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

Optimal. Leaf size=661 \[ \frac{(c+d \tan (e+f x))^3 \left (-3 a^2 b d^2 (3 c C-16 B d)+4 a^3 C d^3+3 a b^2 d \left (20 d^2 (A-C)-5 B c d+2 c^2 C\right )+b^3 \left (-\left (5 c d^2 (A-C)-2 B c^2 d+20 B d^3+c^3 C\right )\right )\right )}{60 d^4 f}+\frac{\log (\cos (e+f x)) \left (3 a^2 b \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )+a^3 \left (-\left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )+3 a b^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )-b^3 \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )\right )}{f}-x \left (3 a^2 b \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )+a^3 \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )-3 a b^2 \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )-b^3 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )+\frac{\left (3 a^2 b (A-C)+a^3 B-3 a b^2 B-b^3 (A-C)\right ) (c+d \tan (e+f x))^2}{2 f}+\frac{d \tan (e+f x) \left (3 a^2 b (A c-B d-c C)+a^3 (d (A-C)+B c)-3 a b^2 (d (A-C)+B c)-b^3 (A c-B d-c C)\right )}{f}+\frac{b \tan (e+f x) (c+d \tan (e+f x))^3 \left (5 b d^2 (a B+A b-b C)+(b c-a d) (-a C d-2 b B d+b c C)\right )}{20 d^3 f}-\frac{(-a C d-2 b B d+b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3}{10 d^2 f}+\frac{C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^3}{6 d f} \]

[Out]

-((a^3*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) - 3*a*b^2*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) + 3*a^2*b
*(2*c*(A - C)*d + B*(c^2 - d^2)) - b^3*(2*c*(A - C)*d + B*(c^2 - d^2)))*x) + ((3*a^2*b*(c^2*C + 2*B*c*d - C*d^
2 - A*(c^2 - d^2)) - b^3*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) - a^3*(2*c*(A - C)*d + B*(c^2 - d^2)) + 3*a
*b^2*(2*c*(A - C)*d + B*(c^2 - d^2)))*Log[Cos[e + f*x]])/f + (d*(3*a^2*b*(A*c - c*C - B*d) - b^3*(A*c - c*C -
B*d) + a^3*(B*c + (A - C)*d) - 3*a*b^2*(B*c + (A - C)*d))*Tan[e + f*x])/f + ((a^3*B - 3*a*b^2*B + 3*a^2*b*(A -
C) - b^3*(A - C))*(c + d*Tan[e + f*x])^2)/(2*f) + ((4*a^3*C*d^3 - 3*a^2*b*d^2*(3*c*C - 16*B*d) + 3*a*b^2*d*(2
*c^2*C - 5*B*c*d + 20*(A - C)*d^2) - b^3*(c^3*C - 2*B*c^2*d + 5*c*(A - C)*d^2 + 20*B*d^3))*(c + d*Tan[e + f*x]
)^3)/(60*d^4*f) + (b*(5*b*(A*b + a*B - b*C)*d^2 + (b*c - a*d)*(b*c*C - 2*b*B*d - a*C*d))*Tan[e + f*x]*(c + d*T
an[e + f*x])^3)/(20*d^3*f) - ((b*c*C - 2*b*B*d - a*C*d)*(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^3)/(10*d^2
*f) + (C*(a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^3)/(6*d*f)

