3.1.69 \(\int \frac {(c+d \tan (e+f x))^3 (A+B \tan (e+f x)+C \tan ^2(e+f x))}{(a+b \tan (e+f x))^3} \, dx\) [69]

Optimal. Leaf size=798 \[ -\frac {\left (3 a b^2 \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )+a^3 \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-3 a^2 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+b^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) x}{\left (a^2+b^2\right )^3}-\frac {\left (b^3 \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )+3 a^2 b \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )+a^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )-3 a b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \log (\cos (e+f x))}{\left (a^2+b^2\right )^3 f}-\frac {(b c-a d) \left (a^5 b B d^2-3 a^6 C d^2+a^4 b^2 d (B c-9 C d)+a^3 b^3 B \left (c^2+3 d^2\right )-b^6 \left (c (c C+3 B d)-A \left (c^2-3 d^2\right )\right )-a b^5 \left (8 c (A-C) d+3 B \left (c^2-2 d^2\right )\right )+a^2 b^4 \left (3 c^2 C+6 B c d-10 C d^2-A \left (3 c^2-d^2\right )\right )\right ) \log (a+b \tan (e+f x))}{b^4 \left (a^2+b^2\right )^3 f}-\frac {d^2 \left (a^3 b B d-3 a^4 C d-a b^3 (2 A c-2 c C-3 B d)+a^2 b^2 (B c-6 C d)-b^4 (B c+(2 A+C) d)\right ) \tan (e+f x)}{b^3 \left (a^2+b^2\right )^2 f}+\frac {\left (a^3 b B d-3 a^4 C d-b^4 (2 B c+3 A d)-a b^3 (4 A c-4 c C-5 B d)+a^2 b^2 (2 B c+(A-7 C) d)\right ) (c+d \tan (e+f x))^2}{2 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^3}{2 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 1.85, antiderivative size = 798, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3726, 3718, 3707, 3698, 31, 3556} \begin {gather*} -\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^3}{2 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}+\frac {\left (-3 C d a^4+b B d a^3+b^2 (2 B c+(A-7 C) d) a^2-b^3 (4 A c-4 C c-5 B d) a-b^4 (2 B c+3 A d)\right ) (c+d \tan (e+f x))^2}{2 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}-\frac {\left (\left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^3-3 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^2+3 b^2 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a+b^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) x}{\left (a^2+b^2\right )^3}-\frac {\left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^3+3 b \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^2-3 b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^3 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right )\right ) \log (\cos (e+f x))}{\left (a^2+b^2\right )^3 f}-\frac {(b c-a d) \left (-3 C d^2 a^6+b B d^2 a^5+b^2 d (B c-9 C d) a^4+b^3 B \left (c^2+3 d^2\right ) a^3+b^4 \left (3 C c^2+6 B d c-10 C d^2-A \left (3 c^2-d^2\right )\right ) a^2-b^5 \left (8 c (A-C) d+3 B \left (c^2-2 d^2\right )\right ) a-b^6 \left (c (c C+3 B d)-A \left (c^2-3 d^2\right )\right )\right ) \log (a+b \tan (e+f x))}{b^4 \left (a^2+b^2\right )^3 f}-\frac {d^2 \left (-3 C d a^4+b B d a^3+b^2 (B c-6 C d) a^2-b^3 (2 A c-2 C c-3 B d) a-b^4 (B c+(2 A+C) d)\right ) \tan (e+f x)}{b^3 \left (a^2+b^2\right )^2 f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(((3*a*b^2*(A*c^3 - c^3*C - 3*B*c^2*d - 3*A*c*d^2 + 3*c*C*d^2 + B*d^3) + a^3*(c^3*C + 3*B*c^2*d - 3*c*C*d^2 -
B*d^3 - A*(c^3 - 3*c*d^2)) - 3*a^2*b*((A - C)*d*(3*c^2 - d^2) + B*(c^3 - 3*c*d^2)) + b^3*((A - C)*d*(3*c^2 -
d^2) + B*(c^3 - 3*c*d^2)))*x)/(a^2 + b^2)^3) - ((b^3*(A*c^3 - c^3*C - 3*B*c^2*d - 3*A*c*d^2 + 3*c*C*d^2 + B*d^
3) + 3*a^2*b*(c^3*C + 3*B*c^2*d - 3*c*C*d^2 - B*d^3 - A*(c^3 - 3*c*d^2)) + a^3*((A - C)*d*(3*c^2 - d^2) + B*(c
^3 - 3*c*d^2)) - 3*a*b^2*((A - C)*d*(3*c^2 - d^2) + B*(c^3 - 3*c*d^2)))*Log[Cos[e + f*x]])/((a^2 + b^2)^3*f) -
((b*c - a*d)*(a^5*b*B*d^2 - 3*a^6*C*d^2 + a^4*b^2*d*(B*c - 9*C*d) + a^3*b^3*B*(c^2 + 3*d^2) - b^6*(c*(c*C + 3
*B*d) - A*(c^2 - 3*d^2)) - a*b^5*(8*c*(A - C)*d + 3*B*(c^2 - 2*d^2)) + a^2*b^4*(3*c^2*C + 6*B*c*d - 10*C*d^2 -
A*(3*c^2 - d^2)))*Log[a + b*Tan[e + f*x]])/(b^4*(a^2 + b^2)^3*f) - (d^2*(a^3*b*B*d - 3*a^4*C*d - a*b^3*(2*A*c
- 2*c*C - 3*B*d) + a^2*b^2*(B*c - 6*C*d) - b^4*(B*c + (2*A + C)*d))*Tan[e + f*x])/(b^3*(a^2 + b^2)^2*f) + ((a
^3*b*B*d - 3*a^4*C*d - b^4*(2*B*c + 3*A*d) - a*b^3*(4*A*c - 4*c*C - 5*B*d) + a^2*b^2*(2*B*c + (A - 7*C)*d))*(c
+ d*Tan[e + f*x])^2)/(2*b^2*(a^2 + b^2)^2*f*(a + b*Tan[e + f*x])) - ((A*b^2 - a*(b*B - a*C))*(c + d*Tan[e + f
*x])^3)/(2*b*(a^2 + b^2)*f*(a + b*Tan[e + f*x])^2)

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 3556

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

Rule 3698

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[
A/(b*f), Subst[Int[(a + x)^m, x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, A, C, m}, x] && EqQ[A, C]

Rule 3707

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

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 3726

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*d^2 + c*(c*C - B*d))*(a + b*Ta
n[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Dist[1/(d*(n + 1)*(c^2 + d^2)), I
nt[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)*Simp[A*d*(b*d*m - a*c*(n + 1)) + (c*C - B*d)*(b*c
*m + a*d*(n + 1)) - d*(n + 1)*((A - C)*(b*c - a*d) + B*(a*c + b*d))*Tan[e + f*x] - b*(d*(B*c - A*d)*(m + n + 1
) - C*(c^2*m - d^2*(n + 1)))*Tan[e + f*x]^2, 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] && GtQ[m, 0] && LtQ[n, -1]

