3.88 \(\int \frac{A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^3} \, dx\)

Optimal. Leaf size=487 \[ -\frac{\left (a^2 d^3 \left (d (A-C) \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )-a b d^2 \left (8 c^3 d (A-C)-B \left (-6 c^2 d^2+3 c^4-d^4\right )\right )+b^2 \left (3 c^4 d^2 (2 A-C)+3 A c^2 d^4+A d^6+B c^3 d^3-3 B c^5 d+c^6 C\right )\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{f \left (c^2+d^2\right )^3 (b c-a d)^3}-\frac{x \left (a \left (-A \left (c^3-3 c d^2\right )-3 B c^2 d+B d^3+c^3 C-3 c C d^2\right )+b \left (d (A-C) \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )\right )}{\left (a^2+b^2\right ) \left (c^2+d^2\right )^3}+\frac{b^2 \left (A b^2-a (b B-a C)\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{f \left (a^2+b^2\right ) (b c-a d)^3}+\frac{b \left (c^2 d^2 (3 A-C)+A d^4-2 B c^3 d+c^4 C\right )-a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )}{f \left (c^2+d^2\right )^2 (b c-a d)^2 (c+d \tan (e+f x))}+\frac{A d^2-B c d+c^2 C}{2 f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))^2} \]

[Out]

-(((a*(c^3*C - 3*B*c^2*d - 3*c*C*d^2 + B*d^3 - A*(c^3 - 3*c*d^2)) + b*((A - C)*d*(3*c^2 - d^2) - B*(c^3 - 3*c*
d^2)))*x)/((a^2 + b^2)*(c^2 + d^2)^3)) + (b^2*(A*b^2 - a*(b*B - a*C))*Log[a*Cos[e + f*x] + b*Sin[e + f*x]])/((
a^2 + b^2)*(b*c - a*d)^3*f) - ((b^2*(c^6*C - 3*B*c^5*d + 3*c^4*(2*A - C)*d^2 + B*c^3*d^3 + 3*A*c^2*d^4 + A*d^6
) + a^2*d^3*((A - C)*d*(3*c^2 - d^2) - B*(c^3 - 3*c*d^2)) - a*b*d^2*(8*c^3*(A - C)*d - B*(3*c^4 - 6*c^2*d^2 -
d^4)))*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/((b*c - a*d)^3*(c^2 + d^2)^3*f) + (c^2*C - B*c*d + A*d^2)/(2*(b*c
- a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^2) + (b*(c^4*C - 2*B*c^3*d + c^2*(3*A - C)*d^2 + A*d^4) - a*d^2*(2*
c*(A - C)*d - B*(c^2 - d^2)))/((b*c - a*d)^2*(c^2 + d^2)^2*f*(c + d*Tan[e + f*x]))

________________________________________________________________________________________

Rubi [A]  time = 1.82961, antiderivative size = 487, 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} \[ -\frac{\left (a^2 d^3 \left (d (A-C) \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )-a b d^2 \left (8 c^3 d (A-C)-B \left (-6 c^2 d^2+3 c^4-d^4\right )\right )+b^2 \left (3 c^4 d^2 (2 A-C)+3 A c^2 d^4+A d^6+B c^3 d^3-3 B c^5 d+c^6 C\right )\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{f \left (c^2+d^2\right )^3 (b c-a d)^3}-\frac{x \left (a \left (-A \left (c^3-3 c d^2\right )-3 B c^2 d+B d^3+c^3 C-3 c C d^2\right )+b \left (d (A-C) \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )\right )}{\left (a^2+b^2\right ) \left (c^2+d^2\right )^3}+\frac{b^2 \left (A b^2-a (b B-a C)\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{f \left (a^2+b^2\right ) (b c-a d)^3}+\frac{b \left (c^2 d^2 (3 A-C)+A d^4-2 B c^3 d+c^4 C\right )-a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )}{f \left (c^2+d^2\right )^2 (b c-a d)^2 (c+d \tan (e+f x))}+\frac{A d^2-B c d+c^2 C}{2 f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))^2} \]

Antiderivative was successfully verified.

[In]

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])^3),x]

[Out]

