a^4*b^2 - 3*B*a^3*b^3 + 4*A*a^2*b^4 - B*a*b^5 + 2*A*b^6)*c^2*d^4 + (B*a^2*b^4 - 2*(A - C)*a*b^5 - B*b^6)
*c*d^5 + (2*C*a^4*b^2 - 3*B*a^3*b^3 + 4*A*a^2*b^4 - B*a*b^5 + 2*A*b^6)*d^6)*tan(f*x + e)^2 + ((B*a^2*b^4 - 2*(
A - C)*a*b^5 - B*b^6)*c^6 + 2*(C*a^4*b^2 - B*a^3*b^3 + (A + C)*a^2*b^4 - B*a*b^5 + A*b^6)*c^5*d + (2*C*a^5*b -
3*B*a^4*b^2 + 4*A*a^3*b^3 + B*a^2*b^4 - 2*(A - 2*C)*a*b^5 - 2*B*b^6)*c^4*d^2 + 4*(C*a^4*b^2 - B*a^3*b^3 + (A
+ C)*a^2*b^4 - B*a*b^5 + A*b^6)*c^3*d^3 + (4*C*a^5*b - 6*B*a^4*b^2 + 8*A*a^3*b^3 - B*a^2*b^4 + 2*(A + C)*a*b^5
- B*b^6)*c^2*d^4 + 2*(C*a^4*b^2 - B*a^3*b^3 + (A + C)*a^2*b^4 - B*a*b^5 + A*b^6)*c*d^5 + (2*C*a^5*b - 3*B*a^4
*b^2 + 4*A*a^3*b^3 - B*a^2*b^4 + 2*A*a*b^5)*d^6)*tan(f*x + e))*log((b^2*tan(f*x + e)^2 + 2*a*b*tan(f*x + e) +
a^2)/(tan(f*x + e)^2 + 1)) - (2*(C*a^5*b + 2*C*a^3*b^3 + C*a*b^5)*c^5*d - 3*(B*a^5*b + 2*B*a^3*b^3 + B*a*b^5)*
c^4*d^2 + (B*a^6 + 4*A*a^5*b + 2*B*a^4*b^2 + 8*A*a^3*b^3 + B*a^2*b^4 + 4*A*a*b^5)*c^3*d^3 - (2*(A - C)*a^6 + B
*a^5*b + 4*(A - C)*a^4*b^2 + 2*B*a^3*b^3 + 2*(A - C)*a^2*b^4 + B*a*b^5)*c^2*d^4 - (B*a^6 - 2*A*a^5*b + 2*B*a^4
*b^2 - 4*A*a^3*b^3 + B*a^2*b^4 - 2*A*a*b^5)*c*d^5 + (2*(C*a^4*b^2 + 2*C*a^2*b^4 + C*b^6)*c^4*d^2 - 3*(B*a^4*b^
2 + 2*B*a^2*b^4 + B*b^6)*c^3*d^3 + (B*a^5*b + 4*A*a^4*b^2 + 2*B*a^3*b^3 + 8*A*a^2*b^4 + B*a*b^5 + 4*A*b^6)*c^2
*d^4 - (2*(A - C)*a^5*b + B*a^4*b^2 + 4*(A - C)*a^3*b^3 + 2*B*a^2*b^4 + 2*(A - C)*a*b^5 + B*b^6)*c*d^5 - (B*a^
5*b - 2*A*a^4*b^2 + 2*B*a^3*b^3 - 4*A*a^2*b^4 + B*a*b^5 - 2*A*b^6)*d^6)*tan(f*x + e)^2 + (2*(C*a^4*b^2 + 2*C*a
^2*b^4 + C*b^6)*c^5*d + (2*C*a^5*b - 3*B*a^4*b^2 + 4*C*a^3*b^3 - 6*B*a^2*b^4 + 2*C*a*b^5 - 3*B*b^6)*c^4*d^2 -
2*(B*a^5*b - 2*A*a^4*b^2 + 2*B*a^3*b^3 - 4*A*a^2*b^4 + B*a*b^5 - 2*A*b^6)*c^3*d^3 + (B*a^6 + 2*(A + C)*a^5*b +
B*a^4*b^2 + 4*(A + C)*a^3*b^3 - B*a^2*b^4 + 2*(A + C)*a*b^5 - B*b^6)*c^2*d^4 - 2*((A - C)*a^6 + B*a^5*b + (A
- 2*C)*a^4*b^2 + 2*B*a^3*b^3 - (A + C)*a^2*b^4 + B*a*b^5 - A*b^6)*c*d^5 - (B*a^6 - 2*A*a^5*b + 2*B*a^4*b^2 - 4
*A*a^3*b^3 + B*a^2*b^4 - 2*A*a*b^5)*d^6)*tan(f*x + e))*log((d^2*tan(f*x + e)^2 + 2*c*d*tan(f*x + e) + c^2)/(ta
n(f*x + e)^2 + 1)) - 2*((C*a^3*b^3 - B*a^2*b^4 + A*a*b^5)*c^6 - (C*a^4*b^2 - B*a^3*b^3 + (A + C)*a^2*b^4 - B*a
