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

Optimal. Leaf size=597 \[ \frac{\left (-a^3 b^3 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )-3 a^2 b^4 \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-2 C d^2\right )+3 a^4 b^2 C d^2+a^6 C d^2+3 a b^5 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )+b^6 \left (c (2 B d+c C)-A \left (c^2-d^2\right )\right )\right ) \log (a+b \tan (e+f x))}{b^3 f \left (a^2+b^2\right )^3}-\frac{\log (\cos (e+f x)) \left (3 a^2 b \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )+a^3 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )-3 a b^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )-b^3 \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )\right )}{f \left (a^2+b^2\right )^3}-\frac{x \left (-3 a^2 b \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )+a^3 \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )-3 a b^2 \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )+b^3 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )}{\left (a^2+b^2\right )^3}-\frac{\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^2}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}-\frac{(b c-a d) \left (-a^2 b^2 (d (A-3 C)+B c)+a^4 C d+2 a b^3 (A c-B d-c C)+b^4 (A d+B c)\right )}{b^3 f \left (a^2+b^2\right )^2 (a+b \tan (e+f x))} \]

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

________________________________________________________________________________________

Rubi [A]  time = 1.29003, antiderivative size = 597, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {3645, 3635, 3626, 3617, 31, 3475} \[ \frac{\left (-a^3 b^3 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )-3 a^2 b^4 \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-2 C d^2\right )+3 a^4 b^2 C d^2+a^6 C d^2+3 a b^5 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )+b^6 \left (c (2 B d+c C)-A \left (c^2-d^2\right )\right )\right ) \log (a+b \tan (e+f x))}{b^3 f \left (a^2+b^2\right )^3}-\frac{\log (\cos (e+f x)) \left (3 a^2 b \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )+a^3 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )-3 a b^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )-b^3 \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )\right )}{f \left (a^2+b^2\right )^3}-\frac{x \left (-3 a^2 b \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )+a^3 \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )-3 a b^2 \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )+b^3 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )}{\left (a^2+b^2\right )^3}-\frac{\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^2}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}-\frac{(b c-a d) \left (-a^2 b^2 (d (A-3 C)+B c)+a^4 C d+2 a b^3 (A c-B d-c C)+b^4 (A d+B c)\right )}{b^3 f \left (a^2+b^2\right )^2 (a+b \tan (e+f x))} \]

Antiderivative was successfully verified.

[In]

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

Rule 3645

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*T
an[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 3635