-(((a*(c^3*C - 3*B*c^2*d - 3*c*C*d^2 + B*d^3 - A*(c^3 - 3*c*d^2)) + b*((A - C)*d*(3*c^2 - d^2) - B*(c^3 - 3*c*
d^2)))*x)/((a^2 + b^2)*(c^2 + d^2)^3)) + (b^2*(A*b^2 - a*(b*B - a*C))*Log[a*Cos[e + f*x] + b*Sin[e + f*x]])/((
a^2 + b^2)*(b*c - a*d)^3*f) - ((b^2*(c^6*C - 3*B*c^5*d + 3*c^4*(2*A - C)*d^2 + B*c^3*d^3 + 3*A*c^2*d^4 + A*d^6
) + a^2*d^3*((A - C)*d*(3*c^2 - d^2) - B*(c^3 - 3*c*d^2)) - a*b*d^2*(8*c^3*(A - C)*d - B*(3*c^4 - 6*c^2*d^2 -
d^4)))*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/((b*c - a*d)^3*(c^2 + d^2)^3*f) + (c^2*C - B*c*d + A*d^2)/(2*(b*c
- a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^2) + (b*(c^4*C - 2*B*c^3*d + c^2*(3*A - C)*d^2 + A*d^4) - a*d^2*(2*
c*(A - C)*d - B*(c^2 - d^2)))/((b*c - a*d)^2*(c^2 + d^2)^2*f*(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)) (c+d \tan (e+f x))^3} \, dx &=\frac{c^2 C-B c d+A d^2}{2 (b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}+\frac{\int \frac{-2 \left (a A c d-a d (c C-B d)-A b \left (c^2+d^2\right )\right )+2 (b c-a d) (B c-(A-C) d) \tan (e+f x)+2 b \left (c^2 C-B c d+A d^2\right ) \tan ^2(e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^2} \, dx}{2 (b c-a d) \left (c^2+d^2\right )}\\ &=\frac{c^2 C-B c d+A d^2}{2 (b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}+\frac{b \left (c^4 C-2 B c^3 d+c^2 (3 A-C) d^2+A d^4\right )-a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )}{(b c-a d)^2 \left (c^2+d^2\right )^2 f (c+d \tan (e+f x))}+\frac{\int \frac{-2 \left (A \left (2 a b c^3 d-a^2 d^2 \left (c^2-d^2\right )-b^2 \left (c^2+d^2\right )^2\right )+a d \left (a d \left (c^2 C-2 B c d-C d^2\right )-b \left (2 c^3 C-3 B c^2 d-B d^3\right )\right )\right )-2 (b c-a d)^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) \tan (e+f x)+2 b \left (b \left (c^4 C-2 B c^3 d+c^2 (3 A-C) d^2+A d^4\right )-a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \tan ^2(e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))} \, dx}{2 (b c-a d)^2 \left (c^2+d^2\right )^2}\\ &=-\frac{\left (b (A-C) d \left (3 c^2-d^2\right )-b B \left (c^3-3 c d^2\right )-a \left (A c^3-c^3 C+3 B c^2 d-3 A c d^2+3 c C d^2-B d^3\right )\right ) x}{\left (a^2+b^2\right ) \left (c^2+d^2\right )^3}+\frac{c^2 C-B c d+A d^2}{2 (b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}+\frac{b \left (c^4 C-2 B c^3 d+c^2 (3 A-C) d^2+A d^4\right )-a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )}{(b c-a d)^2 \left (c^2+d^2\right )^2 f (c+d \tan (e+f x))}+\frac{\left (b^2 \left (A b^2-a (b B-a C)\right )\right ) \int \frac{b-a \tan (e+f x)}{a+b \tan (e+f x)} \, dx}{\left (a^2+b^2\right ) (b c-a d)^3}-\frac{\left (b^2 \left (c^6 C-3 B c^5 d+3 c^4 (2 A-C) d^2+B c^3 d^3+3 A c^2 d^4+A d^6\right )+a^2 d^3 \left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )-a b d^2 \left (8 c^3 (A-C) d-B \left (3 c^4-6 c^2 d^2-d^4\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 )^3}\\ &=-\frac{\left (b (A-C) d \left (3 c^2-d^2\right )-b B \left (c^3-3 c d^2\right )-a \left (A c^3-c^3 C+3 B c^2 d-3 A c d^2+3 c C d^2-B d^3\right )\right ) x}{\left (a^2+b^2\right ) \left (c^2+d^2\right )^3}+\frac{b^2 \left (A b^2-a (b B-a C)\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{\left (a^2+b^2\right ) (b c-a d)^3 f}-\frac{\left (b^2 \left (c^6 C-3 B c^5 d+3 c^4 (2 A-C) d^2+B c^3 d^3+3 A c^2 d^4+A d^6\right )+a^2 d^3 \left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )-a b d^2 \left (8 c^3 (A-C) d-B \left (3 c^4-6 c^2 d^2-d^4\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 )^3 f}+\frac{c^2 C-B c d+A d^2}{2 (b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}+\frac{b \left (c^4 C-2 B c^3 d+c^2 (3 A-C) d^2+A d^4\right )-a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )}{(b c-a d)^2 \left (c^2+d^2\right )^2 f (c+d \tan (e+f x))}\\ \end{align*}

Mathematica [A]  time = 8.88245, size = 912, normalized size = 1.87 \[ -\frac{A d^2-c (B d-c C)}{2 (a d-b c) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}-\frac{-\frac{-2 \left (a A c d-a (c C-B d) d-A b \left (c^2+d^2\right )\right ) d^2-c \left (2 d (b c-a d) (B c-(A-C) d)-2 b c \left (C c^2-B d c+A d^2\right )\right )}{(a d-b c) \left (c^2+d^2\right ) f (c+d \tan (e+f x))}-\frac{\frac{2 \left (A b^2-a (b B-a C)\right ) \left (c^2+d^2\right )^2 \log (a+b \tan (e+f x)) b^3}{\left (a^2+b^2\right ) (b c-a d)}-\frac{(b c-a d)^2 \left (A b c^3-a B c^3-b C c^3+3 a A d c^2+3 b B d c^2-3 a C d c^2-3 A b d^2 c+3 a B d^2 c+3 b C d^2 c-a A d^3-b B d^3+a C d^3-\frac{\sqrt{-b^2} \left (a \left (C c^3-3 B d c^2-3 C d^2 c+B d^3-A \left (c^3-3 c d^2\right )\right )+b \left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )\right )}{b}\right ) \log \left (\sqrt{-b^2}-b \tan (e+f x)\right ) b}{\left (a^2+b^2\right ) \left (c^2+d^2\right )}-\frac{(b c-a d)^2 \left (A b c^3-a B c^3-b C c^3+3 a A d c^2+3 b B d c^2-3 a C d c^2-3 A b d^2 c+3 a B d^2 c+3 b C d^2 c-a A d^3-b B d^3+a C d^3+\frac{\sqrt{-b^2} \left (b (A-C) d \left (3 c^2-d^2\right )-b B \left (c^3-3 c d^2\right )-a \left (A c^3-C c^3+3 B d c^2-3 A d^2 c+3 C d^2 c-B d^3\right )\right )}{b}\right ) \log \left (b \tan (e+f x)+\sqrt{-b^2}\right ) b}{\left (a^2+b^2\right ) \left (c^2+d^2\right )}-\frac{2 \left (a^2 \left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right ) d^3-a b \left (8 c^3 (A-C) d-B \left (3 c^4-6 d^2 c^2-d^4\right )\right ) d^2+b^2 \left (C c^6-3 B d c^5+3 (2 A-C) d^2 c^4+B d^3 c^3+3 A d^4 c^2+A d^6\right )\right ) \log (c+d \tan (e+f x)) b}{(b c-a d) \left (c^2+d^2\right )}}{b (a d-b c) \left (c^2+d^2\right ) f}}{2 (a d-b c) \left (c^2+d^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^3),x]

