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

Optimal. Leaf size=603 \[ \frac{(c+d \tan (e+f x))^4 \left (5 a^2 C d^2-6 a b d (c C-5 B d)+b^2 \left (15 d^2 (A-C)-3 B c d+c^2 C\right )\right )}{60 d^3 f}-\frac{d \tan (e+f x) \left (a^2 \left (-\left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )+2 a b \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )+b^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )}{f}+\frac{\log (\cos (e+f x)) \left (a^2 \left (-\left (d (A-C) \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )+2 a b \left (-A \left (c^3-3 c d^2\right )+3 B c^2 d-B d^3+c^3 C-3 c C d^2\right )+b^2 \left (d (A-C) \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )}{f}+x \left (a^2 \left (A c^3-3 A c d^2-3 B c^2 d+B d^3-c^3 C+3 c C d^2\right )-2 a b \left (d (A-C) \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+b^2 \left (-A \left (c^3-3 c d^2\right )+3 B c^2 d-B d^3+c^3 C-3 c C d^2\right )\right )+\frac{\left (a^2 B+2 a b (A-C)-b^2 B\right ) (c+d \tan (e+f x))^3}{3 f}+\frac{(c+d \tan (e+f x))^2 \left (a^2 (d (A-C)+B c)+2 a b (A c-B d-c C)-b^2 (d (A-C)+B c)\right )}{2 f}-\frac{b \tan (e+f x) (-a C d-3 b B d+b c C) (c+d \tan (e+f x))^4}{15 d^2 f}+\frac{C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^4}{6 d f} \]

[Out]

(a^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) + b^2*(c^3*C + 3*B*c^2*d - 3*c*C*d^2 - B*d^3
- A*(c^3 - 3*c*d^2)) - 2*a*b*((A - C)*d*(3*c^2 - d^2) + B*(c^3 - 3*c*d^2)))*x + ((2*a*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^2*((A - C)*d*(3*c^2 - d^2) + B*(c^3 - 3*c*d^2)) + b^2*((A - C)*d*(3*
c^2 - d^2) + B*(c^3 - 3*c*d^2)))*Log[Cos[e + f*x]])/f - (d*(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)))*Tan[e + f*x])/f + ((2*a*b*(A*c - c*
C - B*d) + a^2*(B*c + (A - C)*d) - b^2*(B*c + (A - C)*d))*(c + d*Tan[e + f*x])^2)/(2*f) + ((a^2*B - b^2*B + 2*
a*b*(A - C))*(c + d*Tan[e + f*x])^3)/(3*f) + ((5*a^2*C*d^2 - 6*a*b*d*(c*C - 5*B*d) + b^2*(c^2*C - 3*B*c*d + 15
*(A - C)*d^2))*(c + d*Tan[e + f*x])^4)/(60*d^3*f) - (b*(b*c*C - 3*b*B*d - a*C*d)*Tan[e + f*x]*(c + d*Tan[e + f
*x])^4)/(15*d^2*f) + (C*(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^4)/(6*d*f)

