\(\int \frac {(a+b \tan (e+f x))^3 (A+B \tan (e+f x)+C \tan ^2(e+f x))}{(c+d \tan (e+f x))^3} \, dx\) [84]

Optimal result

Integrand size = 45, antiderivative size = 804 \[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^3} \, dx=-\frac {\left (3 a b^2 \left (A c^3-c^3 C+3 B c^2 d-3 A c d^2+3 c C d^2-B d^3\right )+a^3 \left (c^3 C-3 B c^2 d-3 c C d^2+B d^3-A \left (c^3-3 c d^2\right )\right )-3 a^2 b \left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )+b^3 \left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )\right ) x}{\left (c^2+d^2\right )^3}-\frac {\left (3 a^2 b \left (A c^3-c^3 C+3 B c^2 d-3 A c d^2+3 c C d^2-B d^3\right )-b^3 \left (A c^3-c^3 C+3 B c^2 d-3 A c d^2+3 c C d^2-B d^3\right )-a^3 \left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )+3 a b^2 \left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )\right ) \log (\cos (e+f x))}{\left (c^2+d^2\right )^3 f}-\frac {(b c-a d) \left (b^2 \left (3 c^6 C-B c^5 d+9 c^4 C d^2-3 B c^3 d^3-c^2 (A-10 C) d^4-6 B c d^5+3 A d^6\right )+a^2 d^3 \left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )+a b d^2 \left (8 c (A-C) d^3-B \left (c^4+6 c^2 d^2-3 d^4\right )\right )\right ) \log (c+d \tan (e+f x))}{d^4 \left (c^2+d^2\right )^3 f}+\frac {b^2 \left (b \left (3 c^4 C-B c^3 d+6 c^2 C d^2-3 B c d^3+(2 A+C) d^4\right )+a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \tan (e+f x)}{d^3 \left (c^2+d^2\right )^2 f}-\frac {\left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^3}{2 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}-\frac {\left (b \left (3 c^4 C-B c^3 d-c^2 (A-7 C) d^2-5 B c d^3+3 A d^4\right )+2 a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) (a+b \tan (e+f x))^2}{2 d^2 \left (c^2+d^2\right )^2 f (c+d \tan (e+f x))} \]

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

Rubi [A] (verified)

Time = 2.97 (sec) , antiderivative size = 804, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3726, 3718, 3707, 3698, 31, 3556} \[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^3} \, dx=-\frac {\left (C c^2-B d c+A d^2\right ) (a+b \tan (e+f x))^3}{2 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}-\frac {\left (2 a \left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) d^2+b \left (3 C c^4-B d c^3-(A-7 C) d^2 c^2-5 B d^3 c+3 A d^4\right )\right ) (a+b \tan (e+f x))^2}{2 d^2 \left (c^2+d^2\right )^2 f (c+d \tan (e+f x))}-\frac {\left (\left (C c^3-3 B d c^2-3 C d^2 c+B d^3-A \left (c^3-3 c d^2\right )\right ) a^3-3 b \left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right ) a^2+3 b^2 \left (A c^3-C c^3+3 B d c^2-3 A d^2 c+3 C d^2 c-B d^3\right ) a+b^3 \left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )\right ) x}{\left (c^2+d^2\right )^3}-\frac {\left (-\left (\left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right ) a^3\right )+3 b \left (A c^3-C c^3+3 B d c^2-3 A d^2 c+3 C d^2 c-B d^3\right ) a^2+3 b^2 \left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right ) a-b^3 \left (A c^3-C c^3+3 B d c^2-3 A d^2 c+3 C d^2 c-B d^3\right )\right ) \log (\cos (e+f x))}{\left (c^2+d^2\right )^3 f}-\frac {(b c-a d) \left (a^2 \left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right ) d^3+a b \left (8 c (A-C) d^3-B \left (c^4+6 d^2 c^2-3 d^4\right )\right ) d^2+b^2 \left (3 C c^6-B d c^5+9 C d^2 c^4-3 B d^3 c^3-(A-10 C) d^4 c^2-6 B d^5 c+3 A d^6\right )\right ) \log (c+d \tan (e+f x))}{d^4 \left (c^2+d^2\right )^3 f}+\frac {b^2 \left (a \left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) d^2+b \left (3 C c^4-B d c^3+6 C d^2 c^2-3 B d^3 c+(2 A+C) d^4\right )\right ) \tan (e+f x)}{d^3 \left (c^2+d^2\right )^2 f} \]

[In]

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

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 7.37 (sec) , antiderivative size = 454, normalized size of antiderivative = 0.56 \[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^3} \, dx=\frac {\frac {(a+i b)^3 (A+i B-C) \log (i-\tan (e+f x))}{(-i c+d)^3}+\frac {(a-i b)^3 (A-i B-C) \log (i+\tan (e+f x))}{(i c+d)^3}+\frac {2 (-b c+a d) \left (b^2 \left (3 c^6 C-B c^5 d+9 c^4 C d^2-3 B c^3 d^3-c^2 (A-10 C) d^4-6 B c d^5+3 A d^6\right )+a^2 d^3 \left (-\left ((A-C) d \left (-3 c^2+d^2\right )\right )-B \left (c^3-3 c d^2\right )\right )-a b d^2 \left (8 c (-A+C) d^3+B \left (c^4+6 c^2 d^2-3 d^4\right )\right )\right ) \log (c+d \tan (e+f x))}{d^4 \left (c^2+d^2\right )^3}+\frac {(b c-a d)^3 \left (3 c^2 C-B c d+(A+2 C) d^2\right )}{d^4 \left (c^2+d^2\right ) (c+d \tan (e+f x))^2}+\frac {2 C (a+b \tan (e+f x))^3}{d (c+d \tan (e+f x))^2}-\frac {2 (b c-a d)^2 \left (b \left (6 c^4 C-2 B c^3 d+c^2 (A+11 C) d^2-4 B c d^3+3 (A+C) d^4\right )+a d^2 \left (2 c (A-C) d+B \left (-c^2+d^2\right )\right )\right )}{d^4 \left (c^2+d^2\right )^2 (c+d \tan (e+f x))}}{2 f} \]

[In]

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

[Out]

(((a + I*b)^3*(A + I*B - C)*Log[I - Tan[e + f*x]])/((-I)*c + d)^3 + ((a - I*b)^3*(A - I*B - C)*Log[I + Tan[e +
f*x]])/(I*c + d)^3 + (2*(-(b*c) + a*d)*(b^2*(3*c^6*C - B*c^5*d + 9*c^4*C*d^2 - 3*B*c^3*d^3 - c^2*(A - 10*C)*d
^4 - 6*B*c*d^5 + 3*A*d^6) + a^2*d^3*(-((A - C)*d*(-3*c^2 + d^2)) - B*(c^3 - 3*c*d^2)) - a*b*d^2*(8*c*(-A + C)*
d^3 + B*(c^4 + 6*c^2*d^2 - 3*d^4)))*Log[c + d*Tan[e + f*x]])/(d^4*(c^2 + d^2)^3) + ((b*c - a*d)^3*(3*c^2*C - B
*c*d + (A + 2*C)*d^2))/(d^4*(c^2 + d^2)*(c + d*Tan[e + f*x])^2) + (2*C*(a + b*Tan[e + f*x])^3)/(d*(c + d*Tan[e
+ f*x])^2) - (2*(b*c - a*d)^2*(b*(6*c^4*C - 2*B*c^3*d + c^2*(A + 11*C)*d^2 - 4*B*c*d^3 + 3*(A + C)*d^4) + a*d
^2*(2*c*(A - C)*d + B*(-c^2 + d^2))))/(d^4*(c^2 + d^2)^2*(c + d*Tan[e + f*x])))/(2*f)

Maple [A] (verified)

Time = 0.58 (sec) , antiderivative size = 1271, normalized size of antiderivative = 1.58

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1271\)
default	\(\text {Expression too large to display}\)	\(1271\)
norman	\(\text {Expression too large to display}\)	\(2076\)
parallelrisch	\(\text {Expression too large to display}\)	\(6687\)
risch	\(\text {Expression too large to display}\)	\(6825\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2490 vs. \(2 (797) = 1594\).

