3.1.83 \(\int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x))^3 (c+d \tan (e+f x))^2} \, dx\) [83]

Optimal. Leaf size=841 \[ -\frac {\left (a^3 \left (c^2 C-2 B c d-C d^2-A \left (c^2-d^2\right )\right )-3 a b^2 \left (c^2 C-2 B c d-C d^2-A \left (c^2-d^2\right )\right )+3 a^2 b \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )-b^3 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) x}{\left (a^2+b^2\right )^3 \left (c^2+d^2\right )^2}-\frac {b \left (6 a^5 b B d^2-3 a^6 C d^2-a^4 b^2 d (4 B c+(10 A-C) d)-b^6 \left (c (c C-2 B d)-A \left (c^2-3 d^2\right )\right )+a b^5 \left (2 c (A-C) d-B \left (3 c^2-d^2\right )\right )+3 a^2 b^4 \left (c (c C+2 B d)-A \left (c^2+3 d^2\right )\right )+a^3 b^3 \left (10 c (A-C) d+B \left (c^2+3 d^2\right )\right )\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{\left (a^2+b^2\right )^3 (b c-a d)^4 f}-\frac {d^2 \left (b \left (3 c^4 C-4 B c^3 d+c^2 (5 A+C) d^2-2 B c d^3+3 A d^4\right )-a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{(b c-a d)^4 \left (c^2+d^2\right )^2 f}-\frac {d \left (3 a^3 b B d \left (c^2+d^2\right )+a b^3 (2 A c-2 c C+B d) \left (c^2+d^2\right )-a^4 d \left (3 c^2 C-B c d+(A+2 C) d^2\right )-a^2 b^2 \left (B c^3+4 A c^2 d+2 c^2 C d-B c d^2+6 A d^3\right )-b^4 \left (d \left (2 A c^2+c^2 C+3 A d^2\right )-B \left (c^3+2 c d^2\right )\right )\right )}{\left (a^2+b^2\right )^2 (b c-a d)^3 \left (c^2+d^2\right ) f (c+d \tan (e+f x))}-\frac {A b^2-a (b B-a C)}{2 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^2 (c+d \tan (e+f x))}-\frac {5 a^3 b B d-3 a^4 C d+b^4 (2 B c-3 A d)+a b^3 (4 A c-4 c C+B d)-a^2 b^2 (2 B c+(7 A-C) d)}{2 \left (a^2+b^2\right )^2 (b c-a d)^2 f (a+b \tan (e+f x)) (c+d \tan (e+f x))} \]

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

________________________________________________________________________________________

Rubi [A]
time = 2.72, antiderivative size = 841, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {3730, 3732, 3611} \begin {gather*} -\frac {\left (b \left (3 C c^4-4 B d c^3+(5 A+C) d^2 c^2-2 B d^3 c+3 A d^4\right )-a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \log (c \cos (e+f x)+d \sin (e+f x)) d^2}{(b c-a d)^4 \left (c^2+d^2\right )^2 f}-\frac {\left (-d \left (3 C c^2-B d c+(A+2 C) d^2\right ) a^4+3 b B d \left (c^2+d^2\right ) a^3-b^2 \left (B c^3+4 A d c^2+2 C d c^2-B d^2 c+6 A d^3\right ) a^2+b^3 (2 A c-2 C c+B d) \left (c^2+d^2\right ) a-b^4 \left (d \left (2 A c^2+C c^2+3 A d^2\right )-B \left (c^3+2 d^2 c\right )\right )\right ) d}{\left (a^2+b^2\right )^2 (b c-a d)^3 \left (c^2+d^2\right ) f (c+d \tan (e+f x))}-\frac {\left (\left (C c^2-2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a^3+3 b \left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) a^2-3 b^2 \left (C c^2-2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a-b^3 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) x}{\left (a^2+b^2\right )^3 \left (c^2+d^2\right )^2}-\frac {b \left (-3 C d^2 a^6+6 b B d^2 a^5-b^2 d (4 B c+(10 A-C) d) a^4+b^3 \left (10 c (A-C) d+B \left (c^2+3 d^2\right )\right ) a^3+3 b^4 \left (c (c C+2 B d)-A \left (c^2+3 d^2\right )\right ) a^2+b^5 \left (2 c (A-C) d-B \left (3 c^2-d^2\right )\right ) a-b^6 \left (c (c C-2 B d)-A \left (c^2-3 d^2\right )\right )\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{\left (a^2+b^2\right )^3 (b c-a d)^4 f}-\frac {-3 C d a^4+5 b B d a^3-b^2 (2 B c+(7 A-C) d) a^2+b^3 (4 A c-4 C c+B d) a+b^4 (2 B c-3 A d)}{2 \left (a^2+b^2\right )^2 (b c-a d)^2 f (a+b \tan (e+f x)) (c+d \tan (e+f x))}-\frac {A b^2-a (b B-a C)}{2 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^2 (c+d \tan (e+f x))} \end {gather*}

Antiderivative was successfully verified.

[In]

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

Rule 3611

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c/(b*f))
*Log[RemoveContent[a*Cos[e + f*x] + b*Sin[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] && EqQ[a*c + b*d, 0]

Rule 3730

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*b^2 - a*(b*B - a*C))*(a + b*Ta
n[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Dist[1/((m + 1)*(
b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1)
- b^2*d*(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - a*B - b*C)*Tan[e
+ f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*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] && LtQ[m, -1] &&  !(ILtQ[n, -1] && ( !I
ntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rule 3732

Int[((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2)/(((a_) + (b_.)*tan[(e_.) + (f_.)
*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[(a*(A*c - c*C + B*d) + b*(B*c - A*d + C*d)
)*(x/((a^2 + b^2)*(c^2 + d^2))), x] + (Dist[(A*b^2 - a*b*B + a^2*C)/((b*c - a*d)*(a^2 + b^2)), Int[(b - a*Tan[
e + f*x])/(a + b*Tan[e + f*x]), x], x] - Dist[(c^2*C - B*c*d + A*d^2)/((b*c - a*d)*(c^2 + d^2)), Int[(d - c*Ta
n[e + f*x])/(c + d*Tan[e + f*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]

