\(\int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^2 (A+B \tan (e+f x)+C \tan ^2(e+f x)) \, dx\) [58]

Optimal result

Integrand size = 45, antiderivative size = 443 \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=-\left (\left (a^2 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-b^2 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )+2 a b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) x\right )+\frac {\left (2 a b \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-a^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )+b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) \log (\cos (e+f x))}{f}+\frac {d \left (2 a b (A c-c C-B d)+a^2 (B c+(A-C) d)-b^2 (B c+(A-C) d)\right ) \tan (e+f x)}{f}+\frac {\left (a^2 B-b^2 B+2 a b (A-C)\right ) (c+d \tan (e+f x))^2}{2 f}+\frac {\left (8 a^2 C d^2-10 a b d (c C-4 B d)+b^2 \left (2 c^2 C-5 B c d+20 (A-C) d^2\right )\right ) (c+d \tan (e+f x))^3}{60 d^3 f}-\frac {b (2 b c C-5 b B d-2 a C d) \tan (e+f x) (c+d \tan (e+f x))^3}{20 d^2 f}+\frac {C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3}{5 d f} \]

[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+(2*a*b*(c^2*C+2*B*c*d-C*d^2-A*(c^2-d^2))-a^2*(2*c*(A-C)*d+B*(c^2-d^2))+b^2*(2*c*(A-C)*d+B*(c^2-d^2)))*ln(co
s(f*x+e))/f+d*(2*a*b*(A*c-B*d-C*c)+a^2*(B*c+(A-C)*d)-b^2*(B*c+(A-C)*d))*tan(f*x+e)/f+1/2*(B*a^2-B*b^2+2*a*b*(A
-C))*(c+d*tan(f*x+e))^2/f+1/60*(8*a^2*C*d^2-10*a*b*d*(-4*B*d+C*c)+b^2*(2*c^2*C-5*B*c*d+20*(A-C)*d^2))*(c+d*tan
(f*x+e))^3/d^3/f-1/20*b*(-5*B*b*d-2*C*a*d+2*C*b*c)*tan(f*x+e)*(c+d*tan(f*x+e))^3/d^2/f+1/5*C*(a+b*tan(f*x+e))^
2*(c+d*tan(f*x+e))^3/d/f

Rubi [A] (verified)

Time = 1.35 (sec) , antiderivative size = 443, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3728, 3718, 3711, 3609, 3606, 3556} \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\frac {\log (\cos (e+f x)) \left (-\left (a^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )+2 a b \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )+b^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )}{f}-x \left (a^2 \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )+2 a b \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )-b^2 \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )\right )+\frac {(c+d \tan (e+f x))^3 \left (8 a^2 C d^2-10 a b d (c C-4 B d)+b^2 \left (20 d^2 (A-C)-5 B c d+2 c^2 C\right )\right )}{60 d^3 f}+\frac {\left (a^2 B+2 a b (A-C)-b^2 B\right ) (c+d \tan (e+f x))^2}{2 f}+\frac {d \tan (e+f x) \left (a^2 (d (A-C)+B c)+2 a b (A c-B d-c C)-b^2 (d (A-C)+B c)\right )}{f}-\frac {b \tan (e+f x) (-2 a C d-5 b B d+2 b c C) (c+d \tan (e+f x))^3}{20 d^2 f}+\frac {C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3}{5 d f} \]

[In]

Int[(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^2*(A + B*Tan[e + f*x] + C*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) + ((2*a*b*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) - a^2*(2*c*(A - C)*d + B*(c
^2 - d^2)) + b^2*(2*c*(A - C)*d + B*(c^2 - d^2)))*Log[Cos[e + f*x]])/f + (d*(2*a*b*(A*c - c*C - B*d) + a^2*(B*
c + (A - C)*d) - b^2*(B*c + (A - C)*d))*Tan[e + f*x])/f + ((a^2*B - b^2*B + 2*a*b*(A - C))*(c + d*Tan[e + f*x]
)^2)/(2*f) + ((8*a^2*C*d^2 - 10*a*b*d*(c*C - 4*B*d) + b^2*(2*c^2*C - 5*B*c*d + 20*(A - C)*d^2))*(c + d*Tan[e +
f*x])^3)/(60*d^3*f) - (b*(2*b*c*C - 5*b*B*d - 2*a*C*d)*Tan[e + f*x]*(c + d*Tan[e + f*x])^3)/(20*d^2*f) + (C*(
a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^3)/(5*d*f)

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3606

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(a*c - b
*d)*x, x] + (Dist[b*c + a*d, Int[Tan[e + f*x], x], x] + Simp[b*d*(Tan[e + f*x]/f), x]) /; FreeQ[{a, b, c, d, e
, f}, x] && NeQ[b*c - a*d, 0] && NeQ[b*c + a*d, 0]

Rule 3609

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d*
((a + b*Tan[e + f*x])^m/(f*m)), x] + Int[(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 0]

Rule 3711

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[C*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f*x])
^m*Simp[A - C + B*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0]
&&  !LeQ[m, -1]

Rule 3718

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*tan[(e
_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[b*C*Tan[e + f*x]*((c + d*Tan[e + f*x])
^(n + 1)/(d*f*(n + 2))), x] - Dist[1/(d*(n + 2)), Int[(c + d*Tan[e + f*x])^n*Simp[b*c*C - a*A*d*(n + 2) - (A*b
+ a*B - b*C)*d*(n + 2)*Tan[e + f*x] - (a*C*d*(n + 2) - b*(c*C - B*d*(n + 2)))*Tan[e + f*x]^2, x], x], x] /; F
reeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[c^2 + d^2, 0] &&  !LtQ[n, -1]

Rule 3728

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*
tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*(a + b*Tan[e + f*x])^m*((c + d
*Tan[e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Dist[1/(d*(m + n + 1)), Int[(a + b*Tan[e + f*x])^(m - 1)*(c +
d*Tan[e + f*x])^n*Simp[a*A*d*(m + n + 1) - C*(b*c*m + a*d*(n + 1)) + d*(A*b + a*B - b*C)*(m + n + 1)*Tan[e + f
*x] - (C*m*(b*c - a*d) - b*B*d*(m + n + 1))*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] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !Intege
rQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rubi steps \begin{align*} \text {integral}& = \frac {C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3}{5 d f}+\frac {\int (a+b \tan (e+f x)) (c+d \tan (e+f x))^2 \left (-2 b c C+a (5 A-3 C) d+5 (A b+a B-b C) d \tan (e+f x)-(2 b c C-5 b B d-2 a C d) \tan ^2(e+f x)\right ) \, dx}{5 d} \\ & = -\frac {b (2 b c C-5 b B d-2 a C d) \tan (e+f x) (c+d \tan (e+f x))^3}{20 d^2 f}+\frac {C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3}{5 d f}-\frac {\int (c+d \tan (e+f x))^2 \left (10 a b c C d-4 a^2 (5 A-3 C) d^2-b^2 c (2 c C-5 B d)-20 \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2 \tan (e+f x)-\left (8 a^2 C d^2-10 a b d (c C-4 B d)+b^2 \left (2 c^2 C-5 B c d+20 (A-C) d^2\right )\right ) \tan ^2(e+f x)\right ) \, dx}{20 d^2} \\ & = \frac {\left (8 a^2 C d^2-10 a b d (c C-4 B d)+b^2 \left (2 c^2 C-5 B c d+20 (A-C) d^2\right )\right ) (c+d \tan (e+f x))^3}{60 d^3 f}-\frac {b (2 b c C-5 b B d-2 a C d) \tan (e+f x) (c+d \tan (e+f x))^3}{20 d^2 f}+\frac {C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3}{5 d f}-\frac {\int (c+d \tan (e+f x))^2 \left (20 \left (2 a b B-a^2 (A-C)+b^2 (A-C)\right ) d^2-20 \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2 \tan (e+f x)\right ) \, dx}{20 d^2} \\ & = \frac {\left (a^2 B-b^2 B+2 a b (A-C)\right ) (c+d \tan (e+f x))^2}{2 f}+\frac {\left (8 a^2 C d^2-10 a b d (c C-4 B d)+b^2 \left (2 c^2 C-5 B c d+20 (A-C) d^2\right )\right ) (c+d \tan (e+f x))^3}{60 d^3 f}-\frac {b (2 b c C-5 b B d-2 a C d) \tan (e+f x) (c+d \tan (e+f x))^3}{20 d^2 f}+\frac {C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3}{5 d f}-\frac {\int (c+d \tan (e+f x)) \left (-20 d^2 \left (a^2 (A c-c C-B d)-b^2 (A c-c C-B d)-2 a b (B c+(A-C) d)\right )-20 d^2 \left (2 a b (A c-c C-B d)+a^2 (B c+(A-C) d)-b^2 (B c+(A-C) d)\right ) \tan (e+f x)\right ) \, dx}{20 d^2} \\ & = -\left (\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\right )+\frac {d \left (2 a b (A c-c C-B d)+a^2 (B c+(A-C) d)-b^2 (B c+(A-C) d)\right ) \tan (e+f x)}{f}+\frac {\left (a^2 B-b^2 B+2 a b (A-C)\right ) (c+d \tan (e+f x))^2}{2 f}+\frac {\left (8 a^2 C d^2-10 a b d (c C-4 B d)+b^2 \left (2 c^2 C-5 B c d+20 (A-C) d^2\right )\right ) (c+d \tan (e+f x))^3}{60 d^3 f}-\frac {b (2 b c C-5 b B d-2 a C d) \tan (e+f x) (c+d \tan (e+f x))^3}{20 d^2 f}+\frac {C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3}{5 d f}-\left (2 a b \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-a^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )+b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) \int \tan (e+f x) \, dx \\ & = -\left (\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\right )+\frac {\left (2 a b \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-a^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )+b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) \log (\cos (e+f x))}{f}+\frac {d \left (2 a b (A c-c C-B d)+a^2 (B c+(A-C) d)-b^2 (B c+(A-C) d)\right ) \tan (e+f x)}{f}+\frac {\left (a^2 B-b^2 B+2 a b (A-C)\right ) (c+d \tan (e+f x))^2}{2 f}+\frac {\left (8 a^2 C d^2-10 a b d (c C-4 B d)+b^2 \left (2 c^2 C-5 B c d+20 (A-C) d^2\right )\right ) (c+d \tan (e+f x))^3}{60 d^3 f}-\frac {b (2 b c C-5 b B d-2 a C d) \tan (e+f x) (c+d \tan (e+f x))^3}{20 d^2 f}+\frac {C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3}{5 d f} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.55 (sec) , antiderivative size = 383, normalized size of antiderivative = 0.86 \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\frac {C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3}{5 d f}+\frac {\frac {b (-2 b c C+5 b B d+2 a C d) \tan (e+f x) (c+d \tan (e+f x))^3}{4 d f}-\frac {\frac {\left (-8 a^2 C d^2+10 a b d (c C-4 B d)-b^2 \left (2 c^2 C-5 B c d+20 (A-C) d^2\right )\right ) (c+d \tan (e+f x))^3}{3 d f}-\frac {10 \left (d \left (2 a b (A c-c C+B d)+a^2 (B c-(A-C) d)-b^2 (B c-(A-C) d)\right ) \left (i (c+i d)^2 \log (i-\tan (e+f x))-i (c-i d)^2 \log (i+\tan (e+f x))-2 d^2 \tan (e+f x)\right )+\left (a^2 B-b^2 B+2 a b (A-C)\right ) d \left ((i c-d)^3 \log (i-\tan (e+f x))-(i c+d)^3 \log (i+\tan (e+f x))+6 c d^2 \tan (e+f x)+d^3 \tan ^2(e+f x)\right )\right )}{f}}{4 d}}{5 d} \]