Rubi steps

\begin {align*} \int \frac {(c+d \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^3} \, dx &=-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^3}{2 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}+\frac {\int \frac {(c+d \tan (e+f x))^2 \left ((b B-a C) (2 b c-3 a d)+A b (2 a c+3 b d)-2 b ((A-C) (b c-a d)-B (a c+b d)) \tan (e+f x)+\left (A b^2-a b B+3 a^2 C+2 b^2 C\right ) d \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^2} \, dx}{2 b \left (a^2+b^2\right )}\\ &=\frac {\left (a^3 b B d-3 a^4 C d-b^4 (2 B c+3 A d)-a b^3 (4 A c-4 c C-5 B d)+a^2 b^2 (2 B c+(A-7 C) d)\right ) (c+d \tan (e+f x))^2}{2 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^3}{2 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}+\frac {\int \frac {(c+d \tan (e+f x)) \left (b (a c+2 b d) ((b B-a C) (2 b c-3 a d)+A b (2 a c+3 b d))+(b c-2 a d) \left (a^2 b B d-3 a^3 C d-A b^2 (2 b c-a d)+2 b^3 (c C+B d)+2 a b^2 (B c-2 C d)\right )-2 b^2 ((a c+b d) ((A-C) (b c-a d)-B (a c+b d))+(b c-a d) (b B c+b (A-C) d+a (A c-c C-B d))) \tan (e+f x)-2 d \left (a^3 b B d-3 a^4 C d-a b^3 (2 A c-2 c C-3 B d)+a^2 b^2 (B c-6 C d)-b^4 (B c+(2 A+C) d)\right ) \tan ^2(e+f x)\right )}{a+b \tan (e+f x)} \, dx}{2 b^2 \left (a^2+b^2\right )^2}\\ &=-\frac {d^2 \left (a^3 b B d-3 a^4 C d-a b^3 (2 A c-2 c C-3 B d)+a^2 b^2 (B c-6 C d)-b^4 (B c+(2 A+C) d)\right ) \tan (e+f x)}{b^3 \left (a^2+b^2\right )^2 f}+\frac {\left (a^3 b B d-3 a^4 C d-b^4 (2 B c+3 A d)-a b^3 (4 A c-4 c C-5 B d)+a^2 b^2 (2 B c+(A-7 C) d)\right ) (c+d \tan (e+f x))^2}{2 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^3}{2 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {\int \frac {2 \left (3 a^5 C d^3+6 a^3 b^2 C d^3-a^4 b d^2 (3 c C+B d)-a^2 b^3 \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+9 c C d^2+3 B d^3\right )-a b^4 \left (2 B c^3+6 A c^2 d-6 c^2 C d-6 B c d^2-2 A d^3-C d^3\right )-b^5 c \left (c (c C+3 B d)-A \left (c^2-3 d^2\right )\right )\right )-2 b^3 \left (2 a b \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )+a^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )-b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x)-2 \left (a^2+b^2\right )^2 d^2 (3 b c C+b B d-3 a C d) \tan ^2(e+f x)}{a+b \tan (e+f x)} \, dx}{2 b^3 \left (a^2+b^2\right )^2}\\ &=-\frac {\left (3 a b^2 \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )+a^3 \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-3 a^2 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+b^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) x}{\left (a^2+b^2\right )^3}-\frac {d^2 \left (a^3 b B d-3 a^4 C d-a b^3 (2 A c-2 c C-3 B d)+a^2 b^2 (B c-6 C d)-b^4 (B c+(2 A+C) d)\right ) \tan (e+f x)}{b^3 \left (a^2+b^2\right )^2 f}+\frac {\left (a^3 b B d-3 a^4 C d-b^4 (2 B c+3 A d)-a b^3 (4 A c-4 c C-5 B d)+a^2 b^2 (2 B c+(A-7 C) d)\right ) (c+d \tan (e+f x))^2}{2 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^3}{2 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {\left ((b c-a d) \left (a^5 b B d^2-3 a^6 C d^2+a^4 b^2 d (B c-9 C d)+a^3 b^3 B \left (c^2+3 d^2\right )-b^6 \left (c (c C+3 B d)-A \left (c^2-3 d^2\right )\right )-a b^5 \left (8 c (A-C) d+3 B \left (c^2-2 d^2\right )\right )+a^2 b^4 \left (3 c^2 C+6 B c d-10 C d^2-A \left (3 c^2-d^2\right )\right )\right )\right ) \int \frac {1+\tan ^2(e+f x)}{a+b \tan (e+f x)} \, dx}{b^3 \left (a^2+b^2\right )^3}+\frac {\left (2 b \left (a^2+b^2\right )^2 d^2 (3 b c C+b B d-3 a C d)+2 b \left (3 a^5 C d^3+6 a^3 b^2 C d^3-a^4 b d^2 (3 c C+B d)-a^2 b^3 \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+9 c C d^2+3 B d^3\right )-a b^4 \left (2 B c^3+6 A c^2 d-6 c^2 C d-6 B c d^2-2 A d^3-C d^3\right )-b^5 c \left (c (c C+3 B d)-A \left (c^2-3 d^2\right )\right )\right )+2 a b^3 \left (2 a b \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )+a^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )-b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )\right ) \int \tan (e+f x) \, dx}{2 b^3 \left (a^2+b^2\right )^3}\\ &=-\frac {\left (3 a b^2 \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )+a^3 \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-3 a^2 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+b^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) x}{\left (a^2+b^2\right )^3}+\frac {\left (3 a^2 b \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )+b^3 \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-a^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+3 a b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \log (\cos (e+f x))}{\left (a^2+b^2\right )^3 f}-\frac {d^2 \left (a^3 b B d-3 a^4 C d-a b^3 (2 A c-2 c C-3 B d)+a^2 b^2 (B c-6 C d)-b^4 (B c+(2 A+C) d)\right ) \tan (e+f x)}{b^3 \left (a^2+b^2\right )^2 f}+\frac {\left (a^3 b B d-3 a^4 C d-b^4 (2 B c+3 A d)-a b^3 (4 A c-4 c C-5 B d)+a^2 b^2 (2 B c+(A-7 C) d)\right ) (c+d \tan (e+f x))^2}{2 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^3}{2 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {\left ((b c-a d) \left (a^5 b B d^2-3 a^6 C d^2+a^4 b^2 d (B c-9 C d)+a^3 b^3 B \left (c^2+3 d^2\right )-b^6 \left (c (c C+3 B d)-A \left (c^2-3 d^2\right )\right )-a b^5 \left (8 c (A-C) d+3 B \left (c^2-2 d^2\right )\right )+a^2 b^4 \left (3 c^2 C+6 B c d-10 C d^2-A \left (3 c^2-d^2\right )\right )\right )\right ) \text {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \tan (e+f x)\right )}{b^4 \left (a^2+b^2\right )^3 f}\\ &=-\frac {\left (3 a b^2 \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )+a^3 \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-3 a^2 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+b^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) x}{\left (a^2+b^2\right )^3}+\frac {\left (3 a^2 b \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )+b^3 \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-a^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+3 a b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \log (\cos (e+f x))}{\left (a^2+b^2\right )^3 f}-\frac {(b c-a d) \left (a^5 b B d^2-3 a^6 C d^2+a^4 b^2 d (B c-9 C d)+a^3 b^3 B \left (c^2+3 d^2\right )-b^6 \left (c (c C+3 B d)-A \left (c^2-3 d^2\right )\right )-a b^5 \left (8 c (A-C) d+3 B \left (c^2-2 d^2\right )\right )+a^2 b^4 \left (3 c^2 C+6 B c d-10 C d^2-A \left (3 c^2-d^2\right )\right )\right ) \log (a+b \tan (e+f x))}{b^4 \left (a^2+b^2\right )^3 f}-\frac {d^2 \left (a^3 b B d-3 a^4 C d-a b^3 (2 A c-2 c C-3 B d)+a^2 b^2 (B c-6 C d)-b^4 (B c+(2 A+C) d)\right ) \tan (e+f x)}{b^3 \left (a^2+b^2\right )^2 f}+\frac {\left (a^3 b B d-3 a^4 C d-b^4 (2 B c+3 A d)-a b^3 (4 A c-4 c C-5 B d)+a^2 b^2 (2 B c+(A-7 C) d)\right ) (c+d \tan (e+f x))^2}{2 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^3}{2 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}\\ \end {align*}