[Out]

-(A*d^2 - c*(-(c*C) + B*d))/(2*(-(b*c) + a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^2) - (-((-((b*(b*c - a*d)^2*(
A*b*c^3 - a*B*c^3 - b*c^3*C + 3*a*A*c^2*d + 3*b*B*c^2*d - 3*a*c^2*C*d - 3*A*b*c*d^2 + 3*a*B*c*d^2 + 3*b*c*C*d^
2 - a*A*d^3 - b*B*d^3 + a*C*d^3 - (Sqrt[-b^2]*(a*(c^3*C - 3*B*c^2*d - 3*c*C*d^2 + B*d^3 - A*(c^3 - 3*c*d^2)) +
b*((A - C)*d*(3*c^2 - d^2) - B*(c^3 - 3*c*d^2))))/b)*Log[Sqrt[-b^2] - b*Tan[e + f*x]])/((a^2 + b^2)*(c^2 + d^
2))) + (2*b^3*(A*b^2 - a*(b*B - a*C))*(c^2 + d^2)^2*Log[a + b*Tan[e + f*x]])/((a^2 + b^2)*(b*c - a*d)) - (b*(b
*c - a*d)^2*(A*b*c^3 - a*B*c^3 - b*c^3*C + 3*a*A*c^2*d + 3*b*B*c^2*d - 3*a*c^2*C*d - 3*A*b*c*d^2 + 3*a*B*c*d^2
+ 3*b*c*C*d^2 - a*A*d^3 - b*B*d^3 + a*C*d^3 + (Sqrt[-b^2]*(b*(A - C)*d*(3*c^2 - d^2) - b*B*(c^3 - 3*c*d^2) -
a*(A*c^3 - c^3*C + 3*B*c^2*d - 3*A*c*d^2 + 3*c*C*d^2 - B*d^3)))/b)*Log[Sqrt[-b^2] + b*Tan[e + f*x]])/((a^2 + b
^2)*(c^2 + d^2)) - (2*b*(b^2*(c^6*C - 3*B*c^5*d + 3*c^4*(2*A - C)*d^2 + B*c^3*d^3 + 3*A*c^2*d^4 + A*d^6) + a^2
*d^3*((A - C)*d*(3*c^2 - d^2) - B*(c^3 - 3*c*d^2)) - a*b*d^2*(8*c^3*(A - C)*d - B*(3*c^4 - 6*c^2*d^2 - d^4)))*
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)) - (-2*d^2*(a*A*c*d - a*d
*(c*C - B*d) - A*b*(c^2 + d^2)) - c*(2*d*(b*c - a*d)*(B*c - (A - C)*d) - 2*b*c*(c^2*C - B*c*d + A*d^2)))/((-(b
*c) + a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x])))/(2*(-(b*c) + a*d)*(c^2 + d^2))

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Maple [B]  time = 0.117, size = 2298, normalized size = 4.7 \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))/(c+d*tan(f*x+e))^3,x)

[Out]

