3.1.68 \(\int \frac {(c+d \tan (e+f x))^3 (A+B \tan (e+f x)+C \tan ^2(e+f x))}{(a+b \tan (e+f x))^2} \, dx\) [68]

Optimal. Leaf size=574 \[ -\frac {\left (b^2 \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )+a^2 \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-2 a b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) x}{\left (a^2+b^2\right )^2}+\frac {\left (2 a b \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )-a^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \log (\cos (e+f x))}{\left (a^2+b^2\right )^2 f}-\frac {(b c-a d)^2 \left (2 a^3 b B d-3 a^4 C d-b^4 (B c+3 A d)-2 a b^3 (A c-c C-2 B d)+a^2 b^2 (B c-(A+5 C) d)\right ) \log (a+b \tan (e+f x))}{b^4 \left (a^2+b^2\right )^2 f}-\frac {d^2 \left (3 a^3 C d-A b^2 (b c-a d)-b^3 (2 c C+B d)-a^2 b (3 c C+2 B d)+a b^2 (B c+2 C d)\right ) \tan (e+f x)}{b^3 \left (a^2+b^2\right ) f}+\frac {\left (2 A b^2-2 a b B+3 a^2 C+b^2 C\right ) d (c+d \tan (e+f x))^2}{2 b^2 \left (a^2+b^2\right ) f}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^3}{b \left (a^2+b^2\right ) f (a+b \tan (e+f x))} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 1.52, antiderivative size = 574, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {3726, 3728, 3718, 3707, 3698, 31, 3556} \begin {gather*} \frac {\log (\cos (e+f x)) \left (-\left (a^2 \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 c^3-3 A c d^2-3 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 \left (a^2+b^2\right )^2}-\frac {x \left (a^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 )-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 c^3-3 A c d^2-3 B c^2 d+B d^3-c^3 C+3 c C d^2\right )\right )}{\left (a^2+b^2\right )^2}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^3}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}+\frac {d \left (3 a^2 C-2 a b B+2 A b^2+b^2 C\right ) (c+d \tan (e+f x))^2}{2 b^2 f \left (a^2+b^2\right )}-\frac {d^2 \tan (e+f x) \left (3 a^3 C d-a^2 b (2 B d+3 c C)-A b^2 (b c-a d)+a b^2 (B c+2 C d)-b^3 (B d+2 c C)\right )}{b^3 f \left (a^2+b^2\right )}-\frac {(b c-a d)^2 \left (-3 a^4 C d+2 a^3 b B d+a^2 b^2 (B c-d (A+5 C))-2 a b^3 (A c-2 B d-c C)-b^4 (3 A d+B c)\right ) \log (a+b \tan (e+f x))}{b^4 f \left (a^2+b^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3728

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])))

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

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Mathematica [C] Result contains complex when optimal does not.
time = 7.75, size = 2467, normalized size = 4.30 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[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])^2,x]

[Out]

((a^2*A*c^3 - A*b^2*c^3 + 2*a*b*B*c^3 - a^2*c^3*C + b^2*c^3*C + 6*a*A*b*c^2*d - 3*a^2*B*c^2*d + 3*b^2*B*c^2*d
- 6*a*b*c^2*C*d - 3*a^2*A*c*d^2 + 3*A*b^2*c*d^2 - 6*a*b*B*c*d^2 + 3*a^2*c*C*d^2 - 3*b^2*c*C*d^2 - 2*a*A*b*d^3
+ a^2*B*d^3 - b^2*B*d^3 + 2*a*b*C*d^3)*(e + f*x)*Cos[e + f*x]*(a*Cos[e + f*x] + b*Sin[e + f*x])^2*(c + d*Tan[e