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Mathematica [A]
time = 13.87, size = 1451, normalized size = 1.82 \begin {gather*} \frac {\left (3 a b^2 \left (-A c^3+c^3 C+3 B c^2 d+3 A c d^2-3 c C d^2-B d^3\right )+a^3 \left (-c^3 C-3 B c^2 d+3 c C d^2+B d^3+A \left (c^3-3 c d^2\right )\right )+b^3 \left ((A-C) d \left (-3 c^2+d^2\right )-B \left (c^3-3 c d^2\right )\right )+3 a^2 b \left (-\left ((A-C) d \left (-3 c^2+d^2\right )\right )+B \left (c^3-3 c d^2\right )\right )\right ) (e+f x) (a \cos (e+f x)+b \sin (e+f x))^3 (c+d \tan (e+f x))^3}{\left (a^2+b^2\right )^3 f (c \cos (e+f x)+d \sin (e+f x))^3 (a+b \tan (e+f x))^3}-\frac {d^2 (3 b c C+b B d-3 a C d) \log \left (1-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) (a \cos (e+f x)+b \sin (e+f x))^3 (c+d \tan (e+f x))^3}{b^4 f (c \cos (e+f x)+d \sin (e+f x))^3 (a+b \tan (e+f x))^3}+\frac {\left (3 a^2 b \left (-A c^3+c^3 C+3 B c^2 d+3 A c d^2-3 c C d^2-B d^3\right )+b^3 \left (-c^3 C-3 B c^2 d+3 c C d^2+B d^3+A \left (c^3-3 c d^2\right )\right )+a^3 \left (-\left ((A-C) d \left (-3 c^2+d^2\right )\right )+B \left (c^3-3 c d^2\right )\right )-3 a b^2 \left (-\left ((A-C) d \left (-3 c^2+d^2\right )\right )+B \left (c^3-3 c d^2\right )\right )\right ) \log \left (1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) (a \cos (e+f x)+b \sin (e+f x))^3 (c+d \tan (e+f x))^3}{\left (a^2+b^2\right )^3 f (c \cos (e+f x)+d \sin (e+f x))^3 (a+b \tan (e+f x))^3}-\frac {(b c-a d) \left (a^5 b B d^2-3 a^6 C d^2+a^4 b^2 d (B c-9 C d)+a^3 b^3 B \left (c^2+3 d^2\right )+b^6 \left (-c (c C+3 B d)+A \left (c^2-3 d^2\right )\right )+a b^5 \left (8 c (-A+C) d-3 B \left (c^2-2 d^2\right )\right )+a^2 b^4 \left (3 c^2 C+6 B c d-10 C d^2+A \left (-3 c^2+d^2\right )\right )\right ) \log \left (-2 b \tan \left (\frac {1}{2} (e+f x)\right )+a \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) (a \cos (e+f x)+b \sin (e+f x))^3 (c+d \tan (e+f x))^3}{b^4 \left (a^2+b^2\right )^3 f (c \cos (e+f x)+d \sin (e+f x))^3 (a+b \tan (e+f x))^3}-\frac {2 C d^3 (a \cos (e+f x)+b \sin (e+f x))^3 \tan \left (\frac {1}{2} (e+f x)\right ) (c+d \tan (e+f x))^3}{b^3 f (c \cos (e+f x)+d \sin (e+f x))^3 \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) (a+b \tan (e+f x))^3}+\frac {2 \left (A b^2+a (-b B+a C)\right ) (-b c+a d)^3 (a \cos (e+f x)+b \sin (e+f x))^3 \left (a+2 b \tan \left (\frac {1}{2} (e+f x)\right )\right ) (c+d \tan (e+f x))^3}{a^3 b^2 \left (a^2+b^2\right ) f (c \cos (e+f x)+d \sin (e+f x))^3 \left (a+2 b \tan \left (\frac {1}{2} (e+f x)\right )-a \tan ^2\left (\frac {1}{2} (e+f x)\right )\right )^2 (a+b \tan (e+f x))^3}-\frac {2 (b c-a d)^2 (a \cos (e+f x)+b \sin (e+f x))^3 \left (A b^6 c+2 a^6 C d \tan \left (\frac {1}{2} (e+f x)\right )-a b^5 \left (B c+A \left (d-c \tan \left (\frac {1}{2} (e+f x)\right )\right )\right )-a^5 b \left (B d \tan \left (\frac {1}{2} (e+f x)\right )+C \left (d-c \tan \left (\frac {1}{2} (e+f x)\right )\right )\right )+a^4 b^2 \left (c \left (C-2 B \tan \left (\frac {1}{2} (e+f x)\right )\right )+d \left (B+4 C \tan \left (\frac {1}{2} (e+f x)\right )\right )\right )+a^2 b^4 \left (c C+B d+A \left (c+2 d \tan \left (\frac {1}{2} (e+f x)\right )\right )\right )-a^3 b^3 \left (A d+C d-3 A c \tan \left (\frac {1}{2} (e+f x)\right )+c C \tan \left (\frac {1}{2} (e+f x)\right )+B \left (c+3 d \tan \left (\frac {1}{2} (e+f x)\right )\right )\right )\right ) (c+d \tan (e+f x))^3}{a^3 b^3 \left (a^2+b^2\right )^2 f (c \cos (e+f x)+d \sin (e+f x))^3 \left (-2 b \tan \left (\frac {1}{2} (e+f x)\right )+a \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) (a+b \tan (e+f x))^3} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((3*a*b^2*(-(A*c^3) + c^3*C + 3*B*c^2*d + 3*A*c*d^2 - 3*c*C*d^2 - B*d^3) + a^3*(-(c^3*C) - 3*B*c^2*d + 3*c*C*d
^2 + B*d^3 + A*(c^3 - 3*c*d^2)) + b^3*((A - C)*d*(-3*c^2 + d^2) - B*(c^3 - 3*c*d^2)) + 3*a^2*b*(-((A - C)*d*(-
3*c^2 + d^2)) + B*(c^3 - 3*c*d^2)))*(e + f*x)*(a*Cos[e + f*x] + b*Sin[e + f*x])^3*(c + d*Tan[e + f*x])^3)/((a^
2 + b^2)^3*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^3*(a + b*Tan[e + f*x])^3) - (d^2*(3*b*c*C + b*B*d - 3*a*C*d)*Lo
g[1 - Tan[(e + f*x)/2]^2]*(a*Cos[e + f*x] + b*Sin[e + f*x])^3*(c + d*Tan[e + f*x])^3)/(b^4*f*(c*Cos[e + f*x] +
d*Sin[e + f*x])^3*(a + b*Tan[e + f*x])^3) + ((3*a^2*b*(-(A*c^3) + c^3*C + 3*B*c^2*d + 3*A*c*d^2 - 3*c*C*d^2 -
B*d^3) + b^3*(-(c^3*C) - 3*B*c^2*d + 3*c*C*d^2 + B*d^3 + A*(c^3 - 3*c*d^2)) + a^3*(-((A - C)*d*(-3*c^2 + d^2)
) + B*(c^3 - 3*c*d^2)) - 3*a*b^2*(-((A - C)*d*(-3*c^2 + d^2)) + B*(c^3 - 3*c*d^2)))*Log[1 + Tan[(e + f*x)/2]^2
]*(a*Cos[e + f*x] + b*Sin[e + f*x])^3*(c + d*Tan[e + f*x])^3)/((a^2 + b^2)^3*f*(c*Cos[e + f*x] + d*Sin[e + f*x
])^3*(a + b*Tan[e + f*x])^3) - ((b*c - a*d)*(a^5*b*B*d^2 - 3*a^6*C*d^2 + a^4*b^2*d*(B*c - 9*C*d) + a^3*b^3*B*(
c^2 + 3*d^2) + b^6*(-(c*(c*C + 3*B*d)) + A*(c^2 - 3*d^2)) + a*b^5*(8*c*(-A + C)*d - 3*B*(c^2 - 2*d^2)) + a^2*b
^4*(3*c^2*C + 6*B*c*d - 10*C*d^2 + A*(-3*c^2 + d^2)))*Log[-2*b*Tan[(e + f*x)/2] + a*(-1 + Tan[(e + f*x)/2]^2)]
*(a*Cos[e + f*x] + b*Sin[e + f*x])^3*(c + d*Tan[e + f*x])^3)/(b^4*(a^2 + b^2)^3*f*(c*Cos[e + f*x] + d*Sin[e +
f*x])^3*(a + b*Tan[e + f*x])^3) - (2*C*d^3*(a*Cos[e + f*x] + b*Sin[e + f*x])^3*Tan[(e + f*x)/2]*(c + d*Tan[e +
f*x])^3)/(b^3*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^3*(-1 + Tan[(e + f*x)/2]^2)*(a + b*Tan[e + f*x])^3) + (2*(A
*b^2 + a*(-(b*B) + a*C))*(-(b*c) + a*d)^3*(a*Cos[e + f*x] + b*Sin[e + f*x])^3*(a + 2*b*Tan[(e + f*x)/2])*(c +
d*Tan[e + f*x])^3)/(a^3*b^2*(a^2 + b^2)*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^3*(a + 2*b*Tan[(e + f*x)/2] - a*Ta
n[(e + f*x)/2]^2)^2*(a + b*Tan[e + f*x])^3) - (2*(b*c - a*d)^2*(a*Cos[e + f*x] + b*Sin[e + f*x])^3*(A*b^6*c +
2*a^6*C*d*Tan[(e + f*x)/2] - a*b^5*(B*c + A*(d - c*Tan[(e + f*x)/2])) - a^5*b*(B*d*Tan[(e + f*x)/2] + C*(d - c
*Tan[(e + f*x)/2])) + a^4*b^2*(c*(C - 2*B*Tan[(e + f*x)/2]) + d*(B + 4*C*Tan[(e + f*x)/2])) + a^2*b^4*(c*C + B
*d + A*(c + 2*d*Tan[(e + f*x)/2])) - a^3*b^3*(A*d + C*d - 3*A*c*Tan[(e + f*x)/2] + c*C*Tan[(e + f*x)/2] + B*(c
+ 3*d*Tan[(e + f*x)/2])))*(c + d*Tan[e + f*x])^3)/(a^3*b^3*(a^2 + b^2)^2*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^
3*(-2*b*Tan[(e + f*x)/2] + a*(-1 + Tan[(e + f*x)/2]^2))*(a + b*Tan[e + f*x])^3)

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Maple [A]
time = 0.90, size = 1271, normalized size = 1.59 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c+d*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e))^3,x,method=_RETURNVERBOSE)

[Out]

1/f*(C*d^3/b^3*tan(f*x+e)+1/(a^2+b^2)^3*(1/2*(3*A*a^3*c^2*d-A*a^3*d^3-3*A*a^2*b*c^3+9*A*a^2*b*c*d^2-9*A*a*b^2*
c^2*d+3*A*a*b^2*d^3+A*b^3*c^3-3*A*b^3*c*d^2+B*a^3*c^3-3*B*a^3*c*d^2+9*B*a^2*b*c^2*d-3*B*a^2*b*d^3-3*B*a*b^2*c^
3+9*B*a*b^2*c*d^2-3*B*b^3*c^2*d+B*b^3*d^3-3*C*a^3*c^2*d+C*a^3*d^3+3*C*a^2*b*c^3-9*C*a^2*b*c*d^2+9*C*a*b^2*c^2*
d-3*C*a*b^2*d^3-C*b^3*c^3+3*C*b^3*c*d^2)*ln(1+tan(f*x+e)^2)+(A*a^3*c^3-3*A*a^3*c*d^2+9*A*a^2*b*c^2*d-3*A*a^2*b
*d^3-3*A*a*b^2*c^3+9*A*a*b^2*c*d^2-3*A*b^3*c^2*d+A*b^3*d^3-3*B*a^3*c^2*d+B*a^3*d^3+3*B*a^2*b*c^3-9*B*a^2*b*c*d
^2+9*B*a*b^2*c^2*d-3*B*a*b^2*d^3-B*b^3*c^3+3*B*b^3*c*d^2-C*a^3*c^3+3*C*a^3*c*d^2-9*C*a^2*b*c^2*d+3*C*a^2*b*d^3
+3*C*a*b^2*c^3-9*C*a*b^2*c*d^2+3*C*b^3*c^2*d-C*b^3*d^3)*arctan(tan(f*x+e)))-1/2*(-A*a^3*b^2*d^3+3*A*a^2*b^3*c*
d^2-3*A*a*b^4*c^2*d+A*b^5*c^3+B*a^4*b*d^3-3*B*a^3*b^2*c*d^2+3*B*a^2*b^3*c^2*d-B*a*b^4*c^3-C*a^5*d^3+3*C*a^4*b*
c*d^2-3*C*a^3*b^2*c^2*d+C*a^2*b^3*c^3)/b^4/(a^2+b^2)/(a+b*tan(f*x+e))^2-1/b^4*(A*a^4*b^2*d^3-3*A*a^2*b^4*c^2*d
+3*A*a^2*b^4*d^3+2*A*a*b^5*c^3-6*A*a*b^5*c*d^2+3*A*b^6*c^2*d-2*B*a^5*b*d^3+3*B*a^4*b^2*c*d^2-4*B*a^3*b^3*d^3-B
*a^2*b^4*c^3+9*B*a^2*b^4*c*d^2-6*B*a*b^5*c^2*d+B*b^6*c^3+3*C*a^6*d^3-6*C*a^5*b*c*d^2+3*C*a^4*b^2*c^2*d+5*C*a^4
*b^2*d^3-12*C*a^3*b^3*c*d^2+9*C*a^2*b^4*c^2*d-2*C*a*b^5*c^3)/(a^2+b^2)^2/(a+b*tan(f*x+e))+(-3*A*a^3*b^4*c^2*d+
A*a^3*b^4*d^3+3*A*a^2*b^5*c^3-9*A*a^2*b^5*c*d^2+9*A*a*b^6*c^2*d-3*A*a*b^6*d^3-A*b^7*c^3+3*A*b^7*c*d^2+B*a^6*b*
d^3+3*B*a^4*b^3*d^3-B*a^3*b^4*c^3+3*B*a^3*b^4*c*d^2-9*B*a^2*b^5*c^2*d+6*B*a^2*b^5*d^3+3*B*a*b^6*c^3-9*B*a*b^6*
c*d^2+3*B*b^7*c^2*d-3*C*a^7*d^3+3*C*a^6*b*c*d^2-9*C*a^5*b^2*d^3+9*C*a^4*b^3*c*d^2+3*C*a^3*b^4*c^2*d-10*C*a^3*b
^4*d^3-3*C*a^2*b^5*c^3+18*C*a^2*b^5*c*d^2-9*C*a*b^6*c^2*d+C*b^7*c^3)/b^4/(a^2+b^2)^3*ln(a+b*tan(f*x+e)))

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Maxima [A]
time = 0.57, size = 1126, normalized size = 1.41 \begin {gather*} \frac {\frac {2 \, C d^{3} \tan \left (f x + e\right )}{b^{3}} + \frac {2 \, {\left ({\left ({\left (A - C\right )} a^{3} + 3 \, B a^{2} b - 3 \, {\left (A - C\right )} a b^{2} - B b^{3}\right )} c^{3} - 3 \, {\left (B a^{3} - 3 \, {\left (A - C\right )} a^{2} b - 3 \, B a b^{2} + {\left (A - C\right )} b^{3}\right )} c^{2} d - 3 \, {\left ({\left (A - C\right )} a^{3} + 3 \, B a^{2} b - 3 \, {\left (A - C\right )} a b^{2} - B b^{3}\right )} c d^{2} + {\left (B a^{3} - 3 \, {\left (A - C\right )} a^{2} b - 3 \, B a b^{2} + {\left (A - C\right )} b^{3}\right )} d^{3}\right )} {\left (f x + e\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {2 \, {\left ({\left (B a^{3} b^{4} - 3 \, {\left (A - C\right )} a^{2} b^{5} - 3 \, B a b^{6} + {\left (A - C\right )} b^{7}\right )} c^{3} + 3 \, {\left ({\left (A - C\right )} a^{3} b^{4} + 3 \, B a^{2} b^{5} - 3 \, {\left (A - C\right )} a b^{6} - B b^{7}\right )} c^{2} d - 3 \, {\left (C a^{6} b + 3 \, C a^{4} b^{3} + B a^{3} b^{4} - 3 \, {\left (A - 2 \, C\right )} a^{2} b^{5} - 3 \, B a b^{6} + A b^{7}\right )} c d^{2} + {\left (3 \, C a^{7} - B a^{6} b + 9 \, C a^{5} b^{2} - 3 \, B a^{4} b^{3} - {\left (A - 10 \, C\right )} a^{3} b^{4} - 6 \, B a^{2} b^{5} + 3 \, A a b^{6}\right )} d^{3}\right )} \log \left (b \tan \left (f x + e\right ) + a\right )}{a^{6} b^{4} + 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} + b^{10}} + \frac {{\left ({\left (B a^{3} - 3 \, {\left (A - C\right )} a^{2} b - 3 \, B a b^{2} + {\left (A - C\right )} b^{3}\right )} c^{3} + 3 \, {\left ({\left (A - C\right )} a^{3} + 3 \, B a^{2} b - 3 \, {\left (A - C\right )} a b^{2} - B b^{3}\right )} c^{2} d - 3 \, {\left (B a^{3} - 3 \, {\left (A - C\right )} a^{2} b - 3 \, B a b^{2} + {\left (A - C\right )} b^{3}\right )} c d^{2} - {\left ({\left (A - C\right )} a^{3} + 3 \, B a^{2} b - 3 \, {\left (A - C\right )} a b^{2} - B b^{3}\right )} d^{3}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {{\left (C a^{4} b^{3} - 3 \, B a^{3} b^{4} + {\left (5 \, A - 3 \, C\right )} a^{2} b^{5} + B a b^{6} + A b^{7}\right )} c^{3} + 3 \, {\left (C a^{5} b^{2} + B a^{4} b^{3} - {\left (3 \, A - 5 \, C\right )} a^{3} b^{4} - 3 \, B a^{2} b^{5} + A a b^{6}\right )} c^{2} d - 3 \, {\left (3 \, C a^{6} b - B a^{5} b^{2} - {\left (A - 7 \, C\right )} a^{4} b^{3} - 5 \, B a^{3} b^{4} + 3 \, A a^{2} b^{5}\right )} c d^{2} + {\left (5 \, C a^{7} - 3 \, B a^{6} b + {\left (A + 9 \, C\right )} a^{5} b^{2} - 7 \, B a^{4} b^{3} + 5 \, A a^{3} b^{4}\right )} d^{3} - 2 \, {\left ({\left (B a^{2} b^{5} - 2 \, {\left (A - C\right )} a b^{6} - B b^{7}\right )} c^{3} - 3 \, {\left (C a^{4} b^{3} - {\left (A - 3 \, C\right )} a^{2} b^{5} - 2 \, B a b^{6} + A b^{7}\right )} c^{2} d + 3 \, {\left (2 \, C a^{5} b^{2} - B a^{4} b^{3} + 4 \, C a^{3} b^{4} - 3 \, B a^{2} b^{5} + 2 \, A a b^{6}\right )} c d^{2} - {\left (3 \, C a^{6} b - 2 \, B a^{5} b^{2} + {\left (A + 5 \, C\right )} a^{4} b^{3} - 4 \, B a^{3} b^{4} + 3 \, A a^{2} b^{5}\right )} d^{3}\right )} \tan \left (f x + e\right )}{a^{6} b^{4} + 2 \, a^{4} b^{6} + a^{2} b^{8} + {\left (a^{4} b^{6} + 2 \, a^{2} b^{8} + b^{10}\right )} \tan \left (f x + e\right )^{2} + 2 \, {\left (a^{5} b^{5} + 2 \, a^{3} b^{7} + a b^{9}\right )} \tan \left (f x + e\right )}}{2 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e))^3,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2562 vs. \(2 (801) = 1602\).
time = 4.55, size = 2562, normalized size = 3.21 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e))^3,x, algorithm="fricas")

[Out]

1/2*(2*(C*a^6*b^3 + 3*C*a^4*b^5 + 3*C*a^2*b^7 + C*b^9)*d^3*tan(f*x + e)^3 - (3*C*a^4*b^5 - 5*B*a^3*b^6 + (7*A
- 3*C)*a^2*b^7 + B*a*b^8 + A*b^9)*c^3 + 3*(C*a^5*b^4 - 3*B*a^4*b^5 + 5*(A - C)*a^3*b^6 + 3*B*a^2*b^7 - A*a*b^8
)*c^2*d + 3*(C*a^6*b^3 + B*a^5*b^4 - (3*A - 7*C)*a^4*b^5 - 5*B*a^3*b^6 + 3*A*a^2*b^7)*c*d^2 - (3*C*a^7*b^2 - B
*a^6*b^3 - (A - 9*C)*a^5*b^4 - 7*B*a^4*b^5 + 5*A*a^3*b^6)*d^3 + 2*(((A - C)*a^5*b^4 + 3*B*a^4*b^5 - 3*(A - C)*
a^3*b^6 - B*a^2*b^7)*c^3 - 3*(B*a^5*b^4 - 3*(A - C)*a^4*b^5 - 3*B*a^3*b^6 + (A - C)*a^2*b^7)*c^2*d - 3*((A - C
)*a^5*b^4 + 3*B*a^4*b^5 - 3*(A - C)*a^3*b^6 - B*a^2*b^7)*c*d^2 + (B*a^5*b^4 - 3*(A - C)*a^4*b^5 - 3*B*a^3*b^6
+ (A - C)*a^2*b^7)*d^3)*f*x + ((C*a^4*b^5 - 3*B*a^3*b^6 + 5*(A - C)*a^2*b^7 + 3*B*a*b^8 - A*b^9)*c^3 + 3*(C*a^
5*b^4 + B*a^4*b^5 - (3*A - 7*C)*a^3*b^6 - 5*B*a^2*b^7 + 3*A*a*b^8)*c^2*d - 3*(3*C*a^6*b^3 - B*a^5*b^4 - (A - 9
*C)*a^4*b^5 - 7*B*a^3*b^6 + 5*A*a^2*b^7)*c*d^2 + (9*C*a^7*b^2 - 3*B*a^6*b^3 + (A + 23*C)*a^5*b^4 - 9*B*a^4*b^5
+ (7*A + 12*C)*a^3*b^6 + 4*C*a*b^8)*d^3 + 2*(((A - C)*a^3*b^6 + 3*B*a^2*b^7 - 3*(A - C)*a*b^8 - B*b^9)*c^3 -
3*(B*a^3*b^6 - 3*(A - C)*a^2*b^7 - 3*B*a*b^8 + (A - C)*b^9)*c^2*d - 3*((A - C)*a^3*b^6 + 3*B*a^2*b^7 - 3*(A -
C)*a*b^8 - B*b^9)*c*d^2 + (B*a^3*b^6 - 3*(A - C)*a^2*b^7 - 3*B*a*b^8 + (A - C)*b^9)*d^3)*f*x)*tan(f*x + e)^2 -
((B*a^5*b^4 - 3*(A - C)*a^4*b^5 - 3*B*a^3*b^6 + (A - C)*a^2*b^7)*c^3 + 3*((A - C)*a^5*b^4 + 3*B*a^4*b^5 - 3*(
A - C)*a^3*b^6 - B*a^2*b^7)*c^2*d - 3*(C*a^8*b + 3*C*a^6*b^3 + B*a^5*b^4 - 3*(A - 2*C)*a^4*b^5 - 3*B*a^3*b^6 +
A*a^2*b^7)*c*d^2 + (3*C*a^9 - B*a^8*b + 9*C*a^7*b^2 - 3*B*a^6*b^3 - (A - 10*C)*a^5*b^4 - 6*B*a^4*b^5 + 3*A*a^
3*b^6)*d^3 + ((B*a^3*b^6 - 3*(A - C)*a^2*b^7 - 3*B*a*b^8 + (A - C)*b^9)*c^3 + 3*((A - C)*a^3*b^6 + 3*B*a^2*b^7
- 3*(A - C)*a*b^8 - B*b^9)*c^2*d - 3*(C*a^6*b^3 + 3*C*a^4*b^5 + B*a^3*b^6 - 3*(A - 2*C)*a^2*b^7 - 3*B*a*b^8 +
A*b^9)*c*d^2 + (3*C*a^7*b^2 - B*a^6*b^3 + 9*C*a^5*b^4 - 3*B*a^4*b^5 - (A - 10*C)*a^3*b^6 - 6*B*a^2*b^7 + 3*A*
a*b^8)*d^3)*tan(f*x + e)^2 + 2*((B*a^4*b^5 - 3*(A - C)*a^3*b^6 - 3*B*a^2*b^7 + (A - C)*a*b^8)*c^3 + 3*((A - C)
*a^4*b^5 + 3*B*a^3*b^6 - 3*(A - C)*a^2*b^7 - B*a*b^8)*c^2*d - 3*(C*a^7*b^2 + 3*C*a^5*b^4 + B*a^4*b^5 - 3*(A -
2*C)*a^3*b^6 - 3*B*a^2*b^7 + A*a*b^8)*c*d^2 + (3*C*a^8*b - B*a^7*b^2 + 9*C*a^6*b^3 - 3*B*a^5*b^4 - (A - 10*C)*
a^4*b^5 - 6*B*a^3*b^6 + 3*A*a^2*b^7)*d^3)*tan(f*x + e))*log((b^2*tan(f*x + e)^2 + 2*a*b*tan(f*x + e) + a^2)/(t
an(f*x + e)^2 + 1)) - (3*(C*a^8*b + 3*C*a^6*b^3 + 3*C*a^4*b^5 + C*a^2*b^7)*c*d^2 - (3*C*a^9 - B*a^8*b + 9*C*a^
7*b^2 - 3*B*a^6*b^3 + 9*C*a^5*b^4 - 3*B*a^4*b^5 + 3*C*a^3*b^6 - B*a^2*b^7)*d^3 + (3*(C*a^6*b^3 + 3*C*a^4*b^5 +
3*C*a^2*b^7 + C*b^9)*c*d^2 - (3*C*a^7*b^2 - B*a^6*b^3 + 9*C*a^5*b^4 - 3*B*a^4*b^5 + 9*C*a^3*b^6 - 3*B*a^2*b^7
+ 3*C*a*b^8 - B*b^9)*d^3)*tan(f*x + e)^2 + 2*(3*(C*a^7*b^2 + 3*C*a^5*b^4 + 3*C*a^3*b^6 + C*a*b^8)*c*d^2 - (3*
C*a^8*b - B*a^7*b^2 + 9*C*a^6*b^3 - 3*B*a^5*b^4 + 9*C*a^4*b^5 - 3*B*a^3*b^6 + 3*C*a^2*b^7 - B*a*b^8)*d^3)*tan(
f*x + e))*log(1/(tan(f*x + e)^2 + 1)) + 2*((C*a^5*b^4 - 2*B*a^4*b^5 + 3*(A - C)*a^3*b^6 + 3*B*a^2*b^7 - (3*A -
2*C)*a*b^8 - B*b^9)*c^3 + 3*(B*a^5*b^4 - (2*A - 3*C)*a^4*b^5 - 3*B*a^3*b^6 + 3*(A - C)*a^2*b^7 + 2*B*a*b^8 -
A*b^9)*c^2*d - 3*(C*a^7*b^2 - (A - 3*C)*a^5*b^4 - 3*B*a^4*b^5 + (3*A - 4*C)*a^3*b^6 + 3*B*a^2*b^7 - 2*A*a*b^8)
*c*d^2 + (3*C*a^8*b - B*a^7*b^2 + 6*C*a^6*b^3 - 3*B*a^5*b^4 + (3*A - 2*C)*a^4*b^5 + 4*B*a^3*b^6 - (3*A - C)*a^
2*b^7)*d^3 + 2*(((A - C)*a^4*b^5 + 3*B*a^3*b^6 - 3*(A - C)*a^2*b^7 - B*a*b^8)*c^3 - 3*(B*a^4*b^5 - 3*(A - C)*a
^3*b^6 - 3*B*a^2*b^7 + (A - C)*a*b^8)*c^2*d - 3*((A - C)*a^4*b^5 + 3*B*a^3*b^6 - 3*(A - C)*a^2*b^7 - B*a*b^8)*
c*d^2 + (B*a^4*b^5 - 3*(A - C)*a^3*b^6 - 3*B*a^2*b^7 + (A - C)*a*b^8)*d^3)*f*x)*tan(f*x + e))/((a^6*b^6 + 3*a^
4*b^8 + 3*a^2*b^10 + b^12)*f*tan(f*x + e)^2 + 2*(a^7*b^5 + 3*a^5*b^7 + 3*a^3*b^9 + a*b^11)*f*tan(f*x + e) + (a
^8*b^4 + 3*a^6*b^6 + 3*a^4*b^8 + a^2*b^10)*f)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: AttributeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*tan(f*x+e))**3*(A+B*tan(f*x+e)+C*tan(f*x+e)**2)/(a+b*tan(f*x+e))**3,x)

[Out]

Exception raised: AttributeError >> 'NoneType' object has no attribute 'primitive'

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2505 vs. \(2 (801) = 1602\).
time = 1.50, size = 2505, normalized size = 3.14 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e))^3,x, algorithm="giac")

[Out]

