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

Optimal. Leaf size=477 \[ \frac{\left (-a^3 b^3 \left (8 c d (A-C)+B \left (c^2-d^2\right )\right )-3 a^2 b^4 \left (c (2 B d+c C)-A \left (c^2+d^2\right )\right )+3 a^4 b^2 d (2 A d+B c-C d)-3 a^5 b B d^2+a^6 C d^2+3 a b^5 B c^2+b^6 \left (c (c C-B d)-A \left (c^2-d^2\right )\right )\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{f \left (a^2+b^2\right )^3 (b c-a d)^3}+\frac{x \left (3 a^2 b (B c-d (A-C))+a^3 (A c+B d-c C)-3 a b^2 (A c+B d-c C)-b^3 (B c-d (A-C))\right )}{\left (a^2+b^2\right )^3 \left (c^2+d^2\right )}-\frac{-a^2 b^2 (3 A d+B c-C d)+2 a^3 b B d+a^4 (-C) d+2 a b^3 c (A-C)+b^4 (B c-A d)}{f \left (a^2+b^2\right )^2 (b c-a d)^2 (a+b \tan (e+f x))}-\frac{A b^2-a (b B-a C)}{2 f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))^2}-\frac{d^2 \left (A d^2-B c d+c^2 C\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{f \left (c^2+d^2\right ) (b c-a d)^3} \]

[Out]

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

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Rubi [A]  time = 1.78584, antiderivative size = 477, 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} \[ \frac{\left (-a^3 b^3 \left (8 c d (A-C)+B \left (c^2-d^2\right )\right )-3 a^2 b^4 \left (c (2 B d+c C)-A \left (c^2+d^2\right )\right )+3 a^4 b^2 d (2 A d+B c-C d)-3 a^5 b B d^2+a^6 C d^2+3 a b^5 B c^2+b^6 \left (c (c C-B d)-A \left (c^2-d^2\right )\right )\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{f \left (a^2+b^2\right )^3 (b c-a d)^3}+\frac{x \left (3 a^2 b (B c-d (A-C))+a^3 (A c+B d-c C)-3 a b^2 (A c+B d-c C)-b^3 (B c-d (A-C))\right )}{\left (a^2+b^2\right )^3 \left (c^2+d^2\right )}-\frac{-a^2 b^2 (3 A d+B c-C d)+2 a^3 b B d+a^4 (-C) d+2 a b^3 c (A-C)+b^4 (B c-A d)}{f \left (a^2+b^2\right )^2 (b c-a d)^2 (a+b \tan (e+f x))}-\frac{A b^2-a (b B-a C)}{2 f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))^2}-\frac{d^2 \left (A d^2-B c d+c^2 C\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{f \left (c^2+d^2\right ) (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 8.87886, size = 898, normalized size = 1.88 \[ -\frac{A b^2-a (b B-a C)}{2 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^2}-\frac{-\frac{-2 \left (-A d a^2+b c (A-C) a+b^2 (B c-A d)\right ) b^2-a \left (2 b (A b-C b-a B) (b c-a d)-2 a \left (A b^2-a (b B-a C)\right ) d\right )}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))}-\frac{-\frac{2 b \left (a^2+b^2\right )^2 \left (C c^2-B d c+A d^2\right ) \log (c+d \tan (e+f x)) d^2}{(b c-a d) \left (c^2+d^2\right )}-\frac{b (b c-a d)^2 \left (-B c a^3+A d a^3-C d a^3+3 A b c a^2-3 b c C a^2+3 b B d a^2+3 b^2 B c a-3 A b^2 d a+3 b^2 C d a-A b^3 c+b^3 c C-b^3 B d+\frac{\sqrt{-b^2} \left ((A c-C c+B d) a^3+3 b (B c-(A-C) d) a^2-3 b^2 (A c-C c+B d) a-b^3 (B c-(A-C) d)\right )}{b}\right ) \log \left (\sqrt{-b^2}-b \tan (e+f x)\right )}{\left (a^2+b^2\right ) \left (c^2+d^2\right )}+\frac{2 b \left (C d^2 a^6-3 b B d^2 a^5+3 b^2 d (B c+2 A d-C d) a^4-b^3 \left (8 c (A-C) d+B \left (c^2-d^2\right )\right ) a^3-3 b^4 \left (c (c C+2 B d)-A \left (c^2+d^2\right )\right ) a^2+3 b^5 B c^2 a+b^6 \left (c (c C-B d)-A \left (c^2-d^2\right )\right )\right ) \log (a+b \tan (e+f x))}{\left (a^2+b^2\right ) (b c-a d)}-\frac{b (b c-a d)^2 \left (-B c a^3+A d a^3-C d a^3+3 A b c a^2-3 b c C a^2+3 b B d a^2+3 b^2 B c a-3 A b^2 d a+3 b^2 C d a-A b^3 c+b^3 c C-b^3 B d-\frac{\sqrt{-b^2} \left ((A c-C c+B d) a^3+3 b (B c-(A-C) d) a^2-3 b^2 (A c-C c+B d) a-b^3 (B c-(A-C) d)\right )}{b}\right ) \log \left (b \tan (e+f x)+\sqrt{-b^2}\right )}{\left (a^2+b^2\right ) \left (c^2+d^2\right )}}{b \left (a^2+b^2\right ) (b c-a d) f}}{2 \left (a^2+b^2\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.111, size = 2298, normalized size = 4.8 \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))^3/(c+d*tan(f*x+e)),x)

[Out]

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

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Maxima [B]  time = 1.80378, size = 1480, normalized size = 3.1 \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))^3/(c+d*tan(f*x+e)),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 26.3859, size = 7380, normalized size = 15.47 \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))^3/(c+d*tan(f*x+e)),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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Giac [B]  time = 1.91993, size = 2871, normalized size = 6.02 \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))^3/(c+d*tan(f*x+e)),x, algorithm="giac")

[Out]

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