________________________________________________________________________________________

Rubi [A]  time = 2.38359, antiderivative size = 661, 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} \[ \frac{(c+d \tan (e+f x))^3 \left (-3 a^2 b d^2 (3 c C-16 B d)+4 a^3 C d^3+3 a b^2 d \left (20 d^2 (A-C)-5 B c d+2 c^2 C\right )+b^3 \left (-\left (5 c d^2 (A-C)-2 B c^2 d+20 B d^3+c^3 C\right )\right )\right )}{60 d^4 f}+\frac{\log (\cos (e+f x)) \left (3 a^2 b \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )+a^3 \left (-\left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )+3 a b^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )-b^3 \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )\right )}{f}-x \left (3 a^2 b \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )+a^3 \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )-3 a b^2 \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )-b^3 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )+\frac{\left (3 a^2 b (A-C)+a^3 B-3 a b^2 B-b^3 (A-C)\right ) (c+d \tan (e+f x))^2}{2 f}+\frac{d \tan (e+f x) \left (3 a^2 b (A c-B d-c C)+a^3 (d (A-C)+B c)-3 a b^2 (d (A-C)+B c)-b^3 (A c-B d-c C)\right )}{f}+\frac{b \tan (e+f x) (c+d \tan (e+f x))^3 \left (5 b d^2 (a B+A b-b C)+(b c-a d) (-a C d-2 b B d+b c C)\right )}{20 d^3 f}-\frac{(-a C d-2 b B d+b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3}{10 d^2 f}+\frac{C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^3}{6 d f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a^3*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) - 3*a*b^2*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) + 3*a^2*b
*(2*c*(A - C)*d + B*(c^2 - d^2)) - b^3*(2*c*(A - C)*d + B*(c^2 - d^2)))*x) + ((3*a^2*b*(c^2*C + 2*B*c*d - C*d^
2 - A*(c^2 - d^2)) - b^3*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) - a^3*(2*c*(A - C)*d + B*(c^2 - d^2)) + 3*a
*b^2*(2*c*(A - C)*d + B*(c^2 - d^2)))*Log[Cos[e + f*x]])/f + (d*(3*a^2*b*(A*c - c*C - B*d) - b^3*(A*c - c*C -
B*d) + a^3*(B*c + (A - C)*d) - 3*a*b^2*(B*c + (A - C)*d))*Tan[e + f*x])/f + ((a^3*B - 3*a*b^2*B + 3*a^2*b*(A -
C) - b^3*(A - C))*(c + d*Tan[e + f*x])^2)/(2*f) + ((4*a^3*C*d^3 - 3*a^2*b*d^2*(3*c*C - 16*B*d) + 3*a*b^2*d*(2
*c^2*C - 5*B*c*d + 20*(A - C)*d^2) - b^3*(c^3*C - 2*B*c^2*d + 5*c*(A - C)*d^2 + 20*B*d^3))*(c + d*Tan[e + f*x]
)^3)/(60*d^4*f) + (b*(5*b*(A*b + a*B - b*C)*d^2 + (b*c - a*d)*(b*c*C - 2*b*B*d - a*C*d))*Tan[e + f*x]*(c + d*T
an[e + f*x])^3)/(20*d^3*f) - ((b*c*C - 2*b*B*d - a*C*d)*(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^3)/(10*d^2
*f) + (C*(a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^3)/(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))^3 (c+d \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx &=\frac{C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^3}{6 d f}+\frac{\int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^2 \left (-3 (b c C-a (2 A-C) d)+6 (A b+a B-b C) d \tan (e+f x)-3 (b c C-2 b B d-a C d) \tan ^2(e+f x)\right ) \, dx}{6 d}\\ &=-\frac{(b c C-2 b B d-a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3}{10 d^2 f}+\frac{C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^3}{6 d f}+\frac{\int (a+b \tan (e+f x)) (c+d \tan (e+f x))^2 \left (6 \left (a^2 (5 A-4 C) d^2+b^2 c (c C-2 B d)-a b d (2 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)+6 \left (5 b (A b+a B-b C) d^2+(b c-a d) (b c C-2 b B d-a C d)\right ) \tan ^2(e+f x)\right ) \, dx}{30 d^2}\\ &=\frac{b \left (5 b (A b+a B-b C) d^2+(b c-a d) (b c C-2 b B d-a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^3}{20 d^3 f}-\frac{(b c C-2 b B d-a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3}{10 d^2 f}+\frac{C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^3}{6 d f}-\frac{\int (c+d \tan (e+f x))^2 \left (-6 \left (4 a^3 (5 A-4 C) d^3+3 a b^2 c d (2 c C-5 B d)-3 a^2 b d^2 (3 c C+4 B d)-b^3 c \left (c^2 C-2 B c d+5 (A-C) d^2\right )\right )-120 \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) d^3 \tan (e+f x)-6 \left (4 a^3 C d^3-3 a^2 b d^2 (3 c C-16 B d)+3 a b^2 d \left (2 c^2 C-5 B c d+20 (A-C) d^2\right )-b^3 \left (c^3 C-2 B c^2 