________________________________________________________________________________________

Rubi [A]  time = 1.53279, antiderivative size = 603, 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 = {3647, 3637, 3630, 3528, 3525, 3475} \[ \frac{(c+d \tan (e+f x))^4 \left (5 a^2 C d^2-6 a b d (c C-5 B d)+b^2 \left (15 d^2 (A-C)-3 B c d+c^2 C\right )\right )}{60 d^3 f}-\frac{d \tan (e+f x) \left (a^2 \left (-\left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )+2 a b \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )+b^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )}{f}+\frac{\log (\cos (e+f x)) \left (a^2 \left (-\left (d (A-C) \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )+2 a b \left (-A \left (c^3-3 c d^2\right )+3 B c^2 d-B d^3+c^3 C-3 c C d^2\right )+b^2 \left (d (A-C) \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )}{f}+x \left (a^2 \left (A c^3-3 A c d^2-3 B c^2 d+B d^3-c^3 C+3 c C d^2\right )-2 a b \left (d (A-C) \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+b^2 \left (-A \left (c^3-3 c d^2\right )+3 B c^2 d-B d^3+c^3 C-3 c C d^2\right )\right )+\frac{\left (a^2 B+2 a b (A-C)-b^2 B\right ) (c+d \tan (e+f x))^3}{3 f}+\frac{(c+d \tan (e+f x))^2 \left (a^2 (d (A-C)+B c)+2 a b (A c-B d-c C)-b^2 (d (A-C)+B c)\right )}{2 f}-\frac{b \tan (e+f x) (-a C d-3 b B d+b c C) (c+d \tan (e+f x))^4}{15 d^2 f}+\frac{C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^4}{6 d f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^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) + b^2*(c^3*C + 3*B*c^2*d - 3*c*C*d^2 - B*d^3
- A*(c^3 - 3*c*d^2)) - 2*a*b*((A - C)*d*(3*c^2 - d^2) + B*(c^3 - 3*c*d^2)))*x + ((2*a*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^2*((A - C)*d*(3*c^2 - d^2) + B*(c^3 - 3*c*d^2)) + b^2*((A - C)*d*(3*
c^2 - d^2) + B*(c^3 - 3*c*d^2)))*Log[Cos[e + f*x]])/f - (d*(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)))*Tan[e + f*x])/f + ((2*a*b*(A*c - c*
C - B*d) + a^2*(B*c + (A - C)*d) - b^2*(B*c + (A - C)*d))*(c + d*Tan[e + f*x])^2)/(2*f) + ((a^2*B - b^2*B + 2*
a*b*(A - C))*(c + d*Tan[e + f*x])^3)/(3*f) + ((5*a^2*C*d^2 - 6*a*b*d*(c*C - 5*B*d) + b^2*(c^2*C - 3*B*c*d + 15
*(A - C)*d^2))*(c + d*Tan[e + f*x])^4)/(60*d^3*f) - (b*(b*c*C - 3*b*B*d - a*C*d)*Tan[e + f*x]*(c + d*Tan[e + f
*x])^4)/(15*d^2*f) + (C*(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^4)/(6*d*f)

Rule 3647

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])))