[In]

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

[Out]

(C*(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^3)/(5*d*f) + ((b*(-2*b*c*C + 5*b*B*d + 2*a*C*d)*Tan[e + f*x]*(c
+ d*Tan[e + f*x])^3)/(4*d*f) - (((-8*a^2*C*d^2 + 10*a*b*d*(c*C - 4*B*d) - b^2*(2*c^2*C - 5*B*c*d + 20*(A - C)
*d^2))*(c + d*Tan[e + f*x])^3)/(3*d*f) - (10*(d*(2*a*b*(A*c - c*C + B*d) + a^2*(B*c - (A - C)*d) - b^2*(B*c -
(A - C)*d))*(I*(c + I*d)^2*Log[I - Tan[e + f*x]] - I*(c - I*d)^2*Log[I + Tan[e + f*x]] - 2*d^2*Tan[e + f*x]) +
(a^2*B - b^2*B + 2*a*b*(A - C))*d*((I*c - d)^3*Log[I - Tan[e + f*x]] - (I*c + d)^3*Log[I + Tan[e + f*x]] + 6*
c*d^2*Tan[e + f*x] + d^3*Tan[e + f*x]^2)))/f)/(4*d))/(5*d)

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 392, normalized size of antiderivative = 0.88

method	result	size
parts	\(\frac {\left (2 A \,a^{2} c d +2 A a b \,c^{2}+B \,a^{2} c^{2}\right ) \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2 f}+\frac {\left (B \,b^{2} d^{2}+2 C a b \,d^{2}+2 C \,b^{2} c d \right ) \left (\frac {\tan \left (f x +e \right )^{4}}{4}-\frac {\tan \left (f x +e \right )^{2}}{2}+\frac {\ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}\right )}{f}+\frac {\left (A \,b^{2} d^{2}+2 B a b \,d^{2}+2 B \,b^{2} c d +a^{2} C \,d^{2}+4 C a b c d +C \,b^{2} c^{2}\right ) \left (\frac {\tan \left (f x +e \right )^{3}}{3}-\tan \left (f x +e \right )+\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}+\frac {\left (A \,a^{2} d^{2}+4 A a b c d +A \,b^{2} c^{2}+2 B \,a^{2} c d +2 B a b \,c^{2}+C \,a^{2} c^{2}\right ) \left (\tan \left (f x +e \right )-\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}+\frac {\left (2 A a b \,d^{2}+2 A \,b^{2} c d +B \,a^{2} d^{2}+4 B a b c d +B \,b^{2} c^{2}+2 C \,a^{2} c d +2 C a b \,c^{2}\right ) \left (\frac {\tan \left (f x +e \right )^{2}}{2}-\frac {\ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}\right )}{f}+A \,a^{2} c^{2} x +\frac {C \,b^{2} d^{2} \left (\frac {\tan \left (f x +e \right )^{5}}{5}-\frac {\tan \left (f x +e \right )^{3}}{3}+\tan \left (f x +e \right )-\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}\)	\(392\)
norman	\(\left (A \,a^{2} c^{2}-A \,a^{2} d^{2}-4 A a b c d -A \,b^{2} c^{2}+A \,b^{2} d^{2}-2 B \,a^{2} c d -2 B a b \,c^{2}+2 B a b \,d^{2}+2 B \,b^{2} c d -C \,a^{2} c^{2}+a^{2} C \,d^{2}+4 C a b c d +C \,b^{2} c^{2}-C \,b^{2} d^{2}\right ) x +\frac {\left (A \,a^{2} d^{2}+4 A a b c d +A \,b^{2} c^{2}-A \,b^{2} d^{2}+2 B \,a^{2} c d +2 B a b \,c^{2}-2 B a b \,d^{2}-2 B \,b^{2} c d +C \,a^{2} c^{2}-a^{2} C \,d^{2}-4 C a b c d -C \,b^{2} c^{2}+C \,b^{2} d^{2}\right ) \tan \left (f x +e \right )}{f}+\frac {\left (A \,b^{2} d^{2}+2 B a b \,d^{2}+2 B \,b^{2} c d +a^{2} C \,d^{2}+4 C a b c d +C \,b^{2} c^{2}-C \,b^{2} d^{2}\right ) \tan \left (f x +e \right )^{3}}{3 f}+\frac {\left (2 A a b \,d^{2}+2 A \,b^{2} c d +B \,a^{2} d^{2}+4 B a b c d +B \,b^{2} c^{2}-B \,b^{2} d^{2}+2 C \,a^{2} c d +2 C a b \,c^{2}-2 C a b \,d^{2}-2 C \,b^{2} c d \right ) \tan \left (f x +e \right )^{2}}{2 f}+\frac {C \,b^{2} d^{2} \tan \left (f x +e \right )^{5}}{5 f}+\frac {b d \left (b d B +2 C a d +2 C b c \right ) \tan \left (f x +e \right )^{4}}{4 f}+\frac {\left (2 A \,a^{2} c d +2 A a b \,c^{2}-2 A a b \,d^{2}-2 A \,b^{2} c d +B \,a^{2} c^{2}-B \,a^{2} d^{2}-4 B a b c d -B \,b^{2} c^{2}+B \,b^{2} d^{2}-2 C \,a^{2} c d -2 C a b \,c^{2}+2 C a b \,d^{2}+2 C \,b^{2} c d \right ) \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2 f}\)	\(571\)
derivativedivides	\(\frac {\frac {C \,a^{2} d^{2} \tan \left (f x +e \right )^{3}}{3}+\frac {C \,b^{2} c^{2} \tan \left (f x +e \right )^{3}}{3}-\frac {C \,b^{2} d^{2} \tan \left (f x +e \right )^{3}}{3}+\frac {B \,a^{2} d^{2} \tan \left (f x +e \right )^{2}}{2}+\frac {B \,b^{2} c^{2} \tan \left (f x +e \right )^{2}}{2}-\frac {B \,b^{2} d^{2} \tan \left (f x +e \right )^{2}}{2}-\tan \left (f x +e \right ) A \,b^{2} d^{2}-\tan \left (f x +e \right ) C \,b^{2} c^{2}+\tan \left (f x +e \right ) C \,b^{2} d^{2}-\tan \left (f x +e \right ) a^{2} C \,d^{2}+\frac {B \,b^{2} d^{2} \tan \left (f x +e \right )^{4}}{4}+\frac {A \,b^{2} d^{2} \tan \left (f x +e \right )^{3}}{3}+\frac {C \,b^{2} d^{2} \tan \left (f x +e \right )^{5}}{5}+\tan \left (f x +e \right ) A \,b^{2} c^{2}+\tan \left (f x +e \right ) C \,a^{2} c^{2}+\tan \left (f x +e \right ) A \,a^{2} d^{2}+\frac {\left (2 A \,a^{2} c d +2 A a b \,c^{2}-2 A a b \,d^{2}-2 A \,b^{2} c d +B \,a^{2} c^{2}-B \,a^{2} d^{2}-4 B a b c d -B \,b^{2} c^{2}+B \,b^{2} d^{2}-2 C \,a^{2} c d -2 C a b \,c^{2}+2 C a b \,d^{2}+2 C \,b^{2} c d \right ) \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}+\left (A \,a^{2} c^{2}-A \,a^{2} d^{2}-4 A a b c d -A \,b^{2} c^{2}+A \,b^{2} d^{2}-2 B \,a^{2} c d -2 B a b \,c^{2}+2 B a b \,d^{2}+2 B \,b^{2} c d -C \,a^{2} c^{2}+a^{2} C \,d^{2}+4 C a b c d +C \,b^{2} c^{2}-C \,b^{2} d^{2}\right ) \arctan \left (\tan \left (f x +e \right )\right )+2 B a b c d \tan \left (f x +e \right )^{2}+\frac {4 C a b c d \tan \left (f x +e \right )^{3}}{3}+4 \tan \left (f x +e \right ) A a b c d -4 \tan \left (f x +e \right ) C a b c d +C a b \,c^{2} \tan \left (f x +e \right )^{2}+A \,b^{2} c d \tan \left (f x +e \right )^{2}+C \,a^{2} c d \tan \left (f x +e \right )^{2}+A a b \,d^{2} \tan \left (f x +e \right )^{2}+2 \tan \left (f x +e \right ) B a b \,c^{2}-2 \tan \left (f x +e \right ) B a b \,d^{2}-2 \tan \left (f x +e \right ) B \,b^{2} c d +\frac {C a b \,d^{2} \tan \left (f x +e \right )^{4}}{2}+\frac {C \,b^{2} c d \tan \left (f x +e \right )^{4}}{2}+\frac {2 B a b \,d^{2} \tan \left (f x +e \right )^{3}}{3}+\frac {2 B \,b^{2} c d \tan \left (f x +e \right )^{3}}{3}-C a b \,d^{2} \tan \left (f x +e \right )^{2}-C \,b^{2} c d \tan \left (f x +e \right )^{2}+2 \tan \left (f x +e \right ) B \,a^{2} c d}{f}\)	\(770\)
default	\(\frac {\frac {C \,a^{2} d^{2} \tan \left (f x +e \right )^{3}}{3}+\frac {C \,b^{2} c^{2} \tan \left (f x +e \right )^{3}}{3}-\frac {C \,b^{2} d^{2} \tan \left (f x +e \right )^{3}}{3}+\frac {B \,a^{2} d^{2} \tan \left (f x +e \right )^{2}}{2}+\frac {B \,b^{2} c^{2} \tan \left (f x +e \right )^{2}}{2}-\frac {B \,b^{2} d^{2} \tan \left (f x +e \right )^{2}}{2}-\tan \left (f x +e \right ) A \,b^{2} d^{2}-\tan \left (f x +e \right ) C \,b^{2} c^{2}+\tan \left (f x +e \right ) C \,b^{2} d^{2}-\tan \left (f x +e \right ) a^{2} C \,d^{2}+\frac {B \,b^{2} d^{2} \tan \left (f x +e \right )^{4}}{4}+\frac {A \,b^{2} d^{2} \tan \left (f x +e \right )^{3}}{3}+\frac {C \,b^{2} d^{2} \tan \left (f x +e \right )^{5}}{5}+\tan \left (f x +e \right ) A \,b^{2} c^{2}+\tan \left (f x +e \right ) C \,a^{2} c^{2}+\tan \left (f x +e \right ) A \,a^{2} d^{2}+\frac {\left (2 A \,a^{2} c d +2 A a b \,c^{2}-2 A a b \,d^{2}-2 A \,b^{2} c d +B \,a^{2} c^{2}-B \,a^{2} d^{2}-4 B a b c d -B \,b^{2} c^{2}+B \,b^{2} d^{2}-2 C \,a^{2} c d -2 C a b \,c^{2}+2 C a b \,d^{2}+2 C \,b^{2} c d \right ) \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}+\left (A \,a^{2} c^{2}-A \,a^{2} d^{2}-4 A a b c d -A \,b^{2} c^{2}+A \,b^{2} d^{2}-2 B \,a^{2} c d -2 B a b \,c^{2}+2 B a b \,d^{2}+2 B \,b^{2} c d -C \,a^{2} c^{2}+a^{2} C \,d^{2}+4 C a b c d +C \,b^{2} c^{2}-C \,b^{2} d^{2}\right ) \arctan \left (\tan \left (f x +e \right )\right )+2 B a b c d \tan \left (f x +e \right )^{2}+\frac {4 C a b c d \tan \left (f x +e \right )^{3}}{3}+4 \tan \left (f x +e \right ) A a b c d -4 \tan \left (f x +e \right ) C a b c d +C a b \,c^{2} \tan \left (f x +e \right )^{2}+A \,b^{2} c d \tan \left (f x +e \right )^{2}+C \,a^{2} c d \tan \left (f x +e \right )^{2}+A a b \,d^{2} \tan \left (f x +e \right )^{2}+2 \tan \left (f x +e \right ) B a b \,c^{2}-2 \tan \left (f x +e \right ) B a b \,d^{2}-2 \tan \left (f x +e \right ) B \,b^{2} c d +\frac {C a b \,d^{2} \tan \left (f x +e \right )^{4}}{2}+\frac {C \,b^{2} c d \tan \left (f x +e \right )^{4}}{2}+\frac {2 B a b \,d^{2} \tan \left (f x +e \right )^{3}}{3}+\frac {2 B \,b^{2} c d \tan \left (f x +e \right )^{3}}{3}-C a b \,d^{2} \tan \left (f x +e \right )^{2}-C \,b^{2} c d \tan \left (f x +e \right )^{2}+2 \tan \left (f x +e \right ) B \,a^{2} c d}{f}\)	\(770\)
parallelrisch	\(\frac {20 C \,a^{2} d^{2} \tan \left (f x +e \right )^{3}+20 C \,b^{2} c^{2} \tan \left (f x +e \right )^{3}-20 C \,b^{2} d^{2} \tan \left (f x +e \right )^{3}+30 B \,a^{2} d^{2} \tan \left (f x +e \right )^{2}+30 B \,b^{2} c^{2} \tan \left (f x +e \right )^{2}-30 B \,b^{2} d^{2} \tan \left (f x +e \right )^{2}-60 \tan \left (f x +e \right ) A \,b^{2} d^{2}-60 \tan \left (f x +e \right ) C \,b^{2} c^{2}+60 \tan \left (f x +e \right ) C \,b^{2} d^{2}-60 \tan \left (f x +e \right ) a^{2} C \,d^{2}+15 B \,b^{2} d^{2} \tan \left (f x +e \right )^{4}+20 A \,b^{2} d^{2} \tan \left (f x +e \right )^{3}+12 C \,b^{2} d^{2} \tan \left (f x +e \right )^{5}+60 \tan \left (f x +e \right ) A \,b^{2} c^{2}+60 \tan \left (f x +e \right ) C \,a^{2} c^{2}+60 \tan \left (f x +e \right ) A \,a^{2} d^{2}+30 B \ln \left (1+\tan \left (f x +e \right )^{2}\right ) a^{2} c^{2}-30 B \ln \left (1+\tan \left (f x +e \right )^{2}\right ) a^{2} d^{2}-30 B \ln \left (1+\tan \left (f x +e \right )^{2}\right ) b^{2} c^{2}+30 B \ln \left (1+\tan \left (f x +e \right )^{2}\right ) b^{2} d^{2}+120 B a b c d \tan \left (f x +e \right )^{2}+80 C a b c d \tan \left (f x +e \right )^{3}+240 \tan \left (f x +e \right ) A a b c d -240 \tan \left (f x +e \right ) C a b c d +60 A \ln \left (1+\tan \left (f x +e \right )^{2}\right ) a^{2} c d +60 A \ln \left (1+\tan \left (f x +e \right )^{2}\right ) a b \,c^{2}-60 A \ln \left (1+\tan \left (f x +e \right )^{2}\right ) a b \,d^{2}-60 A \ln \left (1+\tan \left (f x +e \right )^{2}\right ) b^{2} c d -60 C \ln \left (1+\tan \left (f x +e \right )^{2}\right ) a^{2} c d -60 C \ln \left (1+\tan \left (f x +e \right )^{2}\right ) a b \,c^{2}+60 C \ln \left (1+\tan \left (f x +e \right )^{2}\right ) a b \,d^{2}+60 C \ln \left (1+\tan \left (f x +e \right )^{2}\right ) b^{2} c d +60 C a b \,c^{2} \tan \left (f x +e \right )^{2}+60 A \,b^{2} c d \tan \left (f x +e \right )^{2}+60 C \,a^{2} c d \tan \left (f x +e \right )^{2}+60 A a b \,d^{2} \tan \left (f x +e \right )^{2}+120 \tan \left (f x +e \right ) B a b \,c^{2}-120 \tan \left (f x +e \right ) B a b \,d^{2}-120 \tan \left (f x +e \right ) B \,b^{2} c d +30 C a b \,d^{2} \tan \left (f x +e \right )^{4}+30 C \,b^{2} c d \tan \left (f x +e \right )^{4}+40 B a b \,d^{2} \tan \left (f x +e \right )^{3}+40 B \,b^{2} c d \tan \left (f x +e \right )^{3}-60 C a b \,d^{2} \tan \left (f x +e \right )^{2}-60 C \,b^{2} c d \tan \left (f x +e \right )^{2}+120 \tan \left (f x +e \right ) B \,a^{2} c d -120 B \,a^{2} c d f x -120 B \ln \left (1+\tan \left (f x +e \right )^{2}\right ) a b c d -240 A a b c d f x +240 C a b c d f x -120 B a b \,c^{2} f x +120 B a b \,d^{2} f x +120 B \,b^{2} c d f x +60 A \,a^{2} c^{2} f x -60 A \,a^{2} d^{2} f x -60 A \,b^{2} c^{2} f x +60 A \,b^{2} d^{2} f x -60 C \,a^{2} c^{2} f x +60 C \,a^{2} d^{2} f x +60 C \,b^{2} c^{2} f x -60 C \,b^{2} d^{2} f x}{60 f}\)	\(933\)
risch	\(\text {Expression too large to display}\)	\(2496\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 462, normalized size of antiderivative = 1.04 \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\frac {12 \, C b^{2} d^{2} \tan \left (f x + e\right )^{5} + 15 \, {\left (2 \, C b^{2} c d + {\left (2 \, C a b + B b^{2}\right )} d^{2}\right )} \tan \left (f x + e\right )^{4} + 20 \, {\left (C b^{2} c^{2} + 2 \, {\left (2 \, C a b + B b^{2}\right )} c d + {\left (C a^{2} + 2 \, B a b + {\left (A - C\right )} b^{2}\right )} d^{2}\right )} \tan \left (f x + e\right )^{3} + 60 \, {\left ({\left ({\left (A - C\right )} a^{2} - 2 \, B a b - {\left (A - C\right )} b^{2}\right )} c^{2} - 2 \, {\left (B a^{2} + 2 \, {\left (A - C\right )} a b - B b^{2}\right )} c d - {\left ({\left (A - C\right )} a^{2} - 2 \, B a b - {\left (A - C\right )} b^{2}\right )} d^{2}\right )} f x + 30 \, {\left ({\left (2 \, C a b + B b^{2}\right )} c^{2} + 2 \, {\left (C a^{2} + 2 \, B a b + {\left (A - C\right )} b^{2}\right )} c d + {\left (B a^{2} + 2 \, {\left (A - C\right )} a b - B b^{2}\right )} d^{2}\right )} \tan \left (f x + e\right )^{2} - 30 \, {\left ({\left (B a^{2} + 2 \, {\left (A - C\right )} a b - B b^{2}\right )} c^{2} + 2 \, {\left ({\left (A - C\right )} a^{2} - 2 \, B a b - {\left (A - C\right )} b^{2}\right )} c d - {\left (B a^{2} + 2 \, {\left (A - C\right )} a b - B b^{2}\right )} d^{2}\right )} \log \left (\frac {1}{\tan \left (f x + e\right )^{2} + 1}\right ) + 60 \, {\left ({\left (C a^{2} + 2 \, B a b + {\left (A - C\right )} b^{2}\right )} c^{2} + 2 \, {\left (B a^{2} + 2 \, {\left (A - C\right )} a b - B b^{2}\right )} c d + {\left ({\left (A - C\right )} a^{2} - 2 \, B a b - {\left (A - C\right )} b^{2}\right )} d^{2}\right )} \tan \left (f x + e\right )}{60 \, f} \]

[In]

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

[Out]

1/60*(12*C*b^2*d^2*tan(f*x + e)^5 + 15*(2*C*b^2*c*d + (2*C*a*b + B*b^2)*d^2)*tan(f*x + e)^4 + 20*(C*b^2*c^2 +
2*(2*C*a*b + B*b^2)*c*d + (C*a^2 + 2*B*a*b + (A - C)*b^2)*d^2)*tan(f*x + e)^3 + 60*(((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 + 30*
((2*C*a*b + B*b^2)*c^2 + 2*(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)*tan(f*x
+ e)^2 - 30*((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(1/(tan(f*x + e)^2 + 1)) + 60*((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)*tan(f*x + e))/f

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1134 vs. \(2 (396) = 792\).

Time = 0.30 (sec) , antiderivative size = 1134, normalized size of antiderivative = 2.56 \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((A*a**2*c**2*x + A*a**2*c*d*log(tan(e + f*x)**2 + 1)/f - A*a**2*d**2*x + A*a**2*d**2*tan(e + f*x)/f
+ A*a*b*c**2*log(tan(e + f*x)**2 + 1)/f - 4*A*a*b*c*d*x + 4*A*a*b*c*d*tan(e + f*x)/f - A*a*b*d**2*log(tan(e +
f*x)**2 + 1)/f + A*a*b*d**2*tan(e + f*x)**2/f - A*b**2*c**2*x + A*b**2*c**2*tan(e + f*x)/f - A*b**2*c*d*log(ta
n(e + f*x)**2 + 1)/f + A*b**2*c*d*tan(e + f*x)**2/f + A*b**2*d**2*x + A*b**2*d**2*tan(e + f*x)**3/(3*f) - A*b*
*2*d**2*tan(e + f*x)/f + B*a**2*c**2*log(tan(e + f*x)**2 + 1)/(2*f) - 2*B*a**2*c*d*x + 2*B*a**2*c*d*tan(e + f*
x)/f - B*a**2*d**2*log(tan(e + f*x)**2 + 1)/(2*f) + B*a**2*d**2*tan(e + f*x)**2/(2*f) - 2*B*a*b*c**2*x + 2*B*a
*b*c**2*tan(e + f*x)/f - 2*B*a*b*c*d*log(tan(e + f*x)**2 + 1)/f + 2*B*a*b*c*d*tan(e + f*x)**2/f + 2*B*a*b*d**2
*x + 2*B*a*b*d**2*tan(e + f*x)**3/(3*f) - 2*B*a*b*d**2*tan(e + f*x)/f - B*b**2*c**2*log(tan(e + f*x)**2 + 1)/(
2*f) + B*b**2*c**2*tan(e + f*x)**2/(2*f) + 2*B*b**2*c*d*x + 2*B*b**2*c*d*tan(e + f*x)**3/(3*f) - 2*B*b**2*c*d*
tan(e + f*x)/f + B*b**2*d**2*log(tan(e + f*x)**2 + 1)/(2*f) + B*b**2*d**2*tan(e + f*x)**4/(4*f) - B*b**2*d**2*
tan(e + f*x)**2/(2*f) - C*a**2*c**2*x + C*a**2*c**2*tan(e + f*x)/f - C*a**2*c*d*log(tan(e + f*x)**2 + 1)/f + C
*a**2*c*d*tan(e + f*x)**2/f + C*a**2*d**2*x + C*a**2*d**2*tan(e + f*x)**3/(3*f) - C*a**2*d**2*tan(e + f*x)/f -
C*a*b*c**2*log(tan(e + f*x)**2 + 1)/f + C*a*b*c**2*tan(e + f*x)**2/f + 4*C*a*b*c*d*x + 4*C*a*b*c*d*tan(e + f*
x)**3/(3*f) - 4*C*a*b*c*d*tan(e + f*x)/f + C*a*b*d**2*log(tan(e + f*x)**2 + 1)/f + C*a*b*d**2*tan(e + f*x)**4/
(2*f) - C*a*b*d**2*tan(e + f*x)**2/f + C*b**2*c**2*x + C*b**2*c**2*tan(e + f*x)**3/(3*f) - C*b**2*c**2*tan(e +
f*x)/f + C*b**2*c*d*log(tan(e + f*x)**2 + 1)/f + C*b**2*c*d*tan(e + f*x)**4/(2*f) - C*b**2*c*d*tan(e + f*x)**
2/f - C*b**2*d**2*x + C*b**2*d**2*tan(e + f*x)**5/(5*f) - C*b**2*d**2*tan(e + f*x)**3/(3*f) + C*b**2*d**2*tan(
e + f*x)/f, Ne(f, 0)), (x*(a + b*tan(e))**2*(c + d*tan(e))**2*(A + B*tan(e) + C*tan(e)**2), True))

Maxima [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 463, normalized size of antiderivative = 1.05 \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\frac {12 \, C b^{2} d^{2} \tan \left (f x + e\right )^{5} + 15 \, {\left (2 \, C b^{2} c d + {\left (2 \, C a b + B b^{2}\right )} d^{2}\right )} \tan \left (f x + e\right )^{4} + 20 \, {\left (C b^{2} c^{2} + 2 \, {\left (2 \, C a b + B b^{2}\right )} c d + {\left (C a^{2} + 2 \, B a b + {\left (A - C\right )} b^{2}\right )} d^{2}\right )} \tan \left (f x + e\right )^{3} + 30 \, {\left ({\left (2 \, C a b + B b^{2}\right )} c^{2} + 2 \, {\left (C a^{2} + 2 \, B a b + {\left (A - C\right )} b^{2}\right )} c d + {\left (B a^{2} + 2 \, {\left (A - C\right )} a b - B b^{2}\right )} d^{2}\right )} \tan \left (f x + e\right )^{2} + 60 \, {\left ({\left ({\left (A - C\right )} a^{2} - 2 \, B a b - {\left (A - C\right )} b^{2}\right )} c^{2} - 2 \, {\left (B a^{2} + 2 \, {\left (A - C\right )} a b - B b^{2}\right )} c d - {\left ({\left (A - C\right )} a^{2} - 2 \, B a b - {\left (A - C\right )} b^{2}\right )} d^{2}\right )} {\left (f x + e\right )} + 30 \, {\left ({\left (B a^{2} + 2 \, {\left (A - C\right )} a b - B b^{2}\right )} c^{2} + 2 \, {\left ({\left (A - C\right )} a^{2} - 2 \, B a b - {\left (A - C\right )} b^{2}\right )} c d - {\left (B a^{2} + 2 \, {\left (A - C\right )} a b - B b^{2}\right )} d^{2}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right ) + 60 \, {\left ({\left (C a^{2} + 2 \, B a b + {\left (A - C\right )} b^{2}\right )} c^{2} + 2 \, {\left (B a^{2} + 2 \, {\left (A - C\right )} a b - B b^{2}\right )} c d + {\left ({\left (A - C\right )} a^{2} - 2 \, B a b - {\left (A - C\right )} b^{2}\right )} d^{2}\right )} \tan \left (f x + e\right )}{60 \, f} \]

[In]

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

[Out]

1/60*(12*C*b^2*d^2*tan(f*x + e)^5 + 15*(2*C*b^2*c*d + (2*C*a*b + B*b^2)*d^2)*tan(f*x + e)^4 + 20*(C*b^2*c^2 +
2*(2*C*a*b + B*b^2)*c*d + (C*a^2 + 2*B*a*b + (A - C)*b^2)*d^2)*tan(f*x + e)^3 + 30*((2*C*a*b + B*b^2)*c^2 + 2*
(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)*tan(f*x + e)^2 + 60*(((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) + 30*((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) + 60*((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)*tan(f*x + e))/f

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 11957 vs. \(2 (436) = 872\).

Time = 10.52 (sec) , antiderivative size = 11957, normalized size of antiderivative = 26.99 \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/60*(60*A*a^2*c^2*f*x*tan(f*x)^5*tan(e)^5 - 60*C*a^2*c^2*f*x*tan(f*x)^5*tan(e)^5 - 120*B*a*b*c^2*f*x*tan(f*x)
^5*tan(e)^5 - 60*A*b^2*c^2*f*x*tan(f*x)^5*tan(e)^5 + 60*C*b^2*c^2*f*x*tan(f*x)^5*tan(e)^5 - 120*B*a^2*c*d*f*x*
tan(f*x)^5*tan(e)^5 - 240*A*a*b*c*d*f*x*tan(f*x)^5*tan(e)^5 + 240*C*a*b*c*d*f*x*tan(f*x)^5*tan(e)^5 + 120*B*b^
2*c*d*f*x*tan(f*x)^5*tan(e)^5 - 60*A*a^2*d^2*f*x*tan(f*x)^5*tan(e)^5 + 60*C*a^2*d^2*f*x*tan(f*x)^5*tan(e)^5 +
120*B*a*b*d^2*f*x*tan(f*x)^5*tan(e)^5 + 60*A*b^2*d^2*f*x*tan(f*x)^5*tan(e)^5 - 60*C*b^2*d^2*f*x*tan(f*x)^5*tan
(e)^5 - 30*B*a^2*c^2*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + t
an(e)^2 + 1))*tan(f*x)^5*tan(e)^5 - 60*A*a*b*c^2*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)
^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(f*x)^5*tan(e)^5 + 60*C*a*b*c^2*log(4*(tan(f*x)^2*tan(e)^2 - 2*ta
n(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(f*x)^5*tan(e)^5 + 30*B*b^2*c^2*log(4
*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(f*x)^5*t
an(e)^5 - 60*A*a^2*c*d*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 +
tan(e)^2 + 1))*tan(f*x)^5*tan(e)^5 + 60*C*a^2*c*d*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*
x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(f*x)^5*tan(e)^5 + 120*B*a*b*c*d*log(4*(tan(f*x)^2*tan(e)^2 - 2
*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(f*x)^5*tan(e)^5 + 60*A*b^2*c*d*lo
g(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(f*x)^
5*tan(e)^5 - 60*C*b^2*c*d*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^
2 + tan(e)^2 + 1))*tan(f*x)^5*tan(e)^5 + 30*B*a^2*d^2*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan
(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(f*x)^5*tan(e)^5 + 60*A*a*b*d^2*log(4*(tan(f*x)^2*tan(e)^2 -
2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(f*x)^5*tan(e)^5 - 60*C*a*b*d^2*
log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(f*x
)^5*tan(e)^5 - 30*B*b^2*d^2*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x
)^2 + tan(e)^2 + 1))*tan(f*x)^5*tan(e)^5 - 300*A*a^2*c^2*f*x*tan(f*x)^4*tan(e)^4 + 300*C*a^2*c^2*f*x*tan(f*x)^
4*tan(e)^4 + 600*B*a*b*c^2*f*x*tan(f*x)^4*tan(e)^4 + 300*A*b^2*c^2*f*x*tan(f*x)^4*tan(e)^4 - 300*C*b^2*c^2*f*x
*tan(f*x)^4*tan(e)^4 + 600*B*a^2*c*d*f*x*tan(f*x)^4*tan(e)^4 + 1200*A*a*b*c*d*f*x*tan(f*x)^4*tan(e)^4 - 1200*C
*a*b*c*d*f*x*tan(f*x)^4*tan(e)^4 - 600*B*b^2*c*d*f*x*tan(f*x)^4*tan(e)^4 + 300*A*a^2*d^2*f*x*tan(f*x)^4*tan(e)
^4 - 300*C*a^2*d^2*f*x*tan(f*x)^4*tan(e)^4 - 600*B*a*b*d^2*f*x*tan(f*x)^4*tan(e)^4 - 300*A*b^2*d^2*f*x*tan(f*x
)^4*tan(e)^4 + 300*C*b^2*d^2*f*x*tan(f*x)^4*tan(e)^4 + 60*C*a*b*c^2*tan(f*x)^5*tan(e)^5 + 30*B*b^2*c^2*tan(f*x
)^5*tan(e)^5 + 60*C*a^2*c*d*tan(f*x)^5*tan(e)^5 + 120*B*a*b*c*d*tan(f*x)^5*tan(e)^5 + 60*A*b^2*c*d*tan(f*x)^5*
tan(e)^5 - 90*C*b^2*c*d*tan(f*x)^5*tan(e)^5 + 30*B*a^2*d^2*tan(f*x)^5*tan(e)^5 + 60*A*a*b*d^2*tan(f*x)^5*tan(e
)^5 - 90*C*a*b*d^2*tan(f*x)^5*tan(e)^5 - 45*B*b^2*d^2*tan(f*x)^5*tan(e)^5 + 150*B*a^2*c^2*log(4*(tan(f*x)^2*ta
n(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(f*x)^4*tan(e)^4 + 300*A
*a*b*c^2*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1)
)*tan(f*x)^4*tan(e)^4 - 300*C*a*b*c^2*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2
+ tan(f*x)^2 + tan(e)^2 + 1))*tan(f*x)^4*tan(e)^4 - 150*B*b^2*c^2*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan
(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(f*x)^4*tan(e)^4 + 300*A*a^2*c*d*log(4*(tan(f*x
)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(f*x)^4*tan(e)^4 -
300*C*a^2*c*d*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^
2 + 1))*tan(f*x)^4*tan(e)^4 - 600*B*a*b*c*d*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*ta
n(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(f*x)^4*tan(e)^4 - 300*A*b^2*c*d*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*
x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(f*x)^4*tan(e)^4 + 300*C*b^2*c*d*log(4*(t
an(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(f*x)^4*tan(
e)^4 - 150*B*a^2*d^2*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + t
an(e)^2 + 1))*tan(f*x)^4*tan(e)^4 - 300*A*a*b*d^2*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x
)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(f*x)^4*tan(e)^4 + 300*C*a*b*d^2*log(4*(tan(f*x)^2*tan(e)^2 - 2*
tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(f*x)^4*tan(e)^4 + 150*B*b^2*d^2*lo
g(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(f*x)^
4*tan(e)^4 - 60*C*a^2*c^2*tan(f*x)^5*tan(e)^4 - 120*B*a*b*c^2*tan(f*x)^5*tan(e)^4 - 60*A*b^2*c^2*tan(f*x)^5*ta
n(e)^4 + 60*C*b^2*c^2*tan(f*x)^5*tan(e)^4 - 120*B*a^2*c*d*tan(f*x)^5*tan(e)^4 - 240*A*a*b*c*d*tan(f*x)^5*tan(e
)^4 + 240*C*a*b*c*d*tan(f*x)^5*tan(e)^4 + 120*B*b^2*c*d*tan(f*x)^5*tan(e)^4 - 60*A*a^2*d^2*tan(f*x)^5*tan(e)^4
+ 60*C*a^2*d^2*tan(f*x)^5*tan(e)^4 + 120*B*a*b*d^2*tan(f*x)^5*tan(e)^4 + 60*A*b^2*d^2*tan(f*x)^5*tan(e)^4 - 6
0*C*b^2*d^2*tan(f*x)^5*tan(e)^4 - 60*C*a^2*c^2*tan(f*x)^4*tan(e)^5 - 120*B*a*b*c^2*tan(f*x)^4*tan(e)^5 - 60*A*
b^2*c^2*tan(f*x)^4*tan(e)^5 + 60*C*b^2*c^2*tan(f*x)^4*tan(e)^5 - 120*B*a^2*c*d*tan(f*x)^4*tan(e)^5 - 240*A*a*b
*c*d*tan(f*x)^4*tan(e)^5 + 240*C*a*b*c*d*tan(f*x)^4*tan(e)^5 + 120*B*b^2*c*d*tan(f*x)^4*tan(e)^5 - 60*A*a^2*d^
2*tan(f*x)^4*tan(e)^5 + 60*C*a^2*d^2*tan(f*x)^4*tan(e)^5 + 120*B*a*b*d^2*tan(f*x)^4*tan(e)^5 + 60*A*b^2*d^2*ta
n(f*x)^4*tan(e)^5 - 60*C*b^2*d^2*tan(f*x)^4*tan(e)^5 + 600*A*a^2*c^2*f*x*tan(f*x)^3*tan(e)^3 - 600*C*a^2*c^2*f
*x*tan(f*x)^3*tan(e)^3 - 1200*B*a*b*c^2*f*x*tan(f*x)^3*tan(e)^3 - 600*A*b^2*c^2*f*x*tan(f*x)^3*tan(e)^3 + 600*
C*b^2*c^2*f*x*tan(f*x)^3*tan(e)^3 - 1200*B*a^2*c*d*f*x*tan(f*x)^3*tan(e)^3 - 2400*A*a*b*c*d*f*x*tan(f*x)^3*tan
(e)^3 + 2400*C*a*b*c*d*f*x*tan(f*x)^3*tan(e)^3 + 1200*B*b^2*c*d*f*x*tan(f*x)^3*tan(e)^3 - 600*A*a^2*d^2*f*x*ta
n(f*x)^3*tan(e)^3 + 600*C*a^2*d^2*f*x*tan(f*x)^3*tan(e)^3 + 1200*B*a*b*d^2*f*x*tan(f*x)^3*tan(e)^3 + 600*A*b^2
*d^2*f*x*tan(f*x)^3*tan(e)^3 - 600*C*b^2*d^2*f*x*tan(f*x)^3*tan(e)^3 + 60*C*a*b*c^2*tan(f*x)^5*tan(e)^3 + 30*B
*b^2*c^2*tan(f*x)^5*tan(e)^3 + 60*C*a^2*c*d*tan(f*x)^5*tan(e)^3 + 120*B*a*b*c*d*tan(f*x)^5*tan(e)^3 + 60*A*b^2
*c*d*tan(f*x)^5*tan(e)^3 - 60*C*b^2*c*d*tan(f*x)^5*tan(e)^3 + 30*B*a^2*d^2*tan(f*x)^5*tan(e)^3 + 60*A*a*b*d^2*
tan(f*x)^5*tan(e)^3 - 60*C*a*b*d^2*tan(f*x)^5*tan(e)^3 - 30*B*b^2*d^2*tan(f*x)^5*tan(e)^3 - 180*C*a*b*c^2*tan(
f*x)^4*tan(e)^4 - 90*B*b^2*c^2*tan(f*x)^4*tan(e)^4 - 180*C*a^2*c*d*tan(f*x)^4*tan(e)^4 - 360*B*a*b*c*d*tan(f*x
)^4*tan(e)^4 - 180*A*b^2*c*d*tan(f*x)^4*tan(e)^4 + 330*C*b^2*c*d*tan(f*x)^4*tan(e)^4 - 90*B*a^2*d^2*tan(f*x)^4
*tan(e)^4 - 180*A*a*b*d^2*tan(f*x)^4*tan(e)^4 + 330*C*a*b*d^2*tan(f*x)^4*tan(e)^4 + 165*B*b^2*d^2*tan(f*x)^4*t
an(e)^4 + 60*C*a*b*c^2*tan(f*x)^3*tan(e)^5 + 30*B*b^2*c^2*tan(f*x)^3*tan(e)^5 + 60*C*a^2*c*d*tan(f*x)^3*tan(e)
^5 + 120*B*a*b*c*d*tan(f*x)^3*tan(e)^5 + 60*A*b^2*c*d*tan(f*x)^3*tan(e)^5 - 60*C*b^2*c*d*tan(f*x)^3*tan(e)^5 +
30*B*a^2*d^2*tan(f*x)^3*tan(e)^5 + 60*A*a*b*d^2*tan(f*x)^3*tan(e)^5 - 60*C*a*b*d^2*tan(f*x)^3*tan(e)^5 - 30*B
*b^2*d^2*tan(f*x)^3*tan(e)^5 - 20*C*b^2*c^2*tan(f*x)^5*tan(e)^2 - 80*C*a*b*c*d*tan(f*x)^5*tan(e)^2 - 40*B*b^2*
c*d*tan(f*x)^5*tan(e)^2 - 20*C*a^2*d^2*tan(f*x)^5*tan(e)^2 - 40*B*a*b*d^2*tan(f*x)^5*tan(e)^2 - 20*A*b^2*d^2*t
an(f*x)^5*tan(e)^2 + 20*C*b^2*d^2*tan(f*x)^5*tan(e)^2 - 300*B*a^2*c^2*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*
tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(f*x)^3*tan(e)^3 - 600*A*a*b*c^2*log(4*(tan(
f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(f*x)^3*tan(e)^
3 + 600*C*a*b*c^2*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(
e)^2 + 1))*tan(f*x)^3*tan(e)^3 + 300*B*b^2*c^2*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2
*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(f*x)^3*tan(e)^3 - 600*A*a^2*c*d*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan
(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(f*x)^3*tan(e)^3 + 600*C*a^2*c*d*log(4
*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(f*x)^3*t
an(e)^3 + 1200*B*a*b*c*d*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2
+ tan(e)^2 + 1))*tan(f*x)^3*tan(e)^3 + 600*A*b^2*c*d*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan
(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(f*x)^3*tan(e)^3 - 600*C*b^2*c*d*log(4*(tan(f*x)^2*tan(e)^2
- 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(f*x)^3*tan(e)^3 + 300*B*a^2*d^
2*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(f
*x)^3*tan(e)^3 + 600*A*a*b*d^2*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(
f*x)^2 + tan(e)^2 + 1))*tan(f*x)^3*tan(e)^3 - 600*C*a*b*d^2*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1
)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(f*x)^3*tan(e)^3 - 300*B*b^2*d^2*log(4*(tan(f*x)^2*tan
(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(f*x)^3*tan(e)^3 + 240*C*
a^2*c^2*tan(f*x)^4*tan(e)^3 + 480*B*a*b*c^2*tan(f*x)^4*tan(e)^3 + 240*A*b^2*c^2*tan(f*x)^4*tan(e)^3 - 300*C*b^
2*c^2*tan(f*x)^4*tan(e)^3 + 480*B*a^2*c*d*tan(f*x)^4*tan(e)^3 + 960*A*a*b*c*d*tan(f*x)^4*tan(e)^3 - 1200*C*a*b
*c*d*tan(f*x)^4*tan(e)^3 - 600*B*b^2*c*d*tan(f*x)^4*tan(e)^3 + 240*A*a^2*d^2*tan(f*x)^4*tan(e)^3 - 300*C*a^2*d
^2*tan(f*x)^4*tan(e)^3 - 600*B*a*b*d^2*tan(f*x)^4*tan(e)^3 - 300*A*b^2*d^2*tan(f*x)^4*tan(e)^3 + 300*C*b^2*d^2
*tan(f*x)^4*tan(e)^3 + 240*C*a^2*c^2*tan(f*x)^3*tan(e)^4 + 480*B*a*b*c^2*tan(f*x)^3*tan(e)^4 + 240*A*b^2*c^2*t
an(f*x)^3*tan(e)^4 - 300*C*b^2*c^2*tan(f*x)^3*tan(e)^4 + 480*B*a^2*c*d*tan(f*x)^3*tan(e)^4 + 960*A*a*b*c*d*tan
(f*x)^3*tan(e)^4 - 1200*C*a*b*c*d*tan(f*x)^3*tan(e)^4 - 600*B*b^2*c*d*tan(f*x)^3*tan(e)^4 + 240*A*a^2*d^2*tan(
f*x)^3*tan(e)^4 - 300*C*a^2*d^2*tan(f*x)^3*tan(e)^4 - 600*B*a*b*d^2*tan(f*x)^3*tan(e)^4 - 300*A*b^2*d^2*tan(f*
x)^3*tan(e)^4 + 300*C*b^2*d^2*tan(f*x)^3*tan(e)^4 - 20*C*b^2*c^2*tan(f*x)^2*tan(e)^5 - 80*C*a*b*c*d*tan(f*x)^2
*tan(e)^5 - 40*B*b^2*c*d*tan(f*x)^2*tan(e)^5 - 20*C*a^2*d^2*tan(f*x)^2*tan(e)^5 - 40*B*a*b*d^2*tan(f*x)^2*tan(
e)^5 - 20*A*b^2*d^2*tan(f*x)^2*tan(e)^5 + 20*C*b^2*d^2*tan(f*x)^2*tan(e)^5 + 30*C*b^2*c*d*tan(f*x)^5*tan(e) +
30*C*a*b*d^2*tan(f*x)^5*tan(e) + 15*B*b^2*d^2*tan(f*x)^5*tan(e) - 600*A*a^2*c^2*f*x*tan(f*x)^2*tan(e)^2 + 600*
C*a^2*c^2*f*x*tan(f*x)^2*tan(e)^2 + 1200*B*a*b*c^2*f*x*tan(f*x)^2*tan(e)^2 + 600*A*b^2*c^2*f*x*tan(f*x)^2*tan(
e)^2 - 600*C*b^2*c^2*f*x*tan(f*x)^2*tan(e)^2 + 1200*B*a^2*c*d*f*x*tan(f*x)^2*tan(e)^2 + 2400*A*a*b*c*d*f*x*tan
(f*x)^2*tan(e)^2 - 2400*C*a*b*c*d*f*x*tan(f*x)^2*tan(e)^2 - 1200*B*b^2*c*d*f*x*tan(f*x)^2*tan(e)^2 + 600*A*a^2
*d^2*f*x*tan(f*x)^2*tan(e)^2 - 600*C*a^2*d^2*f*x*tan(f*x)^2*tan(e)^2 - 1200*B*a*b*d^2*f*x*tan(f*x)^2*tan(e)^2
- 600*A*b^2*d^2*f*x*tan(f*x)^2*tan(e)^2 + 600*C*b^2*d^2*f*x*tan(f*x)^2*tan(e)^2 - 180*C*a*b*c^2*tan(f*x)^4*tan
(e)^2 - 90*B*b^2*c^2*tan(f*x)^4*tan(e)^2 - 180*C*a^2*c*d*tan(f*x)^4*tan(e)^2 - 360*B*a*b*c*d*tan(f*x)^4*tan(e)
^2 - 180*A*b^2*c*d*tan(f*x)^4*tan(e)^2 + 300*C*b^2*c*d*tan(f*x)^4*tan(e)^2 - 90*B*a^2*d^2*tan(f*x)^4*tan(e)^2
- 180*A*a*b*d^2*tan(f*x)^4*tan(e)^2 + 300*C*a*b*d^2*tan(f*x)^4*tan(e)^2 + 150*B*b^2*d^2*tan(f*x)^4*tan(e)^2 +
240*C*a*b*c^2*tan(f*x)^3*tan(e)^3 + 120*B*b^2*c^2*tan(f*x)^3*tan(e)^3 + 240*C*a^2*c*d*tan(f*x)^3*tan(e)^3 + 48
0*B*a*b*c*d*tan(f*x)^3*tan(e)^3 + 240*A*b^2*c*d*tan(f*x)^3*tan(e)^3 - 360*C*b^2*c*d*tan(f*x)^3*tan(e)^3 + 120*
B*a^2*d^2*tan(f*x)^3*tan(e)^3 + 240*A*a*b*d^2*tan(f*x)^3*tan(e)^3 - 360*C*a*b*d^2*tan(f*x)^3*tan(e)^3 - 180*B*
b^2*d^2*tan(f*x)^3*tan(e)^3 - 180*C*a*b*c^2*tan(f*x)^2*tan(e)^4 - 90*B*b^2*c^2*tan(f*x)^2*tan(e)^4 - 180*C*a^2
*c*d*tan(f*x)^2*tan(e)^4 - 360*B*a*b*c*d*tan(f*x)^2*tan(e)^4 - 180*A*b^2*c*d*tan(f*x)^2*tan(e)^4 + 300*C*b^2*c
*d*tan(f*x)^2*tan(e)^4 - 90*B*a^2*d^2*tan(f*x)^2*tan(e)^4 - 180*A*a*b*d^2*tan(f*x)^2*tan(e)^4 + 300*C*a*b*d^2*
tan(f*x)^2*tan(e)^4 + 150*B*b^2*d^2*tan(f*x)^2*tan(e)^4 + 30*C*b^2*c*d*tan(f*x)*tan(e)^5 + 30*C*a*b*d^2*tan(f*
x)*tan(e)^5 + 15*B*b^2*d^2*tan(f*x)*tan(e)^5 - 12*C*b^2*d^2*tan(f*x)^5 + 40*C*b^2*c^2*tan(f*x)^4*tan(e) + 160*
C*a*b*c*d*tan(f*x)^4*tan(e) + 80*B*b^2*c*d*tan(f*x)^4*tan(e) + 40*C*a^2*d^2*tan(f*x)^4*tan(e) + 80*B*a*b*d^2*t
an(f*x)^4*tan(e) + 40*A*b^2*d^2*tan(f*x)^4*tan(e) - 100*C*b^2*d^2*tan(f*x)^4*tan(e) + 300*B*a^2*c^2*log(4*(tan
(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(f*x)^2*tan(e)
^2 + 600*A*a*b*c^2*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan
(e)^2 + 1))*tan(f*x)^2*tan(e)^2 - 600*C*a*b*c^2*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^
2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(f*x)^2*tan(e)^2 - 300*B*b^2*c^2*log(4*(tan(f*x)^2*tan(e)^2 - 2*ta
n(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(f*x)^2*tan(e)^2 + 600*A*a^2*c*d*log(
4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(f*x)^2*
tan(e)^2 - 600*C*a^2*c*d*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2
+ tan(e)^2 + 1))*tan(f*x)^2*tan(e)^2 - 1200*B*a*b*c*d*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(ta
n(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(f*x)^2*tan(e)^2 - 600*A*b^2*c*d*log(4*(tan(f*x)^2*tan(e)^2
- 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(f*x)^2*tan(e)^2 + 600*C*b^2*c
*d*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(
f*x)^2*tan(e)^2 - 300*B*a^2*d^2*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan
(f*x)^2 + tan(e)^2 + 1))*tan(f*x)^2*tan(e)^2 - 600*A*a*b*d^2*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) +
1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(f*x)^2*tan(e)^2 + 600*C*a*b*d^2*log(4*(tan(f*x)^2*ta
n(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(f*x)^2*tan(e)^2 + 300*B
*b^2*d^2*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1)
)*tan(f*x)^2*tan(e)^2 - 360*C*a^2*c^2*tan(f*x)^3*tan(e)^2 - 720*B*a*b*c^2*tan(f*x)^3*tan(e)^2 - 360*A*b^2*c^2*
tan(f*x)^3*tan(e)^2 + 480*C*b^2*c^2*tan(f*x)^3*tan(e)^2 - 720*B*a^2*c*d*tan(f*x)^3*tan(e)^2 - 1440*A*a*b*c*d*t
an(f*x)^3*tan(e)^2 + 1920*C*a*b*c*d*tan(f*x)^3*tan(e)^2 + 960*B*b^2*c*d*tan(f*x)^3*tan(e)^2 - 360*A*a^2*d^2*ta
n(f*x)^3*tan(e)^2 + 480*C*a^2*d^2*tan(f*x)^3*tan(e)^2 + 960*B*a*b*d^2*tan(f*x)^3*tan(e)^2 + 480*A*b^2*d^2*tan(
f*x)^3*tan(e)^2 - 600*C*b^2*d^2*tan(f*x)^3*tan(e)^2 - 360*C*a^2*c^2*tan(f*x)^2*tan(e)^3 - 720*B*a*b*c^2*tan(f*
x)^2*tan(e)^3 - 360*A*b^2*c^2*tan(f*x)^2*tan(e)^3 + 480*C*b^2*c^2*tan(f*x)^2*tan(e)^3 - 720*B*a^2*c*d*tan(f*x)
^2*tan(e)^3 - 1440*A*a*b*c*d*tan(f*x)^2*tan(e)^3 + 1920*C*a*b*c*d*tan(f*x)^2*tan(e)^3 + 960*B*b^2*c*d*tan(f*x)
^2*tan(e)^3 - 360*A*a^2*d^2*tan(f*x)^2*tan(e)^3 + 480*C*a^2*d^2*tan(f*x)^2*tan(e)^3 + 960*B*a*b*d^2*tan(f*x)^2
*tan(e)^3 + 480*A*b^2*d^2*tan(f*x)^2*tan(e)^3 - 600*C*b^2*d^2*tan(f*x)^2*tan(e)^3 + 40*C*b^2*c^2*tan(f*x)*tan(
e)^4 + 160*C*a*b*c*d*tan(f*x)*tan(e)^4 + 80*B*b^2*c*d*tan(f*x)*tan(e)^4 + 40*C*a^2*d^2*tan(f*x)*tan(e)^4 + 80*
B*a*b*d^2*tan(f*x)*tan(e)^4 + 40*A*b^2*d^2*tan(f*x)*tan(e)^4 - 100*C*b^2*d^2*tan(f*x)*tan(e)^4 - 12*C*b^2*d^2*
tan(e)^5 - 30*C*b^2*c*d*tan(f*x)^4 - 30*C*a*b*d^2*tan(f*x)^4 - 15*B*b^2*d^2*tan(f*x)^4 + 300*A*a^2*c^2*f*x*tan
(f*x)*tan(e) - 300*C*a^2*c^2*f*x*tan(f*x)*tan(e) - 600*B*a*b*c^2*f*x*tan(f*x)*tan(e) - 300*A*b^2*c^2*f*x*tan(f