1/2*(2*C*d^3*tan(f*x + e)/b^3 + 2*(A*a^3*c^3 - C*a^3*c^3 + 3*B*a^2*b*c^3 - 3*A*a*b^2*c^3 + 3*C*a*b^2*c^3 - B*b
^3*c^3 - 3*B*a^3*c^2*d + 9*A*a^2*b*c^2*d - 9*C*a^2*b*c^2*d + 9*B*a*b^2*c^2*d - 3*A*b^3*c^2*d + 3*C*b^3*c^2*d -
3*A*a^3*c*d^2 + 3*C*a^3*c*d^2 - 9*B*a^2*b*c*d^2 + 9*A*a*b^2*c*d^2 - 9*C*a*b^2*c*d^2 + 3*B*b^3*c*d^2 + B*a^3*d
^3 - 3*A*a^2*b*d^3 + 3*C*a^2*b*d^3 - 3*B*a*b^2*d^3 + A*b^3*d^3 - C*b^3*d^3)*(f*x + e)/(a^6 + 3*a^4*b^2 + 3*a^2
*b^4 + b^6) + (B*a^3*c^3 - 3*A*a^2*b*c^3 + 3*C*a^2*b*c^3 - 3*B*a*b^2*c^3 + A*b^3*c^3 - C*b^3*c^3 + 3*A*a^3*c^2
*d - 3*C*a^3*c^2*d + 9*B*a^2*b*c^2*d - 9*A*a*b^2*c^2*d + 9*C*a*b^2*c^2*d - 3*B*b^3*c^2*d - 3*B*a^3*c*d^2 + 9*A
*a^2*b*c*d^2 - 9*C*a^2*b*c*d^2 + 9*B*a*b^2*c*d^2 - 3*A*b^3*c*d^2 + 3*C*b^3*c*d^2 - A*a^3*d^3 + C*a^3*d^3 - 3*B
*a^2*b*d^3 + 3*A*a*b^2*d^3 - 3*C*a*b^2*d^3 + B*b^3*d^3)*log(tan(f*x + e)^2 + 1)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 +
b^6) - 2*(B*a^3*b^4*c^3 - 3*A*a^2*b^5*c^3 + 3*C*a^2*b^5*c^3 - 3*B*a*b^6*c^3 + A*b^7*c^3 - C*b^7*c^3 + 3*A*a^3
*b^4*c^2*d - 3*C*a^3*b^4*c^2*d + 9*B*a^2*b^5*c^2*d - 9*A*a*b^6*c^2*d + 9*C*a*b^6*c^2*d - 3*B*b^7*c^2*d - 3*C*a
^6*b*c*d^2 - 9*C*a^4*b^3*c*d^2 - 3*B*a^3*b^4*c*d^2 + 9*A*a^2*b^5*c*d^2 - 18*C*a^2*b^5*c*d^2 + 9*B*a*b^6*c*d^2
- 3*A*b^7*c*d^2 + 3*C*a^7*d^3 - B*a^6*b*d^3 + 9*C*a^5*b^2*d^3 - 3*B*a^4*b^3*d^3 - A*a^3*b^4*d^3 + 10*C*a^3*b^4
*d^3 - 6*B*a^2*b^5*d^3 + 3*A*a*b^6*d^3)*log(abs(b*tan(f*x + e) + a))/(a^6*b^4 + 3*a^4*b^6 + 3*a^2*b^8 + b^10)
+ (3*B*a^3*b^6*c^3*tan(f*x + e)^2 - 9*A*a^2*b^7*c^3*tan(f*x + e)^2 + 9*C*a^2*b^7*c^3*tan(f*x + e)^2 - 9*B*a*b^
8*c^3*tan(f*x + e)^2 + 3*A*b^9*c^3*tan(f*x + e)^2 - 3*C*b^9*c^3*tan(f*x + e)^2 + 9*A*a^3*b^6*c^2*d*tan(f*x + e
)^2 - 9*C*a^3*b^6*c^2*d*tan(f*x + e)^2 + 27*B*a^2*b^7*c^2*d*tan(f*x + e)^2 - 27*A*a*b^8*c^2*d*tan(f*x + e)^2 +
27*C*a*b^8*c^2*d*tan(f*x + e)^2 - 9*B*b^9*c^2*d*tan(f*x + e)^2 - 9*C*a^6*b^3*c*d^2*tan(f*x + e)^2 - 27*C*a^4*
b^5*c*d^2*tan(f*x + e)^2 - 9*B*a^3*b^6*c*d^2*tan(f*x + e)^2 + 27*A*a^2*b^7*c*d^2*tan(f*x + e)^2 - 54*C*a^2*b^7
*c*d^2*tan(f*x + e)^2 + 27*B*a*b^8*c*d^2*tan(f*x + e)^2 - 9*A*b^9*c*d^2*tan(f*x + e)^2 + 9*C*a^7*b^2*d^3*tan(f
*x + e)^2 - 3*B*a^6*b^3*d^3*tan(f*x + e)^2 + 27*C*a^5*b^4*d^3*tan(f*x + e)^2 - 9*B*a^4*b^5*d^3*tan(f*x + e)^2
- 3*A*a^3*b^6*d^3*tan(f*x + e)^2 + 30*C*a^3*b^6*d^3*tan(f*x + e)^2 - 18*B*a^2*b^7*d^3*tan(f*x + e)^2 + 9*A*a*b
^8*d^3*tan(f*x + e)^2 + 8*B*a^4*b^5*c^3*tan(f*x + e) - 22*A*a^3*b^6*c^3*tan(f*x + e) + 22*C*a^3*b^6*c^3*tan(f*
x + e) - 18*B*a^2*b^7*c^3*tan(f*x + e) + 2*A*a*b^8*c^3*tan(f*x + e) - 2*C*a*b^8*c^3*tan(f*x + e) - 2*B*b^9*c^3
*tan(f*x + e) - 6*C*a^6*b^3*c^2*d*tan(f*x + e) + 24*A*a^4*b^5*c^2*d*tan(f*x + e) - 42*C*a^4*b^5*c^2*d*tan(f*x
+ e) + 66*B*a^3*b^6*c^2*d*tan(f*x + e) - 54*A*a^2*b^7*c^2*d*tan(f*x + e) + 36*C*a^2*b^7*c^2*d*tan(f*x + e) - 6
*B*a*b^8*c^2*d*tan(f*x + e) - 6*A*b^9*c^2*d*tan(f*x + e) - 6*C*a^7*b^2*c*d^2*tan(f*x + e) - 6*B*a^6*b^3*c*d^2*
tan(f*x + e) - 18*C*a^5*b^4*c*d^2*tan(f*x + e) - 42*B*a^4*b^5*c*d^2*tan(f*x + e) + 66*A*a^3*b^6*c*d^2*tan(f*x
+ e) - 84*C*a^3*b^6*c*d^2*tan(f*x + e) + 36*B*a^2*b^7*c*d^2*tan(f*x + e) - 6*A*a*b^8*c*d^2*tan(f*x + e) + 12*C
*a^8*b*d^3*tan(f*x + e) - 2*B*a^7*b^2*d^3*tan(f*x + e) - 2*A*a^6*b^3*d^3*tan(f*x + e) + 38*C*a^6*b^3*d^3*tan(f
*x + e) - 6*B*a^5*b^4*d^3*tan(f*x + e) - 14*A*a^4*b^5*d^3*tan(f*x + e) + 50*C*a^4*b^5*d^3*tan(f*x + e) - 28*B*
a^3*b^6*d^3*tan(f*x + e) + 12*A*a^2*b^7*d^3*tan(f*x + e) - C*a^6*b^3*c^3 + 6*B*a^5*b^4*c^3 - 14*A*a^4*b^5*c^3
+ 11*C*a^4*b^5*c^3 - 7*B*a^3*b^6*c^3 - 3*A*a^2*b^7*c^3 - B*a*b^8*c^3 - A*b^9*c^3 - 3*C*a^7*b^2*c^2*d - 3*B*a^6
*b^3*c^2*d + 18*A*a^5*b^4*c^2*d - 27*C*a^5*b^4*c^2*d + 33*B*a^4*b^5*c^2*d - 21*A*a^3*b^6*c^2*d + 12*C*a^3*b^6*
c^2*d - 3*A*a*b^8*c^2*d - 3*B*a^7*b^2*c*d^2 - 3*A*a^6*b^3*c*d^2 + 3*C*a^6*b^3*c*d^2 - 27*B*a^5*b^4*c*d^2 + 33*
A*a^4*b^5*c*d^2 - 33*C*a^4*b^5*c*d^2 + 12*B*a^3*b^6*c*d^2 + 4*C*a^9*d^3 - A*a^7*b^2*d^3 + 13*C*a^7*b^2*d^3 + B
*a^6*b^3*d^3 - 9*A*a^5*b^4*d^3 + 21*C*a^5*b^4*d^3 - 11*B*a^4*b^5*d^3 + 4*A*a^3*b^6*d^3)/((a^6*b^4 + 3*a^4*b^6
+ 3*a^2*b^8 + b^10)*(b*tan(f*x + e) + a)^2))/f