Rubi steps

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1758\) vs. \(2(841)=1682\).
time = 7.70, size = 1758, normalized size = 2.09 \begin {gather*} -\frac {A b^2-a (b B-a C)}{2 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^2 (c+d \tan (e+f x))}-\frac {-\frac {-a \left (-3 a \left (A b^2-a (b B-a C)\right ) d+2 b (A b-a B-b C) (b c-a d)\right )+b^2 \left (3 A b^2 d-2 a A (b c-a d)-(b B-a C) (2 b c+a d)\right )}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) (c+d \tan (e+f x))}-\frac {-\frac {-\frac {(b c-a d)^3 \left (-b^2 \left (-3 a^2 A b c^2+A b^3 c^2+a^3 B c^2-3 a b^2 B c^2+3 a^2 b c^2 C-b^3 c^2 C-2 a^3 A c d+6 a A b^2 c d-6 a^2 b B c d+2 b^3 B c d+2 a^3 c C d-6 a b^2 c C d+3 a^2 A b d^2-A b^3 d^2-a^3 B d^2+3 a b^2 B d^2-3 a^2 b C d^2+b^3 C d^2\right )+\sqrt {-b^2} \left (a^3 A b c^2-3 a A b^3 c^2+3 a^2 b^2 B c^2-b^4 B c^2-a^3 b c^2 C+3 a b^3 c^2 C-6 a^2 A b^2 c d+2 A b^4 c d+2 a^3 b B c d-6 a b^3 B c d+6 a^2 b^2 c C d-2 b^4 c C d-a^3 A b d^2+3 a A b^3 d^2-3 a^2 b^2 B d^2+b^4 B d^2+a^3 b C d^2-3 a b^3 C d^2\right )\right ) \log \left (\sqrt {-b^2}-b \tan (e+f x)\right )}{b \left (a^2+b^2\right ) \left (c^2+d^2\right )}-\frac {2 b^2 \left (c^2+d^2\right ) \left (6 a^5 b B d^2-3 a^6 C d^2-a^4 b^2 d (4 B c+(10 A-C) d)-b^6 \left (c (c C-2 B d)-A \left (c^2-3 d^2\right )\right )+a b^5 \left (2 c (A-C) d-B \left (3 c^2-d^2\right )\right )+3 a^2 b^4 \left (c (c C+2 B d)-A \left (c^2+3 d^2\right )\right )+a^3 b^3 \left (10 c (A-C) d+B \left (c^2+3 d^2\right )\right )\right ) \log (a+b \tan (e+f x))}{\left (a^2+b^2\right ) (b c-a d)}+\frac {(b c-a d)^3 \left (b^2 \left (-3 a^2 A b c^2+A b^3 c^2+a^3 B c^2-3 a b^2 B c^2+3 a^2 b c^2 C-b^3 c^2 C-2 a^3 A c d+6 a A b^2 c d-6 a^2 b B c d+2 b^3 B c d+2 a^3 c C d-6 a b^2 c C d+3 a^2 A b d^2-A b^3 d^2-a^3 B d^2+3 a b^2 B d^2-3 a^2 b C d^2+b^3 C d^2\right )+\sqrt {-b^2} \left (a^3 A b c^2-3 a A b^3 c^2+3 a^2 b^2 B c^2-b^4 B c^2-a^3 b c^2 C+3 a b^3 c^2 C-6 a^2 A b^2 c d+2 A b^4 c d+2 a^3 b B c d-6 a b^3 B c d+6 a^2 b^2 c C d-2 b^4 c C d-a^3 A b d^2+3 a A b^3 d^2-3 a^2 b^2 B d^2+b^4 B d^2+a^3 b C d^2-3 a b^3 C d^2\right )\right ) \log \left (\sqrt {-b^2}+b \tan (e+f x)\right )}{b \left (a^2+b^2\right ) \left (c^2+d^2\right )}-\frac {2 b \left (a^2+b^2\right )^2 d^2 \left (b \left (3 c^4 C-4 B c^3 d+c^2 (5 A+C) d^2-2 B c d^3+3 A d^4\right )-a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \log (c+d \tan (e+f x))}{(b c-a d) \left (c^2+d^2\right )}}{b (-b c+a d) \left (c^2+d^2\right ) f}-\frac {d^2 \left ((-b c-a d) \left (-3 a \left (A b^2-a (b B-a C)\right ) d+2 b (A b-a B-b C) (b c-a d)\right )+\left (2 b^2 d-a (b c-a d)\right ) \left (3 A b^2 d-2 a A (b c-a d)-(b B-a C) (2 b c+a d)\right )\right )-c \left (d (b c-a d) \left (-3 b \left (A b^2-a (b B-a C)\right ) d-2 a (A b-a B-b C) (b c-a d)+b \left (3 A b^2 d-2 a A (b c-a d)-(b B-a C) (2 b c+a d)\right )\right )-2 c d \left (-a \left (-3 a \left (A b^2-a (b B-a C)\right ) d+2 b (A b-a B-b C) (b c-a d)\right )+b^2 \left (3 A b^2 d-2 a A (b c-a d)-(b B-a C) (2 b c+a d)\right )\right )\right )}{(-b c+a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))}}{\left (a^2+b^2\right ) (b c-a d)}}{2 \left (a^2+b^2\right ) (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(A*b^2 - a*(b*B - a*C))/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])) - (-((-(a
*(-3*a*(A*b^2 - a*(b*B - a*C))*d + 2*b*(A*b - a*B - b*C)*(b*c - a*d))) + b^2*(3*A*b^2*d - 2*a*A*(b*c - a*d) -
(b*B - a*C)*(2*b*c + a*d)))/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x]))) - (-((-(((b
*c - a*d)^3*(-(b^2*(-3*a^2*A*b*c^2 + A*b^3*c^2 + a^3*B*c^2 - 3*a*b^2*B*c^2 + 3*a^2*b*c^2*C - b^3*c^2*C - 2*a^3
*A*c*d + 6*a*A*b^2*c*d - 6*a^2*b*B*c*d + 2*b^3*B*c*d + 2*a^3*c*C*d - 6*a*b^2*c*C*d + 3*a^2*A*b*d^2 - A*b^3*d^2
- a^3*B*d^2 + 3*a*b^2*B*d^2 - 3*a^2*b*C*d^2 + b^3*C*d^2)) + Sqrt[-b^2]*(a^3*A*b*c^2 - 3*a*A*b^3*c^2 + 3*a^2*b
^2*B*c^2 - b^4*B*c^2 - a^3*b*c^2*C + 3*a*b^3*c^2*C - 6*a^2*A*b^2*c*d + 2*A*b^4*c*d + 2*a^3*b*B*c*d - 6*a*b^3*B
*c*d + 6*a^2*b^2*c*C*d - 2*b^4*c*C*d - a^3*A*b*d^2 + 3*a*A*b^3*d^2 - 3*a^2*b^2*B*d^2 + b^4*B*d^2 + a^3*b*C*d^2
- 3*a*b^3*C*d^2))*Log[Sqrt[-b^2] - b*Tan[e + f*x]])/(b*(a^2 + b^2)*(c^2 + d^2))) - (2*b^2*(c^2 + d^2)*(6*a^5*
b*B*d^2 - 3*a^6*C*d^2 - a^4*b^2*d*(4*B*c + (10*A - C)*d) - b^6*(c*(c*C - 2*B*d) - A*(c^2 - 3*d^2)) + a*b^5*(2*
c*(A - C)*d - B*(3*c^2 - d^2)) + 3*a^2*b^4*(c*(c*C + 2*B*d) - A*(c^2 + 3*d^2)) + a^3*b^3*(10*c*(A - C)*d + B*(
c^2 + 3*d^2)))*Log[a + b*Tan[e + f*x]])/((a^2 + b^2)*(b*c - a*d)) + ((b*c - a*d)^3*(b^2*(-3*a^2*A*b*c^2 + A*b^
3*c^2 + a^3*B*c^2 - 3*a*b^2*B*c^2 + 3*a^2*b*c^2*C - b^3*c^2*C - 2*a^3*A*c*d + 6*a*A*b^2*c*d - 6*a^2*b*B*c*d +
2*b^3*B*c*d + 2*a^3*c*C*d - 6*a*b^2*c*C*d + 3*a^2*A*b*d^2 - A*b^3*d^2 - a^3*B*d^2 + 3*a*b^2*B*d^2 - 3*a^2*b*C*
d^2 + b^3*C*d^2) + Sqrt[-b^2]*(a^3*A*b*c^2 - 3*a*A*b^3*c^2 + 3*a^2*b^2*B*c^2 - b^4*B*c^2 - a^3*b*c^2*C + 3*a*b
^3*c^2*C - 6*a^2*A*b^2*c*d + 2*A*b^4*c*d + 2*a^3*b*B*c*d - 6*a*b^3*B*c*d + 6*a^2*b^2*c*C*d - 2*b^4*c*C*d - a^3
*A*b*d^2 + 3*a*A*b^3*d^2 - 3*a^2*b^2*B*d^2 + b^4*B*d^2 + a^3*b*C*d^2 - 3*a*b^3*C*d^2))*Log[Sqrt[-b^2] + b*Tan[
e + f*x]])/(b*(a^2 + b^2)*(c^2 + d^2)) - (2*b*(a^2 + b^2)^2*d^2*(b*(3*c^4*C - 4*B*c^3*d + c^2*(5*A + C)*d^2 -
2*B*c*d^3 + 3*A*d^4) - a*d^2*(2*c*(A - C)*d - B*(c^2 - d^2)))*Log[c + d*Tan[e + f*x]])/((b*c - a*d)*(c^2 + d^2
)))/(b*(-(b*c) + a*d)*(c^2 + d^2)*f)) - (d^2*((-(b*c) - a*d)*(-3*a*(A*b^2 - a*(b*B - a*C))*d + 2*b*(A*b - a*B
- b*C)*(b*c - a*d)) + (2*b^2*d - a*(b*c - a*d))*(3*A*b^2*d - 2*a*A*(b*c - a*d) - (b*B - a*C)*(2*b*c + a*d))) -
c*(d*(b*c - a*d)*(-3*b*(A*b^2 - a*(b*B - a*C))*d - 2*a*(A*b - a*B - b*C)*(b*c - a*d) + b*(3*A*b^2*d - 2*a*A*(
b*c - a*d) - (b*B - a*C)*(2*b*c + a*d))) - 2*c*d*(-(a*(-3*a*(A*b^2 - a*(b*B - a*C))*d + 2*b*(A*b - a*B - b*C)*
(b*c - a*d))) + b^2*(3*A*b^2*d - 2*a*A*(b*c - a*d) - (b*B - a*C)*(2*b*c + a*d)))))/((-(b*c) + a*d)*(c^2 + d^2)
*f*(c + d*Tan[e + f*x])))/((a^2 + b^2)*(b*c - a*d)))/(2*(a^2 + b^2)*(b*c - a*d))