*x)*tan(e) + 300*C*b^2*c^2*f*x*tan(f*x)*tan(e) - 600*B*a^2*c*d*f*x*tan(f*x)*tan(e) - 1200*A*a*b*c*d*f*x*tan(f*
x)*tan(e) + 1200*C*a*b*c*d*f*x*tan(f*x)*tan(e) + 600*B*b^2*c*d*f*x*tan(f*x)*tan(e) - 300*A*a^2*d^2*f*x*tan(f*x
)*tan(e) + 300*C*a^2*d^2*f*x*tan(f*x)*tan(e) + 600*B*a*b*d^2*f*x*tan(f*x)*tan(e) + 300*A*b^2*d^2*f*x*tan(f*x)*
tan(e) - 300*C*b^2*d^2*f*x*tan(f*x)*tan(e) + 180*C*a*b*c^2*tan(f*x)^3*tan(e) + 90*B*b^2*c^2*tan(f*x)^3*tan(e)
+ 180*C*a^2*c*d*tan(f*x)^3*tan(e) + 360*B*a*b*c*d*tan(f*x)^3*tan(e) + 180*A*b^2*c*d*tan(f*x)^3*tan(e) - 300*C*
b^2*c*d*tan(f*x)^3*tan(e) + 90*B*a^2*d^2*tan(f*x)^3*tan(e) + 180*A*a*b*d^2*tan(f*x)^3*tan(e) - 300*C*a*b*d^2*t
an(f*x)^3*tan(e) - 150*B*b^2*d^2*tan(f*x)^3*tan(e) - 240*C*a*b*c^2*tan(f*x)^2*tan(e)^2 - 120*B*b^2*c^2*tan(f*x
)^2*tan(e)^2 - 240*C*a^2*c*d*tan(f*x)^2*tan(e)^2 - 480*B*a*b*c*d*tan(f*x)^2*tan(e)^2 - 240*A*b^2*c*d*tan(f*x)^
2*tan(e)^2 + 360*C*b^2*c*d*tan(f*x)^2*tan(e)^2 - 120*B*a^2*d^2*tan(f*x)^2*tan(e)^2 - 240*A*a*b*d^2*tan(f*x)^2*
tan(e)^2 + 360*C*a*b*d^2*tan(f*x)^2*tan(e)^2 + 180*B*b^2*d^2*tan(f*x)^2*tan(e)^2 + 180*C*a*b*c^2*tan(f*x)*tan(
e)^3 + 90*B*b^2*c^2*tan(f*x)*tan(e)^3 + 180*C*a^2*c*d*tan(f*x)*tan(e)^3 + 360*B*a*b*c*d*tan(f*x)*tan(e)^3 + 18
0*A*b^2*c*d*tan(f*x)*tan(e)^3 - 300*C*b^2*c*d*tan(f*x)*tan(e)^3 + 90*B*a^2*d^2*tan(f*x)*tan(e)^3 + 180*A*a*b*d
^2*tan(f*x)*tan(e)^3 - 300*C*a*b*d^2*tan(f*x)*tan(e)^3 - 150*B*b^2*d^2*tan(f*x)*tan(e)^3 - 30*C*b^2*c*d*tan(e)
^4 - 30*C*a*b*d^2*tan(e)^4 - 15*B*b^2*d^2*tan(e)^4 - 20*C*b^2*c^2*tan(f*x)^3 - 80*C*a*b*c*d*tan(f*x)^3 - 40*B*
b^2*c*d*tan(f*x)^3 - 20*C*a^2*d^2*tan(f*x)^3 - 40*B*a*b*d^2*tan(f*x)^3 - 20*A*b^2*d^2*tan(f*x)^3 + 20*C*b^2*d^
2*tan(f*x)^3 - 150*B*a^2*c^2*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*
x)^2 + tan(e)^2 + 1))*tan(f*x)*tan(e) - 300*A*a*b*c^2*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan
(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(f*x)*tan(e) + 300*C*a*b*c^2*log(4*(tan(f*x)^2*tan(e)^2 - 2*
tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(f*x)*tan(e) + 150*B*b^2*c^2*log(4*
(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(f*x)*tan(
e) - 300*A*a^2*c*d*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan
(e)^2 + 1))*tan(f*x)*tan(e) + 300*C*a^2*c*d*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*ta
n(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(f*x)*tan(e) + 600*B*a*b*c*d*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*t
an(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(f*x)*tan(e) + 300*A*b^2*c*d*log(4*(tan(f*x)^
2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(f*x)*tan(e) - 300*C
*b^2*c*d*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1)
)*tan(f*x)*tan(e) + 150*B*a^2*d^2*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + t
an(f*x)^2 + tan(e)^2 + 1))*tan(f*x)*tan(e) + 300*A*a*b*d^2*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)
/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(f*x)*tan(e) - 300*C*a*b*d^2*log(4*(tan(f*x)^2*tan(e)^2
- 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(f*x)*tan(e) - 150*B*b^2*d^2*l
og(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1))*tan(f*x)
*tan(e) + 240*C*a^2*c^2*tan(f*x)^2*tan(e) + 480*B*a*b*c^2*tan(f*x)^2*tan(e) + 240*A*b^2*c^2*tan(f*x)^2*tan(e)
- 300*C*b^2*c^2*tan(f*x)^2*tan(e) + 480*B*a^2*c*d*tan(f*x)^2*tan(e) + 960*A*a*b*c*d*tan(f*x)^2*tan(e) - 1200*C
*a*b*c*d*tan(f*x)^2*tan(e) - 600*B*b^2*c*d*tan(f*x)^2*tan(e) + 240*A*a^2*d^2*tan(f*x)^2*tan(e) - 300*C*a^2*d^2
*tan(f*x)^2*tan(e) - 600*B*a*b*d^2*tan(f*x)^2*tan(e) - 300*A*b^2*d^2*tan(f*x)^2*tan(e) + 300*C*b^2*d^2*tan(f*x
)^2*tan(e) + 240*C*a^2*c^2*tan(f*x)*tan(e)^2 + 480*B*a*b*c^2*tan(f*x)*tan(e)^2 + 240*A*b^2*c^2*tan(f*x)*tan(e)
^2 - 300*C*b^2*c^2*tan(f*x)*tan(e)^2 + 480*B*a^2*c*d*tan(f*x)*tan(e)^2 + 960*A*a*b*c*d*tan(f*x)*tan(e)^2 - 120
0*C*a*b*c*d*tan(f*x)*tan(e)^2 - 600*B*b^2*c*d*tan(f*x)*tan(e)^2 + 240*A*a^2*d^2*tan(f*x)*tan(e)^2 - 300*C*a^2*
d^2*tan(f*x)*tan(e)^2 - 600*B*a*b*d^2*tan(f*x)*tan(e)^2 - 300*A*b^2*d^2*tan(f*x)*tan(e)^2 + 300*C*b^2*d^2*tan(
f*x)*tan(e)^2 - 20*C*b^2*c^2*tan(e)^3 - 80*C*a*b*c*d*tan(e)^3 - 40*B*b^2*c*d*tan(e)^3 - 20*C*a^2*d^2*tan(e)^3
- 40*B*a*b*d^2*tan(e)^3 - 20*A*b^2*d^2*tan(e)^3 + 20*C*b^2*d^2*tan(e)^3 - 60*A*a^2*c^2*f*x + 60*C*a^2*c^2*f*x
+ 120*B*a*b*c^2*f*x + 60*A*b^2*c^2*f*x - 60*C*b^2*c^2*f*x + 120*B*a^2*c*d*f*x + 240*A*a*b*c*d*f*x - 240*C*a*b*
c*d*f*x - 120*B*b^2*c*d*f*x + 60*A*a^2*d^2*f*x - 60*C*a^2*d^2*f*x - 120*B*a*b*d^2*f*x - 60*A*b^2*d^2*f*x + 60*
C*b^2*d^2*f*x - 60*C*a*b*c^2*tan(f*x)^2 - 30*B*b^2*c^2*tan(f*x)^2 - 60*C*a^2*c*d*tan(f*x)^2 - 120*B*a*b*c*d*ta
n(f*x)^2 - 60*A*b^2*c*d*tan(f*x)^2 + 60*C*b^2*c*d*tan(f*x)^2 - 30*B*a^2*d^2*tan(f*x)^2 - 60*A*a*b*d^2*tan(f*x)
^2 + 60*C*a*b*d^2*tan(f*x)^2 + 30*B*b^2*d^2*tan(f*x)^2 + 180*C*a*b*c^2*tan(f*x)*tan(e) + 90*B*b^2*c^2*tan(f*x)
*tan(e) + 180*C*a^2*c*d*tan(f*x)*tan(e) + 360*B*a*b*c*d*tan(f*x)*tan(e) + 180*A*b^2*c*d*tan(f*x)*tan(e) - 330*
C*b^2*c*d*tan(f*x)*tan(e) + 90*B*a^2*d^2*tan(f*x)*tan(e) + 180*A*a*b*d^2*tan(f*x)*tan(e) - 330*C*a*b*d^2*tan(f
*x)*tan(e) - 165*B*b^2*d^2*tan(f*x)*tan(e) - 60*C*a*b*c^2*tan(e)^2 - 30*B*b^2*c^2*tan(e)^2 - 60*C*a^2*c*d*tan(
e)^2 - 120*B*a*b*c*d*tan(e)^2 - 60*A*b^2*c*d*tan(e)^2 + 60*C*b^2*c*d*tan(e)^2 - 30*B*a^2*d^2*tan(e)^2 - 60*A*a
*b*d^2*tan(e)^2 + 60*C*a*b*d^2*tan(e)^2 + 30*B*b^2*d^2*tan(e)^2 + 30*B*a^2*c^2*log(4*(tan(f*x)^2*tan(e)^2 - 2*
tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1)) + 60*A*a*b*c^2*log(4*(tan(f*x)^2*tan(e
)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1)) - 60*C*a*b*c^2*log(4*(tan(f*x)
^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1)) - 30*B*b^2*c^2*log(4*(
tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1)) + 60*A*a^2*c*d
*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1)) - 60*C
*a^2*c*d*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1)
) - 120*B*a*b*c*d*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(
e)^2 + 1)) - 60*A*b^2*c*d*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^
2 + tan(e)^2 + 1)) + 60*C*b^2*c*d*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(e)^2 + t
an(f*x)^2 + tan(e)^2 + 1)) - 30*B*a^2*d^2*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x)^2*tan(
e)^2 + tan(f*x)^2 + tan(e)^2 + 1)) - 60*A*a*b*d^2*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(f*x
)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1)) + 60*C*a*b*d^2*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(e) + 1)/
(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1)) + 30*B*b^2*d^2*log(4*(tan(f*x)^2*tan(e)^2 - 2*tan(f*x)*tan(
e) + 1)/(tan(f*x)^2*tan(e)^2 + tan(f*x)^2 + tan(e)^2 + 1)) - 60*C*a^2*c^2*tan(f*x) - 120*B*a*b*c^2*tan(f*x) -
60*A*b^2*c^2*tan(f*x) + 60*C*b^2*c^2*tan(f*x) - 120*B*a^2*c*d*tan(f*x) - 240*A*a*b*c*d*tan(f*x) + 240*C*a*b*c*
d*tan(f*x) + 120*B*b^2*c*d*tan(f*x) - 60*A*a^2*d^2*tan(f*x) + 60*C*a^2*d^2*tan(f*x) + 120*B*a*b*d^2*tan(f*x) +
60*A*b^2*d^2*tan(f*x) - 60*C*b^2*d^2*tan(f*x) - 60*C*a^2*c^2*tan(e) - 120*B*a*b*c^2*tan(e) - 60*A*b^2*c^2*tan
(e) + 60*C*b^2*c^2*tan(e) - 120*B*a^2*c*d*tan(e) - 240*A*a*b*c*d*tan(e) + 240*C*a*b*c*d*tan(e) + 120*B*b^2*c*d
*tan(e) - 60*A*a^2*d^2*tan(e) + 60*C*a^2*d^2*tan(e) + 120*B*a*b*d^2*tan(e) + 60*A*b^2*d^2*tan(e) - 60*C*b^2*d^
2*tan(e) - 60*C*a*b*c^2 - 30*B*b^2*c^2 - 60*C*a^2*c*d - 120*B*a*b*c*d - 60*A*b^2*c*d + 90*C*b^2*c*d - 30*B*a^2
*d^2 - 60*A*a*b*d^2 + 90*C*a*b*d^2 + 45*B*b^2*d^2)/(f*tan(f*x)^5*tan(e)^5 - 5*f*tan(f*x)^4*tan(e)^4 + 10*f*tan
(f*x)^3*tan(e)^3 - 10*f*tan(f*x)^2*tan(e)^2 + 5*f*tan(f*x)*tan(e) - f)