-3/f/(a*d-b*c)^3/(c^2+d^2)^3*ln(c+d*tan(f*x+e))*C*a^2*c^2*d^4-3/f/(a*d-b*c)^3/(c^2+d^2)^3*ln(c+d*tan(f*x+e))*C
*b^2*c^4*d^2+1/f/(a*d-b*c)^3/(c^2+d^2)^3*ln(c+d*tan(f*x+e))*B*b^2*c^3*d^3+3/f/(a*d-b*c)^3/(c^2+d^2)^3*ln(c+d*t
an(f*x+e))*A*b^2*c^2*d^4-1/f/(a*d-b*c)^3/(c^2+d^2)^3*ln(c+d*tan(f*x+e))*B*a^2*c^3*d^3+1/f/(a*d-b*c)^2/(c^2+d^2
)^2/(c+d*tan(f*x+e))*B*a*c^2*d^2-2/f/(a*d-b*c)^2/(c^2+d^2)^2/(c+d*tan(f*x+e))*B*b*c^3*d+2/f/(a*d-b*c)^2/(c^2+d
^2)^2/(c+d*tan(f*x+e))*C*a*c*d^3-2/f/(a*d-b*c)^2/(c^2+d^2)^2/(c+d*tan(f*x+e))*A*a*c*d^3+3/f/(a*d-b*c)^2/(c^2+d
^2)^2/(c+d*tan(f*x+e))*A*b*c^2*d^2+3/f/(a^2+b^2)/(c^2+d^2)^3*B*arctan(tan(f*x+e))*a*c^2*d-3/f/(a^2+b^2)/(c^2+d
^2)^3*B*arctan(tan(f*x+e))*b*c*d^2-3/2/f/(a^2+b^2)/(c^2+d^2)^3*ln(1+tan(f*x+e)^2)*B*a*c*d^2+1/2/f/(a^2+b^2)/(c
^2+d^2)^3*ln(1+tan(f*x+e)^2)*B*b*d^3-1/2/f/(a^2+b^2)/(c^2+d^2)^3*ln(1+tan(f*x+e)^2)*C*a*d^3+3/f/(a*d-b*c)^3/(c
^2+d^2)^3*ln(c+d*tan(f*x+e))*A*a^2*c^2*d^4-3/2/f/(a^2+b^2)/(c^2+d^2)^3*ln(1+tan(f*x+e)^2)*B*b*c^2*d+3/2/f/(a^2
+b^2)/(c^2+d^2)^3*ln(1+tan(f*x+e)^2)*C*a*c^2*d-3/2/f/(a^2+b^2)/(c^2+d^2)^3*ln(1+tan(f*x+e)^2)*C*b*c*d^2-3/f/(a
^2+b^2)/(c^2+d^2)^3*A*arctan(tan(f*x+e))*a*c*d^2+3/f/(a^2+b^2)/(c^2+d^2)^3*C*arctan(tan(f*x+e))*b*c^2*d+3/f/(a
^2+b^2)/(c^2+d^2)^3*C*arctan(tan(f*x+e))*a*c*d^2-3/f/(a^2+b^2)/(c^2+d^2)^3*A*arctan(tan(f*x+e))*b*c^2*d+6/f/(a
*d-b*c)^3/(c^2+d^2)^3*ln(c+d*tan(f*x+e))*A*b^2*c^4*d^2+3/f/(a*d-b*c)^3/(c^2+d^2)^3*ln(c+d*tan(f*x+e))*B*a^2*c*
d^5-1/f/(a*d-b*c)^3/(c^2+d^2)^3*ln(c+d*tan(f*x+e))*B*a*b*d^6-1/f*b^4/(a*d-b*c)^3/(a^2+b^2)*ln(a+b*tan(f*x+e))*
A-1/2/f/(a*d-b*c)/(c^2+d^2)/(c+d*tan(f*x+e))^2*A*d^2-1/2/f/(a*d-b*c)/(c^2+d^2)/(c+d*tan(f*x+e))^2*c^2*C-3/f/(a
*d-b*c)^3/(c^2+d^2)^3*ln(c+d*tan(f*x+e))*B*b^2*c^5*d+8/f/(a*d-b*c)^3/(c^2+d^2)^3*ln(c+d*tan(f*x+e))*C*a*b*c^3*
d^3-1/f/(a*d-b*c)^2/(c^2+d^2)^2/(c+d*tan(f*x+e))*C*b*c^2*d^2-3/2/f/(a^2+b^2)/(c^2+d^2)^3*ln(1+tan(f*x+e)^2)*A*
a*c^2*d+3/2/f/(a^2+b^2)/(c^2+d^2)^3*ln(1+tan(f*x+e)^2)*A*b*c*d^2-1/f*b^2/(a*d-b*c)^3/(a^2+b^2)*ln(a+b*tan(f*x+
e))*C*a^2-1/2/f/(a^2+b^2)/(c^2+d^2)^3*ln(1+tan(f*x+e)^2)*A*b*c^3+1/2/f/(a^2+b^2)/(c^2+d^2)^3*ln(1+tan(f*x+e)^2
)*B*a*c^3+1/2/f/(a^2+b^2)/(c^2+d^2)^3*ln(1+tan(f*x+e)^2)*C*b*c^3+1/f/(a^2+b^2)/(c^2+d^2)^3*A*arctan(tan(f*x+e)
)*a*c^3+1/f/(a^2+b^2)/(c^2+d^2)^3*A*arctan(tan(f*x+e))*b*d^3-1/f/(a^2+b^2)/(c^2+d^2)^3*B*arctan(tan(f*x+e))*a*
d^3+1/f/(a^2+b^2)/(c^2+d^2)^3*B*arctan(tan(f*x+e))*b*c^3-1/f/(a^2+b^2)/(c^2+d^2)^3*C*arctan(tan(f*x+e))*a*c^3-
1/f/(a^2+b^2)/(c^2+d^2)^3*C*arctan(tan(f*x+e))*b*d^3+1/f/(a*d-b*c)^2/(c^2+d^2)^2/(c+d*tan(f*x+e))*A*b*d^4-6/f/
(a*d-b*c)^3/(c^2+d^2)^3*ln(c+d*tan(f*x+e))*B*a*b*c^2*d^4+3/f/(a*d-b*c)^3/(c^2+d^2)^3*ln(c+d*tan(f*x+e))*B*a*b*
c^4*d^2-1/f/(a*d-b*c)^2/(c^2+d^2)^2/(c+d*tan(f*x+e))*B*a*d^4+1/f/(a*d-b*c)^2/(c^2+d^2)^2/(c+d*tan(f*x+e))*C*b*
c^4-1/f/(a*d-b*c)^3/(c^2+d^2)^3*ln(c+d*tan(f*x+e))*A*a^2*d^6+1/2/f/(a*d-b*c)/(c^2+d^2)/(c+d*tan(f*x+e))^2*B*c*
d+1/f*b^3/(a*d-b*c)^3/(a^2+b^2)*ln(a+b*tan(f*x+e))*B*a+1/f/(a*d-b*c)^3/(c^2+d^2)^3*ln(c+d*tan(f*x+e))*A*b^2*d^
6+1/f/(a*d-b*c)^3/(c^2+d^2)^3*ln(c+d*tan(f*x+e))*C*a^2*d^6+1/f/(a*d-b*c)^3/(c^2+d^2)^3*ln(c+d*tan(f*x+e))*C*b^
2*c^6+1/2/f/(a^2+b^2)/(c^2+d^2)^3*ln(1+tan(f*x+e)^2)*A*a*d^3-8/f/(a*d-b*c)^3/(c^2+d^2)^3*ln(c+d*tan(f*x+e))*A*
a*b*c^3*d^3

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Maxima [B]  time = 1.84013, size = 1455, normalized size = 2.99 \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))/(c+d*tan(f*x+e))^3,x, algorithm="maxima")

[Out]