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Maple [A]
time = 3.59, size = 951, normalized size = 1.13

method	result	size
derivativedivides	\(\frac {\frac {d^{2} \left (2 A a c \,d^{3}-5 A b \,c^{2} d^{2}-3 A b \,d^{4}-B a \,c^{2} d^{2}+B a \,d^{4}+4 B b \,c^{3} d +2 B b c \,d^{3}-2 C a c \,d^{3}-3 C b \,c^{4}-C b \,c^{2} d^{2}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{\left (a d -b c \right )^{4} \left (c^{2}+d^{2}\right )^{2}}-\frac {\left (A \,d^{2}-B c d +c^{2} C \right ) d^{2}}{\left (a d -b c \right )^{3} \left (c^{2}+d^{2}\right ) \left (c +d \tan \left (f x +e \right )\right )}+\frac {\frac {\left (-2 A \,a^{3} c d -3 A \,a^{2} b \,c^{2}+3 A \,a^{2} b \,d^{2}+6 A a \,b^{2} c d +A \,b^{3} c^{2}-A \,b^{3} d^{2}+B \,a^{3} c^{2}-B \,a^{3} d^{2}-6 B \,a^{2} b c d -3 B a \,b^{2} c^{2}+3 B a \,b^{2} d^{2}+2 B \,b^{3} c d +2 C \,a^{3} c d +3 C \,a^{2} b \,c^{2}-3 C \,a^{2} b \,d^{2}-6 C a \,b^{2} c d -C \,b^{3} c^{2}+C \,b^{3} d^{2}\right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (A \,a^{3} c^{2}-A \,a^{3} d^{2}-6 A \,a^{2} b c d -3 A a \,b^{2} c^{2}+3 A a \,b^{2} d^{2}+2 A \,b^{3} c d +2 B \,a^{3} c d +3 B \,a^{2} b \,c^{2}-3 B \,a^{2} b \,d^{2}-6 B a \,b^{2} c d -B \,b^{3} c^{2}+B \,b^{3} d^{2}-C \,a^{3} c^{2}+C \,a^{3} d^{2}+6 C \,a^{2} b c d +3 C a \,b^{2} c^{2}-3 C a \,b^{2} d^{2}-2 C \,b^{3} c d \right ) \arctan \left (\tan \left (f x +e \right )\right )}{\left (a^{2}+b^{2}\right )^{3} \left (c^{2}+d^{2}\right )^{2}}-\frac {b \left (A \,b^{2}-B a b +C \,a^{2}\right )}{2 \left (a d -b c \right )^{2} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (f x +e \right )\right )^{2}}-\frac {b \left (4 A \,a^{2} b^{2} d -2 A a \,b^{3} c +2 A \,b^{4} d -3 a^{3} b B d +B \,a^{2} b^{2} c -B a \,b^{3} d -B \,b^{4} c +2 a^{4} C d +2 C a \,b^{3} c \right )}{\left (a d -b c \right )^{3} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (f x +e \right )\right )}+\frac {b \left (10 A \,a^{4} b^{2} d^{2}-10 A \,a^{3} b^{3} c d +3 A \,a^{2} b^{4} c^{2}+9 A \,a^{2} b^{4} d^{2}-2 A a \,b^{5} c d -A \,b^{6} c^{2}+3 A \,b^{6} d^{2}-6 a^{5} b B \,d^{2}+4 B \,a^{4} b^{2} c d -B \,a^{3} b^{3} c^{2}-3 B \,a^{3} b^{3} d^{2}-6 B \,a^{2} b^{4} c d +3 a \,b^{5} B \,c^{2}-B a \,b^{5} d^{2}-2 B \,b^{6} c d +3 a^{6} C \,d^{2}-a^{4} b^{2} C \,d^{2}+10 C \,a^{3} b^{3} c d -3 C \,a^{2} b^{4} c^{2}+2 C a \,b^{5} c d +C \,b^{6} c^{2}\right ) \ln \left (a +b \tan \left (f x +e \right )\right )}{\left (a d -b c \right )^{4} \left (a^{2}+b^{2}\right )^{3}}}{f}\)	\(951\)
default	\(\frac {\frac {d^{2} \left (2 A a c \,d^{3}-5 A b \,c^{2} d^{2}-3 A b \,d^{4}-B a \,c^{2} d^{2}+B a \,d^{4}+4 B b \,c^{3} d +2 B b c \,d^{3}-2 C a c \,d^{3}-3 C b \,c^{4}-C b \,c^{2} d^{2}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{\left (a d -b c \right )^{4} \left (c^{2}+d^{2}\right )^{2}}-\frac {\left (A \,d^{2}-B c d +c^{2} C \right ) d^{2}}{\left (a d -b c \right )^{3} \left (c^{2}+d^{2}\right ) \left (c +d \tan \left (f x +e \right )\right )}+\frac {\frac {\left (-2 A \,a^{3} c d -3 A \,a^{2} b \,c^{2}+3 A \,a^{2} b \,d^{2}+6 A a \,b^{2} c d +A \,b^{3} c^{2}-A \,b^{3} d^{2}+B \,a^{3} c^{2}-B \,a^{3} d^{2}-6 B \,a^{2} b c d -3 B a \,b^{2} c^{2}+3 B a \,b^{2} d^{2}+2 B \,b^{3} c d +2 C \,a^{3} c d +3 C \,a^{2} b \,c^{2}-3 C \,a^{2} b \,d^{2}-6 C a \,b^{2} c d -C \,b^{3} c^{2}+C \,b^{3} d^{2}\right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (A \,a^{3} c^{2}-A \,a^{3} d^{2}-6 A \,a^{2} b c d -3 A a \,b^{2} c^{2}+3 A a \,b^{2} d^{2}+2 A \,b^{3} c d +2 B \,a^{3} c d +3 B \,a^{2} b \,c^{2}-3 B \,a^{2} b \,d^{2}-6 B a \,b^{2} c d -B \,b^{3} c^{2}+B \,b^{3} d^{2}-C \,a^{3} c^{2}+C \,a^{3} d^{2}+6 C \,a^{2} b c d +3 C a \,b^{2} c^{2}-3 C a \,b^{2} d^{2}-2 C \,b^{3} c d \right ) \arctan \left (\tan \left (f x +e \right )\right )}{\left (a^{2}+b^{2}\right )^{3} \left (c^{2}+d^{2}\right )^{2}}-\frac {b \left (A \,b^{2}-B a b +C \,a^{2}\right )}{2 \left (a d -b c \right )^{2} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (f x +e \right )\right )^{2}}-\frac {b \left (4 A \,a^{2} b^{2} d -2 A a \,b^{3} c +2 A \,b^{4} d -3 a^{3} b B d +B \,a^{2} b^{2} c -B a \,b^{3} d -B \,b^{4} c +2 a^{4} C d +2 C a \,b^{3} c \right )}{\left (a d -b c \right )^{3} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (f x +e \right )\right )}+\frac {b \left (10 A \,a^{4} b^{2} d^{2}-10 A \,a^{3} b^{3} c d +3 A \,a^{2} b^{4} c^{2}+9 A \,a^{2} b^{4} d^{2}-2 A a \,b^{5} c d -A \,b^{6} c^{2}+3 A \,b^{6} d^{2}-6 a^{5} b B \,d^{2}+4 B \,a^{4} b^{2} c d -B \,a^{3} b^{3} c^{2}-3 B \,a^{3} b^{3} d^{2}-6 B \,a^{2} b^{4} c d +3 a \,b^{5} B \,c^{2}-B a \,b^{5} d^{2}-2 B \,b^{6} c d +3 a^{6} C \,d^{2}-a^{4} b^{2} C \,d^{2}+10 C \,a^{3} b^{3} c d -3 C \,a^{2} b^{4} c^{2}+2 C a \,b^{5} c d +C \,b^{6} c^{2}\right ) \ln \left (a +b \tan \left (f x +e \right )\right )}{\left (a d -b c \right )^{4} \left (a^{2}+b^{2}\right )^{3}}}{f}\)	\(951\)
norman	\(\text {Expression too large to display}\)	\(3278\)
risch	\(\text {Expression too large to display}\)	\(26447\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 2528 vs. \(2 (844) = 1688\).
time = 0.72, size = 2528, normalized size = 3.01 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 9612 vs. \(2 (844) = 1688\).
time = 20.93, size = 9612, normalized size = 11.43 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*((3*C*a^4*b^5 - 5*B*a^3*b^6 + (7*A - 3*C)*a^2*b^7 + B*a*b^8 + A*b^9)*c^7 - 2*(5*C*a^5*b^4 - 7*B*a^4*b^5 +
(9*A - C)*a^3*b^6 - B*a^2*b^7 + 3*A*a*b^8)*c^6*d + (7*C*a^6*b^3 - 9*B*a^5*b^4 + (11*A + 7*C)*a^4*b^5 - 13*B*a
^3*b^6 + (19*A - 6*C)*a^2*b^7 + 2*B*a*b^8 + 2*A*b^9)*c^5*d^2 - 4*(5*C*a^5*b^4 - 7*B*a^4*b^5 + (9*A - C)*a^3*b^
6 - B*a^2*b^7 + 3*A*a*b^8)*c^4*d^3 - (2*C*a^8*b - 8*C*a^6*b^3 + 18*B*a^5*b^4 - (22*A - C)*a^4*b^5 + 11*B*a^3*b
^6 - (17*A - 5*C)*a^2*b^7 - B*a*b^8 - A*b^9)*c^3*d^4 + 2*(C*a^9 + B*a^8*b + 3*C*a^7*b^2 + 3*B*a^6*b^3 - 2*C*a^
5*b^4 + 10*B*a^4*b^5 - (9*A - 2*C)*a^3*b^6 + 2*B*a^2*b^7 - 3*A*a*b^8)*c^2*d^5 - (2*B*a^9 + 2*A*a^8*b + 6*B*a^7
*b^2 + (6*A - 7*C)*a^6*b^3 + 15*B*a^5*b^4 - (5*A + C)*a^4*b^5 + 5*B*a^3*b^6 - 3*A*a^2*b^7)*c*d^6 + 2*(A*a^9 +
3*A*a^7*b^2 + 3*A*a^5*b^4 + A*a^3*b^6)*d^7 - ((C*a^4*b^5 - 3*B*a^3*b^6 + 5*(A - C)*a^2*b^7 + 3*B*a*b^8 - A*b^9
)*c^6*d - 2*(3*C*a^5*b^4 - 5*B*a^4*b^5 + (7*A - 3*C)*a^3*b^6 + B*a^2*b^7 + A*a*b^8)*c^5*d^2 + (3*C*a^6*b^3 - 7
*B*a^5*b^4 + (9*A - 5*C)*a^4*b^5 - 7*B*a^3*b^6 + (13*A - 16*C)*a^2*b^7 + 6*B*a*b^8 - 2*(A + C)*b^9)*c^4*d^3 +
2*(C*a^7*b^2 + B*a^6*b^3 - 3*C*a^5*b^4 + 13*B*a^4*b^5 - (14*A - 9*C)*a^3*b^6 + B*a^2*b^7 - (2*A - C)*a*b^8 + B
*b^9)*c^3*d^4 - (2*B*a^7*b^2 + 2*(A - 5*C)*a^6*b^3 + 20*B*a^5*b^4 - (12*A - C)*a^4*b^5 + 11*B*a^3*b^6 - 5*(A -
C)*a^2*b^7 - B*a*b^8 + 3*A*b^9)*c^2*d^5 + 2*(A*a^7*b^2 + 3*(A - C)*a^5*b^4 + 5*B*a^4*b^5 - (4*A - 3*C)*a^3*b^
6 - B*a^2*b^7)*c*d^6 + (5*C*a^6*b^3 - 7*B*a^5*b^4 + (9*A - C)*a^4*b^5 - B*a^3*b^6 + 3*A*a^2*b^7)*d^7 + 2*(((A