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 3626

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 3617

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 31

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

Rule 3475

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

Rubi steps

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

Mathematica [C]  time = 7.9052, size = 2499, normalized size = 4.19 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-(A*b^4*c^2) + a*b^3*B*c^2 - a^2*b^2*c^2*C + 2*a*A*b^3*c*d - 2*a^2*b^2*B*c*d + 2*a^3*b*c*C*d - a^2*A*b^2*d^2
+ a^3*b*B*d^2 - a^4*C*d^2)*Sec[e + f*x]*(a*Cos[e + f*x] + b*Sin[e + f*x])*(c + d*Tan[e + f*x])^2)/(2*(a - I*b
)^2*(a + I*b)^2*b*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^2*(a + b*Tan[e + f*x])^3) + ((a^3*A*c^2 - 3*a*A*b^2*c^2
+ 3*a^2*b*B*c^2 - b^3*B*c^2 - a^3*c^2*C + 3*a*b^2*c^2*C + 6*a^2*A*b*c*d - 2*A*b^3*c*d - 2*a^3*B*c*d + 6*a*b^2*
B*c*d - 6*a^2*b*c*C*d + 2*b^3*c*C*d - a^3*A*d^2 + 3*a*A*b^2*d^2 - 3*a^2*b*B*d^2 + b^3*B*d^2 + a^3*C*d^2 - 3*a*
b^2*C*d^2)*(e + f*x)*Sec[e + f*x]*(a*Cos[e + f*x] + b*Sin[e + f*x])^3*(c + d*Tan[e + f*x])^2)/((a - I*b)^3*(a
+ I*b)^3*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^2*(a + b*Tan[e + f*x])^3) + (((3*I)*a^9*A*b^6*c^2 + 3*a^8*A*b^7*c
^2 + (5*I)*a^7*A*b^8*c^2 + 5*a^6*A*b^9*c^2 + I*a^5*A*b^10*c^2 + a^4*A*b^11*c^2 - I*a^3*A*b^12*c^2 - a^2*A*b^13
*c^2 - I*a^10*b^5*B*c^2 - a^9*b^6*B*c^2 + I*a^8*b^7*B*c^2 + a^7*b^8*B*c^2 + (5*I)*a^6*b^9*B*c^2 + 5*a^5*b^10*B
*c^2 + (3*I)*a^4*b^11*B*c^2 + 3*a^3*b^12*B*c^2 - (3*I)*a^9*b^6*c^2*C - 3*a^8*b^7*c^2*C - (5*I)*a^7*b^8*c^2*C -
5*a^6*b^9*c^2*C - I*a^5*b^10*c^2*C - a^4*b^11*c^2*C + I*a^3*b^12*c^2*C + a^2*b^13*c^2*C - (2*I)*a^10*A*b^5*c*
d - 2*a^9*A*b^6*c*d + (2*I)*a^8*A*b^7*c*d + 2*a^7*A*b^8*c*d + (10*I)*a^6*A*b^9*c*d + 10*a^5*A*b^10*c*d + (6*I)
*a^4*A*b^11*c*d + 6*a^3*A*b^12*c*d - (6*I)*a^9*b^6*B*c*d - 6*a^8*b^7*B*c*d - (10*I)*a^7*b^8*B*c*d - 10*a^6*b^9
*B*c*d - (2*I)*a^5*b^10*B*c*d - 2*a^4*b^11*B*c*d + (2*I)*a^3*b^12*B*c*d + 2*a^2*b^13*B*c*d + (2*I)*a^10*b^5*c*
C*d + 2*a^9*b^6*c*C*d - (2*I)*a^8*b^7*c*C*d - 2*a^7*b^8*c*C*d - (10*I)*a^6*b^9*c*C*d - 10*a^5*b^10*c*C*d - (6*
I)*a^4*b^11*c*C*d - 6*a^3*b^12*c*C*d - (3*I)*a^9*A*b^6*d^2 - 3*a^8*A*b^7*d^2 - (5*I)*a^7*A*b^8*d^2 - 5*a^6*A*b
^9*d^2 - I*a^5*A*b^10*d^2 - a^4*A*b^11*d^2 + I*a^3*A*b^12*d^2 + a^2*A*b^13*d^2 + I*a^10*b^5*B*d^2 + a^9*b^6*B*
d^2 - I*a^8*b^7*B*d^2 - a^7*b^8*B*d^2 - (5*I)*a^6*b^9*B*d^2 - 5*a^5*b^10*B*d^2 - (3*I)*a^4*b^11*B*d^2 - 3*a^3*
b^12*B*d^2 + I*a^13*b^2*C*d^2 + a^12*b^3*C*d^2 + (5*I)*a^11*b^4*C*d^2 + 5*a^10*b^5*C*d^2 + (13*I)*a^9*b^6*C*d^
2 + 13*a^8*b^7*C*d^2 + (15*I)*a^7*b^8*C*d^2 + 15*a^6*b^9*C*d^2 + (6*I)*a^5*b^10*C*d^2 + 6*a^4*b^11*C*d^2)*(e +
