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

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

[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*(a^2*B-b^2*B+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]
time = 0.83, antiderivative size = 443, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3728, 3718, 3711, 3609, 3606, 3556} \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[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*} \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 {\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 (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+\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 (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+\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] Result contains complex when optimal does not.
time = 5.18, size = 352, normalized size = 0.79 \begin {gather*} \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 b d (-2 b c C+5 b B d+2 a C d) \tan (e+f x) (c+d \tan (e+f x))^3+12 C d^2 (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3+30 d \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 \left ((c+i d)^2 \log (i-\tan (e+f x))-(c-i d)^2 \log (i+\tan (e+f x))\right )-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 )}{60 d^3 f} \end {gather*}

Antiderivative was successfully verified.

[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]

((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*b*d*(-2*b*c*C + 5*b*B*d + 2*a*C*d)*Tan[e + f*x]*(c + d*Tan[e + f*x])^3 + 12*C*d^2*(a + b*Tan[e + f*x])^2*(c
+ d*Tan[e + f*x])^3 + 30*d*(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]] - (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)))/(60*d^3*f)

________________________________________________________________________________________

Maple [A]
time = 0.18, size = 770, normalized size = 1.74 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

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

________________________________________________________________________________________

Maxima [A]
time = 0.51, size = 470, normalized size = 1.06 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

