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

Optimal. Leaf size=415 \[ -\frac{\log (\cos (e+f x)) \left (a^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )+2 a b \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )-b^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )}{f \left (a^2+b^2\right )^2}-\frac{x \left (a^2 \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )-2 a b \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )-b^2 \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )\right )}{\left (a^2+b^2\right )^2}-\frac{\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^2}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}-\frac{(b c-a d) \left (a^2 b^2 (B c-4 C d)+a^3 b B d-2 a^4 C d-a b^3 (2 A c-3 B d-2 c C)-b^4 (2 A d+B c)\right ) \log (a+b \tan (e+f x))}{b^3 f \left (a^2+b^2\right )^2}+\frac{d^2 \tan (e+f x) \left (2 a^2 C-a b B+A b^2+b^2 C\right )}{b^2 f \left (a^2+b^2\right )} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.05285, antiderivative size = 415, 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, 3637, 3626, 3617, 31, 3475} \[ -\frac{\log (\cos (e+f x)) \left (a^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )+2 a b \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )-b^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )}{f \left (a^2+b^2\right )^2}-\frac{x \left (a^2 \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )-2 a b \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )-b^2 \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )\right )}{\left (a^2+b^2\right )^2}-\frac{\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^2}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}-\frac{(b c-a d) \left (a^2 b^2 (B c-4 C d)+a^3 b B d-2 a^4 C d-a b^3 (2 A c-3 B d-2 c C)-b^4 (2 A d+B c)\right ) \log (a+b \tan (e+f x))}{b^3 f \left (a^2+b^2\right )^2}+\frac{d^2 \tan (e+f x) \left (2 a^2 C-a b B+A b^2+b^2 C\right )}{b^2 f \left (a^2+b^2\right )} \]

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])^2,x]

[Out]

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

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 3637

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*tan[(e
_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(b*C*Tan[e + f*x]*(c + d*Tan[e + f*x])
^(n + 1))/(d*f*(n + 2)), x] - Dist[1/(d*(n + 2)), Int[(c + d*Tan[e + f*x])^n*Simp[b*c*C - a*A*d*(n + 2) - (A*b
+ a*B - b*C)*d*(n + 2)*Tan[e + f*x] - (a*C*d*(n + 2) - b*(c*C - B*d*(n + 2)))*Tan[e + f*x]^2, x], x], x] /; F
reeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[c^2 + d^2, 0] &&  !LtQ[n, -1]

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

Mathematica [C]  time = 7.78334, size = 2640, normalized size = 6.36 \[ \text{Result too large to show} \]

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])^2,x]

[Out]