f*x)*Sec[e + f*x]*(a*Cos[e + f*x] + b*Sin[e + f*x])^3*(c + d*Tan[e + f*x])^2)/(a^2*(a - I*b)^6*(a + I*b)^5*b^
5*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^2*(a + b*Tan[e + f*x])^3) - (I*(3*a^2*A*b^4*c^2 - A*b^6*c^2 - a^3*b^3*B*
c^2 + 3*a*b^5*B*c^2 - 3*a^2*b^4*c^2*C + b^6*c^2*C - 2*a^3*A*b^3*c*d + 6*a*A*b^5*c*d - 6*a^2*b^4*B*c*d + 2*b^6*
B*c*d + 2*a^3*b^3*c*C*d - 6*a*b^5*c*C*d - 3*a^2*A*b^4*d^2 + A*b^6*d^2 + a^3*b^3*B*d^2 - 3*a*b^5*B*d^2 + a^6*C*
d^2 + 3*a^4*b^2*C*d^2 + 6*a^2*b^4*C*d^2)*ArcTan[Tan[e + f*x]]*Sec[e + f*x]*(a*Cos[e + f*x] + b*Sin[e + f*x])^3
*(c + d*Tan[e + f*x])^2)/(b^3*(a^2 + b^2)^3*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^2*(a + b*Tan[e + f*x])^3) - (C
*d^2*Log[Cos[e + f*x]]*Sec[e + f*x]*(a*Cos[e + f*x] + b*Sin[e + f*x])^3*(c + d*Tan[e + f*x])^2)/(b^3*f*(c*Cos[
e + f*x] + d*Sin[e + f*x])^2*(a + b*Tan[e + f*x])^3) + ((3*a^2*A*b^4*c^2 - A*b^6*c^2 - a^3*b^3*B*c^2 + 3*a*b^5
*B*c^2 - 3*a^2*b^4*c^2*C + b^6*c^2*C - 2*a^3*A*b^3*c*d + 6*a*A*b^5*c*d - 6*a^2*b^4*B*c*d + 2*b^6*B*c*d + 2*a^3
*b^3*c*C*d - 6*a*b^5*c*C*d - 3*a^2*A*b^4*d^2 + A*b^6*d^2 + a^3*b^3*B*d^2 - 3*a*b^5*B*d^2 + a^6*C*d^2 + 3*a^4*b
^2*C*d^2 + 6*a^2*b^4*C*d^2)*Log[(a*Cos[e + f*x] + b*Sin[e + f*x])^2]*Sec[e + f*x]*(a*Cos[e + f*x] + b*Sin[e +
f*x])^3*(c + d*Tan[e + f*x])^2)/(2*b^3*(a^2 + b^2)^3*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^2*(a + b*Tan[e + f*x]
)^3) + (Sec[e + f*x]*(a*Cos[e + f*x] + b*Sin[e + f*x])^2*(3*a*A*b^4*c^2*Sin[e + f*x] - 2*a^2*b^3*B*c^2*Sin[e +
f*x] + b^5*B*c^2*Sin[e + f*x] + a^3*b^2*c^2*C*Sin[e + f*x] - 2*a*b^4*c^2*C*Sin[e + f*x] - 4*a^2*A*b^3*c*d*Sin
[e + f*x] + 2*A*b^5*c*d*Sin[e + f*x] + 2*a^3*b^2*B*c*d*Sin[e + f*x] - 4*a*b^4*B*c*d*Sin[e + f*x] + 6*a^2*b^3*c
*C*d*Sin[e + f*x] + a^3*A*b^2*d^2*Sin[e + f*x] - 2*a*A*b^4*d^2*Sin[e + f*x] + 3*a^2*b^3*B*d^2*Sin[e + f*x] - a
^5*C*d^2*Sin[e + f*x] - 4*a^3*b^2*C*d^2*Sin[e + f*x])*(c + d*Tan[e + f*x])^2)/(a*(a - I*b)^2*(a + I*b)^2*b^2*f
*(c*Cos[e + f*x] + d*Sin[e + f*x])^2*(a + b*Tan[e + f*x])^3)

________________________________________________________________________________________

Maple [B]  time = 0.077, size = 2465, normalized size = 4.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.59634, size = 1133, normalized size = 1.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Fricas [B]  time = 4.65581, size = 3513, normalized size = 5.88 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: AttributeError

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Giac [B]  time = 1.87047, size = 2314, normalized size = 3.88 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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