1/2*(2*(((A - C)*a + B*b)*c^3 + 3*(B*a - (A - C)*b)*c^2*d - 3*((A - C)*a + B*b)*c*d^2 - (B*a - (A - C)*b)*d^3)
*(f*x + e)/((a^2 + b^2)*c^6 + 3*(a^2 + b^2)*c^4*d^2 + 3*(a^2 + b^2)*c^2*d^4 + (a^2 + b^2)*d^6) + 2*(C*a^2*b^2
- B*a*b^3 + A*b^4)*log(b*tan(f*x + e) + a)/((a^2*b^3 + b^5)*c^3 - 3*(a^3*b^2 + a*b^4)*c^2*d + 3*(a^4*b + a^2*b
^3)*c*d^2 - (a^5 + a^3*b^2)*d^3) - 2*(C*b^2*c^6 - 3*B*b^2*c^5*d + 3*B*a^2*c*d^5 + 3*(B*a*b + (2*A - C)*b^2)*c^
4*d^2 - (B*a^2 + 8*(A - C)*a*b - B*b^2)*c^3*d^3 + 3*((A - C)*a^2 - 2*B*a*b + A*b^2)*c^2*d^4 - ((A - C)*a^2 + B
*a*b - A*b^2)*d^6)*log(d*tan(f*x + e) + c)/(b^3*c^9 - 3*a*b^2*c^8*d + 3*a^2*b*c*d^8 - a^3*d^9 + 3*(a^2*b + b^3
)*c^7*d^2 - (a^3 + 9*a*b^2)*c^6*d^3 + 3*(3*a^2*b + b^3)*c^5*d^4 - 3*(a^3 + 3*a*b^2)*c^4*d^5 + (9*a^2*b + b^3)*
c^3*d^6 - 3*(a^3 + a*b^2)*c^2*d^7) + ((B*a - (A - C)*b)*c^3 - 3*((A - C)*a + B*b)*c^2*d - 3*(B*a - (A - C)*b)*
c*d^2 + ((A - C)*a + B*b)*d^3)*log(tan(f*x + e)^2 + 1)/((a^2 + b^2)*c^6 + 3*(a^2 + b^2)*c^4*d^2 + 3*(a^2 + b^2
)*c^2*d^4 + (a^2 + b^2)*d^6) + (3*C*b*c^5 - A*a*d^5 - (C*a + 5*B*b)*c^4*d + (3*B*a + (7*A - C)*b)*c^3*d^2 - ((
5*A - 3*C)*a + B*b)*c^2*d^3 - (B*a - 3*A*b)*c*d^4 + 2*(C*b*c^4*d - 2*B*b*c^3*d^2 - 2*(A - C)*a*c*d^4 + (B*a +
(3*A - C)*b)*c^2*d^3 - (B*a - A*b)*d^5)*tan(f*x + e))/(b^2*c^8 - 2*a*b*c^7*d - 4*a*b*c^5*d^3 - 2*a*b*c^3*d^5 +
a^2*c^2*d^6 + (a^2 + 2*b^2)*c^6*d^2 + (2*a^2 + b^2)*c^4*d^4 + (b^2*c^6*d^2 - 2*a*b*c^5*d^3 - 4*a*b*c^3*d^5 -
2*a*b*c*d^7 + a^2*d^8 + (a^2 + 2*b^2)*c^4*d^4 + (2*a^2 + b^2)*c^2*d^6)*tan(f*x + e)^2 + 2*(b^2*c^7*d - 2*a*b*c
^6*d^2 - 4*a*b*c^4*d^4 - 2*a*b*c^2*d^6 + a^2*c*d^7 + (a^2 + 2*b^2)*c^5*d^3 + (2*a^2 + b^2)*c^3*d^5)*tan(f*x +
e)))/f

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Fricas [B]  time = 38.5076, size = 7109, normalized size = 14.6 \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))/(c+d*tan(f*x+e))^3,x, algorithm="fricas")

[Out]