d+5 c (A-C) d^2+20 B d^3\right )\right ) \tan ^2(e+f x)\right ) \, dx}{120 d^3}\\ &=\frac{\left (4 a^3 C d^3-3 a^2 b d^2 (3 c C-16 B d)+3 a b^2 d \left (2 c^2 C-5 B c d+20 (A-C) d^2\right )-b^3 \left (c^3 C-2 B c^2 d+5 c (A-C) d^2+20 B d^3\right )\right ) (c+d \tan (e+f x))^3}{60 d^4 f}+\frac{b \left (5 b (A b+a B-b C) d^2+(b c-a d) (b c C-2 b B d-a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^3}{20 d^3 f}-\frac{(b c C-2 b B d-a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3}{10 d^2 f}+\frac{C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^3}{6 d f}-\frac{\int (c+d \tan (e+f x))^2 \left (120 \left (3 a^2 b B-b^3 B-a^3 (A-C)+3 a b^2 (A-C)\right ) d^3-120 \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) d^3 \tan (e+f x)\right ) \, dx}{120 d^3}\\ &=\frac{\left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) (c+d \tan (e+f x))^2}{2 f}+\frac{\left (4 a^3 C d^3-3 a^2 b d^2 (3 c C-16 B d)+3 a b^2 d \left (2 c^2 C-5 B c d+20 (A-C) d^2\right )-b^3 \left (c^3 C-2 B c^2 d+5 c (A-C) d^2+20 B d^3\right )\right ) (c+d \tan (e+f x))^3}{60 d^4 f}+\frac{b \left (5 b (A b+a B-b C) d^2+(b c-a d) (b c C-2 b B d-a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^3}{20 d^3 f}-\frac{(b c C-2 b B d-a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3}{10 d^2 f}+\frac{C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^3}{6 d f}-\frac{\int (c+d \tan (e+f x)) \left (-120 d^3 \left (a^3 (A c-c C-B d)-3 a b^2 (A c-c C-B d)-3 a^2 b (B c+(A-C) d)+b^3 (B c+(A-C) d)\right )-120 d^3 \left (3 a^2 b (A c-c C-B d)-b^3 (A c-c C-B d)+a^3 (B c+(A-C) d)-3 a b^2 (B c+(A-C) d)\right ) \tan (e+f x)\right ) \, dx}{120 d^3}\\ &=-\left (a^3 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-3 a b^2 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )+3 a^2 b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )-b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) x+\frac{d \left (3 a^2 b (A c-c C-B d)-b^3 (A c-c C-B d)+a^3 (B c+(A-C) d)-3 a b^2 (B c+(A-C) d)\right ) \tan (e+f x)}{f}+\frac{\left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) (c+d \tan (e+f x))^2}{2 f}+\frac{\left (4 a^3 C d^3-3 a^2 b d^2 (3 c C-16 B d)+3 a b^2 d \left (2 c^2 C-5 B c d+20 (A-C) d^2\right )-b^3 \left (c^3 C-2 B c^2 d+5 c (A-C) d^2+20 B d^3\right )\right ) (c+d \tan (e+f x))^3}{60 d^4 f}+\frac{b \left (5 b (A b+a B-b C) d^2+(b c-a d) (b c C-2 b B d-a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^3}{20 d^3 f}-\frac{(b c C-2 b B d-a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3}{10 d^2 f}+\frac{C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^3}{6 d f}-\left (3 a^2 b \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-b^3 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-a^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )+3 a b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) \int \tan (e+f x) \, dx\\ &=-\left (a^3 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-3 a b^2 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )+3 a^2 b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )-b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) x+\frac{\left (3 a^2 b \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-b^3 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-a^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )+3 a b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) \log (\cos (e+f x))}{f}+\frac{d \left (3 a^2 b (A c-c C-B d)-b^3 (A c-c C-B d)+a^3 (B c+(A-C) d)-3 a b^2 (B c+(A-C) d)\right ) \tan (e+f x)}{f}+\frac{\left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) (c+d \tan (e+f x))^2}{2 f}+\frac{\left (4 a^3 C d^3-3 a^2 b d^2 (3 c C-16 B d)+3 a b^2 d \left (2 c^2 C-5 B c d+20 (A-C) d^2\right )-b^3 \left (c^3 C-2 B c^2 d+5 c (A-C) d^2+20 B d^3\right )\right ) (c+d \tan (e+f x))^3}{60 d^4 f}+\frac{b \left (5 b (A b+a B-b C) d^2+(b c-a d) (b c C-2 b B d-a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^3}{20 d^3 f}-\frac{(b c C-2 b B d-a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3}{10 d^2 f}+\frac{C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^3}{6 d f}\\ \end{align*}