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 3630

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 3528

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 3525

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 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 (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3 \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))^4}{6 d f}+\frac{\int (a+b \tan (e+f x)) (c+d \tan (e+f x))^3 \left (-2 (b c C-3 a A d+2 a C d)+6 (A b+a B-b C) d \tan (e+f x)-2 (b c C-3 b B d-a C d) \tan ^2(e+f x)\right ) \, dx}{6 d}\\ &=-\frac{b (b c C-3 b B d-a C d) \tan (e+f x) (c+d \tan (e+f x))^4}{15 d^2 f}+\frac{C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^4}{6 d f}-\frac{\int (c+d \tan (e+f x))^3 \left (2 \left (6 a b c C d-5 a^2 (3 A-2 C) d^2-b^2 c (c C-3 B d)\right )-30 \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2 \tan (e+f x)-2 \left (5 a^2 C d^2-6 a b d (c C-5 B d)+b^2 \left (c^2 C-3 B c d+15 (A-C) d^2\right )\right ) \tan ^2(e+f x)\right ) \, dx}{30 d^2}\\ &=\frac{\left (5 a^2 C d^2-6 a b d (c C-5 B d)+b^2 \left (c^2 C-3 B c d+15 (A-C) d^2\right )\right ) (c+d \tan (e+f x))^4}{60 d^3 f}-\frac{b (b c C-3 b B d-a C d) \tan (e+f x) (c+d \tan (e+f x))^4}{15 d^2 f}+\frac{C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^4}{6 d f}-\frac{\int (c+d \tan (e+f x))^3 \left (30 \left (2 a b B-a^2 (A-C)+b^2 (A-C)\right ) d^2-30 \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2 \tan (e+f x)\right ) \, dx}{30 d^2}\\ &=\frac{\left (a^2 B-b^2 B+2 a b (A-C)\right ) (c+d \tan (e+f x))^3}{3 f}+\frac{\left (5 a^2 C d^2-6 a b d (c C-5 B d)+b^2 \left (c^2 C-3 B c d+15 (A-C) d^2\right )\right ) (c+d \tan (e+f x))^4}{60 d^3 f}-\frac{b (b c C-3 b B d-a C d) \tan (e+f x) (c+d \tan (e+f x))^4}{15 d^2 f}+\frac{C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^4}{6 d f}-\frac{\int (c+d \tan (e+f x))^2 \left (-30 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 )-30 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}{30 d^2}\\ &=\frac{\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 ) (c+d \tan (e+f x))^2}{2 f}+\frac{\left (a^2 B-b^2 B+2 a b (A-C)\right ) (c+d \tan (e+f x))^3}{3 f}+\frac{\left (5 a^2 C d^2-6 a b d (c C-5 B d)+b^2 \left (c^2 C-3 B c d+15 (A-C) d^2\right )\right ) (c+d \tan (e+f x))^4}{60 d^3 f}-\frac{b (b c C-3 b B d-a C d) \tan (e+f x) (c+d \tan (e+f x))^4}{15 d^2 f}+\frac{C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^4}{6 d f}-\frac{\int (c+d \tan (e+f x)) \left (30 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 )+30 d^2 \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 ) \tan (e+f x)\right ) \, dx}{30 d^2}\\ &=\left (a^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 )+b^2 \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 )-2 a b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) x-\frac{d \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 ) \tan (e+f x)}{f}+\frac{\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 ) (c+d \tan (e+f x))^2}{2 f}+\frac{\left (a^2 B-b^2 B+2 a b (A-C)\right ) (c+d \tan (e+f x))^3}{3 f}+\frac{\left (5 a^2 C d^2-6 a b d (c C-5 B d)+b^2 \left (c^2 C-3 B c d+15 (A-C) d^2\right )\right ) (c+d \tan (e+f x))^4}{60 d^3 f}-\frac{b (b c C-3 b B d-a C d) \tan (e+f x) (c+d \tan (e+f x))^4}{15 d^2 f}+\frac{C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^4}{6 d f}-\frac{\left (30 d^3 \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 )+30 c d^2 \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 )\right ) \int \tan (e+f x) \, dx}{30 d^2}\\ &=\left (a^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 )+b^2 \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 )-2 a b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) x+\frac{\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 ) \log (\cos (e+f x))}{f}-\frac{d \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 ) \tan (e+f x)}{f}+\frac{\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 ) (c+d \tan (e+f x))^2}{2 f}+\frac{\left (a^2 B-b^2 B+2 a b (A-C)\right ) (c+d \tan (e+f x))^3}{3 f}+\frac{\left (5 a^2 C d^2-6 a b d (c C-5 B d)+b^2 \left (c^2 C-3 B c d+15 (A-C) d^2\right )\right ) (c+d \tan (e+f x))^4}{60 d^3 f}-\frac{b (b c C-3 b B d-a C d) \tan (e+f x) (c+d \tan (e+f x))^4}{15 d^2 f}+\frac{C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^4}{6 d f}\\ \end{align*}

Mathematica [C]  time = 6.58338, size = 419, normalized size = 0.69 \[ \frac{C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^4}{6 d f}+\frac{-\frac{2 b \tan (e+f x) (-a C d-3 b B d+b c C) (c+d \tan (e+f x))^4}{5 d f}-\frac{-\frac{(c+d \tan (e+f x))^4 \left (5 a^2 C d^2-6 a b d (c C-5 B d)+b^2 \left (15 d^2 (A-C)-3 B c d+c^2 C\right )\right )}{2 d f}+\frac{5 \left (d \left (a^2 B+2 a b (A-C)-b^2 B\right ) \left (-6 d^2 \left (6 c^2-d^2\right ) \tan (e+f x)-12 c d^3 \tan ^2(e+f x)-3 i (c-i d)^4 \log (\tan (e+f x)+i)+3 i (c+i d)^4 \log (-\tan (e+f x)+i)-2 d^4 \tan ^3(e+f x)\right )+3 d \left (a^2 (B c-d (A-C))+2 a b (A c+B d-c C)-b^2 (B c-d (A-C))\right ) \left (6 c d^2 \tan (e+f x)+(-d+i c)^3 \log (-\tan (e+f x)+i)-(d+i c)^3 \log (\tan (e+f x)+i)+d^3 \tan ^2(e+f x)\right )\right )}{f}}{5 d}}{6 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(C*(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^4)/(6*d*f) + ((-2*b*(b*c*C - 3*b*B*d - a*C*d)*Tan[e + f*x]*(c +
d*Tan[e + f*x])^4)/(5*d*f) - (-((5*a^2*C*d^2 - 6*a*b*d*(c*C - 5*B*d) + b^2*(c^2*C - 3*B*c*d + 15*(A - C)*d^2)
)*(c + d*Tan[e + f*x])^4)/(2*d*f) + (5*(3*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 - 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*Ta
n[e + f*x]^2) + (a^2*B - b^2*B + 2*a*b*(A - C))*d*((3*I)*(c + I*d)^4*Log[I - Tan[e + f*x]] - (3*I)*(c - I*d)^4
*Log[I + Tan[e + f*x]] - 6*d^2*(6*c^2 - d^2)*Tan[e + f*x] - 12*c*d^3*Tan[e + f*x]^2 - 2*d^4*Tan[e + f*x]^3)))/
f)/(5*d))/(6*d)

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Maple [B]  time = 0.021, size = 1807, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.49778, size = 918, normalized size = 1.52 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(10*C*b^2*d^3*tan(f*x + e)^6 + 12*(3*C*b^2*c*d^2 + (2*C*a*b + B*b^2)*d^3)*tan(f*x + e)^5 + 15*(3*C*b^2*c^
2*d + 3*(2*C*a*b + B*b^2)*c*d^2 + (C*a^2 + 2*B*a*b + (A - C)*b^2)*d^3)*tan(f*x + e)^4 + 20*(C*b^2*c^3 + 3*(2*C
*a*b + B*b^2)*c^2*d + 3*(C*a^2 + 2*B*a*b + (A - C)*b^2)*c*d^2 + (B*a^2 + 2*(A - C)*a*b - B*b^2)*d^3)*tan(f*x +
e)^3 + 30*((2*C*a*b + B*b^2)*c^3 + 3*(C*a^2 + 2*B*a*b + (A - C)*b^2)*c^2*d + 3*(B*a^2 + 2*(A - C)*a*b - B*b^2
)*c*d^2 + ((A - C)*a^2 - 2*B*a*b - (A - C)*b^2)*d^3)*tan(f*x + e)^2 + 60*(((A - C)*a^2 - 2*B*a*b - (A - C)*b^2
)*c^3 - 3*(B*a^2 + 2*(A - C)*a*b - B*b^2)*c^2*d - 3*((A - C)*a^2 - 2*B*a*b - (A - C)*b^2)*c*d^2 + (B*a^2 + 2*(
A - C)*a*b - B*b^2)*d^3)*(f*x + e) + 30*((B*a^2 + 2*(A - C)*a*b - B*b^2)*c^3 + 3*((A - C)*a^2 - 2*B*a*b - (A -
C)*b^2)*c^2*d - 3*(B*a^2 + 2*(A - C)*a*b - B*b^2)*c*d^2 - ((A - C)*a^2 - 2*B*a*b - (A - C)*b^2)*d^3)*log(tan(
f*x + e)^2 + 1) + 60*((C*a^2 + 2*B*a*b + (A - C)*b^2)*c^3 + 3*(B*a^2 + 2*(A - C)*a*b - B*b^2)*c^2*d + 3*((A -
C)*a^2 - 2*B*a*b - (A - C)*b^2)*c*d^2 - (B*a^2 + 2*(A - C)*a*b - B*b^2)*d^3)*tan(f*x + e))/f