1/2*(5*(C*a^2*b^2 + C*b^4)*c^6*d^2 - (8*C*a^3*b + 7*B*a^2*b^2 + 8*C*a*b^3 + 7*B*b^4)*c^5*d^3 + (3*C*a^4 + 12*B
*a^3*b + (9*A + 2*C)*a^2*b^2 + 12*B*a*b^3 + (9*A - C)*b^4)*c^4*d^4 - (5*B*a^4 + 4*(4*A - C)*a^3*b + 6*B*a^2*b^
2 + 4*(4*A - C)*a*b^3 + B*b^4)*c^3*d^5 + ((7*A - 3*C)*a^4 + (10*A - 3*C)*a^2*b^2 + 3*A*b^4)*c^2*d^6 + (B*a^4 -
4*A*a^3*b + B*a^2*b^2 - 4*A*a*b^3)*c*d^7 + (A*a^4 + A*a^2*b^2)*d^8 + 2*(((A - C)*a*b^3 + B*b^4)*c^8 - 3*((A -
C)*a^2*b^2 + (A - C)*b^4)*c^7*d + 3*((A - C)*a^3*b - 2*B*a^2*b^2 + 2*(A - C)*a*b^3 - B*b^4)*c^6*d^2 - ((A - C
)*a^4 - 8*B*a^3*b - 8*B*a*b^3 - (A - C)*b^4)*c^5*d^3 - 3*(B*a^4 + 2*(A - C)*a^3*b + 2*B*a^2*b^2 + (A - C)*a*b^
3)*c^4*d^4 + 3*((A - C)*a^4 + (A - C)*a^2*b^2)*c^3*d^5 + (B*a^4 - (A - C)*a^3*b)*c^2*d^6)*f*x - (3*(C*a^2*b^2
+ C*b^4)*c^6*d^2 - (4*C*a^3*b + 5*B*a^2*b^2 + 4*C*a*b^3 + 5*B*b^4)*c^5*d^3 + (C*a^4 + 8*B*a^3*b + (7*A - 2*C)*
a^2*b^2 + 8*B*a*b^3 + (7*A - 3*C)*b^4)*c^4*d^4 - (3*B*a^4 + 4*(3*A - 2*C)*a^3*b + 2*B*a^2*b^2 + 4*(3*A - 2*C)*
a*b^3 - B*b^4)*c^3*d^5 + (5*(A - C)*a^4 - 4*B*a^3*b + (6*A - 5*C)*a^2*b^2 - 4*B*a*b^3 + A*b^4)*c^2*d^6 + 3*(B*
a^4 + B*a^2*b^2)*c*d^7 - (A*a^4 + A*a^2*b^2)*d^8 - 2*(((A - C)*a*b^3 + B*b^4)*c^6*d^2 - 3*((A - C)*a^2*b^2 + (
A - C)*b^4)*c^5*d^3 + 3*((A - C)*a^3*b - 2*B*a^2*b^2 + 2*(A - C)*a*b^3 - B*b^4)*c^4*d^4 - ((A - C)*a^4 - 8*B*a
^3*b - 8*B*a*b^3 - (A - C)*b^4)*c^3*d^5 - 3*(B*a^4 + 2*(A - C)*a^3*b + 2*B*a^2*b^2 + (A - C)*a*b^3)*c^2*d^6 +
3*((A - C)*a^4 + (A - C)*a^2*b^2)*c*d^7 + (B*a^4 - (A - C)*a^3*b)*d^8)*f*x)*tan(f*x + e)^2 + ((C*a^2*b^2 - B*a
*b^3 + A*b^4)*c^8 + 3*(C*a^2*b^2 - B*a*b^3 + A*b^4)*c^6*d^2 + 3*(C*a^2*b^2 - B*a*b^3 + A*b^4)*c^4*d^4 + (C*a^2
*b^2 - B*a*b^3 + A*b^4)*c^2*d^6 + ((C*a^2*b^2 - B*a*b^3 + A*b^4)*c^6*d^2 + 3*(C*a^2*b^2 - B*a*b^3 + A*b^4)*c^4
*d^4 + 3*(C*a^2*b^2 - B*a*b^3 + A*b^4)*c^2*d^6 + (C*a^2*b^2 - B*a*b^3 + A*b^4)*d^8)*tan(f*x + e)^2 + 2*((C*a^2
*b^2 - B*a*b^3 + A*b^4)*c^7*d + 3*(C*a^2*b^2 - B*a*b^3 + A*b^4)*c^5*d^3 + 3*(C*a^2*b^2 - B*a*b^3 + A*b^4)*c^3*
d^5 + (C*a^2*b^2 - B*a*b^3 + A*b^4)*c*d^7)*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)) - ((C*a^2*b^2 + C*b^4)*c^8 - 3*(B*a^2*b^2 + B*b^4)*c^7*d + 3*(B*a^3*b + (2*A - C)*a^2*b^2
+ B*a*b^3 + (2*A - C)*b^4)*c^6*d^2 - (B*a^4 + 8*(A - C)*a^3*b + 8*(A - C)*a*b^3 - B*b^4)*c^5*d^3 + 3*((A - C)
*a^4 - 2*B*a^3*b + (2*A - C)*a^2*b^2 - 2*B*a*b^3 + A*b^4)*c^4*d^4 + 3*(B*a^4 + B*a^2*b^2)*c^3*d^5 - ((A - C)*a
^4 + B*a^3*b - C*a^2*b^2 + B*a*b^3 - A*b^4)*c^2*d^6 + ((C*a^2*b^2 + C*b^4)*c^6*d^2 - 3*(B*a^2*b^2 + B*b^4)*c^5
*d^3 + 3*(B*a^3*b + (2*A - C)*a^2*b^2 + B*a*b^3 + (2*A - C)*b^4)*c^4*d^4 - (B*a^4 + 8*(A - C)*a^3*b + 8*(A - C
