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

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

[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)

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Rubi [A]  time = 2.83884, antiderivative size = 798, normalized size of antiderivative = 1., 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 = {3645, 3637, 3626, 3617, 31, 3475} \[ -\frac{\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^3}{2 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}+\frac{\left (-3 C d a^4+b B d a^3+b^2 (2 B c+(A-7 C) d) a^2-b^3 (4 A c-4 C c-5 B d) a-b^4 (2 B c+3 A d)\right ) (c+d \tan (e+f x))^2}{2 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}-\frac{\left (\left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^3-3 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^2+3 b^2 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a+b^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) x}{\left (a^2+b^2\right )^3}-\frac{\left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^3+3 b \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^2-3 b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^3 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right )\right ) \log (\cos (e+f x))}{\left (a^2+b^2\right )^3 f}-\frac{(b c-a d) \left (-3 C d^2 a^6+b B d^2 a^5+b^2 d (B c-9 C d) a^4+b^3 B \left (c^2+3 d^2\right ) a^3+b^4 \left (3 C c^2+6 B d c-10 C d^2-A \left (3 c^2-d^2\right )\right ) a^2-b^5 \left (8 c (A-C) d+3 B \left (c^2-2 d^2\right )\right ) a-b^6 \left (c (c C+3 B d)-A \left (c^2-3 d^2\right )\right )\right ) \log (a+b \tan (e+f x))}{b^4 \left (a^2+b^2\right )^3 f}-\frac{d^2 \left (-3 C d a^4+b B d a^3+b^2 (B c-6 C d) a^2-b^3 (2 A c-2 C c-3 B d) a-b^4 (B c+(2 A+C) d)\right ) \tan (e+f x)}{b^3 \left (a^2+b^2\right )^2 f} \]

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 3645

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*T
an[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]

Rule 3637

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 3626

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 3617

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 31

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

Rule 3475

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

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 ) \operatorname{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*}

Mathematica [A]  time = 15.0509, size = 1451, normalized size = 1.82 \[ \text{result too large to display} \]

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)

________________________________________________________________________________________

Maple [B]  time = 0.079, size = 3522, normalized size = 4.4 \begin{align*} \text{output too large to display} \end{align*}

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)

[Out]

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

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Maxima [A]  time = 1.77136, size = 1511, normalized size = 1.89 \begin{align*} \text{result too large to display} \end{align*}

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]  time = 10.9242, size = 5261, normalized size = 6.59 \begin{align*} \text{result too large to display} \end{align*}

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(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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]

Timed out

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Giac [B]  time = 2.48784, size = 3382, normalized size = 4.24 \begin{align*} \text{result too large to display} \end{align*}

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