Time = 1.33 (sec) , antiderivative size = 2490, normalized size of antiderivative = 3.10 \[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/2*(3*C*b^3*c^7*d^2 + A*a^3*d^9 - (3*C*a*b^2 + B*b^3)*c^6*d^3 - (3*C*a^2*b + 3*B*a*b^2 + (A - 9*C)*b^3)*c^5*
d^4 + (3*C*a^3 + 9*B*a^2*b + 3*(3*A - 7*C)*a*b^2 - 7*B*b^3)*c^4*d^5 - 5*(B*a^3 + 3*(A - C)*a^2*b - 3*B*a*b^2 -
A*b^3)*c^3*d^6 + ((7*A - 3*C)*a^3 - 9*B*a^2*b - 9*A*a*b^2)*c^2*d^7 + (B*a^3 + 3*A*a^2*b)*c*d^8 - 2*(C*b^3*c^6
*d^3 + 3*C*b^3*c^4*d^5 + 3*C*b^3*c^2*d^7 + C*b^3*d^9)*tan(f*x + e)^3 - 2*(((A - C)*a^3 - 3*B*a^2*b - 3*(A - C)
*a*b^2 + B*b^3)*c^5*d^4 + 3*(B*a^3 + 3*(A - C)*a^2*b - 3*B*a*b^2 - (A - C)*b^3)*c^4*d^5 - 3*((A - C)*a^3 - 3*B
*a^2*b - 3*(A - C)*a*b^2 + B*b^3)*c^3*d^6 - (B*a^3 + 3*(A - C)*a^2*b - 3*B*a*b^2 - (A - C)*b^3)*c^2*d^7)*f*x -
(9*C*b^3*c^7*d^2 - A*a^3*d^9 - 3*(3*C*a*b^2 + B*b^3)*c^6*d^3 + (3*C*a^2*b + 3*B*a*b^2 + (A + 23*C)*b^3)*c^5*d
^4 + (C*a^3 + 3*B*a^2*b + 3*(A - 9*C)*a*b^2 - 9*B*b^3)*c^4*d^5 - (3*B*a^3 + 3*(3*A - 7*C)*a^2*b - 21*B*a*b^2 -
(7*A + 12*C)*b^3)*c^3*d^6 + 5*((A - C)*a^3 - 3*B*a^2*b - 3*A*a*b^2)*c^2*d^7 + (3*B*a^3 + 9*A*a^2*b + 4*C*b^3)
*c*d^8 + 2*(((A - C)*a^3 - 3*B*a^2*b - 3*(A - C)*a*b^2 + B*b^3)*c^3*d^6 + 3*(B*a^3 + 3*(A - C)*a^2*b - 3*B*a*b
^2 - (A - C)*b^3)*c^2*d^7 - 3*((A - C)*a^3 - 3*B*a^2*b - 3*(A - C)*a*b^2 + B*b^3)*c*d^8 - (B*a^3 + 3*(A - C)*a
^2*b - 3*B*a*b^2 - (A - C)*b^3)*d^9)*f*x)*tan(f*x + e)^2 + (3*C*b^3*c^9 + 9*C*b^3*c^7*d^2 - (3*C*a*b^2 + B*b^3
)*c^8*d - 3*(3*C*a*b^2 + B*b^3)*c^6*d^3 + (B*a^3 + 3*(A - C)*a^2*b - 3*B*a*b^2 - (A - 10*C)*b^3)*c^5*d^4 - 3*(
(A - C)*a^3 - 3*B*a^2*b - 3*(A - 2*C)*a*b^2 + 2*B*b^3)*c^4*d^5 - 3*(B*a^3 + 3*(A - C)*a^2*b - 3*B*a*b^2 - A*b^
3)*c^3*d^6 + ((A - C)*a^3 - 3*B*a^2*b - 3*A*a*b^2)*c^2*d^7 + (3*C*b^3*c^7*d^2 + 9*C*b^3*c^5*d^4 - (3*C*a*b^2 +
B*b^3)*c^6*d^3 - 3*(3*C*a*b^2 + B*b^3)*c^4*d^5 + (B*a^3 + 3*(A - C)*a^2*b - 3*B*a*b^2 - (A - 10*C)*b^3)*c^3*d
^6 - 3*((A - C)*a^3 - 3*B*a^2*b - 3*(A - 2*C)*a*b^2 + 2*B*b^3)*c^2*d^7 - 3*(B*a^3 + 3*(A - C)*a^2*b - 3*B*a*b^