)*a*b^3 - B*b^4)*c^3*d^5 + 3*((A - C)*a^4 - 2*B*a^3*b + (2*A - C)*a^2*b^2 - 2*B*a*b^3 + A*b^4)*c^2*d^6 + 3*(B*
a^4 + B*a^2*b^2)*c*d^7 - ((A - C)*a^4 + B*a^3*b - C*a^2*b^2 + B*a*b^3 - A*b^4)*d^8)*tan(f*x + e)^2 + 2*((C*a^2
*b^2 + C*b^4)*c^7*d - 3*(B*a^2*b^2 + B*b^4)*c^6*d^2 + 3*(B*a^3*b + (2*A - C)*a^2*b^2 + B*a*b^3 + (2*A - C)*b^4
)*c^5*d^3 - (B*a^4 + 8*(A - C)*a^3*b + 8*(A - C)*a*b^3 - B*b^4)*c^4*d^4 + 3*((A - C)*a^4 - 2*B*a^3*b + (2*A -
C)*a^2*b^2 - 2*B*a*b^3 + A*b^4)*c^3*d^5 + 3*(B*a^4 + B*a^2*b^2)*c^2*d^6 - ((A - C)*a^4 + B*a^3*b - C*a^2*b^2 +
B*a*b^3 - A*b^4)*c*d^7)*tan(f*x + e))*log((d^2*tan(f*x + e)^2 + 2*c*d*tan(f*x + e) + c^2)/(tan(f*x + e)^2 + 1
)) - 2*(2*(C*a^2*b^2 + C*b^4)*c^7*d - 3*(C*a^3*b + B*a^2*b^2 + C*a*b^3 + B*b^4)*c^6*d^2 + (C*a^4 + 5*B*a^3*b +
2*(2*A - C)*a^2*b^2 + 5*B*a*b^3 + (4*A - 3*C)*b^4)*c^5*d^3 - (2*B*a^4 + (7*A - 6*C)*a^3*b - B*a^2*b^2 + (7*A
- 6*C)*a*b^3 - 3*B*b^4)*c^4*d^4 + (3*(A - C)*a^4 - 6*B*a^3*b - 2*C*a^2*b^2 - 6*B*a*b^3 - (3*A - C)*b^4)*c^3*d^
5 + 3*(B*a^4 + (2*A - C)*a^3*b + B*a^2*b^2 + (2*A - C)*a*b^3)*c^2*d^6 - ((3*A - 2*C)*a^4 - B*a^3*b + 2*(2*A -
C)*a^2*b^2 - B*a*b^3 + A*b^4)*c*d^7 - (B*a^4 - A*a^3*b + B*a^2*b^2 - A*a*b^3)*d^8 - 2*(((A - C)*a*b^3 + B*b^4)
*c^7*d - 3*((A - C)*a^2*b^2 + (A - C)*b^4)*c^6*d^2 + 3*((A - C)*a^3*b - 2*B*a^2*b^2 + 2*(A - C)*a*b^3 - B*b^4)
*c^5*d^3 - ((A - C)*a^4 - 8*B*a^3*b - 8*B*a*b^3 - (A - C)*b^4)*c^4*d^4 - 3*(B*a^4 + 2*(A - C)*a^3*b + 2*B*a^2*
b^2 + (A - C)*a*b^3)*c^3*d^5 + 3*((A - C)*a^4 + (A - C)*a^2*b^2)*c^2*d^6 + (B*a^4 - (A - C)*a^3*b)*c*d^7)*f*x)
*tan(f*x + e))/(((a^2*b^3 + b^5)*c^9*d^2 - 3*(a^3*b^2 + a*b^4)*c^8*d^3 + 3*(a^4*b + 2*a^2*b^3 + b^5)*c^7*d^4 -
(a^5 + 10*a^3*b^2 + 9*a*b^4)*c^6*d^5 + 3*(3*a^4*b + 4*a^2*b^3 + b^5)*c^5*d^6 - 3*(a^5 + 4*a^3*b^2 + 3*a*b^4)*
c^4*d^7 + (9*a^4*b + 10*a^2*b^3 + b^5)*c^3*d^8 - 3*(a^5 + 2*a^3*b^2 + a*b^4)*c^2*d^9 + 3*(a^4*b + a^2*b^3)*c*d
^10 - (a^5 + a^3*b^2)*d^11)*f*tan(f*x + e)^2 + 2*((a^2*b^3 + b^5)*c^10*d - 3*(a^3*b^2 + a*b^4)*c^9*d^2 + 3*(a^
4*b + 2*a^2*b^3 + b^5)*c^8*d^3 - (a^5 + 10*a^3*b^2 + 9*a*b^4)*c^7*d^4 + 3*(3*a^4*b + 4*a^2*b^3 + b^5)*c^6*d^5
- 3*(a^5 + 4*a^3*b^2 + 3*a*b^4)*c^5*d^6 + (9*a^4*b + 10*a^2*b^3 + b^5)*c^4*d^7 - 3*(a^5 + 2*a^3*b^2 + a*b^4)*c
^3*d^8 + 3*(a^4*b + a^2*b^3)*c^2*d^9 - (a^5 + a^3*b^2)*c*d^10)*f*tan(f*x + e) + ((a^2*b^3 + b^5)*c^11 - 3*(a^3
*b^2 + a*b^4)*c^10*d + 3*(a^4*b + 2*a^2*b^3 + b^5)*c^9*d^2 - (a^5 + 10*a^3*b^2 + 9*a*b^4)*c^8*d^3 + 3*(3*a^4*b
+ 4*a^2*b^3 + b^5)*c^7*d^4 - 3*(a^5 + 4*a^3*b^2 + 3*a*b^4)*c^6*d^5 + (9*a^4*b + 10*a^2*b^3 + b^5)*c^5*d^6 - 3
*(a^5 + 2*a^3*b^2 + a*b^4)*c^4*d^7 + 3*(a^4*b + a^2*b^3)*c^3*d^8 - (a^5 + a^3*b^2)*c^2*d^9)*f)