Fricas [A]
time = 5.08, size = 468, normalized size = 1.06 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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] Leaf count of result is larger than twice the leaf count of optimal. 1134 vs. \(2 (396) = 792\).
time = 0.35, size = 1134, normalized size = 2.56 \begin {gather*} \begin {cases} A a^{2} c^{2} x + \frac {A a^{2} c d \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{f} - A a^{2} d^{2} x + \frac {A a^{2} d^{2} \tan {\left (e + f x \right )}}{f} + \frac {A a b c^{2} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{f} - 4 A a b c d x + \frac {4 A a b c d \tan {\left (e + f x \right )}}{f} - \frac {A a b d^{2} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{f} + \frac {A a b d^{2} \tan ^{2}{\left (e + f x \right )}}{f} - A b^{2} c^{2} x + \frac {A b^{2} c^{2} \tan {\left (e + f x \right )}}{f} - \frac {A b^{2} c d \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{f} + \frac {A b^{2} c d \tan ^{2}{\left (e + f x \right )}}{f} + A b^{2} d^{2} x + \frac {A b^{2} d^{2} \tan ^{3}{\left (e + f x \right )}}{3 f} - \frac {A b^{2} d^{2} \tan {\left (e + f x \right )}}{f} + \frac {B a^{2} c^{2} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} - 2 B a^{2} c d x + \frac {2 B a^{2} c d \tan {\left (e + f x \right )}}{f} - \frac {B a^{2} d^{2} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {B a^{2} d^{2} \tan ^{2}{\left (e + f x \right )}}{2 f} - 2 B a b c^{2} x + \frac {2 B a b c^{2} \tan {\left (e + f x \right )}}{f} - \frac {2 B a b c d \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{f} + \frac {2 B a b c d \tan ^{2}{\left (e + f x \right )}}{f} + 2 B a b d^{2} x + \frac {2 B a b d^{2} \tan ^{3}{\left (e + f x \right )}}{3 f} - \frac {2 B a b d^{2} \tan {\left (e + f x \right )}}{f} - \frac {B b^{2} c^{2} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {B b^{2} c^{2} \tan ^{2}{\left (e + f x \right )}}{2 f} + 2 B b^{2} c d x + \frac {2 B b^{2} c d \tan ^{3}{\left (e + f x \right )}}{3 f} - \frac {2 B b^{2} c d \tan {\left (e + f x \right )}}{f} + \frac {B b^{2} d^{2} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {B b^{2} d^{2} \tan ^{4}{\left (e + f x \right )}}{4 f} - \frac {B b^{2} d^{2} \tan ^{2}{\left (e + f x \right )}}{2 f} - C a^{2} c^{2} x + \frac {C a^{2} c^{2} \tan {\left (e + f x \right )}}{f} - \frac {C a^{2} c d \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{f} + \frac {C a^{2} c d \tan ^{2}{\left (e + f x \right )}}{f} + C a^{2} d^{2} x + \frac {C a^{2} d^{2} \tan ^{3}{\left (e + f x \right )}}{3 f} - \frac {C a^{2} d^{2} \tan {\left (e + f x \right )}}{f} - \frac {C a b c^{2} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{f} + \frac {C a b c^{2} \tan ^{2}{\left (e + f x \right )}}{f} + 4 C a b c d x + \frac {4 C a b c d \tan ^{3}{\left (e + f x \right )}}{3 f} - \frac {4 C a b c d \tan {\left (e + f x \right )}}{f} + \frac {C a b d^{2} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{f} + \frac {C a b d^{2} \tan ^{4}{\left (e + f x \right )}}{2 f} - \frac {C a b d^{2} \tan ^{2}{\left (e + f x \right )}}{f} + C b^{2} c^{2} x + \frac {C b^{2} c^{2} \tan ^{3}{\left (e + f x \right )}}{3 f} - \frac {C b^{2} c^{2} \tan {\left (e + f x \right )}}{f} + \frac {C b^{2} c d \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{f} + \frac {C b^{2} c d \tan ^{4}{\left (e + f x \right )}}{2 f} - \frac {C b^{2} c d \tan ^{2}{\left (e + f x \right )}}{f} - C b^{2} d^{2} x + \frac {C b^{2} d^{2} \tan ^{5}{\left (e + f x \right )}}{5 f} - \frac {C b^{2} d^{2} \tan ^{3}{\left (e + f x \right )}}{3 f} + \frac {C b^{2} d^{2} \tan {\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\x \left (a + b \tan {\left (e \right )}\right )^{2} \left (c + d \tan {\left (e \right )}\right )^{2} \left (A + B \tan {\left (e \right )} + C \tan ^{2}{\left (e \right )}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 13549 vs. \(2 (444) = 888\).
time = 9.57, size = 13549, normalized size = 30.58 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*t
an(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^5*tan(e)^5 - 60*A*a*b*c^2*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)
^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^5*tan(e)^5 + 60
*C*a*b*c^2*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*ta
n(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^5*tan(e)^5 + 30*B*b^2*c^2*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e)
+ tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^5*tan(e)^5 - 60*A*a^2*c*d
*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/
(tan(e)^2 + 1))*tan(f*x)^5*tan(e)^5 + 60*C*a^2*c*d*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)
^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^5*tan(e)^5 + 120*B*a*b*c*d*log(4*(t
an(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2
+ 1))*tan(f*x)^5*tan(e)^5 + 60*A*b^2*c*d*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)
^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^5*tan(e)^5 - 60*C*b^2*c*d*log(4*(tan(f*x)^4*
tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan
(f*x)^5*tan(e)^5 + 30*B*a^2*d^2*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f
*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^5*tan(e)^5 + 60*A*a*b*d^2*log(4*(tan(f*x)^4*tan(e)^2 -
2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^5*ta
n(e)^5 - 60*C*a*b*d^2*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*
tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^5*tan(e)^5 - 30*B*b^2*d^2*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x
)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^5*tan(e)^5 - 3
00*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*ta
n(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)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*ta
n(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^4*tan(e)^4 + 300*A*a*b*c^2*log(4*(tan(f*
x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1)
)*tan(f*x)^4*tan(e)^4 - 300*C*a*b*c^2*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 +
tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^4*tan(e)^4 - 150*B*b^2*c^2*log(4*(tan(f*x)^4*tan
(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*
x)^4*tan(e)^4 + 300*A*a^2*c*d*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x
)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^4*tan(e)^4 - 300*C*a^2*c*d*log(4*(tan(f*x)^4*tan(e)^2 -
2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^4*tan
(e)^4 - 600*B*a*b*c*d*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*
tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^4*tan(e)^4 - 300*A*b^2*c*d*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*
x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^4*tan(e)^4 +
300*C*b^2*c*d*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)
*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^4*tan(e)^...

________________________________________________________________________________________

Mupad [B]
time = 9.12, size = 561, normalized size = 1.27 \begin {gather*} 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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________