2 - A*b^3)*c*d^8 + ((A - C)*a^3 - 3*B*a^2*b - 3*A*a*b^2)*d^9)*tan(f*x + e)^2 + 2*(3*C*b^3*c^8*d + 9*C*b^3*c^6*
d^3 - (3*C*a*b^2 + B*b^3)*c^7*d^2 - 3*(3*C*a*b^2 + B*b^3)*c^5*d^4 + (B*a^3 + 3*(A - C)*a^2*b - 3*B*a*b^2 - (A
- 10*C)*b^3)*c^4*d^5 - 3*((A - C)*a^3 - 3*B*a^2*b - 3*(A - 2*C)*a*b^2 + 2*B*b^3)*c^3*d^6 - 3*(B*a^3 + 3*(A - C
)*a^2*b - 3*B*a*b^2 - A*b^3)*c^2*d^7 + ((A - C)*a^3 - 3*B*a^2*b - 3*A*a*b^2)*c*d^8)*tan(f*x + e))*log((d^2*tan
(f*x + e)^2 + 2*c*d*tan(f*x + e) + c^2)/(tan(f*x + e)^2 + 1)) - (3*C*b^3*c^9 + 9*C*b^3*c^7*d^2 + 9*C*b^3*c^5*d
^4 + 3*C*b^3*c^3*d^6 - (3*C*a*b^2 + B*b^3)*c^8*d - 3*(3*C*a*b^2 + B*b^3)*c^6*d^3 - 3*(3*C*a*b^2 + B*b^3)*c^4*d
^5 - (3*C*a*b^2 + B*b^3)*c^2*d^7 + (3*C*b^3*c^7*d^2 + 9*C*b^3*c^5*d^4 + 9*C*b^3*c^3*d^6 + 3*C*b^3*c*d^8 - (3*C
*a*b^2 + B*b^3)*c^6*d^3 - 3*(3*C*a*b^2 + B*b^3)*c^4*d^5 - 3*(3*C*a*b^2 + B*b^3)*c^2*d^7 - (3*C*a*b^2 + B*b^3)*
d^9)*tan(f*x + e)^2 + 2*(3*C*b^3*c^8*d + 9*C*b^3*c^6*d^3 + 9*C*b^3*c^4*d^5 + 3*C*b^3*c^2*d^7 - (3*C*a*b^2 + B*
b^3)*c^7*d^2 - 3*(3*C*a*b^2 + B*b^3)*c^5*d^4 - 3*(3*C*a*b^2 + B*b^3)*c^3*d^6 - (3*C*a*b^2 + B*b^3)*c*d^8)*tan(
f*x + e))*log(1/(tan(f*x + e)^2 + 1)) - 2*(3*C*b^3*c^8*d + 6*C*b^3*c^6*d^3 - (3*C*a*b^2 + B*b^3)*c^7*d^2 + (C*
a^3 + 3*B*a^2*b + 3*(A - 3*C)*a*b^2 - 3*B*b^3)*c^5*d^4 - (2*B*a^3 + 3*(2*A - 3*C)*a^2*b - 9*B*a*b^2 - (3*A - 2
*C)*b^3)*c^4*d^5 + (3*(A - C)*a^3 - 9*B*a^2*b - 3*(3*A - 4*C)*a*b^2 + 4*B*b^3)*c^3*d^6 + (3*B*a^3 + 9*(A - C)*
a^2*b - 9*B*a*b^2 - (3*A - C)*b^3)*c^2*d^7 - ((3*A - 2*C)*a^3 - 6*B*a^2*b - 6*A*a*b^2)*c*d^8 - (B*a^3 + 3*A*a^
2*b)*d^9 + 2*(((A - C)*a^3 - 3*B*a^2*b - 3*(A - C)*a*b^2 + B*b^3)*c^4*d^5 + 3*(B*a^3 + 3*(A - C)*a^2*b - 3*B*a
*b^2 - (A - C)*b^3)*c^3*d^6 - 3*((A - C)*a^3 - 3*B*a^2*b - 3*(A - C)*a*b^2 + B*b^3)*c^2*d^7 - (B*a^3 + 3*(A -
C)*a^2*b - 3*B*a*b^2 - (A - C)*b^3)*c*d^8)*f*x)*tan(f*x + e))/((c^6*d^6 + 3*c^4*d^8 + 3*c^2*d^10 + d^12)*f*tan
(f*x + e)^2 + 2*(c^7*d^5 + 3*c^5*d^7 + 3*c^3*d^9 + c*d^11)*f*tan(f*x + e) + (c^8*d^4 + 3*c^6*d^6 + 3*c^4*d^8 +
c^2*d^10)*f)