Mathematica [C]  time = 6.64793, size = 573, normalized size = 0.87 \[ \frac{C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^3}{6 d f}+\frac{-\frac{3 (-a C d-2 b B d+b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3}{5 d f}+\frac{\frac{3 b \tan (e+f x) (c+d \tan (e+f x))^3 \left (5 b d^2 (a B+A b-b C)+(b c-a d) (-a C d-2 b B d+b c C)\right )}{2 d f}-\frac{\frac{(c+d \tan (e+f x))^3 \left (b \left (6 c \left (5 b d^2 (a B+A b-b C)+(b c-a d) (-a C d-2 b B d+b c C)\right )-120 d^3 \left (a^2 B+2 a b (A-C)-b^2 B\right )\right )-24 a d \left (5 b d^2 (a B+A b-b C)+(b c-a d) (-a C d-2 b B d+b c C)\right )\right )}{3 d f}-\frac{60 \left (d^2 \left (3 a^2 b (A-C)+a^3 B-3 a b^2 B-b^3 (A-C)\right ) \left (6 c d^2 \tan (e+f x)+(-d+i c)^3 \log (-\tan (e+f x)+i)-(d+i c)^3 \log (\tan (e+f x)+i)+d^3 \tan ^2(e+f x)\right )+d^2 \left (3 a^2 b (A c+B d-c C)+a^3 (B c-d (A-C))-3 a b^2 (B c-d (A-C))-b^3 (A c+B d-c C)\right ) \left (-i (c-i d)^2 \log (\tan (e+f x)+i)+i (c+i d)^2 \log (-\tan (e+f x)+i)-2 d^2 \tan (e+f x)\right )\right )}{f}}{4 d}}{5 d}}{6 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(C*(a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^3)/(6*d*f) + ((-3*(b*c*C - 2*b*B*d - a*C*d)*(a + b*Tan[e + f*x]
)^2*(c + d*Tan[e + f*x])^3)/(5*d*f) + ((3*b*(5*b*(A*b + a*B - b*C)*d^2 + (b*c - a*d)*(b*c*C - 2*b*B*d - a*C*d)
)*Tan[e + f*x]*(c + d*Tan[e + f*x])^3)/(2*d*f) - (((-24*a*d*(5*b*(A*b + a*B - b*C)*d^2 + (b*c - a*d)*(b*c*C -
2*b*B*d - a*C*d)) + b*(-120*(a^2*B - b^2*B + 2*a*b*(A - C))*d^3 + 6*c*(5*b*(A*b + a*B - b*C)*d^2 + (b*c - a*d)
*(b*c*C - 2*b*B*d - a*C*d))))*(c + d*Tan[e + f*x])^3)/(3*d*f) - (60*(d^2*(3*a^2*b*(A*c - c*C + B*d) - b^3*(A*c
- c*C + B*d) + a^3*(B*c - (A - C)*d) - 3*a*b^2*(B*c - (A - C)*d))*(I*(c + I*d)^2*Log[I - Tan[e + f*x]] - I*(c
- I*d)^2*Log[I + Tan[e + f*x]] - 2*d^2*Tan[e + f*x]) + (a^3*B - 3*a*b^2*B + 3*a^2*b*(A - C) - b^3*(A - C))*d^
2*((I*c - d)^3*Log[I - Tan[e + f*x]] - (I*c + d)^3*Log[I + Tan[e + f*x]] + 6*c*d^2*Tan[e + f*x] + d^3*Tan[e +
f*x]^2)))/f)/(4*d))/(5*d))/(6*d)