+ f*x])^3)/((a - I*b)^2*(a + I*b)^2*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^3*(a + b*Tan[e + f*x])^2) - (I*(-2*a^
6*A*b^8*c^3 + (2*I)*a^5*A*b^9*c^3 - 2*a^4*A*b^10*c^3 + (2*I)*a^3*A*b^11*c^3 + a^7*b^7*B*c^3 - I*a^6*b^8*B*c^3
- a^3*b^11*B*c^3 + I*a^2*b^12*B*c^3 + 2*a^6*b^8*c^3*C - (2*I)*a^5*b^9*c^3*C + 2*a^4*b^10*c^3*C - (2*I)*a^3*b^1
1*c^3*C + 3*a^7*A*b^7*c^2*d - (3*I)*a^6*A*b^8*c^2*d - 3*a^3*A*b^11*c^2*d + (3*I)*a^2*A*b^12*c^2*d + 6*a^6*b^8*
B*c^2*d - (6*I)*a^5*b^9*B*c^2*d + 6*a^4*b^10*B*c^2*d - (6*I)*a^3*b^11*B*c^2*d - 3*a^9*b^5*c^2*C*d + (3*I)*a^8*
b^6*c^2*C*d - 12*a^7*b^7*c^2*C*d + (12*I)*a^6*b^8*c^2*C*d - 9*a^5*b^9*c^2*C*d + (9*I)*a^4*b^10*c^2*C*d + 6*a^6
*A*b^8*c*d^2 - (6*I)*a^5*A*b^9*c*d^2 + 6*a^4*A*b^10*c*d^2 - (6*I)*a^3*A*b^11*c*d^2 - 3*a^9*b^5*B*c*d^2 + (3*I)
*a^8*b^6*B*c*d^2 - 12*a^7*b^7*B*c*d^2 + (12*I)*a^6*b^8*B*c*d^2 - 9*a^5*b^9*B*c*d^2 + (9*I)*a^4*b^10*B*c*d^2 +
6*a^10*b^4*c*C*d^2 - (6*I)*a^9*b^5*c*C*d^2 + 18*a^8*b^6*c*C*d^2 - (18*I)*a^7*b^7*c*C*d^2 + 12*a^6*b^8*c*C*d^2
- (12*I)*a^5*b^9*c*C*d^2 - a^9*A*b^5*d^3 + I*a^8*A*b^6*d^3 - 4*a^7*A*b^7*d^3 + (4*I)*a^6*A*b^8*d^3 - 3*a^5*A*b
^9*d^3 + (3*I)*a^4*A*b^10*d^3 + 2*a^10*b^4*B*d^3 - (2*I)*a^9*b^5*B*d^3 + 6*a^8*b^6*B*d^3 - (6*I)*a^7*b^7*B*d^3
+ 4*a^6*b^8*B*d^3 - (4*I)*a^5*b^9*B*d^3 - 3*a^11*b^3*C*d^3 + (3*I)*a^10*b^4*C*d^3 - 8*a^9*b^5*C*d^3 + (8*I)*a
^8*b^6*C*d^3 - 5*a^7*b^7*C*d^3 + (5*I)*a^6*b^8*C*d^3)*(e + f*x)*Cos[e + f*x]*(a*Cos[e + f*x] + b*Sin[e + f*x])
^2*(c + d*Tan[e + f*x])^3)/(a^2*(a - I*b)^4*(a + I*b)^3*b^7*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^3*(a + b*Tan[e
+ f*x])^2) - (I*(2*a*A*b^5*c^3 - a^2*b^4*B*c^3 + b^6*B*c^3 - 2*a*b^5*c^3*C - 3*a^2*A*b^4*c^2*d + 3*A*b^6*c^2*
d - 6*a*b^5*B*c^2*d + 3*a^4*b^2*c^2*C*d + 9*a^2*b^4*c^2*C*d - 6*a*A*b^5*c*d^2 + 3*a^4*b^2*B*c*d^2 + 9*a^2*b^4*
B*c*d^2 - 6*a^5*b*c*C*d^2 - 12*a^3*b^3*c*C*d^2 + a^4*A*b^2*d^3 + 3*a^2*A*b^4*d^3 - 2*a^5*b*B*d^3 - 4*a^3*b^3*B
*d^3 + 3*a^6*C*d^3 + 5*a^4*b^2*C*d^3)*ArcTan[Tan[e + f*x]]*Cos[e + f*x]*(a*Cos[e + f*x] + b*Sin[e + f*x])^2*(c
+ d*Tan[e + f*x])^3)/(b^4*(a^2 + b^2)^2*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^3*(a + b*Tan[e + f*x])^2) + ((-3*
b^2*c^2*C*d - 3*b^2*B*c*d^2 + 6*a*b*c*C*d^2 - A*b^2*d^3 + 2*a*b*B*d^3 - 3*a^2*C*d^3 + b^2*C*d^3)*Cos[e + f*x]*
Log[Cos[e + f*x]]*(a*Cos[e + f*x] + b*Sin[e + f*x])^2*(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])^2) + ((2*a*A*b^5*c^3 - a^2*b^4*B*c^3 + b^6*B*c^3 - 2*a*b^5*c^3*C - 3*a^2*A*b^4
*c^2*d + 3*A*b^6*c^2*d - 6*a*b^5*B*c^2*d + 3*a^4*b^2*c^2*C*d + 9*a^2*b^4*c^2*C*d - 6*a*A*b^5*c*d^2 + 3*a^4*b^2
*B*c*d^2 + 9*a^2*b^4*B*c*d^2 - 6*a^5*b*c*C*d^2 - 12*a^3*b^3*c*C*d^2 + a^4*A*b^2*d^3 + 3*a^2*A*b^4*d^3 - 2*a^5*
b*B*d^3 - 4*a^3*b^3*B*d^3 + 3*a^6*C*d^3 + 5*a^4*b^2*C*d^3)*Cos[e + f*x]*Log[(a*Cos[e + f*x] + b*Sin[e + f*x])^
2]*(a*Cos[e + f*x] + b*Sin[e + f*x])^2*(c + d*Tan[e + f*x])^3)/(2*b^4*(a^2 + b^2)^2*f*(c*Cos[e + f*x] + d*Sin[
e + f*x])^3*(a + b*Tan[e + f*x])^2) + (C*d^3*Sec[e + f*x]*(a*Cos[e + f*x] + b*Sin[e + f*x])^2*(c + d*Tan[e + f
*x])^3)/(2*b^2*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^3*(a + b*Tan[e + f*x])^2) + ((a*Cos[e + f*x] + b*Sin[e + f*