Sympy [F(-2)]

Exception generated. \[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^3} \, dx=\text {Exception raised: AttributeError} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 1110, normalized size of antiderivative = 1.38 \[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2441 vs. \(2 (797) = 1594\).

Time = 1.39 (sec) , antiderivative size = 2441, normalized size of antiderivative = 3.04 \[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 18.17 (sec) , antiderivative size = 1172, normalized size of antiderivative = 1.46 \[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^3} \, dx=\frac {\ln \left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )\,\left (d^3\,\left (3\,B\,b^3\,c^4+9\,C\,a\,b^2\,c^4\right )-d^6\,\left (3\,A\,b^3\,c-3\,B\,a^3\,c-9\,A\,a^2\,b\,c+9\,B\,a\,b^2\,c+9\,C\,a^2\,b\,c\right )+d^5\,\left (3\,A\,a^3\,c^2+6\,B\,b^3\,c^2-3\,C\,a^3\,c^2-9\,A\,a\,b^2\,c^2-9\,B\,a^2\,b\,c^2+18\,C\,a\,b^2\,c^2\right )+d^4\,\left (A\,b^3\,c^3-B\,a^3\,c^3-10\,C\,b^3\,c^3-3\,A\,a^2\,b\,c^3+3\,B\,a\,b^2\,c^3+3\,C\,a^2\,b\,c^3\right )+d^7\,\left (C\,a^3-A\,a^3+3\,A\,a\,b^2+3\,B\,a^2\,b\right )+d\,\left (B\,b^3\,c^6+3\,C\,a\,b^2\,c^6\right )-3\,C\,b^3\,c^7-9\,C\,b^3\,c^5\,d^2\right )}{f\,\left (c^6\,d^4+3\,c^4\,d^6+3\,c^2\,d^8+d^{10}\right )}+\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,\left (A\,a^3+A\,b^3\,1{}\mathrm {i}-B\,a^3\,1{}\mathrm {i}+B\,b^3-C\,a^3-C\,b^3\,1{}\mathrm {i}-3\,A\,a\,b^2-A\,a^2\,b\,3{}\mathrm {i}+B\,a\,b^2\,3{}\mathrm {i}-3\,B\,a^2\,b+3\,C\,a\,b^2+C\,a^2\,b\,3{}\mathrm {i}\right )}{2\,f\,\left (-c^3\,1{}\mathrm {i}-3\,c^2\,d+c\,d^2\,3{}\mathrm {i}+d^3\right )}-\frac {\frac {A\,a^3\,d^7+5\,C\,b^3\,c^7+B\,a^3\,c\,d^6-3\,B\,b^3\,c^6\,d+5\,A\,a^3\,c^2\,d^5+5\,A\,b^3\,c^3\,d^4+A\,b^3\,c^5\,d^2-3\,B\,a^3\,c^3\,d^4-7\,B\,b^3\,c^4\,d^3-3\,C\,a^3\,c^2\,d^5+C\,a^3\,c^4\,d^3+9\,C\,b^3\,c^5\,d^2-9\,A\,a\,b^2\,c^2\,d^5+3\,A\,a\,b^2\,c^4\,d^3-9\,A\,a^2\,b\,c^3\,d^4+15\,B\,a\,b^2\,c^3\,d^4+3\,B\,a\,b^2\,c^5\,d^2-9\,B\,a^2\,b\,c^2\,d^5+3\,B\,a^2\,b\,c^4\,d^3-21\,C\,a\,b^2\,c^4\,d^3+15\,C\,a^2\,b\,c^3\,d^4+3\,C\,a^2\,b\,c^5\,d^2+3\,A\,a^2\,b\,c\,d^6-9\,C\,a\,b^2\,c^6\,d}{2\,d\,\left (c^4+2\,c^2\,d^2+d^4\right )}+\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (B\,a^3\,d^6+3\,C\,b^3\,c^6+3\,A\,a^2\,b\,d^6+2\,A\,a^3\,c\,d^5-2\,B\,b^3\,c^5\,d-2\,C\,a^3\,c\,d^5+3\,A\,b^3\,c^2\,d^4+A\,b^3\,c^4\,d^2-B\,a^3\,c^2\,d^4-4\,B\,b^3\,c^3\,d^3+5\,C\,b^3\,c^4\,d^2-3\,A\,a^2\,b\,c^2\,d^4+9\,B\,a\,b^2\,c^2\,d^4+3\,B\,a\,b^2\,c^4\,d^2-12\,C\,a\,b^2\,c^3\,d^3+9\,C\,a^2\,b\,c^2\,d^4+3\,C\,a^2\,b\,c^4\,d^2-6\,A\,a\,b^2\,c\,d^5-6\,B\,a^2\,b\,c\,d^5-6\,C\,a\,b^2\,c^5\,d\right )}{c^4+2\,c^2\,d^2+d^4}}{f\,\left (c^2\,d^3+2\,c\,d^4\,\mathrm {tan}\left (e+f\,x\right )+d^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\,b^3-B\,a^3-C\,b^3-3\,A\,a^2\,b+3\,B\,a\,b^2+3\,C\,a^2\,b+A\,a^3\,1{}\mathrm {i}+B\,b^3\,1{}\mathrm {i}-C\,a^3\,1{}\mathrm {i}-A\,a\,b^2\,3{}\mathrm {i}-B\,a^2\,b\,3{}\mathrm {i}+C\,a\,b^2\,3{}\mathrm {i}\right )}{2\,f\,\left (-c^3-c^2\,d\,3{}\mathrm {i}+3\,c\,d^2+d^3\,1{}\mathrm {i}\right )}+\frac {C\,b^3\,\mathrm {tan}\left (e+f\,x\right )}{d^3\,f} \]

[In]

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

[Out]

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