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Maple [B]  time = 0.024, size = 1807, normalized size = 2.7 \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))^3*(c+d*tan(f*x+e))^2*(A+B*tan(f*x+e)+C*tan(f*x+e)^2),x)

[Out]

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

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Maxima [A]  time = 1.48842, size = 933, normalized size = 1.41 \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))^3*(c+d*tan(f*x+e))^2*(A+B*tan(f*x+e)+C*tan(f*x+e)^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.28231, size = 1490, normalized size = 2.25 \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))^3*(c+d*tan(f*x+e))^2*(A+B*tan(f*x+e)+C*tan(f*x+e)^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 6.12102, size = 1819, normalized size = 2.75 \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))**3*(c+d*tan(f*x+e))**2*(A+B*tan(f*x+e)+C*tan(f*x+e)**2),x)

[Out]

Piecewise((A*a**3*c**2*x + A*a**3*c*d*log(tan(e + f*x)**2 + 1)/f - A*a**3*d**2*x + A*a**3*d**2*tan(e + f*x)/f
+ 3*A*a**2*b*c**2*log(tan(e + f*x)**2 + 1)/(2*f) - 6*A*a**2*b*c*d*x + 6*A*a**2*b*c*d*tan(e + f*x)/f - 3*A*a**2
*b*d**2*log(tan(e + f*x)**2 + 1)/(2*f) + 3*A*a**2*b*d**2*tan(e + f*x)**2/(2*f) - 3*A*a*b**2*c**2*x + 3*A*a*b**
2*c**2*tan(e + f*x)/f - 3*A*a*b**2*c*d*log(tan(e + f*x)**2 + 1)/f + 3*A*a*b**2*c*d*tan(e + f*x)**2/f + 3*A*a*b
**2*d**2*x + A*a*b**2*d**2*tan(e + f*x)**3/f - 3*A*a*b**2*d**2*tan(e + f*x)/f - A*b**3*c**2*log(tan(e + f*x)**
2 + 1)/(2*f) + A*b**3*c**2*tan(e + f*x)**2/(2*f) + 2*A*b**3*c*d*x + 2*A*b**3*c*d*tan(e + f*x)**3/(3*f) - 2*A*b
**3*c*d*tan(e + f*x)/f + A*b**3*d**2*log(tan(e + f*x)**2 + 1)/(2*f) + A*b**3*d**2*tan(e + f*x)**4/(4*f) - A*b*
*3*d**2*tan(e + f*x)**2/(2*f) + B*a**3*c**2*log(tan(e + f*x)**2 + 1)/(2*f) - 2*B*a**3*c*d*x + 2*B*a**3*c*d*tan
(e + f*x)/f - B*a**3*d**2*log(tan(e + f*x)**2 + 1)/(2*f) + B*a**3*d**2*tan(e + f*x)**2/(2*f) - 3*B*a**2*b*c**2
*x + 3*B*a**2*b*c**2*tan(e + f*x)/f - 3*B*a**2*b*c*d*log(tan(e + f*x)**2 + 1)/f + 3*B*a**2*b*c*d*tan(e + f*x)*