x])^2*(3*b*c*C*d^2*Sin[e + f*x] + b*B*d^3*Sin[e + f*x] - 2*a*C*d^3*Sin[e + f*x])*(c + d*Tan[e + f*x])^3)/(b^3*
f*(c*Cos[e + f*x] + d*Sin[e + f*x])^3*(a + b*Tan[e + f*x])^2) + (Cos[e + f*x]*(a*Cos[e + f*x] + b*Sin[e + f*x]
)*(A*b^5*c^3*Sin[e + f*x] - a*b^4*B*c^3*Sin[e + f*x] + a^2*b^3*c^3*C*Sin[e + f*x] - 3*a*A*b^4*c^2*d*Sin[e + f*
x] + 3*a^2*b^3*B*c^2*d*Sin[e + f*x] - 3*a^3*b^2*c^2*C*d*Sin[e + f*x] + 3*a^2*A*b^3*c*d^2*Sin[e + f*x] - 3*a^3*
b^2*B*c*d^2*Sin[e + f*x] + 3*a^4*b*c*C*d^2*Sin[e + f*x] - a^3*A*b^2*d^3*Sin[e + f*x] + a^4*b*B*d^3*Sin[e + f*x
] - a^5*C*d^3*Sin[e + f*x])*(c + d*Tan[e + f*x])^3)/(a*(a - I*b)*(a + I*b)*b^3*f*(c*Cos[e + f*x] + d*Sin[e + f
*x])^3*(a + b*Tan[e + f*x])^2)

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Maple [A]
time = 0.62, size = 829, normalized size = 1.44 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.54, size = 691, normalized size = 1.20 \begin {gather*} \frac {\frac {2 \, {\left ({\left ({\left (A - C\right )} a^{2} + 2 \, B a b - {\left (A - C\right )} b^{2}\right )} c^{3} - 3 \, {\left (B a^{2} - 2 \, {\left (A - C\right )} a b - B b^{2}\right )} c^{2} d - 3 \, {\left ({\left (A - C\right )} a^{2} + 2 \, B a b - {\left (A - C\right )} b^{2}\right )} c d^{2} + {\left (B a^{2} - 2 \, {\left (A - C\right )} a b - B b^{2}\right )} d^{3}\right )} {\left (f x + e\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {2 \, {\left ({\left (B a^{2} b^{4} - 2 \, {\left (A - C\right )} a b^{5} - B b^{6}\right )} c^{3} - 3 \, {\left (C a^{4} b^{2} - {\left (A - 3 \, C\right )} a^{2} b^{4} - 2 \, B a b^{5} + A b^{6}\right )} c^{2} d + 3 \, {\left (2 \, C a^{5} b - B a^{4} b^{2} + 4 \, C a^{3} b^{3} - 3 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} c d^{2} - {\left (3 \, C a^{6} - 2 \, B a^{5} b + {\left (A + 5 \, C\right )} a^{4} b^{2} - 4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3}\right )} \log \left (b \tan \left (f x + e\right ) + a\right )}{a^{4} b^{4} + 2 \, a^{2} b^{6} + b^{8}} + \frac {{\left ({\left (B a^{2} - 2 \, {\left (A - C\right )} a b - B b^{2}\right )} c^{3} + 3 \, {\left ({\left (A - C\right )} a^{2} + 2 \, B a b - {\left (A - C\right )} b^{2}\right )} c^{2} d - 3 \, {\left (B a^{2} - 2 \, {\left (A - C\right )} a b - B b^{2}\right )} c d^{2} - {\left ({\left (A - C\right )} a^{2} + 2 \, B a b - {\left (A - C\right )} b^{2}\right )} d^{3}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {2 \, {\left ({\left (C a^{2} b^{3} - B a b^{4} + A b^{5}\right )} c^{3} - 3 \, {\left (C a^{3} b^{2} - B a^{2} b^{3} + A a b^{4}\right )} c^{2} d + 3 \, {\left (C a^{4} b - B a^{3} b^{2} + A a^{2} b^{3}\right )} c d^{2} - {\left (C a^{5} - B a^{4} b + A a^{3} b^{2}\right )} d^{3}\right )}}{a^{3} b^{4} + a b^{6} + {\left (a^{2} b^{5} + b^{7}\right )} \tan \left (f x + e\right )} + \frac {C b d^{3} \tan \left (f x + e\right )^{2} + 2 \, {\left (3 \, C b c d^{2} - {\left (2 \, C a - B b\right )} d^{3}\right )} \tan \left (f x + e\right )}{b^{3}}}{2 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1522 vs. \(2 (577) = 1154\).