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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))/(c+d*tan(f*x+e))**3,x)

[Out]

Timed out

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Giac [B]  time = 3.61727, size = 2869, normalized size = 5.89 \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))/(c+d*tan(f*x+e))^3,x, algorithm="giac")

[Out]

1/2*(2*(A*a*c^3 - C*a*c^3 + B*b*c^3 + 3*B*a*c^2*d - 3*A*b*c^2*d + 3*C*b*c^2*d - 3*A*a*c*d^2 + 3*C*a*c*d^2 - 3*
B*b*c*d^2 - B*a*d^3 + A*b*d^3 - C*b*d^3)*(f*x + e)/(a^2*c^6 + b^2*c^6 + 3*a^2*c^4*d^2 + 3*b^2*c^4*d^2 + 3*a^2*
c^2*d^4 + 3*b^2*c^2*d^4 + a^2*d^6 + b^2*d^6) + (B*a*c^3 - A*b*c^3 + C*b*c^3 - 3*A*a*c^2*d + 3*C*a*c^2*d - 3*B*
b*c^2*d - 3*B*a*c*d^2 + 3*A*b*c*d^2 - 3*C*b*c*d^2 + A*a*d^3 - C*a*d^3 + B*b*d^3)*log(tan(f*x + e)^2 + 1)/(a^2*
c^6 + b^2*c^6 + 3*a^2*c^4*d^2 + 3*b^2*c^4*d^2 + 3*a^2*c^2*d^4 + 3*b^2*c^2*d^4 + a^2*d^6 + b^2*d^6) + 2*(C*a^2*
b^3 - B*a*b^4 + A*b^5)*log(abs(b*tan(f*x + e) + a))/(a^2*b^4*c^3 + b^6*c^3 - 3*a^3*b^3*c^2*d - 3*a*b^5*c^2*d +
3*a^4*b^2*c*d^2 + 3*a^2*b^4*c*d^2 - a^5*b*d^3 - a^3*b^3*d^3) - 2*(C*b^2*c^6*d - 3*B*b^2*c^5*d^2 + 3*B*a*b*c^4
*d^3 + 6*A*b^2*c^4*d^3 - 3*C*b^2*c^4*d^3 - B*a^2*c^3*d^4 - 8*A*a*b*c^3*d^4 + 8*C*a*b*c^3*d^4 + B*b^2*c^3*d^4 +
3*A*a^2*c^2*d^5 - 3*C*a^2*c^2*d^5 - 6*B*a*b*c^2*d^5 + 3*A*b^2*c^2*d^5 + 3*B*a^2*c*d^6 - A*a^2*d^7 + C*a^2*d^7
- B*a*b*d^7 + A*b^2*d^7)*log(abs(d*tan(f*x + e) + c))/(b^3*c^9*d - 3*a*b^2*c^8*d^2 + 3*a^2*b*c^7*d^3 + 3*b^3*
c^7*d^3 - a^3*c^6*d^4 - 9*a*b^2*c^6*d^4 + 9*a^2*b*c^5*d^5 + 3*b^3*c^5*d^5 - 3*a^3*c^4*d^6 - 9*a*b^2*c^4*d^6 +
9*a^2*b*c^3*d^7 + b^3*c^3*d^7 - 3*a^3*c^2*d^8 - 3*a*b^2*c^2*d^8 + 3*a^2*b*c*d^9 - a^3*d^10) + (3*C*b^2*c^6*d^2
*tan(f*x + e)^2 - 9*B*b^2*c^5*d^3*tan(f*x + e)^2 + 9*B*a*b*c^4*d^4*tan(f*x + e)^2 + 18*A*b^2*c^4*d^4*tan(f*x +
e)^2 - 9*C*b^2*c^4*d^4*tan(f*x + e)^2 - 3*B*a^2*c^3*d^5*tan(f*x + e)^2 - 24*A*a*b*c^3*d^5*tan(f*x + e)^2 + 24
*C*a*b*c^3*d^5*tan(f*x + e)^2 + 3*B*b^2*c^3*d^5*tan(f*x + e)^2 + 9*A*a^2*c^2*d^6*tan(f*x + e)^2 - 9*C*a^2*c^2*
d^6*tan(f*x + e)^2 - 18*B*a*b*c^2*d^6*tan(f*x + e)^2 + 9*A*b^2*c^2*d^6*tan(f*x + e)^2 + 9*B*a^2*c*d^7*tan(f*x
+ e)^2 - 3*A*a^2*d^8*tan(f*x + e)^2 + 3*C*a^2*d^8*tan(f*x + e)^2 - 3*B*a*b*d^8*tan(f*x + e)^2 + 3*A*b^2*d^8*ta
n(f*x + e)^2 + 8*C*b^2*c^7*d*tan(f*x + e) - 2*C*a*b*c^6*d^2*tan(f*x + e) - 22*B*b^2*c^6*d^2*tan(f*x + e) + 24*
B*a*b*c^5*d^3*tan(f*x + e) + 42*A*b^2*c^5*d^3*tan(f*x + e) - 18*C*b^2*c^5*d^3*tan(f*x + e) - 8*B*a^2*c^4*d^4*t
an(f*x + e) - 58*A*a*b*c^4*d^4*tan(f*x + e) + 52*C*a*b*c^4*d^4*tan(f*x + e) + 2*B*b^2*c^4*d^4*tan(f*x + e) + 2
2*A*a^2*c^3*d^5*tan(f*x + e) - 22*C*a^2*c^3*d^5*tan(f*x + e) - 32*B*a*b*c^3*d^5*tan(f*x + e) + 26*A*b^2*c^3*d^
5*tan(f*x + e) - 2*C*b^2*c^3*d^5*tan(f*x + e) + 18*B*a^2*c^2*d^6*tan(f*x + e) - 12*A*a*b*c^2*d^6*tan(f*x + e)
+ 6*C*a*b*c^2*d^6*tan(f*x + e) - 2*A*a^2*c*d^7*tan(f*x + e) + 2*C*a^2*c*d^7*tan(f*x + e) - 8*B*a*b*c*d^7*tan(f
*x + e) + 8*A*b^2*c*d^7*tan(f*x + e) + 2*B*a^2*d^8*tan(f*x + e) - 2*A*a*b*d^8*tan(f*x + e) + 6*C*b^2*c^8 - 4*C
*a*b*c^7*d - 14*B*b^2*c^7*d + C*a^2*c^6*d^2 + 17*B*a*b*c^6*d^2 + 25*A*b^2*c^6*d^2 - 7*C*b^2*c^6*d^2 - 6*B*a^2*
c^5*d^3 - 36*A*a*b*c^5*d^3 + 24*C*a*b*c^5*d^3 - 3*B*b^2*c^5*d^3 + 14*A*a^2*c^4*d^4 - 11*C*a^2*c^4*d^4 - 10*B*a
*b*c^4*d^4 + 19*A*b^2*c^4*d^4 - C*b^2*c^4*d^4 + 7*B*a^2*c^3*d^5 - 16*A*a*b*c^3*d^5 + 4*C*a*b*c^3*d^5 - B*b^2*c
^3*d^5 + 3*A*a^2*c^2*d^6 - 3*B*a*b*c^2*d^6 + 6*A*b^2*c^2*d^6 + B*a^2*c*d^7 - 4*A*a*b*c*d^7 + A*a^2*d^8)/((b^3*
c^9 - 3*a*b^2*c^8*d + 3*a^2*b*c^7*d^2 + 3*b^3*c^7*d^2 - a^3*c^6*d^3 - 9*a*b^2*c^6*d^3 + 9*a^2*b*c^5*d^4 + 3*b^
3*c^5*d^4 - 3*a^3*c^4*d^5 - 9*a*b^2*c^4*d^5 + 9*a^2*b*c^3*d^6 + b^3*c^3*d^6 - 3*a^3*c^2*d^7 - 3*a*b^2*c^2*d^7
+ 3*a^2*b*c*d^8 - a^3*d^9)*(d*tan(f*x + e) + c)^2))/f