- C)*a^3*b^6 + 3*B*a^2*b^7 - 3*(A - C)*a*b^8 - B*b^9)*c^6*d - 2*(2*(A - C)*a^4*b^5 + 5*B*a^3*b^6 - 3*(A - C)*a
^2*b^7 + B*a*b^8 - (A - C)*b^9)*c^5*d^2 + (6*(A - C)*a^5*b^4 + 10*B*a^4*b^5 + 5*(A - C)*a^3*b^6 + 15*B*a^2*b^7
- 5*(A - C)*a*b^8 + B*b^9)*c^4*d^3 - 4*((A - C)*a^6*b^3 + 5*(A - C)*a^4*b^5 + 5*B*a^3*b^6 + B*a*b^8)*c^3*d^4
+ ((A - C)*a^7*b^2 - 5*B*a^6*b^3 + 15*(A - C)*a^5*b^4 + 5*B*a^4*b^5 + 10*(A - C)*a^3*b^6 + 6*B*a^2*b^7)*c^2*d^
5 + 2*(B*a^7*b^2 - (A - C)*a^6*b^3 + 3*B*a^5*b^4 - 5*(A - C)*a^4*b^5 - 2*B*a^3*b^6)*c*d^6 - ((A - C)*a^7*b^2 +
3*B*a^6*b^3 - 3*(A - C)*a^5*b^4 - B*a^4*b^5)*d^7)*f*x)*tan(f*x + e)^3 - 2*(((A - C)*a^5*b^4 + 3*B*a^4*b^5 - 3
*(A - C)*a^3*b^6 - B*a^2*b^7)*c^7 - 2*(2*(A - C)*a^6*b^3 + 5*B*a^5*b^4 - 3*(A - C)*a^4*b^5 + B*a^3*b^6 - (A -
C)*a^2*b^7)*c^6*d + (6*(A - C)*a^7*b^2 + 10*B*a^6*b^3 + 5*(A - C)*a^5*b^4 + 15*B*a^4*b^5 - 5*(A - C)*a^3*b^6 +
B*a^2*b^7)*c^5*d^2 - 4*((A - C)*a^8*b + 5*(A - C)*a^6*b^3 + 5*B*a^5*b^4 + B*a^3*b^6)*c^4*d^3 + ((A - C)*a^9 -
5*B*a^8*b + 15*(A - C)*a^7*b^2 + 5*B*a^6*b^3 + 10*(A - C)*a^5*b^4 + 6*B*a^4*b^5)*c^3*d^4 + 2*(B*a^9 - (A - C)
*a^8*b + 3*B*a^7*b^2 - 5*(A - C)*a^6*b^3 - 2*B*a^5*b^4)*c^2*d^5 - ((A - C)*a^9 + 3*B*a^8*b - 3*(A - C)*a^7*b^2
- B*a^6*b^3)*c*d^6)*f*x - ((C*a^4*b^5 - 3*B*a^3*b^6 + 5*(A - C)*a^2*b^7 + 3*B*a*b^8 - A*b^9)*c^7 - 2*(2*C*a^5
*b^4 - 3*B*a^4*b^5 + 4*A*a^3*b^6 - 2*B*a^2*b^7 + 2*(2*A - C)*a*b^8 + B*b^9)*c^6*d - (3*C*a^6*b^3 - 5*B*a^5*b^4
+ (7*A - 13*C)*a^4*b^5 + 19*B*a^3*b^6 - (25*A - 14*C)*a^2*b^7 - 6*B*a*b^8 - 2*A*b^9)*c^5*d^2 + 2*(C*a^7*b^2 -
4*B*a^6*b^3 + (5*A - 13*C)*a^5*b^4 + 9*B*a^4*b^5 - (11*A + 6*C)*a^3*b^6 + 5*B*a^2*b^7 - 2*(5*A - C)*a*b^8 - 2
*B*b^9)*c^4*d^3 + (4*C*a^8*b + 4*B*a^7*b^2 + 8*C*a^6*b^3 + 22*B*a^5*b^4 - (14*A - 41*C)*a^4*b^5 - 17*B*a^3*b^6
+ (35*A - 3*C)*a^2*b^7 + 7*B*a*b^8 + (7*A + 2*C)*b^9)*c^3*d^4 - 2*(2*B*a^8*b + (2*A - 5*C)*a^7*b^2 + 15*B*a^6
*b^3 - (4*A - 11*C)*a^5*b^4 + (16*A + 3*C)*a^3*b^6 + B*a^2*b^7 + (10*A - C)*a*b^8 + 2*B*b^9)*c^2*d^5 + (4*A*a^
8*b + 2*B*a^7*b^2 + (14*A - 3*C)*a^6*b^3 + 11*B*a^5*b^4 + 11*(A + C)*a^4*b^5 - 7*B*a^3*b^6 + (25*A - 4*C)*a^2*
b^7 + 2*B*a*b^8 + 6*A*b^9)*c*d^6 - 2*((A - 3*C)*a^7*b^2 + 4*B*a^6*b^3 - (2*A - 3*C)*a^5*b^4 - 3*B*a^4*b^5 + 6*
A*a^3*b^6 - B*a^2*b^7 + 3*A*a*b^8)*d^7 + 2*(((A - C)*a^3*b^6 + 3*B*a^2*b^7 - 3*(A - C)*a*b^8 - B*b^9)*c^7 - 2*
((A - C)*a^4*b^5 + 2*B*a^3*b^6 + 2*B*a*b^8 - (A - C)*b^9)*c^6*d - (2*(A - C)*a^5*b^4 + 10*B*a^4*b^5 - 17*(A -
C)*a^3*b^6 - 11*B*a^2*b^7 + (A - C)*a*b^8 - B*b^9)*c^5*d^2 + 2*(4*(A - C)*a^6*b^3 + 10*B*a^5*b^4 - 5*(A - C)*a
^4*b^5 + 5*B*a^3*b^6 - 5*(A - C)*a^2*b^7 - B*a*b^8)*c^4*d^3 - (7*(A - C)*a^7*b^2 + 5*B*a^6*b^3 + 25*(A - C)*a^
5*b^4 + 35*B*a^4*b^5 - 10*(A - C)*a^3*b^6 + 2*B*a^2*b^7)*c^3*d^4 + 2*((A - C)*a^8*b - 4*B*a^7*b^2 + 14*(A - C)
*a^6*b^3 + 8*B*a^5*b^4 + 5*(A - C)*a^4*b^5 + 4*B*a^3*b^6)*c^2*d^5 + (4*B*a^8*b - 5*(A - C)*a^7*b^2 + 9*B*a^6*b
^3 - 17*(A - C)*a^5*b^4 - 7*B*a^4*b^5)*c*d^6 - 2*((A - C)*a^8*b + 3*B*a^7*b^2 - 3*(A - C)*a^6*b^3 - B*a^5*b^4)
*d^7)*f*x)*tan(f*x + e)^2 + ((B*a^5*b^4 - 3*(A - C)*a^4*b^5 - 3*B*a^3*b^6 + (A - C)*a^2*b^7)*c^7 - 2*(2*B*a^6*
b^3 - 5*(A - C)*a^5*b^4 - 3*B*a^4*b^5 - (A - C)*a^3*b^6 - B*a^2*b^7)*c^6*d - (3*C*a^8*b - 6*B*a^7*b^2 + (10*A
- C)*a^6*b^3 - 5*B*a^5*b^4 + 3*(5*A - 2*C)*a^4*b^5 + 5*B*a^3*b^6 + (A + 2*C)*a^2*b^7)*c^5*d^2 - 4*(2*B*a^6*b^3
- 5*(A - C)*a^5*b^4 - 3*B*a^4*b^5 - (A - C)*a^3*b^6 - B*a^2*b^7)*c^4*d^3 - (6*C*a^8*b - 12*B*a^7*b^2 + 2*(10*
A - C)*a^6*b^3 - 7*B*a^5*b^4 + 3*(7*A - C)*a^4*...