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Fricas [A]  time = 1.31179, size = 1461, normalized size = 2.42 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(10*C*b^2*d^3*tan(f*x + e)^6 + 12*(3*C*b^2*c*d^2 + (2*C*a*b + B*b^2)*d^3)*tan(f*x + e)^5 + 15*(3*C*b^2*c^
2*d + 3*(2*C*a*b + B*b^2)*c*d^2 + (C*a^2 + 2*B*a*b + (A - C)*b^2)*d^3)*tan(f*x + e)^4 + 20*(C*b^2*c^3 + 3*(2*C
*a*b + B*b^2)*c^2*d + 3*(C*a^2 + 2*B*a*b + (A - C)*b^2)*c*d^2 + (B*a^2 + 2*(A - C)*a*b - B*b^2)*d^3)*tan(f*x +
e)^3 + 60*(((A - C)*a^2 - 2*B*a*b - (A - C)*b^2)*c^3 - 3*(B*a^2 + 2*(A - C)*a*b - B*b^2)*c^2*d - 3*((A - C)*a
^2 - 2*B*a*b - (A - C)*b^2)*c*d^2 + (B*a^2 + 2*(A - C)*a*b - B*b^2)*d^3)*f*x + 30*((2*C*a*b + B*b^2)*c^3 + 3*(
C*a^2 + 2*B*a*b + (A - C)*b^2)*c^2*d + 3*(B*a^2 + 2*(A - C)*a*b - B*b^2)*c*d^2 + ((A - C)*a^2 - 2*B*a*b - (A -
C)*b^2)*d^3)*tan(f*x + e)^2 - 30*((B*a^2 + 2*(A - C)*a*b - B*b^2)*c^3 + 3*((A - C)*a^2 - 2*B*a*b - (A - C)*b^
2)*c^2*d - 3*(B*a^2 + 2*(A - C)*a*b - B*b^2)*c*d^2 - ((A - C)*a^2 - 2*B*a*b - (A - C)*b^2)*d^3)*log(1/(tan(f*x
+ e)^2 + 1)) + 60*((C*a^2 + 2*B*a*b + (A - C)*b^2)*c^3 + 3*(B*a^2 + 2*(A - C)*a*b - B*b^2)*c^2*d + 3*((A - C)
*a^2 - 2*B*a*b - (A - C)*b^2)*c*d^2 - (B*a^2 + 2*(A - C)*a*b - B*b^2)*d^3)*tan(f*x + e))/f