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Mupad [B]
time = 19.24, size = 1172, normalized size = 1.47 \begin {gather*} \frac {\ln \left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )\,\left (b^3\,\left (3\,B\,a^4\,d^3+9\,C\,c\,a^4\,d^2\right )-b^6\,\left (3\,A\,a\,d^3-3\,B\,a\,c^3-9\,A\,a\,c^2\,d+9\,B\,a\,c\,d^2+9\,C\,a\,c^2\,d\right )+b^5\,\left (3\,A\,a^2\,c^3+6\,B\,a^2\,d^3-3\,C\,a^2\,c^3-9\,A\,a^2\,c\,d^2-9\,B\,a^2\,c^2\,d+18\,C\,a^2\,c\,d^2\right )+b^4\,\left (A\,a^3\,d^3-B\,a^3\,c^3-10\,C\,a^3\,d^3-3\,A\,a^3\,c^2\,d+3\,B\,a^3\,c\,d^2+3\,C\,a^3\,c^2\,d\right )+b\,\left (B\,a^6\,d^3+3\,C\,c\,a^6\,d^2\right )+b^7\,\left (C\,c^3-A\,c^3+3\,A\,c\,d^2+3\,B\,c^2\,d\right )-3\,C\,a^7\,d^3-9\,C\,a^5\,b^2\,d^3\right )}{f\,\left (a^6\,b^4+3\,a^4\,b^6+3\,a^2\,b^8+b^{10}\right )}+\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,\left (A\,c^3+A\,d^3\,1{}\mathrm {i}-B\,c^3\,1{}\mathrm {i}+B\,d^3-C\,c^3-C\,d^3\,1{}\mathrm {i}-3\,A\,c\,d^2-A\,c^2\,d\,3{}\mathrm {i}+B\,c\,d^2\,3{}\mathrm {i}-3\,B\,c^2\,d+3\,C\,c\,d^2+C\,c^2\,d\,3{}\mathrm {i}\right )}{2\,f\,\left (-a^3\,1{}\mathrm {i}-3\,a^2\,b+a\,b^2\,3{}\mathrm {i}+b^3\right )}-\frac {\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (B\,b^6\,c^3+3\,C\,a^6\,d^3+2\,A\,a\,b^5\,c^3-2\,B\,a^5\,b\,d^3-2\,C\,a\,b^5\,c^3+3\,A\,b^6\,c^2\,d+3\,A\,a^2\,b^4\,d^3+A\,a^4\,b^2\,d^3-B\,a^2\,b^4\,c^3-4\,B\,a^3\,b^3\,d^3+5\,C\,a^4\,b^2\,d^3-3\,A\,a^2\,b^4\,c^2\,d+9\,B\,a^2\,b^4\,c\,d^2+3\,B\,a^4\,b^2\,c\,d^2+9\,C\,a^2\,b^4\,c^2\,d-12\,C\,a^3\,b^3\,c\,d^2+3\,C\,a^4\,b^2\,c^2\,d-6\,A\,a\,b^5\,c\,d^2-6\,B\,a\,b^5\,c^2\,d-6\,C\,a^5\,b\,c\,d^2\right )}{a^4+2\,a^2\,b^2+b^4}+\frac {A\,b^7\,c^3+5\,C\,a^7\,d^3+B\,a\,b^6\,c^3-3\,B\,a^6\,b\,d^3+5\,A\,a^2\,b^5\,c^3+5\,A\,a^3\,b^4\,d^3+A\,a^5\,b^2\,d^3-3\,B\,a^3\,b^4\,c^3-7\,B\,a^4\,b^3\,d^3-3\,C\,a^2\,b^5\,c^3+C\,a^4\,b^3\,c^3+9\,C\,a^5\,b^2\,d^3-9\,A\,a^2\,b^5\,c\,d^2-9\,A\,a^3\,b^4\,c^2\,d+3\,A\,a^4\,b^3\,c\,d^2-9\,B\,a^2\,b^5\,c^2\,d+15\,B\,a^3\,b^4\,c\,d^2+3\,B\,a^4\,b^3\,c^2\,d+3\,B\,a^5\,b^2\,c\,d^2+15\,C\,a^3\,b^4\,c^2\,d-21\,C\,a^4\,b^3\,c\,d^2+3\,C\,a^5\,b^2\,c^2\,d+3\,A\,a\,b^6\,c^2\,d-9\,C\,a^6\,b\,c\,d^2}{2\,b\,\left (a^4+2\,a^2\,b^2+b^4\right )}}{f\,\left (a^2\,b^3+2\,a\,b^4\,\mathrm {tan}\left (e+f\,x\right )+b^5\,{\mathrm {tan}\left (e+f\,x\right )}^2\right )}+\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )-\mathrm {i}\right )\,\left (A\,d^3-B\,c^3-C\,d^3-3\,A\,c^2\,d+3\,B\,c\,d^2+3\,C\,c^2\,d+A\,c^3\,1{}\mathrm {i}+B\,d^3\,1{}\mathrm {i}-C\,c^3\,1{}\mathrm {i}-A\,c\,d^2\,3{}\mathrm {i}-B\,c^2\,d\,3{}\mathrm {i}+C\,c\,d^2\,3{}\mathrm {i}\right )}{2\,f\,\left (-a^3-a^2\,b\,3{}\mathrm {i}+3\,a\,b^2+b^3\,1{}\mathrm {i}\right )}+\frac {C\,d^3\,\mathrm {tan}\left (e+f\,x\right )}{b^3\,f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c + d*tan(e + f*x))^3*(A + B*tan(e + f*x) + C*tan(e + f*x)^2))/(a + b*tan(e + f*x))^3,x)

[Out]

(log(tan(e + f*x) + 1i)*(A*c^3 + A*d^3*1i - B*c^3*1i + B*d^3 - C*c^3 - C*d^3*1i - 3*A*c*d^2 - A*c^2*d*3i + B*c
*d^2*3i - 3*B*c^2*d + 3*C*c*d^2 + C*c^2*d*3i))/(2*f*(a*b^2*3i - 3*a^2*b - a^3*1i + b^3)) - ((tan(e + f*x)*(B*b
^6*c^3 + 3*C*a^6*d^3 + 2*A*a*b^5*c^3 - 2*B*a^5*b*d^3 - 2*C*a*b^5*c^3 + 3*A*b^6*c^2*d + 3*A*a^2*b^4*d^3 + A*a^4
*b^2*d^3 - B*a^2*b^4*c^3 - 4*B*a^3*b^3*d^3 + 5*C*a^4*b^2*d^3 - 3*A*a^2*b^4*c^2*d + 9*B*a^2*b^4*c*d^2 + 3*B*a^4
*b^2*c*d^2 + 9*C*a^2*b^4*c^2*d - 12*C*a^3*b^3*c*d^2 + 3*C*a^4*b^2*c^2*d - 6*A*a*b^5*c*d^2 - 6*B*a*b^5*c^2*d -
6*C*a^5*b*c*d^2))/(a^4 + b^4 + 2*a^2*b^2) + (A*b^7*c^3 + 5*C*a^7*d^3 + B*a*b^6*c^3 - 3*B*a^6*b*d^3 + 5*A*a^2*b
^5*c^3 + 5*A*a^3*b^4*d^3 + A*a^5*b^2*d^3 - 3*B*a^3*b^4*c^3 - 7*B*a^4*b^3*d^3 - 3*C*a^2*b^5*c^3 + C*a^4*b^3*c^3
+ 9*C*a^5*b^2*d^3 - 9*A*a^2*b^5*c*d^2 - 9*A*a^3*b^4*c^2*d + 3*A*a^4*b^3*c*d^2 - 9*B*a^2*b^5*c^2*d + 15*B*a^3*
b^4*c*d^2 + 3*B*a^4*b^3*c^2*d + 3*B*a^5*b^2*c*d^2 + 15*C*a^3*b^4*c^2*d - 21*C*a^4*b^3*c*d^2 + 3*C*a^5*b^2*c^2*
d + 3*A*a*b^6*c^2*d - 9*C*a^6*b*c*d^2)/(2*b*(a^4 + b^4 + 2*a^2*b^2)))/(f*(a^2*b^3 + b^5*tan(e + f*x)^2 + 2*a*b
^4*tan(e + f*x))) + (log(a + b*tan(e + f*x))*(b^3*(3*B*a^4*d^3 + 9*C*a^4*c*d^2) - b^6*(3*A*a*d^3 - 3*B*a*c^3 -
9*A*a*c^2*d + 9*B*a*c*d^2 + 9*C*a*c^2*d) + b^5*(3*A*a^2*c^3 + 6*B*a^2*d^3 - 3*C*a^2*c^3 - 9*A*a^2*c*d^2 - 9*B
*a^2*c^2*d + 18*C*a^2*c*d^2) + b^4*(A*a^3*d^3 - B*a^3*c^3 - 10*C*a^3*d^3 - 3*A*a^3*c^2*d + 3*B*a^3*c*d^2 + 3*C
*a^3*c^2*d) + b*(B*a^6*d^3 + 3*C*a^6*c*d^2) + b^7*(C*c^3 - A*c^3 + 3*A*c*d^2 + 3*B*c^2*d) - 3*C*a^7*d^3 - 9*C*
a^5*b^2*d^3))/(f*(b^10 + 3*a^2*b^8 + 3*a^4*b^6 + a^6*b^4)) + (log(tan(e + f*x) - 1i)*(A*c^3*1i + A*d^3 - B*c^3
+ B*d^3*1i - C*c^3*1i - C*d^3 - A*c*d^2*3i - 3*A*c^2*d + 3*B*c*d^2 - B*c^2*d*3i + C*c*d^2*3i + 3*C*c^2*d))/(2
*f*(3*a*b^2 - a^2*b*3i - a^3 + b^3*1i)) + (C*d^3*tan(e + f*x))/(b^3*f)

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