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError >> no valid subset found

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 3176 vs. \(2 (844) = 1688\).
time = 1.23, size = 3176, normalized size = 3.78 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*(A*a^3*c^2 - C*a^3*c^2 + 3*B*a^2*b*c^2 - 3*A*a*b^2*c^2 + 3*C*a*b^2*c^2 - B*b^3*c^2 + 2*B*a^3*c*d - 6*A*
a^2*b*c*d + 6*C*a^2*b*c*d - 6*B*a*b^2*c*d + 2*A*b^3*c*d - 2*C*b^3*c*d - A*a^3*d^2 + C*a^3*d^2 - 3*B*a^2*b*d^2
+ 3*A*a*b^2*d^2 - 3*C*a*b^2*d^2 + B*b^3*d^2)*(f*x + e)/(a^6*c^4 + 3*a^4*b^2*c^4 + 3*a^2*b^4*c^4 + b^6*c^4 + 2*
a^6*c^2*d^2 + 6*a^4*b^2*c^2*d^2 + 6*a^2*b^4*c^2*d^2 + 2*b^6*c^2*d^2 + a^6*d^4 + 3*a^4*b^2*d^4 + 3*a^2*b^4*d^4
+ b^6*d^4) + (B*a^3*c^2 - 3*A*a^2*b*c^2 + 3*C*a^2*b*c^2 - 3*B*a*b^2*c^2 + A*b^3*c^2 - C*b^3*c^2 - 2*A*a^3*c*d
+ 2*C*a^3*c*d - 6*B*a^2*b*c*d + 6*A*a*b^2*c*d - 6*C*a*b^2*c*d + 2*B*b^3*c*d - B*a^3*d^2 + 3*A*a^2*b*d^2 - 3*C*
a^2*b*d^2 + 3*B*a*b^2*d^2 - A*b^3*d^2 + C*b^3*d^2)*log(tan(f*x + e)^2 + 1)/(a^6*c^4 + 3*a^4*b^2*c^4 + 3*a^2*b^
4*c^4 + b^6*c^4 + 2*a^6*c^2*d^2 + 6*a^4*b^2*c^2*d^2 + 6*a^2*b^4*c^2*d^2 + 2*b^6*c^2*d^2 + a^6*d^4 + 3*a^4*b^2*
d^4 + 3*a^2*b^4*d^4 + b^6*d^4) - 2*(B*a^3*b^5*c^2 - 3*A*a^2*b^6*c^2 + 3*C*a^2*b^6*c^2 - 3*B*a*b^7*c^2 + A*b^8*
c^2 - C*b^8*c^2 - 4*B*a^4*b^4*c*d + 10*A*a^3*b^5*c*d - 10*C*a^3*b^5*c*d + 6*B*a^2*b^6*c*d + 2*A*a*b^7*c*d - 2*
C*a*b^7*c*d + 2*B*b^8*c*d - 3*C*a^6*b^2*d^2 + 6*B*a^5*b^3*d^2 - 10*A*a^4*b^4*d^2 + C*a^4*b^4*d^2 + 3*B*a^3*b^5
*d^2 - 9*A*a^2*b^6*d^2 + B*a*b^7*d^2 - 3*A*b^8*d^2)*log(abs(b*tan(f*x + e) + a))/(a^6*b^5*c^4 + 3*a^4*b^7*c^4
+ 3*a^2*b^9*c^4 + b^11*c^4 - 4*a^7*b^4*c^3*d - 12*a^5*b^6*c^3*d - 12*a^3*b^8*c^3*d - 4*a*b^10*c^3*d + 6*a^8*b^
3*c^2*d^2 + 18*a^6*b^5*c^2*d^2 + 18*a^4*b^7*c^2*d^2 + 6*a^2*b^9*c^2*d^2 - 4*a^9*b^2*c*d^3 - 12*a^7*b^4*c*d^3 -
12*a^5*b^6*c*d^3 - 4*a^3*b^8*c*d^3 + a^10*b*d^4 + 3*a^8*b^3*d^4 + 3*a^6*b^5*d^4 + a^4*b^7*d^4) - 2*(3*C*b*c^4
*d^3 - 4*B*b*c^3*d^4 + B*a*c^2*d^5 + 5*A*b*c^2*d^5 + C*b*c^2*d^5 - 2*A*a*c*d^6 + 2*C*a*c*d^6 - 2*B*b*c*d^6 - B
*a*d^7 + 3*A*b*d^7)*log(abs(d*tan(f*x + e) + c))/(b^4*c^8*d - 4*a*b^3*c^7*d^2 + 6*a^2*b^2*c^6*d^3 + 2*b^4*c^6*
d^3 - 4*a^3*b*c^5*d^4 - 8*a*b^3*c^5*d^4 + a^4*c^4*d^5 + 12*a^2*b^2*c^4*d^5 + b^4*c^4*d^5 - 8*a^3*b*c^3*d^6 - 4
*a*b^3*c^3*d^6 + 2*a^4*c^2*d^7 + 6*a^2*b^2*c^2*d^7 - 4*a^3*b*c*d^8 + a^4*d^9) + 2*(3*C*b*c^4*d^3*tan(f*x + e)
- 4*B*b*c^3*d^4*tan(f*x + e) + B*a*c^2*d^5*tan(f*x + e) + 5*A*b*c^2*d^5*tan(f*x + e) + C*b*c^2*d^5*tan(f*x + e
) - 2*A*a*c*d^6*tan(f*x + e) + 2*C*a*c*d^6*tan(f*x + e) - 2*B*b*c*d^6*tan(f*x + e) - B*a*d^7*tan(f*x + e) + 3*
A*b*d^7*tan(f*x + e) + 4*C*b*c^5*d^2 - C*a*c^4*d^3 - 5*B*b*c^4*d^3 + 2*B*a*c^3*d^4 + 6*A*b*c^3*d^4 + 2*C*b*c^3
*d^4 - 3*A*a*c^2*d^5 + C*a*c^2*d^5 - 3*B*b*c^2*d^5 + 4*A*b*c*d^6 - A*a*d^7)/((b^4*c^8 - 4*a*b^3*c^7*d + 6*a^2*
b^2*c^6*d^2 + 2*b^4*c^6*d^2 - 4*a^3*b*c^5*d^3 - 8*a*b^3*c^5*d^3 + a^4*c^4*d^4 + 12*a^2*b^2*c^4*d^4 + b^4*c^4*d
^4 - 8*a^3*b*c^3*d^5 - 4*a*b^3*c^3*d^5 + 2*a^4*c^2*d^6 + 6*a^2*b^2*c^2*d^6 - 4*a^3*b*c*d^7 + a^4*d^8)*(d*tan(f
*x + e) + c)) + (3*B*a^3*b^6*c^2*tan(f*x + e)^2 - 9*A*a^2*b^7*c^2*tan(f*x + e)^2 + 9*C*a^2*b^7*c^2*tan(f*x + e
)^2 - 9*B*a*b^8*c^2*tan(f*x + e)^2 + 3*A*b^9*c^2*tan(f*x + e)^2 - 3*C*b^9*c^2*tan(f*x + e)^2 - 12*B*a^4*b^5*c*
d*tan(f*x + e)^2 + 30*A*a^3*b^6*c*d*tan(f*x + e)^2 - 30*C*a^3*b^6*c*d*tan(f*x + e)^2 + 18*B*a^2*b^7*c*d*tan(f*
x + e)^2 + 6*A*a*b^8*c*d*tan(f*x + e)^2 - 6*C*a*b^8*c*d*tan(f*x + e)^2 + 6*B*b^9*c*d*tan(f*x + e)^2 - 9*C*a^6*
b^3*d^2*tan(f*x + e)^2 + 18*B*a^5*b^4*d^2*tan(f*x + e)^2 - 30*A*a^4*b^5*d^2*tan(f*x + e)^2 + 3*C*a^4*b^5*d^2*t
an(f*x + e)^2 + 9*B*a^3*b^6*d^2*tan(f*x + e)^2 - 27*A*a^2*b^7*d^2*tan(f*x + e)^2 + 3*B*a*b^8*d^2*tan(f*x + e)^
2 - 9*A*b^9*d^2*tan(f*x + e)^2 + 8*B*a^4*b^5*c^2*tan(f*x + e) - 22*A*a^3*b^6*c^2*tan(f*x + e) + 22*C*a^3*b^6*c
^2*tan(f*x + e) - 18*B*a^2*b^7*c^2*tan(f*x + e) + 2*A*a*b^8*c^2*tan(f*x + e) - 2*C*a*b^8*c^2*tan(f*x + e) - 2*
B*b^9*c^2*tan(f*x + e) + 4*C*a^6*b^3*c*d*tan(f*x + e) - 32*B*a^5*b^4*c*d*tan(f*x + e) + 72*A*a^4*b^5*c*d*tan(f
*x + e) - 60*C*a^4*b^5*c*d*tan(f*x + e) + 28*B*a^3*b^6*c*d*tan(f*x + e) + 28*A*a^2*b^7*c*d*tan(f*x + e) - 16*C
*a^2*b^7*c*d*tan(f*x + e) + 12*B*a*b^8*c*d*tan(f*x + e) + 4*A*b^9*c*d*tan(f*x + e) - 22*C*a^7*b^2*d^2*tan(f*x
+ e) + 42*B*a^6*b^3*d^2*tan(f*x + e) - 68*A*a^5*b^4*d^2*tan(f*x + e) + 2*C*a^5*b^4*d^2*tan(f*x + e) + 26*B*a^4
*b^5*d^2*tan(f*x + e) - 66*A*a^3*b^6*d^2*tan(f*x + e) + 8*B*a^2*b^7*d^2*tan(f*x + e) - 22*A*a*b^8*d^2*tan(f*x
+ e) - C*a^6*b^3*c^2 + 6*B*a^5*b^4*c^2 - 14*A*a^4*b^5*c^2 + 11*C*a^4*b^5*c^2 - 7*B*a^3*b^6*c^2 - 3*A*a^2*b^7*c
^2 - B*a*b^8*c^2 - A*b^9*c^2 + 6*C*a^7*b^2*c*d - 22*B*a^6*b^3*c*d + 44*A*a^5*b^4*c*d - 26*C*a^5*b^4*c*d + 6*B*
a^4*b^5*c*d + 26*A*a^3*b^6*c*d - 8*C*a^3*b^6*c*d + 4*B*a^2*b^7*c*d + 6*A*a*b^8*c*d - 14*C*a^8*b*d^2 + 25*B*a^7
*b^2*d^2 - 39*A*a^6*b^3*d^2 - 3*C*a^6*b^3*d^2 + 19*B*a^5*b^4*d^2 - 41*A*a^4*b^5*d^2 - C*a^4*b^5*d^2 + 6*B*a^3*
b^6*d^2 - 14*A*a^2*b^7*d^2)/((a^6*b^4*c^4 + 3*a^4*b^6*c^4 + 3*a^2*b^8*c^4 + b^10*c^4 - 4*a^7*b^3*c^3*d - 12*a^
5*b^5*c^3*d - 12*a^3*b^7*c^3*d - 4*a*b^9*c^3*d + 6*a^8*b^2*c^2*d^2 + 18*a^6*b^4*c^2*d^2 + 18*a^4*b^6*c^2*d^2 +
6*a^2*b^8*c^2*d^2 - 4*a^9*b*c*d^3 - 12*a^7*b^3*c*d^3 - 12*a^5*b^5*c*d^3 - 4*a^3*b^7*c*d^3 + a^10*d^4 + 3*a^8*
b^2*d^4 + 3*a^6*b^4*d^4 + a^4*b^6*d^4)*(b*tan(f...