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Sympy [A]  time = 7.86021, size = 1819, normalized size = 3.02 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((A*a**2*c**3*x + 3*A*a**2*c**2*d*log(tan(e + f*x)**2 + 1)/(2*f) - 3*A*a**2*c*d**2*x + 3*A*a**2*c*d**
2*tan(e + f*x)/f - A*a**2*d**3*log(tan(e + f*x)**2 + 1)/(2*f) + A*a**2*d**3*tan(e + f*x)**2/(2*f) + A*a*b*c**3
*log(tan(e + f*x)**2 + 1)/f - 6*A*a*b*c**2*d*x + 6*A*a*b*c**2*d*tan(e + f*x)/f - 3*A*a*b*c*d**2*log(tan(e + f*
x)**2 + 1)/f + 3*A*a*b*c*d**2*tan(e + f*x)**2/f + 2*A*a*b*d**3*x + 2*A*a*b*d**3*tan(e + f*x)**3/(3*f) - 2*A*a*
b*d**3*tan(e + f*x)/f - A*b**2*c**3*x + A*b**2*c**3*tan(e + f*x)/f - 3*A*b**2*c**2*d*log(tan(e + f*x)**2 + 1)/
(2*f) + 3*A*b**2*c**2*d*tan(e + f*x)**2/(2*f) + 3*A*b**2*c*d**2*x + A*b**2*c*d**2*tan(e + f*x)**3/f - 3*A*b**2
*c*d**2*tan(e + f*x)/f + A*b**2*d**3*log(tan(e + f*x)**2 + 1)/(2*f) + A*b**2*d**3*tan(e + f*x)**4/(4*f) - A*b*
*2*d**3*tan(e + f*x)**2/(2*f) + B*a**2*c**3*log(tan(e + f*x)**2 + 1)/(2*f) - 3*B*a**2*c**2*d*x + 3*B*a**2*c**2
*d*tan(e + f*x)/f - 3*B*a**2*c*d**2*log(tan(e + f*x)**2 + 1)/(2*f) + 3*B*a**2*c*d**2*tan(e + f*x)**2/(2*f) + B
*a**2*d**3*x + B*a**2*d**3*tan(e + f*x)**3/(3*f) - B*a**2*d**3*tan(e + f*x)/f - 2*B*a*b*c**3*x + 2*B*a*b*c**3*
tan(e + f*x)/f - 3*B*a*b*c**2*d*log(tan(e + f*x)**2 + 1)/f + 3*B*a*b*c**2*d*tan(e + f*x)**2/f + 6*B*a*b*c*d**2
*x + 2*B*a*b*c*d**2*tan(e + f*x)**3/f - 6*B*a*b*c*d**2*tan(e + f*x)/f + B*a*b*d**3*log(tan(e + f*x)**2 + 1)/f
+ B*a*b*d**3*tan(e + f*x)**4/(2*f) - B*a*b*d**3*tan(e + f*x)**2/f - B*b**2*c**3*log(tan(e + f*x)**2 + 1)/(2*f)
+ B*b**2*c**3*tan(e + f*x)**2/(2*f) + 3*B*b**2*c**2*d*x + B*b**2*c**2*d*tan(e + f*x)**3/f - 3*B*b**2*c**2*d*t
an(e + f*x)/f + 3*B*b**2*c*d**2*log(tan(e + f*x)**2 + 1)/(2*f) + 3*B*b**2*c*d**2*tan(e + f*x)**4/(4*f) - 3*B*b
**2*c*d**2*tan(e + f*x)**2/(2*f) - B*b**2*d**3*x + B*b**2*d**3*tan(e + f*x)**5/(5*f) - B*b**2*d**3*tan(e + f*x
)**3/(3*f) + B*b**2*d**3*tan(e + f*x)/f - C*a**2*c**3*x + C*a**2*c**3*tan(e + f*x)/f - 3*C*a**2*c**2*d*log(tan
(e + f*x)**2 + 1)/(2*f) + 3*C*a**2*c**2*d*tan(e + f*x)**2/(2*f) + 3*C*a**2*c*d**2*x + C*a**2*c*d**2*tan(e + f*
x)**3/f - 3*C*a**2*c*d**2*tan(e + f*x)/f + C*a**2*d**3*log(tan(e + f*x)**2 + 1)/(2*f) + C*a**2*d**3*tan(e + f*
x)**4/(4*f) - C*a**2*d**3*tan(e + f*x)**2/(2*f) - C*a*b*c**3*log(tan(e + f*x)**2 + 1)/f + C*a*b*c**3*tan(e + f
*x)**2/f + 6*C*a*b*c**2*d*x + 2*C*a*b*c**2*d*tan(e + f*x)**3/f - 6*C*a*b*c**2*d*tan(e + f*x)/f + 3*C*a*b*c*d**
2*log(tan(e + f*x)**2 + 1)/f + 3*C*a*b*c*d**2*tan(e + f*x)**4/(2*f) - 3*C*a*b*c*d**2*tan(e + f*x)**2/f - 2*C*a
*b*d**3*x + 2*C*a*b*d**3*tan(e + f*x)**5/(5*f) - 2*C*a*b*d**3*tan(e + f*x)**3/(3*f) + 2*C*a*b*d**3*tan(e + f*x
)/f + C*b**2*c**3*x + C*b**2*c**3*tan(e + f*x)**3/(3*f) - C*b**2*c**3*tan(e + f*x)/f + 3*C*b**2*c**2*d*log(tan
(e + f*x)**2 + 1)/(2*f) + 3*C*b**2*c**2*d*tan(e + f*x)**4/(4*f) - 3*C*b**2*c**2*d*tan(e + f*x)**2/(2*f) - 3*C*
b**2*c*d**2*x + 3*C*b**2*c*d**2*tan(e + f*x)**5/(5*f) - C*b**2*c*d**2*tan(e + f*x)**3/f + 3*C*b**2*c*d**2*tan(
e + f*x)/f - C*b**2*d**3*log(tan(e + f*x)**2 + 1)/(2*f) + C*b**2*d**3*tan(e + f*x)**6/(6*f) - C*b**2*d**3*tan(
e + f*x)**4/(4*f) + C*b**2*d**3*tan(e + f*x)**2/(2*f), Ne(f, 0)), (x*(a + b*tan(e))**2*(c + d*tan(e))**3*(A +
B*tan(e) + C*tan(e)**2), True))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out