time = 4.13, size = 1522, normalized size = 2.65 \begin {gather*} \frac {{\left (C a^{4} b^{3} + 2 \, C a^{2} b^{5} + C b^{7}\right )} d^{3} \tan \left (f x + e\right )^{3} - 2 \, {\left (C a^{2} b^{5} - B a b^{6} + A b^{7}\right )} c^{3} + 6 \, {\left (C a^{3} b^{4} - B a^{2} b^{5} + A a b^{6}\right )} c^{2} d - 6 \, {\left (C a^{4} b^{3} - B a^{3} b^{4} + A a^{2} b^{5}\right )} c d^{2} + {\left (3 \, C a^{5} b^{2} - 2 \, B a^{4} b^{3} + 2 \, {\left (A + C\right )} a^{3} b^{4} + C a b^{6}\right )} d^{3} + 2 \, {\left ({\left ({\left (A - C\right )} a^{3} b^{4} + 2 \, B a^{2} b^{5} - {\left (A - C\right )} a b^{6}\right )} c^{3} - 3 \, {\left (B a^{3} b^{4} - 2 \, {\left (A - C\right )} a^{2} b^{5} - B a b^{6}\right )} c^{2} d - 3 \, {\left ({\left (A - C\right )} a^{3} b^{4} + 2 \, B a^{2} b^{5} - {\left (A - C\right )} a b^{6}\right )} c d^{2} + {\left (B a^{3} b^{4} - 2 \, {\left (A - C\right )} a^{2} b^{5} - B a b^{6}\right )} d^{3}\right )} f x + {\left (6 \, {\left (C a^{4} b^{3} + 2 \, C a^{2} b^{5} + C b^{7}\right )} c d^{2} - {\left (3 \, C a^{5} b^{2} - 2 \, B a^{4} b^{3} + 6 \, C a^{3} b^{4} - 4 \, B a^{2} b^{5} + 3 \, C a b^{6} - 2 \, B b^{7}\right )} d^{3}\right )} \tan \left (f x + e\right )^{2} - {\left ({\left (B a^{3} b^{4} - 2 \, {\left (A - C\right )} a^{2} b^{5} - B a b^{6}\right )} c^{3} - 3 \, {\left (C a^{5} b^{2} - {\left (A - 3 \, C\right )} a^{3} b^{4} - 2 \, B a^{2} b^{5} + A a b^{6}\right )} c^{2} d + 3 \, {\left (2 \, C a^{6} b - B a^{5} b^{2} + 4 \, C a^{4} b^{3} - 3 \, B a^{3} b^{4} + 2 \, A a^{2} b^{5}\right )} c d^{2} - {\left (3 \, C a^{7} - 2 \, B a^{6} b + {\left (A + 5 \, C\right )} a^{5} b^{2} - 4 \, B a^{4} b^{3} + 3 \, A a^{3} b^{4}\right )} d^{3} + {\left ({\left (B a^{2} b^{5} - 2 \, {\left (A - C\right )} a b^{6} - B b^{7}\right )} c^{3} - 3 \, {\left (C a^{4} b^{3} - {\left (A - 3 \, C\right )} a^{2} b^{5} - 2 \, B a b^{6} + A b^{7}\right )} c^{2} d + 3 \, {\left (2 \, C a^{5} b^{2} - B a^{4} b^{3} + 4 \, C a^{3} b^{4} - 3 \, B a^{2} b^{5} + 2 \, A a b^{6}\right )} c d^{2} - {\left (3 \, C a^{6} b - 2 \, B a^{5} b^{2} + {\left (A + 5 \, C\right )} a^{4} b^{3} - 4 \, B a^{3} b^{4} + 3 \, A a^{2} b^{5}\right )} d^{3}\right )} \tan \left (f x + e\right )\right )} \log \left (\frac {b^{2} \tan \left (f x + e\right )^{2} + 2 \, a b \tan \left (f x + e\right ) + a^{2}}{\tan \left (f x + e\right )^{2} + 1}\right ) - {\left (3 \, {\left (C a^{5} b^{2} + 2 \, C a^{3} b^{4} + C a b^{6}\right )} c^{2} d - 3 \, {\left (2 \, C a^{6} b - B a^{5} b^{2} + 4 \, C a^{4} b^{3} - 2 \, B a^{3} b^{4} + 2 \, C a^{2} b^{5} - B a b^{6}\right )} c d^{2} + {\left (3 \, C a^{7} - 2 \, B a^{6} b + {\left (A + 5 \, C\right )} a^{5} b^{2} - 4 \, B a^{4} b^{3} + {\left (2 \, A + C\right )} a^{3} b^{4} - 2 \, B a^{2} b^{5} + {\left (A - C\right )} a b^{6}\right )} d^{3} + {\left (3 \, {\left (C a^{4} b^{3} + 2 \, C a^{2} b^{5} + C b^{7}\right )} c^{2} d - 3 \, {\left (2 \, C a^{5} b^{2} - B a^{4} b^{3} + 4 \, C a^{3} b^{4} - 2 \, B a^{2} b^{5} + 2 \, C a b^{6} - B b^{7}\right )} c d^{2} + {\left (3 \, C a^{6} b - 2 \, B a^{5} b^{2} + {\left (A + 5 \, C\right )} a^{4} b^{3} - 4 \, B a^{3} b^{4} + {\left (2 \, A + C\right )} a^{2} b^{5} - 2 \, B a b^{6} + {\left (A - C\right )} b^{7}\right )} d^{3}\right )} \tan \left (f x + e\right )\right )} \log \left (\frac {1}{\tan \left (f x + e\right )^{2} + 1}\right ) + {\left (2 \, {\left (C a^{3} b^{4} - B a^{2} b^{5} + A a b^{6}\right )} c^{3} - 