*2/f + 3*B*a**2*b*d**2*x + B*a**2*b*d**2*tan(e + f*x)**3/f - 3*B*a**2*b*d**2*tan(e + f*x)/f - 3*B*a*b**2*c**2*
log(tan(e + f*x)**2 + 1)/(2*f) + 3*B*a*b**2*c**2*tan(e + f*x)**2/(2*f) + 6*B*a*b**2*c*d*x + 2*B*a*b**2*c*d*tan
(e + f*x)**3/f - 6*B*a*b**2*c*d*tan(e + f*x)/f + 3*B*a*b**2*d**2*log(tan(e + f*x)**2 + 1)/(2*f) + 3*B*a*b**2*d
**2*tan(e + f*x)**4/(4*f) - 3*B*a*b**2*d**2*tan(e + f*x)**2/(2*f) + B*b**3*c**2*x + B*b**3*c**2*tan(e + f*x)**
3/(3*f) - B*b**3*c**2*tan(e + f*x)/f + B*b**3*c*d*log(tan(e + f*x)**2 + 1)/f + B*b**3*c*d*tan(e + f*x)**4/(2*f
) - B*b**3*c*d*tan(e + f*x)**2/f - B*b**3*d**2*x + B*b**3*d**2*tan(e + f*x)**5/(5*f) - B*b**3*d**2*tan(e + f*x
)**3/(3*f) + B*b**3*d**2*tan(e + f*x)/f - C*a**3*c**2*x + C*a**3*c**2*tan(e + f*x)/f - C*a**3*c*d*log(tan(e +
f*x)**2 + 1)/f + C*a**3*c*d*tan(e + f*x)**2/f + C*a**3*d**2*x + C*a**3*d**2*tan(e + f*x)**3/(3*f) - C*a**3*d**
2*tan(e + f*x)/f - 3*C*a**2*b*c**2*log(tan(e + f*x)**2 + 1)/(2*f) + 3*C*a**2*b*c**2*tan(e + f*x)**2/(2*f) + 6*
C*a**2*b*c*d*x + 2*C*a**2*b*c*d*tan(e + f*x)**3/f - 6*C*a**2*b*c*d*tan(e + f*x)/f + 3*C*a**2*b*d**2*log(tan(e
+ f*x)**2 + 1)/(2*f) + 3*C*a**2*b*d**2*tan(e + f*x)**4/(4*f) - 3*C*a**2*b*d**2*tan(e + f*x)**2/(2*f) + 3*C*a*b
**2*c**2*x + C*a*b**2*c**2*tan(e + f*x)**3/f - 3*C*a*b**2*c**2*tan(e + f*x)/f + 3*C*a*b**2*c*d*log(tan(e + f*x
)**2 + 1)/f + 3*C*a*b**2*c*d*tan(e + f*x)**4/(2*f) - 3*C*a*b**2*c*d*tan(e + f*x)**2/f - 3*C*a*b**2*d**2*x + 3*
C*a*b**2*d**2*tan(e + f*x)**5/(5*f) - C*a*b**2*d**2*tan(e + f*x)**3/f + 3*C*a*b**2*d**2*tan(e + f*x)/f + C*b**
3*c**2*log(tan(e + f*x)**2 + 1)/(2*f) + C*b**3*c**2*tan(e + f*x)**4/(4*f) - C*b**3*c**2*tan(e + f*x)**2/(2*f)
- 2*C*b**3*c*d*x + 2*C*b**3*c*d*tan(e + f*x)**5/(5*f) - 2*C*b**3*c*d*tan(e + f*x)**3/(3*f) + 2*C*b**3*c*d*tan(
e + f*x)/f - C*b**3*d**2*log(tan(e + f*x)**2 + 1)/(2*f) + C*b**3*d**2*tan(e + f*x)**6/(6*f) - C*b**3*d**2*tan(
e + f*x)**4/(4*f) + C*b**3*d**2*tan(e + f*x)**2/(2*f), Ne(f, 0)), (x*(a + b*tan(e))**3*(c + d*tan(e))**2*(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))^3*(c+d*tan(f*x+e))^2*(A+B*tan(f*x+e)+C*tan(f*x+e)^2),x, algorithm="giac")

[Out]

Timed out