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Mupad [B]
time = 58.47, size = 2500, normalized size = 2.97 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(symsum(log((24*A^3*a^3*b^7*d^9 + 27*A^3*a^5*b^5*d^9 + B^3*a^2*b^8*d^9 + 4*B^3*a^4*b^6*d^9 + 7*B^3*a^6*b^4*d^9
+ 3*A^3*b^10*c^3*d^6 - A^3*b^10*c^5*d^4 + 4*B^3*b^10*c^2*d^7 + 6*B^3*b^10*c^4*d^5 + C^3*b^10*c^5*d^4 + 9*A^2*
B*b^10*d^9 + 9*A^3*a*b^9*d^9 + 16*A^3*a^2*b^8*c^3*d^6 + 3*A^3*a^2*b^8*c^5*d^4 + 26*A^3*a^3*b^7*c^2*d^7 - 6*A^3
*a^3*b^7*c^4*d^5 - 11*A^3*a^4*b^6*c^3*d^6 + 31*A^3*a^5*b^5*c^2*d^7 + 5*B^3*a^2*b^8*c^2*d^7 + 6*B^3*a^2*b^8*c^4
*d^5 + 28*B^3*a^3*b^7*c^3*d^6 + 7*B^3*a^3*b^7*c^5*d^4 - 14*B^3*a^4*b^6*c^2*d^7 - 20*B^3*a^4*b^6*c^4*d^5 + 19*B
^3*a^5*b^5*c^3*d^6 + 9*B^3*a^6*b^4*c^2*d^7 - 7*C^3*a^2*b^8*c^3*d^6 - 3*C^3*a^2*b^8*c^5*d^4 + C^3*a^3*b^7*c^2*d
^7 + 15*C^3*a^3*b^7*c^4*d^5 + 6*C^3*a^3*b^7*c^6*d^3 - 28*C^3*a^4*b^6*c^3*d^6 - 24*C^3*a^4*b^6*c^5*d^4 - 4*C^3*
a^5*b^5*c^2*d^7 + 3*C^3*a^6*b^4*c^3*d^6 - 9*C^3*a^7*b^3*c^2*d^7 - 9*C^3*a^7*b^3*c^4*d^5 - 6*A*B^2*a*b^9*d^9 -
9*A^2*C*a*b^9*d^9 - 12*A*B^2*b^10*c*d^8 + 4*B^3*a*b^9*c*d^8 - 20*A*B^2*a^3*b^7*d^9 - 28*A*B^2*a^5*b^5*d^9 + 6*
A*B^2*a^7*b^3*d^9 + 21*A^2*B*a^2*b^8*d^9 + 13*A^2*B*a^4*b^6*d^9 - 27*A^2*B*a^6*b^4*d^9 - 3*A*C^2*a^3*b^7*d^9 -
9*A*C^2*a^7*b^3*d^9 - 21*A^2*C*a^3*b^7*d^9 - 27*A^2*C*a^5*b^5*d^9 + 9*A^2*C*a^7*b^3*d^9 - 17*A*B^2*b^10*c^3*d
^6 + 3*A*B^2*b^10*c^5*d^4 + B*C^2*a^4*b^6*d^9 + 3*B*C^2*a^8*b^2*d^9 + 12*A^2*B*b^10*c^2*d^7 - 7*A^2*B*b^10*c^4
*d^5 - B^2*C*a^3*b^7*d^9 - 2*B^2*C*a^5*b^5*d^9 - 9*B^2*C*a^7*b^3*d^9 + 3*A*C^2*b^10*c^3*d^6 - 3*A*C^2*b^10*c^5
*d^4 - 6*A^2*C*b^10*c^3*d^6 + 3*A^2*C*b^10*c^5*d^4 - B*C^2*b^10*c^4*d^5 + 3*B*C^2*b^10*c^6*d^3 - 4*B^2*C*b^10*
c^3*d^6 - 9*B^2*C*b^10*c^5*d^4 + 3*A^3*a*b^9*c^2*d^7 - 10*A^3*a*b^9*c^4*d^5 - 3*A^3*a^2*b^8*c*d^8 - 31*A^3*a^4
*b^6*c*d^8 - 8*A^3*a^6*b^4*c*d^8 + B^3*a*b^9*c^3*d^6 - 5*B^3*a*b^9*c^5*d^4 + 11*B^3*a^3*b^7*c*d^8 + 5*B^3*a^5*
b^5*c*d^8 - 6*B^3*a^7*b^3*c*d^8 - 2*C^3*a*b^9*c^4*d^5 - 6*C^3*a*b^9*c^6*d^3 - 2*C^3*a^4*b^6*c*d^8 - C^3*a^6*b^
4*c*d^8 - 3*C^3*a^8*b^2*c*d^8 - 60*A*B^2*a^2*b^8*c^3*d^6 - 21*A*B^2*a^2*b^8*c^5*d^4 - 4*A*B^2*a^3*b^7*c^2*d^7
+ 44*A*B^2*a^3*b^7*c^4*d^5 + 25*A*B^2*a^4*b^6*c^3*d^6 + 4*A*B^2*a^4*b^6*c^5*d^4 - 77*A*B^2*a^5*b^5*c^2*d^7 - 1
7*A*B^2*a^5*b^5*c^4*d^5 + 28*A*B^2*a^6*b^4*c^3*d^6 - 6*A*B^2*a^7*b^3*c^2*d^7 + 71*A^2*B*a^2*b^8*c^2*d^7 + 16*A
^2*B*a^2*b^8*c^4*d^5 - 116*A^2*B*a^3*b^7*c^3*d^6 - 9*A^2*B*a^3*b^7*c^5*d^4 + 86*A^2*B*a^4*b^6*c^2*d^7 + 35*A^2
*B*a^4*b^6*c^4*d^5 - 37*A^2*B*a^5*b^5*c^3*d^6 - 13*A^2*B*a^6*b^4*c^2*d^7 + 30*A*C^2*a^2*b^8*c^3*d^6 + 9*A*C^2*