Mupad [B] (verification not implemented)

Time = 8.36 (sec) , antiderivative size = 561, normalized size of antiderivative = 1.27 \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=x\,\left (A\,a^2\,c^2-A\,a^2\,d^2-A\,b^2\,c^2+A\,b^2\,d^2-C\,a^2\,c^2+C\,a^2\,d^2+C\,b^2\,c^2-C\,b^2\,d^2-2\,B\,a\,b\,c^2+2\,B\,a\,b\,d^2-2\,B\,a^2\,c\,d+2\,B\,b^2\,c\,d-4\,A\,a\,b\,c\,d+4\,C\,a\,b\,c\,d\right )-\frac {\ln \left ({\mathrm {tan}\left (e+f\,x\right )}^2+1\right )\,\left (\frac {B\,a^2\,d^2}{2}-\frac {B\,a^2\,c^2}{2}+\frac {B\,b^2\,c^2}{2}-\frac {B\,b^2\,d^2}{2}-A\,a\,b\,c^2+A\,a\,b\,d^2-A\,a^2\,c\,d+C\,a\,b\,c^2+A\,b^2\,c\,d-C\,a\,b\,d^2+C\,a^2\,c\,d-C\,b^2\,c\,d+2\,B\,a\,b\,c\,d\right )}{f}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (\frac {B\,a^2\,d^2}{2}+\frac {B\,b^2\,c^2}{2}-\frac {b\,d\,\left (B\,b\,d+2\,C\,a\,d+2\,C\,b\,c\right )}{2}+A\,a\,b\,d^2+C\,a\,b\,c^2+A\,b^2\,c\,d+C\,a^2\,c\,d+2\,B\,a\,b\,c\,d\right )}{f}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^3\,\left (\frac {A\,b^2\,d^2}{3}+\frac {C\,a^2\,d^2}{3}+\frac {C\,b^2\,c^2}{3}-\frac {C\,b^2\,d^2}{3}+\frac {2\,B\,a\,b\,d^2}{3}+\frac {2\,B\,b^2\,c\,d}{3}+\frac {4\,C\,a\,b\,c\,d}{3}\right )}{f}+\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (A\,a^2\,d^2+A\,b^2\,c^2-A\,b^2\,d^2+C\,a^2\,c^2-C\,a^2\,d^2-C\,b^2\,c^2+C\,b^2\,d^2+2\,B\,a\,b\,c^2-2\,B\,a\,b\,d^2+2\,B\,a^2\,c\,d-2\,B\,b^2\,c\,d+4\,A\,a\,b\,c\,d-4\,C\,a\,b\,c\,d\right )}{f}+\frac {b\,d\,{\mathrm {tan}\left (e+f\,x\right )}^4\,\left (B\,b\,d+2\,C\,a\,d+2\,C\,b\,c\right )}{4\,f}+\frac {C\,b^2\,d^2\,{\mathrm {tan}\left (e+f\,x\right )}^5}{5\,f} \]

[In]

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

[Out]

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