6 \, {\left (C a^{4} b^{3} - B a^{3} b^{4} + A a^{2} b^{5}\right )} c^{2} d + 6 \, {\left (2 \, C a^{5} b^{2} - B a^{4} b^{3} + {\left (A + 2 \, C\right )} a^{3} b^{4} + C a b^{6}\right )} c d^{2} - {\left (6 \, C a^{6} b - 4 \, B a^{5} b^{2} + {\left (2 \, A + 7 \, C\right )} a^{4} b^{3} - 4 \, B a^{3} b^{4} + 2 \, C a^{2} b^{5} - 2 \, B a b^{6} - C b^{7}\right )} d^{3} + 2 \, {\left ({\left ({\left (A - C\right )} a^{2} b^{5} + 2 \, B a b^{6} - {\left (A - C\right )} b^{7}\right )} c^{3} - 3 \, {\left (B a^{2} b^{5} - 2 \, {\left (A - C\right )} a b^{6} - B b^{7}\right )} c^{2} d - 3 \, {\left ({\left (A - C\right )} a^{2} b^{5} + 2 \, B a b^{6} - {\left (A - C\right )} b^{7}\right )} c d^{2} + {\left (B a^{2} b^{5} - 2 \, {\left (A - C\right )} a b^{6} - B b^{7}\right )} d^{3}\right )} f x\right )} \tan \left (f x + e\right )}{2 \, {\left ({\left (a^{4} b^{5} + 2 \, a^{2} b^{7} + b^{9}\right )} f \tan \left (f x + e\right ) + {\left (a^{5} b^{4} + 2 \, a^{3} b^{6} + a b^{8}\right )} f\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 33.12, size = 24300, normalized size = 42.33 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*x*(c + d*tan(e))**3*(A + B*tan(e) + C*tan(e)**2)/tan(e)**2, Eq(a, 0) & Eq(b, 0) & Eq(f, 0)), (-
A*c**3*f*x*tan(e + f*x)**2/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2*f*tan(e + f*x) - 4*b**2*f) + 2*I*A*c**3*f*x*ta
n(e + f*x)/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2*f*tan(e + f*x) - 4*b**2*f) + A*c**3*f*x/(4*b**2*f*tan(e + f*x)
**2 - 8*I*b**2*f*tan(e + f*x) - 4*b**2*f) - A*c**3*tan(e + f*x)/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2*f*tan(e +
f*x) - 4*b**2*f) + 2*I*A*c**3/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2*f*tan(e + f*x) - 4*b**2*f) + 3*I*A*c**2*d*
f*x*tan(e + f*x)**2/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2*f*tan(e + f*x) - 4*b**2*f) + 6*A*c**2*d*f*x*tan(e + f
*x)/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2*f*tan(e + f*x) - 4*b**2*f) - 3*I*A*c**2*d*f*x/(4*b**2*f*tan(e + f*x)*
*2 - 8*I*b**2*f*tan(e + f*x) - 4*b**2*f) + 3*I*A*c**2*d*tan(e + f*x)/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2*f*ta
n(e + f*x) - 4*b**2*f) + 3*A*c*d**2*f*x*tan(e + f*x)**2/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2*f*tan(e + f*x) -
4*b**2*f) - 6*I*A*c*d**2*f*x*tan(e + f*x)/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2*f*tan(e + f*x) - 4*b**2*f) - 3*
A*c*d**2*f*x/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2*f*tan(e + f*x) - 4*b**2*f) - 9*A*c*d**2*tan(e + f*x)/(4*b**2
*f*tan(e + f*x)**2 - 8*I*b**2*f*tan(e + f*x) - 4*b**2*f) + 6*I*A*c*d**2/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2*f
*tan(e + f*x) - 4*b**2*f) + 3*I*A*d**3*f*x*tan(e + f*x)**2/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2*f*tan(e + f*x)
- 4*b**2*f) + 6*A*d**3*f*x*tan(e + f*x)/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2*f*tan(e + f*x) - 4*b**2*f) - 3*I
*A*d**3*f*x/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2*f*tan(e + f*x) - 4*b**2*f) + 2*A*d**3*log(tan(e + f*x)**2 + 1
)*tan(e + f*x)**2/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2*f*tan(e + f*x) - 4*b**2*f) - 4*I*A*d**3*log(tan(e + f*x
)**2 + 1)*tan(e + f*x)/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2*f*tan(e + f*x) - 4*b**2*f) - 2*A*d**3*log(tan(e +
f*x)**2 + 1)/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2*f*tan(e + f*x) - 4*b**2*f) - 5*I*A*d**3*tan(e + f*x)/(4*b**2