*b^5 + A*b^6)*c^5*d + (C*a^5*b + 5*C*a^3*b^3 - 3*B*a^2*b^4 + (3*A + C)*a*b^5)*c^4*d^2 - (C*a^6 + B*a^5*b + 5*C
*a^4*b^2 + (2*A + 5*C)*a^2*b^4 - B*a*b^5 + (2*A + C)*b^6)*c^3*d^3 + (B*a^6 + (A + C)*a^5*b + 3*B*a^4*b^2 + (2*
A + 5*C)*a^3*b^3 + (4*A + C)*a*b^5 + B*b^6)*c^2*d^4 - (A*a^6 + B*a^5*b + (3*A + C)*a^4*b^2 + B*a^3*b^3 + (4*A
+ C)*a^2*b^4 + 2*A*b^6)*c*d^5 + (A*a^5*b + (2*A + C)*a^3*b^3 - B*a^2*b^4 + 2*A*a*b^5)*d^6 + (((A - C)*a^2*b^4
+ 2*B*a*b^5 - (A - C)*b^6)*c^6 - 2*((A - C)*a^3*b^3 + B*a^2*b^4 + (A - C)*a*b^5 + B*b^6)*c^5*d - (4*B*a^3*b^3
- 7*(A - C)*a^2*b^4 - 2*B*a*b^5 - (A - C)*b^6)*c^4*d^2 + 2*((A - C)*a^5*b + 2*B*a^4*b^2 + 2*B*a^2*b^4 - (A - C
)*a*b^5)*c^3*d^3 - ((A - C)*a^6 - 2*B*a^5*b + 7*(A - C)*a^4*b^2 + 4*B*a^3*b^3)*c^2*d^4 - 2*(B*a^6 - (A - C)*a^
5*b + B*a^4*b^2 - (A - C)*a^3*b^3)*c*d^5 + ((A - C)*a^6 + 2*B*a^5*b - (A - C)*a^4*b^2)*d^6)*f*x)*tan(f*x + e))
/(((a^4*b^4 + 2*a^2*b^6 + b^8)*c^7*d - 3*(a^5*b^3 + 2*a^3*b^5 + a*b^7)*c^6*d^2 + (3*a^6*b^2 + 8*a^4*b^4 + 7*a^
2*b^6 + 2*b^8)*c^5*d^3 - (a^7*b + 8*a^5*b^3 + 13*a^3*b^5 + 6*a*b^7)*c^4*d^4 + (6*a^6*b^2 + 13*a^4*b^4 + 8*a^2*
b^6 + b^8)*c^3*d^5 - (2*a^7*b + 7*a^5*b^3 + 8*a^3*b^5 + 3*a*b^7)*c^2*d^6 + 3*(a^6*b^2 + 2*a^4*b^4 + a^2*b^6)*c
*d^7 - (a^7*b + 2*a^5*b^3 + a^3*b^5)*d^8)*f*tan(f*x + e)^2 + ((a^4*b^4 + 2*a^2*b^6 + b^8)*c^8 - 2*(a^5*b^3 + 2
*a^3*b^5 + a*b^7)*c^7*d + 2*(a^4*b^4 + 2*a^2*b^6 + b^8)*c^6*d^2 + 2*(a^7*b - 3*a^3*b^5 - 2*a*b^7)*c^5*d^3 - (a
^8 + 2*a^6*b^2 - 2*a^2*b^6 - b^8)*c^4*d^4 + 2*(2*a^7*b + 3*a^5*b^3 - a*b^7)*c^3*d^5 - 2*(a^8 + 2*a^6*b^2 + a^4
*b^4)*c^2*d^6 + 2*(a^7*b + 2*a^5*b^3 + a^3*b^5)*c*d^7 - (a^8 + 2*a^6*b^2 + a^4*b^4)*d^8)*f*tan(f*x + e) + ((a^
5*b^3 + 2*a^3*b^5 + a*b^7)*c^8 - 3*(a^6*b^2 + 2*a^4*b^4 + a^2*b^6)*c^7*d + (3*a^7*b + 8*a^5*b^3 + 7*a^3*b^5 +
2*a*b^7)*c^6*d^2 - (a^8 + 8*a^6*b^2 + 13*a^4*b^4 + 6*a^2*b^6)*c^5*d^3 + (6*a^7*b + 13*a^5*b^3 + 8*a^3*b^5 + a*
b^7)*c^4*d^4 - (2*a^8 + 7*a^6*b^2 + 8*a^4*b^4 + 3*a^2*b^6)*c^3*d^5 + 3*(a^7*b + 2*a^5*b^3 + a^3*b^5)*c^2*d^6 -
(a^8 + 2*a^6*b^2 + a^4*b^4)*c*d^7)*f)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*tan(f*x+e)+C*tan(f*x+e)**2)/(a+b*tan(f*x+e))**2/(c+d*tan(f*x+e))**2,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e))^2/(c+d*tan(f*x+e))^2,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError