\(\int \frac {(a+b \tan (e+f x))^2 (A+B \tan (e+f x)+C \tan ^2(e+f x))}{(c+d \tan (e+f x))^3} \, dx\) [85]

Optimal result

Integrand size = 45, antiderivative size = 597 \[ \int \frac {(a+b \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^3} \, dx=-\frac {\left (b^2 \left (A c^3-c^3 C+3 B c^2 d-3 A c d^2+3 c C d^2-B d^3\right )+a^2 \left (c^3 C-3 B c^2 d-3 c C d^2+B d^3-A \left (c^3-3 c d^2\right )\right )-2 a b \left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )\right ) x}{\left (c^2+d^2\right )^3}-\frac {\left (2 a b \left (A c^3-c^3 C+3 B c^2 d-3 A c d^2+3 c C d^2-B d^3\right )-a^2 \left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )+b^2 \left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )\right ) \log (\cos (e+f x))}{\left (c^2+d^2\right )^3 f}-\frac {\left (2 a b d^3 \left (A c^3-c^3 C+3 B c^2 d-3 A c d^2+3 c C d^2-B d^3\right )-b^2 \left (c^6 C+3 c^4 C d^2+B c^3 d^3-3 c^2 (A-2 C) d^4-3 B c d^5+A d^6\right )-a^2 d^3 \left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )\right ) \log (c+d \tan (e+f x))}{d^3 \left (c^2+d^2\right )^3 f}-\frac {\left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^2}{2 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}+\frac {(b c-a d) \left (b \left (c^4 C-c^2 (A-3 C) d^2-2 B c d^3+A d^4\right )+a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right )}{d^3 \left (c^2+d^2\right )^2 f (c+d \tan (e+f x))} \]

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

Rubi [A] (verified)

Time = 1.45 (sec) , antiderivative size = 597, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3726, 3716, 3707, 3698, 31, 3556} \[ \int \frac {(a+b \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^3} \, dx=-\frac {\log (\cos (e+f x)) \left (-\left (a^2 \left (d (A-C) \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )\right )+2 a b \left (A c^3-3 A c d^2+3 B c^2 d-B d^3-c^3 C+3 c C d^2\right )+b^2 \left (d (A-C) \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )\right )}{f \left (c^2+d^2\right )^3}-\frac {x \left (a^2 \left (-A \left (c^3-3 c d^2\right )-3 B c^2 d+B d^3+c^3 C-3 c C d^2\right )-2 a b \left (d (A-C) \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )+b^2 \left (A c^3-3 A c d^2+3 B c^2 d-B d^3-c^3 C+3 c C d^2\right )\right )}{\left (c^2+d^2\right )^3}-\frac {\left (-a^2 d^3 \left (d (A-C) \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )+2 a b d^3 \left (A c^3-3 A c d^2+3 B c^2 d-B d^3-c^3 C+3 c C d^2\right )-b^2 \left (-3 c^2 d^4 (A-2 C)+A d^6+B c^3 d^3-3 B c d^5+c^6 C+3 c^4 C d^2\right )\right ) \log (c+d \tan (e+f x))}{d^3 f \left (c^2+d^2\right )^3}-\frac {\left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^2}{2 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^2}+\frac {(b c-a d) \left (a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (-c^2 d^2 (A-3 C)+A d^4-2 B c d^3+c^4 C\right )\right )}{d^3 f \left (c^2+d^2\right )^2 (c+d \tan (e+f x))} \]

[In]

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3698

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[
A/(b*f), Subst[Int[(a + x)^m, x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, A, C, m}, x] && EqQ[A, C]

Rule 3707

Int[((A_) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2)/((a_.) + (b_.)*tan[(e_.) + (f_.)*
(x_)]), x_Symbol] :> Simp[(a*A + b*B - a*C)*(x/(a^2 + b^2)), x] + (Dist[(A*b^2 - a*b*B + a^2*C)/(a^2 + b^2), I
nt[(1 + Tan[e + f*x]^2)/(a + b*Tan[e + f*x]), x], x] - Dist[(A*b - a*B - b*C)/(a^2 + b^2), Int[Tan[e + f*x], x
], x]) /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && NeQ[a^2 + b^2, 0] && NeQ[A*b - a
*B - b*C, 0]

Rule 3716

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 - a*d))*(c^2*C - B*c*d + A*d^2)
*((c + d*Tan[e + f*x])^(n + 1)/(d^2*f*(n + 1)*(c^2 + d^2))), x] + Dist[1/(d*(c^2 + d^2)), Int[(c + d*Tan[e + f
*x])^(n + 1)*Simp[a*d*(A*c - c*C + B*d) + b*(c^2*C - B*c*d + A*d^2) + d*(A*b*c + a*B*c - b*c*C - a*A*d + b*B*d
+ a*C*d)*Tan[e + f*x] + b*C*(c^2 + d^2)*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[c^2 + d^2, 0] && LtQ[n, -1]

Rule 3726

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*t
an[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*d^2 + c*(c*C - B*d))*(a + b*Ta
n[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Dist[1/(d*(n + 1)*(c^2 + d^2)), I
nt[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)*Simp[A*d*(b*d*m - a*c*(n + 1)) + (c*C - B*d)*(b*c
*m + a*d*(n + 1)) - d*(n + 1)*((A - C)*(b*c - a*d) + B*(a*c + b*d))*Tan[e + f*x] - b*(d*(B*c - A*d)*(m + n + 1
) - C*(c^2*m - d^2*(n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c -
a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] && LtQ[n, -1]

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.98 (sec) , antiderivative size = 1044, normalized size of antiderivative = 1.75 \[ \int \frac {(a+b \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^3} \, dx=-\frac {\left (-2 a A b c^3-a^2 B c^3+b^2 B c^3+2 a b c^3 C+3 a^2 A c^2 d-3 A b^2 c^2 d-6 a b B c^2 d-3 a^2 c^2 C d+3 b^2 c^2 C d+6 a A b c d^2+3 a^2 B c d^2-3 b^2 B c d^2-6 a b c C d^2-a^2 A d^3+A b^2 d^3+2 a b B d^3+a^2 C d^3-b^2 C d^3+i \left (a^2 A c^3-A b^2 c^3-2 a b B c^3-a^2 c^3 C+b^2 c^3 C+6 a A b c^2 d+3 a^2 B c^2 d-3 b^2 B c^2 d-6 a b c^2 C d-3 a^2 A c d^2+3 A b^2 c d^2+6 a b B c d^2+3 a^2 c C d^2-3 b^2 c C d^2-2 a A b d^3-a^2 B d^3+b^2 B d^3+2 a b C d^3\right )\right ) \log (i-\tan (e+f x))}{2 \left (c^2+d^2\right )^3 f}+\frac {\left (2 a A b c^3+a^2 B c^3-b^2 B c^3-2 a b c^3 C-3 a^2 A c^2 d+3 A b^2 c^2 d+6 a b B c^2 d+3 a^2 c^2 C d-3 b^2 c^2 C d-6 a A b c d^2-3 a^2 B c d^2+3 b^2 B c d^2+6 a b c C d^2+a^2 A d^3-A b^2 d^3-2 a b B d^3-a^2 C d^3+b^2 C d^3+i \left (a^2 A c^3-A b^2 c^3-2 a b B c^3-a^2 c^3 C+b^2 c^3 C+6 a A b c^2 d+3 a^2 B c^2 d-3 b^2 B c^2 d-6 a b c^2 C d-3 a^2 A c d^2+3 A b^2 c d^2+6 a b B c d^2+3 a^2 c C d^2-3 b^2 c C d^2-2 a A b d^3-a^2 B d^3+b^2 B d^3+2 a b C d^3\right )\right ) \log (i+\tan (e+f x))}{2 \left (c^2+d^2\right )^3 f}-\frac {\left (2 a b d^3 \left (A c^3-c^3 C+3 B c^2 d-3 A c d^2+3 c C d^2-B d^3\right )-b^2 \left (c^6 C+3 c^4 C d^2+B c^3 d^3-3 c^2 (A-2 C) d^4-3 B c d^5+A d^6\right )-a^2 d^3 \left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )\right ) \log (c+d \tan (e+f x))}{d^3 \left (c^2+d^2\right )^3 f}-\frac {(b c-a d)^2 \left (c^2 C-B c d+A d^2\right )}{2 d^3 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}+\frac {(b c-a d) \left (b \left (2 c^4 C-B c^3 d+4 c^2 C d^2-3 B c d^3+2 A d^4\right )+a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right )}{d^3 \left (c^2+d^2\right )^2 f (c+d \tan (e+f x))} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 865, normalized size of antiderivative = 1.45

method	result	size
derivativedivides	\(\frac {\frac {\frac {\left (-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}\right ) \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}+\left (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}\right ) \arctan \left (\tan \left (f x +e \right )\right )}{\left (c^{2}+d^{2}\right )^{3}}-\frac {A \,a^{2} d^{4}-2 A a b c \,d^{3}+A \,b^{2} c^{2} d^{2}-B \,a^{2} c \,d^{3}+2 B a b \,c^{2} d^{2}-B \,b^{2} c^{3} d +C \,a^{2} c^{2} d^{2}-2 C a b \,c^{3} d +C \,b^{2} c^{4}}{2 d^{3} \left (c^{2}+d^{2}\right ) \left (c +d \tan \left (f x +e \right )\right )^{2}}-\frac {2 A \,a^{2} c \,d^{4}-2 A a b \,c^{2} d^{3}+2 A a b \,d^{5}-2 A \,b^{2} c \,d^{4}-B \,a^{2} c^{2} d^{3}+B \,a^{2} d^{5}-4 B a b c \,d^{4}+B \,b^{2} c^{4} d +3 B \,b^{2} c^{2} d^{3}-2 C \,a^{2} c \,d^{4}+2 C a b \,c^{4} d +6 C a b \,c^{2} d^{3}-2 C \,b^{2} c^{5}-4 C \,b^{2} c^{3} d^{2}}{d^{3} \left (c^{2}+d^{2}\right )^{2} \left (c +d \tan \left (f x +e \right )\right )}+\frac {\left (3 A \,a^{2} c^{2} d^{4}-A \,a^{2} d^{6}-2 A a b \,c^{3} d^{3}+6 A a b \,d^{5} c -3 A \,b^{2} c^{2} d^{4}+A \,b^{2} d^{6}-B \,a^{2} c^{3} d^{3}+3 a^{2} d^{5} B c -6 B a b \,c^{2} d^{4}+2 B a b \,d^{6}+B \,b^{2} c^{3} d^{3}-3 B \,b^{2} c \,d^{5}-3 C \,a^{2} c^{2} d^{4}+C \,a^{2} d^{6}+2 C a b \,c^{3} d^{3}-6 C a b c \,d^{5}+C \,b^{2} c^{6}+3 C \,b^{2} c^{4} d^{2}+6 C \,b^{2} c^{2} d^{4}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{\left (c^{2}+d^{2}\right )^{3} d^{3}}}{f}\)	\(865\)
default	\(\frac {\frac {\frac {\left (-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}\right ) \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}+\left (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}\right ) \arctan \left (\tan \left (f x +e \right )\right )}{\left (c^{2}+d^{2}\right )^{3}}-\frac {A \,a^{2} d^{4}-2 A a b c \,d^{3}+A \,b^{2} c^{2} d^{2}-B \,a^{2} c \,d^{3}+2 B a b \,c^{2} d^{2}-B \,b^{2} c^{3} d +C \,a^{2} c^{2} d^{2}-2 C a b \,c^{3} d +C \,b^{2} c^{4}}{2 d^{3} \left (c^{2}+d^{2}\right ) \left (c +d \tan \left (f x +e \right )\right )^{2}}-\frac {2 A \,a^{2} c \,d^{4}-2 A a b \,c^{2} d^{3}+2 A a b \,d^{5}-2 A \,b^{2} c \,d^{4}-B \,a^{2} c^{2} d^{3}+B \,a^{2} d^{5}-4 B a b c \,d^{4}+B \,b^{2} c^{4} d +3 B \,b^{2} c^{2} d^{3}-2 C \,a^{2} c \,d^{4}+2 C a b \,c^{4} d +6 C a b \,c^{2} d^{3}-2 C \,b^{2} c^{5}-4 C \,b^{2} c^{3} d^{2}}{d^{3} \left (c^{2}+d^{2}\right )^{2} \left (c +d \tan \left (f x +e \right )\right )}+\frac {\left (3 A \,a^{2} c^{2} d^{4}-A \,a^{2} d^{6}-2 A a b \,c^{3} d^{3}+6 A a b \,d^{5} c -3 A \,b^{2} c^{2} d^{4}+A \,b^{2} d^{6}-B \,a^{2} c^{3} d^{3}+3 a^{2} d^{5} B c -6 B a b \,c^{2} d^{4}+2 B a b \,d^{6}+B \,b^{2} c^{3} d^{3}-3 B \,b^{2} c \,d^{5}-3 C \,a^{2} c^{2} d^{4}+C \,a^{2} d^{6}+2 C a b \,c^{3} d^{3}-6 C a b c \,d^{5}+C \,b^{2} c^{6}+3 C \,b^{2} c^{4} d^{2}+6 C \,b^{2} c^{2} d^{4}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{\left (c^{2}+d^{2}\right )^{3} d^{3}}}{f}\)	\(865\)
norman	\(\text {Expression too large to display}\)	\(1444\)
risch	\(\text {Expression too large to display}\)	\(4025\)
parallelrisch	\(\text {Expression too large to display}\)	\(4621\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1618 vs. \(2 (591) = 1182\).

Time = 0.65 (sec) , antiderivative size = 1618, normalized size of antiderivative = 2.71 \[ \int \frac {(a+b \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Sympy [F(-2)]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 827, normalized size of antiderivative = 1.39 \[ \int \frac {(a+b \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^3} \, dx=\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 )}}{c^{6} + 3 \, c^{4} d^{2} + 3 \, c^{2} d^{4} + d^{6}} + \frac {2 \, {\left (C b^{2} c^{6} + 3 \, C b^{2} c^{4} d^{2} - {\left (B a^{2} + 2 \, {\left (A - C\right )} a b - B b^{2}\right )} c^{3} d^{3} + 3 \, {\left ({\left (A - C\right )} a^{2} - 2 \, B a b - {\left (A - 2 \, C\right )} b^{2}\right )} c^{2} d^{4} + 3 \, {\left (B a^{2} + 2 \, {\left (A - C\right )} a b - B b^{2}\right )} c d^{5} - {\left ({\left (A - C\right )} a^{2} - 2 \, B a b - A b^{2}\right )} d^{6}\right )} \log \left (d \tan \left (f x + e\right ) + c\right )}{c^{6} d^{3} + 3 \, c^{4} d^{5} + 3 \, c^{2} d^{7} + d^{9}} + \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 )}{c^{6} + 3 \, c^{4} d^{2} + 3 \, c^{2} d^{4} + d^{6}} + \frac {3 \, C b^{2} c^{6} - A a^{2} d^{6} - {\left (2 \, C a b + B b^{2}\right )} c^{5} d - {\left (C a^{2} + 2 \, B a b + {\left (A - 7 \, C\right )} b^{2}\right )} c^{4} d^{2} + {\left (3 \, B a^{2} + 2 \, {\left (3 \, A - 5 \, C\right )} a b - 5 \, B b^{2}\right )} c^{3} d^{3} - {\left ({\left (5 \, A - 3 \, C\right )} a^{2} - 6 \, B a b - 3 \, A b^{2}\right )} c^{2} d^{4} - {\left (B a^{2} + 2 \, A a b\right )} c d^{5} + 2 \, {\left (2 \, C b^{2} c^{5} d + 4 \, C b^{2} c^{3} d^{3} - {\left (2 \, C a b + B b^{2}\right )} c^{4} d^{2} + {\left (B a^{2} + 2 \, {\left (A - 3 \, C\right )} a b - 3 \, B b^{2}\right )} c^{2} d^{4} - 2 \, {\left ({\left (A - C\right )} a^{2} - 2 \, B a b - A b^{2}\right )} c d^{5} - {\left (B a^{2} + 2 \, A a b\right )} d^{6}\right )} \tan \left (f x + e\right )}{c^{6} d^{3} + 2 \, c^{4} d^{5} + c^{2} d^{7} + {\left (c^{4} d^{5} + 2 \, c^{2} d^{7} + d^{9}\right )} \tan \left (f x + e\right )^{2} + 2 \, {\left (c^{5} d^{4} + 2 \, c^{3} d^{6} + c d^{8}\right )} \tan \left (f x + e\right )}}{2 \, f} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1663 vs. \(2 (591) = 1182\).

Time = 1.05 (sec) , antiderivative size = 1663, normalized size of antiderivative = 2.79 \[ \int \frac {(a+b \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/2*(2*(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)/(c^6 + 3*c^4*d^2 + 3*c^2*d^4 + d^6) + (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)/(c^6 + 3*c^4*d^2 + 3*c^2*d^4 + d^6) + 2*(C*b^2*c^6 + 3*C*b^2*c^4*
d^2 - B*a^2*c^3*d^3 - 2*A*a*b*c^3*d^3 + 2*C*a*b*c^3*d^3 + B*b^2*c^3*d^3 + 3*A*a^2*c^2*d^4 - 3*C*a^2*c^2*d^4 -
6*B*a*b*c^2*d^4 - 3*A*b^2*c^2*d^4 + 6*C*b^2*c^2*d^4 + 3*B*a^2*c*d^5 + 6*A*a*b*c*d^5 - 6*C*a*b*c*d^5 - 3*B*b^2*
c*d^5 - A*a^2*d^6 + C*a^2*d^6 + 2*B*a*b*d^6 + A*b^2*d^6)*log(abs(d*tan(f*x + e) + c))/(c^6*d^3 + 3*c^4*d^5 + 3
*c^2*d^7 + d^9) - (3*C*b^2*c^6*d*tan(f*x + e)^2 + 9*C*b^2*c^4*d^3*tan(f*x + e)^2 - 3*B*a^2*c^3*d^4*tan(f*x + e
)^2 - 6*A*a*b*c^3*d^4*tan(f*x + e)^2 + 6*C*a*b*c^3*d^4*tan(f*x + e)^2 + 3*B*b^2*c^3*d^4*tan(f*x + e)^2 + 9*A*a
^2*c^2*d^5*tan(f*x + e)^2 - 9*C*a^2*c^2*d^5*tan(f*x + e)^2 - 18*B*a*b*c^2*d^5*tan(f*x + e)^2 - 9*A*b^2*c^2*d^5
*tan(f*x + e)^2 + 18*C*b^2*c^2*d^5*tan(f*x + e)^2 + 9*B*a^2*c*d^6*tan(f*x + e)^2 + 18*A*a*b*c*d^6*tan(f*x + e)
^2 - 18*C*a*b*c*d^6*tan(f*x + e)^2 - 9*B*b^2*c*d^6*tan(f*x + e)^2 - 3*A*a^2*d^7*tan(f*x + e)^2 + 3*C*a^2*d^7*t
an(f*x + e)^2 + 6*B*a*b*d^7*tan(f*x + e)^2 + 3*A*b^2*d^7*tan(f*x + e)^2 + 2*C*b^2*c^7*tan(f*x + e) + 4*C*a*b*c
^6*d*tan(f*x + e) + 2*B*b^2*c^6*d*tan(f*x + e) + 6*C*b^2*c^5*d^2*tan(f*x + e) - 8*B*a^2*c^4*d^3*tan(f*x + e) -
16*A*a*b*c^4*d^3*tan(f*x + e) + 28*C*a*b*c^4*d^3*tan(f*x + e) + 14*B*b^2*c^4*d^3*tan(f*x + e) + 22*A*a^2*c^3*
d^4*tan(f*x + e) - 22*C*a^2*c^3*d^4*tan(f*x + e) - 44*B*a*b*c^3*d^4*tan(f*x + e) - 22*A*b^2*c^3*d^4*tan(f*x +
e) + 28*C*b^2*c^3*d^4*tan(f*x + e) + 18*B*a^2*c^2*d^5*tan(f*x + e) + 36*A*a*b*c^2*d^5*tan(f*x + e) - 24*C*a*b*
c^2*d^5*tan(f*x + e) - 12*B*b^2*c^2*d^5*tan(f*x + e) - 2*A*a^2*c*d^6*tan(f*x + e) + 2*C*a^2*c*d^6*tan(f*x + e)
+ 4*B*a*b*c*d^6*tan(f*x + e) + 2*A*b^2*c*d^6*tan(f*x + e) + 2*B*a^2*d^7*tan(f*x + e) + 4*A*a*b*d^7*tan(f*x +
e) + 2*C*a*b*c^7 + B*b^2*c^7 + C*a^2*c^6*d + 2*B*a*b*c^6*d + A*b^2*c^6*d - C*b^2*c^6*d - 6*B*a^2*c^5*d^2 - 12*
A*a*b*c^5*d^2 + 18*C*a*b*c^5*d^2 + 9*B*b^2*c^5*d^2 + 14*A*a^2*c^4*d^3 - 11*C*a^2*c^4*d^3 - 22*B*a*b*c^4*d^3 -
11*A*b^2*c^4*d^3 + 11*C*b^2*c^4*d^3 + 7*B*a^2*c^3*d^4 + 14*A*a*b*c^3*d^4 - 8*C*a*b*c^3*d^4 - 4*B*b^2*c^3*d^4 +
3*A*a^2*c^2*d^5 + B*a^2*c*d^6 + 2*A*a*b*c*d^6 + A*a^2*d^7)/((c^6*d^2 + 3*c^4*d^4 + 3*c^2*d^6 + d^8)*(d*tan(f*
x + e) + c)^2))/f

Mupad [B] (verification not implemented)

Time = 27.06 (sec) , antiderivative size = 807, normalized size of antiderivative = 1.35 \[ \int \frac {(a+b \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^3} \, dx=-\frac {\frac {A\,a^2\,d^6-3\,C\,b^2\,c^6+B\,a^2\,c\,d^5+B\,b^2\,c^5\,d+5\,A\,a^2\,c^2\,d^4-3\,A\,b^2\,c^2\,d^4+A\,b^2\,c^4\,d^2-3\,B\,a^2\,c^3\,d^3+5\,B\,b^2\,c^3\,d^3-3\,C\,a^2\,c^2\,d^4+C\,a^2\,c^4\,d^2-7\,C\,b^2\,c^4\,d^2+2\,A\,a\,b\,c\,d^5+2\,C\,a\,b\,c^5\,d-6\,A\,a\,b\,c^3\,d^3-6\,B\,a\,b\,c^2\,d^4+2\,B\,a\,b\,c^4\,d^2+10\,C\,a\,b\,c^3\,d^3}{2\,d^3\,\left (c^4+2\,c^2\,d^2+d^4\right )}+\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (B\,a^2\,d^5-2\,C\,b^2\,c^5+2\,A\,a\,b\,d^5+2\,A\,a^2\,c\,d^4-2\,A\,b^2\,c\,d^4+B\,b^2\,c^4\,d-2\,C\,a^2\,c\,d^4-B\,a^2\,c^2\,d^3+3\,B\,b^2\,c^2\,d^3-4\,C\,b^2\,c^3\,d^2-4\,B\,a\,b\,c\,d^4+2\,C\,a\,b\,c^4\,d-2\,A\,a\,b\,c^2\,d^3+6\,C\,a\,b\,c^2\,d^3\right )}{d^2\,\left (c^4+2\,c^2\,d^2+d^4\right )}}{f\,\left (c^2+2\,c\,d\,\mathrm {tan}\left (e+f\,x\right )+d^2\,{\mathrm {tan}\left (e+f\,x\right )}^2\right )}-\frac {\ln \left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )\,\left (\frac {c^2\,\left (d^4\,\left (3\,A\,b^2-3\,A\,a^2+3\,C\,a^2-6\,C\,b^2+6\,B\,a\,b\right )+3\,C\,b^2\,d^4\right )-d^6\,\left (A\,b^2-A\,a^2+C\,a^2+2\,B\,a\,b\right )+C\,b^2\,d^6-c\,d^5\,\left (3\,B\,a^2-3\,B\,b^2+6\,A\,a\,b-6\,C\,a\,b\right )+c^3\,d^3\,\left (B\,a^2-B\,b^2+2\,A\,a\,b-2\,C\,a\,b\right )}{c^6\,d^3+3\,c^4\,d^5+3\,c^2\,d^7+d^9}-\frac {C\,b^2}{d^3}\right )}{f}-\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )-\mathrm {i}\right )\,\left (B\,a^2-B\,b^2+2\,A\,a\,b-2\,C\,a\,b-A\,a^2\,1{}\mathrm {i}+A\,b^2\,1{}\mathrm {i}+C\,a^2\,1{}\mathrm {i}-C\,b^2\,1{}\mathrm {i}+B\,a\,b\,2{}\mathrm {i}\right )}{2\,f\,\left (-c^3-c^2\,d\,3{}\mathrm {i}+3\,c\,d^2+d^3\,1{}\mathrm {i}\right )}-\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,\left (A\,b^2-A\,a^2+B\,a^2\,1{}\mathrm {i}-B\,b^2\,1{}\mathrm {i}+C\,a^2-C\,b^2+A\,a\,b\,2{}\mathrm {i}+2\,B\,a\,b-C\,a\,b\,2{}\mathrm {i}\right )}{2\,f\,\left (-c^3\,1{}\mathrm {i}-3\,c^2\,d+c\,d^2\,3{}\mathrm {i}+d^3\right )} \]

[In]

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

[Out]

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