a^2*b^8*c^5*d^4 - 30*A*C^2*a^3*b^7*c^2*d^7 - 63*A*C^2*a^3*b^7*c^4*d^5 - 12*A*C^2*a^3*b^7*c^6*d^3 + 45*A*C^2*a^
4*b^6*c^3*d^6 + 48*A*C^2*a^4*b^6*c^5*d^4 - 15*A*C^2*a^5*b^5*c^2*d^7 - 27*A*C^2*a^5*b^5*c^4*d^5 - 6*A*C^2*a^6*b
^4*c^3*d^6 + 9*A*C^2*a^7*b^3*c^4*d^5 - 39*A^2*C*a^2*b^8*c^3*d^6 - 9*A^2*C*a^2*b^8*c^5*d^4 + 3*A^2*C*a^3*b^7*c^
2*d^7 + 54*A^2*C*a^3*b^7*c^4*d^5 + 6*A^2*C*a^3*b^7*c^6*d^3 - 6*A^2*C*a^4*b^6*c^3*d^6 - 24*A^2*C*a^4*b^6*c^5*d^
4 - 12*A^2*C*a^5*b^5*c^2*d^7 + 27*A^2*C*a^5*b^5*c^4*d^5 + 3*A^2*C*a^6*b^4*c^3*d^6 + 9*A^2*C*a^7*b^3*c^2*d^7 +
11*B*C^2*a^2*b^8*c^2*d^7 - 17*B*C^2*a^2*b^8*c^4*d^5 - 18*B*C^2*a^2*b^8*c^6*d^3 + 16*B*C^2*a^3*b^7*c^3*d^6 + 39
*B*C^2*a^3*b^7*c^5*d^4 + 47*B*C^2*a^4*b^6*c^2*d^7 + 47*B*C^2*a^4*b^6*c^4*d^5 + 3*B*C^2*a^4*b^6*c^6*d^3 - 25*B*
C^2*a^5*b^5*c^3*d^6 - 12*B*C^2*a^5*b^5*c^5*d^4 + 17*B*C^2*a^6*b^4*c^2*d^7 + 27*B*C^2*a^6*b^4*c^4*d^5 + 12*B*C^
2*a^7*b^3*c^3*d^6 - 3*B*C^2*a^8*b^2*c^2*d^7 + 9*B^2*C*a^2*b^8*c^3*d^6 + 9*B^2*C*a^2*b^8*c^5*d^4 - 35*B^2*C*a^3
*b^7*c^2*d^7 - 68*B^2*C*a^3*b^7*c^4*d^5 - 6*B^2*C*a^3*b^7*c^6*d^3 - 16*B^2*C*a^4*b^6*c^3*d^6 + 14*B^2*C*a^4*b^
6*c^5*d^4 + 26*B^2*C*a^5*b^5*c^2*d^7 - 4*B^2*C*a^5*b^5*c^4*d^5 - 37*B^2*C*a^6*b^4*c^3*d^6 + 3*B^2*C*a^7*b^3*c^
2*d^7 + 6*A*B*C*a^2*b^8*d^9 + 13*A*B*C*a^4*b^6*d^9 + 36*A*B*C*a^6*b^4*d^9 - 3*A*B*C*a^8*b^2*d^9 + 6*A*B*C*b^10
*c^2*d^7 + 17*A*B*C*b^10*c^4*d^5 - 3*A*B*C*b^10*c^6*d^3 - 24*A^2*B*a*b^9*c*d^8 + 11*A*B^2*a*b^9*c^2*d^7 + 25*A
*B^2*a*b^9*c^4*d^5 - 19*A*B^2*a^2*b^8*c*d^8 + 37*A*B^2*a^4*b^6*c*d^8 + 32*A*B^2*a^6*b^4*c*d^8 - 23*A^2*B*a*b^9
*c^3*d^6 + 11*A^2*B*a*b^9*c^5*d^4 - 81*A^2*B*a^3*b^7*c*d^8 - 15*A^2*B*a^5*b^5*c*d^8 + 6*A^2*B*a^7*b^3*c*d^8 -
15*A*C^2*a*b^9*c^2*d^7 - 15*A*C^2*a*b^9*c^4*d^5 + 12*A*C^2*a*b^9*c^6*d^3 - 3*A*C^2*a^2*b^8*c*d^8 - 27*A*C^2*a^
4*b^6*c*d^8 - 6*A*C^2*a^6*b^4*c*d^8 + 6*A*C^2*a^8*b^2*c*d^8 + 12*A^2*C*a*b^9*c^2*d^7 + 27*A^2*C*a*b^9*c^4*d^5
- 6*A^2*C*a*b^9*c^6*d^3 + 6*A^2*C*a^2*b^8*c*d^8 + 60*A^2*C*a^4*b^6*c*d^8 + 15*A^2*C*a^6*b^4*c*d^8 - 3*A^2*C*a^
8*b^2*c*d^8 + 13*B*C^2*a*b^9*c^3*d^6 + 23*B*C^2*a*b^9*c^5*d^4 + 3*B*C^2*a^3*b^7*c*d^8 + 9*B*C^2*a^5*b^5*c*d^8
+ 18*B*C^2*a^7*b^3*c*d^8 - 14*B^2*C*a*b^9*c^2*d^7 - 16*B^2*C*a*b^9*c^4*d^5 + 6*B^2*C*a*b^9*c^6*d^3 - 8*B^2*C*a
^2*b^8*c*d^8 - 28*B^2*C*a^4*b^6*c*d^8 - 29*B^2*C*a^6*b^4*c*d^8 + 3*B^2*C*a^8*b^2*c*d^8 - 28*A*B*C*a^2*b^8*c^2*
d^7 + 28*A*B*C*a^2*b^8*c^4*d^5 + 18*A*B*C*a^2*b^8*c^6*d^3 + 100*A*B*C*a^3*b^7*c^3*d^6 - 30*A*B*C*a^3*b^7*c^5*d
^4 - 79*A*B*C*a^4*b^6*c^2*d^7 - 55*A*B*C*a^4*b^6*c^4*d^5 - 3*A*B*C*a^4*b^6*c^6*d^3 + 62*A*B*C*a^5*b^5*c^3*d^6
+ 12*A*B*C*a^5*b^5*c^5*d^4 + 14*A*B*C*a^6*b^4*c...

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