*f*tan(e + f*x)**2 - 8*I*b**2*f*tan(e + f*x) - 4*b**2*f) - 4*A*d**3/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2*f*tan
(e + f*x) - 4*b**2*f) + I*B*c**3*f*x*tan(e + f*x)**2/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2*f*tan(e + f*x) - 4*b
**2*f) + 2*B*c**3*f*x*tan(e + f*x)/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2*f*tan(e + f*x) - 4*b**2*f) - I*B*c**3*
f*x/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2*f*tan(e + f*x) - 4*b**2*f) + I*B*c**3*tan(e + f*x)/(4*b**2*f*tan(e +
f*x)**2 - 8*I*b**2*f*tan(e + f*x) - 4*b**2*f) + 3*B*c**2*d*f*x*tan(e + f*x)**2/(4*b**2*f*tan(e + f*x)**2 - 8*I
*b**2*f*tan(e + f*x) - 4*b**2*f) - 6*I*B*c**2*d*f*x*tan(e + f*x)/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2*f*tan(e
+ f*x) - 4*b**2*f) - 3*B*c**2*d*f*x/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2*f*tan(e + f*x) - 4*b**2*f) - 9*B*c**2
*d*tan(e + f*x)/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2*f*tan(e + f*x) - 4*b**2*f) + 6*I*B*c**2*d/(4*b**2*f*tan(e
+ f*x)**2 - 8*I*b**2*f*tan(e + f*x) - 4*b**2*f) + 9*I*B*c*d**2*f*x*tan(e + f*x)**2/(4*b**2*f*tan(e + f*x)**2
- 8*I*b**2*f*tan(e + f*x) - 4*b**2*f) + 18*B*c*d**2*f*x*tan(e + f*x)/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2*f*ta
n(e + f*x) - 4*b**2*f) - 9*I*B*c*d**2*f*x/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2*f*tan(e + f*x) - 4*b**2*f) + 6*
B*c*d**2*log(tan(e + f*x)**2 + 1)*tan(e + f*x)**2/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2*f*tan(e + f*x) - 4*b**2
*f) - 12*I*B*c*d**2*log(tan(e + f*x)**2 + 1)*tan(e + f*x)/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2*f*tan(e + f*x)
- 4*b**2*f) - 6*B*c*d**2*log(tan(e + f*x)**2 + 1)/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2*f*tan(e + f*x) - 4*b**2
*f) - 15*I*B*c*d**2*tan(e + f*x)/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2*f*tan(e + f*x) - 4*b**2*f) - 12*B*c*d**2
/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2*f*tan(e + f*x) - 4*b**2*f) - 9*B*d**3*f*x*tan(e + f*x)**2/(4*b**2*f*tan(
e + f*x)**2 - 8*I*b**2*f*tan(e + f*x) - 4*b**2*f) + 18*I*B*d**3*f*x*tan(e + f*x)/(4*b**2*f*tan(e + f*x)**2 - 8
*I*b**2*f*tan(e + f*x) - 4*b**2*f) + 9*B*d**3*f*x/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2*f*tan(e + f*x) - 4*b**2
*f) + 4*I*B*d**3*log(tan(e + f*x)**2 + 1)*tan(e + f*x)**2/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2*f*tan(e + f*x)
- 4*b**2*f) + 8*B*d**3*log(tan(e + f*x)**2 + 1)*tan(e + f*x)/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2*f*tan(e + f*
x) - 4*b**2*f) - 4*I*B*d**3*log(tan(e + f*x)**2 + 1)/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2*f*tan(e + f*x) - 4*b
**2*f) + 4*B*d**3*tan(e + f*x)**3/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2*f*tan(e + f*x) - 4*b**2*f) + 19*B*d**3*
tan(e + f*x)/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2*f*tan(e + f*x) - 4*b**2*f) - 14*I*B*d**3/(4*b**2*f*tan(e + f
*x)**2 - 8*I*b**2*f*tan(e + f*x) - 4*b**2*f) + C*c**3*f*x*tan(e + f*x)**2/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2
*f*tan(e + f*x) - 4*b**2*f) - 2*I*C*c**3*f*x*tan(e + f*x)/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2*f*tan(e + f*x)
- 4*b**2*f) - C*c**3*f*x/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2*f*tan(e + f*x) - 4*b**2*f) - 3*C*c**3*tan(e + f*
x)/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2*f*tan(e + f*x) - 4*b**2*f) + 2*I*C*c**3/(4*b**2*f*tan(e + f*x)**2 - 8*
I*b**2*f*tan(e + f*x) - 4*b**2*f) + 9*I*C*c**2*...

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1357 vs. \(2 (577) = 1154\).
time = 1.24, size = 1357, normalized size = 2.36 \begin {gather*} \frac {\frac {2 \, {\left (A a^{2} c^{3} - C a^{2} c^{3} + 2 \, B a b c^{3} - A b^{2} c^{3} + C b^{2} c^{3} - 3 \, B a^{2} c^{2} d + 6 \, A a b c^{2} d - 6 \, C a b c^{2} d + 3 \, B b^{2} c^{2} d - 3 \, A a^{2} c d^{2} + 3 \, C a^{2} c d^{2} - 6 \, B a b c d^{2} + 3 \, A b^{2} c d^{2} - 3 \, C b^{2} c d^{2} + B a^{2} d^{3} - 2 \, A a b d^{3} + 2 \, C a b d^{3} - B b^{2} d^{3}\right )} {\left (f x + e\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {{\left (B a^{2} c^{3} - 2 \, A a b c^{3} + 2 \, C a b c^{3} - B b^{2} c^{3} + 3 \, A a^{2} c^{2} d - 3 \, C a^{2} c^{2} d + 6 \, B a b c^{2} d - 3 \, A b^{2} c^{2} d + 3 \, C b^{2} c^{2} d - 3 \, B a^{2} c d^{2} + 6 \, A a b c d^{2} - 6 \, C a b c d^{2} + 3 \, B b^{2} c d^{2} - A a^{2} d^{3} + C a^{2} d^{3} - 2 \, B a b d^{3} + A b^{2} d^{3} - C b^{2} d^{3}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {2 \, {\left (B a^{2} b^{4} c^{3} - 2 \, A a b^{5} c^{3} + 2 \, C a b^{5} c^{3} - B b^{6} c^{3} - 3 \, C a^{4} b^{2} c^{2} d + 3 \, A a^{2} b^{4} c^{2} d - 9 \, C a^{2} b^{4} c^{2} d + 6 \, B a b^{5} c^{2} d - 3 \, A b^{6} c^{2} d + 6 \, C a^{5} b c d^{2} - 3 \, B a^{4} b^{2} c d^{2} + 12 \, C a^{3} b^{3} c d^{2} - 9 \, B a^{2} b^{4} c d^{2} + 6 \, A a b^{5} c d^{2} - 3 \, C a^{6} d^{3} + 2 \, B a^{5} b d^{3} - A a^{4} b^{2} d^{3} - 5 \, C a^{4} b^{2} d^{3} + 4 \, B a^{3} b^{3} d^{3} - 3 \, A a^{2} b^{4} d^{3}\right )} \log \left ({\left | b \tan \left (f x + e\right ) + a \right |}\right )}{a^{4} b^{4} + 2 \, a^{2} b^{6} + b^{8}} + \frac {2 \, {\left (B a^{2} b^{5} c^{3} \tan \left (f x + e\right ) - 2 \, A a b^{6} c^{3} \tan \left (f x + e\right ) + 2 \, C a b^{6} c^{3} \tan \left (f x + e\right ) - B b^{7} c^{3} \tan \left (f x + e\right ) - 3 \, C a^{4} b^{3} c^{2} d \tan \left (f x + e\right ) + 3 \, A a^{2} b^{5} c^{2} d \tan \left (f x + e\right ) - 9 \, C a^{2} b^{5} c^{2} d \tan \left (f x + e\right ) + 6 \, B a b^{6} c^{2} d \tan \left (f x + e\right ) - 3 \, A b^{7} c^{2} d \tan \left (f x + e\right ) + 6 \, C a^{5} b^{2} c d^{2} \tan \left (f x + e\right ) - 3 \, B a^{4} b^{3} c d^{2} \tan \left (f x + e\right ) + 12 \, C a^{3} b^{4} c d^{2} \tan \left (f x + e\right ) - 9 \, B a^{2} b^{5} c d^{2} \tan \left (f x + e\right ) + 6 \, A a b^{6} c d^{2} \tan \left (f x + e\right ) - 3 \, C a^{6} b d^{3} \tan \left (f x + e\right ) + 2 \, B a^{5} b^{2} d^{3} \tan \left (f x + e\right ) - A a^{4} b^{3} d^{3} \tan \left (f x + e\right ) - 5 \, C a^{4} b^{3} d^{3} \tan \left (f x + e\right ) + 4 \, B a^{3} b^{4} d^{3} \tan \left (f x + e\right ) - 3 \, A a^{2} b^{5} d^{3} \tan \left (f x + e\right ) - C a^{4} b^{3} c^{3} + 2 \, B a^{3} b^{4} c^{3} - 3 \, A a^{2} b^{5} c^{3} + C a^{2} b^{5} c^{3} - A b^{7} c^{3} - 3 \, B a^{4} b^{3} c^{2} d + 6 \, A a^{3} b^{4} c^{2} d - 6 \, C a^{3} b^{4} c^{2} d + 3 \, B a^{2} b^{5} c^{2} d + 3 \, C a^{6} b c d^{2} - 3 \, A a^{4} b^{3} c d^{2} + 9 \, C a^{4} b^{3} c d^{2} - 6 \, B a^{3} b^{4} c d^{2} + 3 \, A a^{2} b^{5} c d^{2} - 2 \, C a^{7} d^{3} + B a^{6} b d^{3} - 4 \, C a^{5} b^{2} d^{3} + 3 \, B a^{4} b^{3} d^{3} - 2 \, A a^{3} b^{4} d^{3}\right )}}{{\left (a^{4} b^{4} + 2 \, a^{2} b^{6} + b^{8}\right )} {\left (b \tan \left (f x + e\right ) + a\right )}} + \frac {C b^{2} d^{3} \tan \left (f x + e\right )^{2} + 6 \, C b^{2} c d^{2} \tan \left (f x + e\right ) - 4 \, C a b d^{3} \tan \left (f x + e\right ) + 2 \, B b^{2} d^{3} \tan \left (f x + e\right )}{b^{4}}}{2 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 15.70, size = 701, normalized size = 1.22 \begin {gather*} \frac {\mathrm {tan}\left (e+f\,x\right )\,\left (\frac {B\,d^3+3\,C\,c\,d^2}{b^2}-\frac {2\,C\,a\,d^3}{b^3}\right )}{f}-\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,\left (B\,c^3-A\,d^3+C\,d^3+3\,A\,c^2\,d-3\,B\,c\,d^2-3\,C\,c^2\,d+A\,c^3\,1{}\mathrm {i}+B\,d^3\,1{}\mathrm {i}-C\,c^3\,1{}\mathrm {i}-A\,c\,d^2\,3{}\mathrm {i}-B\,c^2\,d\,3{}\mathrm {i}+C\,c\,d^2\,3{}\mathrm {i}\right )}{2\,f\,\left (-a^2+a\,b\,2{}\mathrm {i}+b^2\right )}+\frac {\ln \left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )\,\left (b^4\,\left (3\,A\,a^2\,d^3-B\,a^2\,c^3-3\,A\,a^2\,c^2\,d+9\,B\,a^2\,c\,d^2+9\,C\,a^2\,c^2\,d\right )-b^5\,\left (2\,C\,a\,c^3-2\,A\,a\,c^3+6\,A\,a\,c\,d^2+6\,B\,a\,c^2\,d\right )-b^3\,\left (4\,B\,a^3\,d^3+12\,C\,c\,a^3\,d^2\right )+b^6\,\left (B\,c^3+3\,A\,d\,c^2\right )-b\,\left (2\,B\,a^5\,d^3+6\,C\,c\,a^5\,d^2\right )+b^2\,\left (A\,a^4\,d^3+5\,C\,a^4\,d^3+3\,B\,a^4\,c\,d^2+3\,C\,a^4\,c^2\,d\right )+3\,C\,a^6\,d^3\right )}{f\,\left (a^4\,b^4+2\,a^2\,b^6+b^8\right )}-\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )-\mathrm {i}\right )\,\left (A\,c^3-A\,d^3\,1{}\mathrm {i}+B\,c^3\,1{}\mathrm {i}+B\,d^3-C\,c^3+C\,d^3\,1{}\mathrm {i}-3\,A\,c\,d^2+A\,c^2\,d\,3{}\mathrm {i}-B\,c\,d^2\,3{}\mathrm {i}-3\,B\,c^2\,d+3\,C\,c\,d^2-C\,c^2\,d\,3{}\mathrm {i}\right )}{2\,f\,\left (-a^2\,1{}\mathrm {i}+2\,a\,b+b^2\,1{}\mathrm {i}\right )}-\frac {-C\,a^5\,d^3+3\,C\,a^4\,b\,c\,d^2+B\,a^4\,b\,d^3-3\,C\,a^3\,b^2\,c^2\,d-3\,B\,a^3\,b^2\,c\,d^2-A\,a^3\,b^2\,d^3+C\,a^2\,b^3\,c^3+3\,B\,a^2\,b^3\,c^2\,d+3\,A\,a^2\,b^3\,c\,d^2-B\,a\,b^4\,c^3-3\,A\,a\,b^4\,c^2\,d+A\,b^5\,c^3}{b\,f\,\left (\mathrm {tan}\left (e+f\,x\right )\,b^4+a\,b^3\right )\,\left (a^2+b^2\right )}+\frac {C\,d^3\,{\mathrm {tan}\left (e+f\,x\right )}^2}{2\,b^2\,f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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