((-I)*(-2*a^6*A*b^6*c^2 + (2*I)*a^5*A*b^7*c^2 - 2*a^4*A*b^8*c^2 + (2*I)*a^3*A*b^9*c^2 + a^7*b^5*B*c^2 - I*a^6*
b^6*B*c^2 - a^3*b^9*B*c^2 + I*a^2*b^10*B*c^2 + 2*a^6*b^6*c^2*C - (2*I)*a^5*b^7*c^2*C + 2*a^4*b^8*c^2*C - (2*I)
*a^3*b^9*c^2*C + 2*a^7*A*b^5*c*d - (2*I)*a^6*A*b^6*c*d - 2*a^3*A*b^9*c*d + (2*I)*a^2*A*b^10*c*d + 4*a^6*b^6*B*
c*d - (4*I)*a^5*b^7*B*c*d + 4*a^4*b^8*B*c*d - (4*I)*a^3*b^9*B*c*d - 2*a^9*b^3*c*C*d + (2*I)*a^8*b^4*c*C*d - 8*
a^7*b^5*c*C*d + (8*I)*a^6*b^6*c*C*d - 6*a^5*b^7*c*C*d + (6*I)*a^4*b^8*c*C*d + 2*a^6*A*b^6*d^2 - (2*I)*a^5*A*b^
7*d^2 + 2*a^4*A*b^8*d^2 - (2*I)*a^3*A*b^9*d^2 - a^9*b^3*B*d^2 + I*a^8*b^4*B*d^2 - 4*a^7*b^5*B*d^2 + (4*I)*a^6*
b^6*B*d^2 - 3*a^5*b^7*B*d^2 + (3*I)*a^4*b^8*B*d^2 + 2*a^10*b^2*C*d^2 - (2*I)*a^9*b^3*C*d^2 + 6*a^8*b^4*C*d^2 -
(6*I)*a^7*b^5*C*d^2 + 4*a^6*b^6*C*d^2 - (4*I)*a^5*b^7*C*d^2)*(e + f*x)*(a*Cos[e + f*x] + b*Sin[e + f*x])^2*(c
+ d*Tan[e + f*x])^2)/(a^2*(a - I*b)^4*(a + I*b)^3*b^5*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^2*(a + b*Tan[e + f*
x])^2) - (I*(2*a*A*b^4*c^2 - a^2*b^3*B*c^2 + b^5*B*c^2 - 2*a*b^4*c^2*C - 2*a^2*A*b^3*c*d + 2*A*b^5*c*d - 4*a*b
^4*B*c*d + 2*a^4*b*c*C*d + 6*a^2*b^3*c*C*d - 2*a*A*b^4*d^2 + a^4*b*B*d^2 + 3*a^2*b^3*B*d^2 - 2*a^5*C*d^2 - 4*a
^3*b^2*C*d^2)*ArcTan[Tan[e + f*x]]*(a*Cos[e + f*x] + b*Sin[e + f*x])^2*(c + d*Tan[e + f*x])^2)/(b^3*(a^2 + b^2
)^2*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^2*(a + b*Tan[e + f*x])^2) + ((-2*b*c*C*d - b*B*d^2 + 2*a*C*d^2)*Log[Co
s[e + f*x]]*(a*Cos[e + f*x] + b*Sin[e + f*x])^2*(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])^2) + ((2*a*A*b^4*c^2 - a^2*b^3*B*c^2 + b^5*B*c^2 - 2*a*b^4*c^2*C - 2*a^2*A*b^3*c*d +
2*A*b^5*c*d - 4*a*b^4*B*c*d + 2*a^4*b*c*C*d + 6*a^2*b^3*c*C*d - 2*a*A*b^4*d^2 + a^4*b*B*d^2 + 3*a^2*b^3*B*d^2
- 2*a^5*C*d^2 - 4*a^3*b^2*C*d^2)*Log[(a*Cos[e + f*x] + b*Sin[e + f*x])^2]*(a*Cos[e + f*x] + b*Sin[e + f*x])^2
*(c + d*Tan[e + f*x])^2)/(2*b^3*(a^2 + b^2)^2*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^2*(a + b*Tan[e + f*x])^2) +
(Sec[e + f*x]*(a*Cos[e + f*x] + b*Sin[e + f*x])*(a^5*b*C*d^2 + 2*a^3*b^3*C*d^2 + a*b^5*C*d^2 + a^4*A*b^2*c^2*(
e + f*x) - a^2*A*b^4*c^2*(e + f*x) + 2*a^3*b^3*B*c^2*(e + f*x) - a^4*b^2*c^2*C*(e + f*x) + a^2*b^4*c^2*C*(e +
f*x) + 4*a^3*A*b^3*c*d*(e + f*x) - 2*a^4*b^2*B*c*d*(e + f*x) + 2*a^2*b^4*B*c*d*(e + f*x) - 4*a^3*b^3*c*C*d*(e
+ f*x) - a^4*A*b^2*d^2*(e + f*x) + a^2*A*b^4*d^2*(e + f*x) - 2*a^3*b^3*B*d^2*(e + f*x) + a^4*b^2*C*d^2*(e + f*
x) - a^2*b^4*C*d^2*(e + f*x) - a^5*b*C*d^2*Cos[2*(e + f*x)] - 2*a^3*b^3*C*d^2*Cos[2*(e + f*x)] - a*b^5*C*d^2*C
os[2*(e + f*x)] + a^4*A*b^2*c^2*(e + f*x)*Cos[2*(e + f*x)] - a^2*A*b^4*c^2*(e + f*x)*Cos[2*(e + f*x)] + 2*a^3*
b^3*B*c^2*(e + f*x)*Cos[2*(e + f*x)] - a^4*b^2*c^2*C*(e + f*x)*Cos[2*(e + f*x)] + a^2*b^4*c^2*C*(e + f*x)*Cos[
2*(e + f*x)] + 4*a^3*A*b^3*c*d*(e + f*x)*Cos[2*(e + f*x)] - 2*a^4*b^2*B*c*d*(e + f*x)*Cos[2*(e + f*x)] + 2*a^2
*b^4*B*c*d*(e + f*x)*Cos[2*(e + f*x)] - 4*a^3*b^3*c*C*d*(e + f*x)*Cos[2*(e + f*x)] - a^4*A*b^2*d^2*(e + f*x)*C
os[2*(e + f*x)] + a^2*A*b^4*d^2*(e + f*x)*Cos[2*(e + f*x)] - 2*a^3*b^3*B*d^2*(e + f*x)*Cos[2*(e + f*x)] + a^4*
b^2*C*d^2*(e + f*x)*Cos[2*(e + f*x)] - a^2*b^4*C*d^2*(e + f*x)*Cos[2*(e + f*x)] + a^2*A*b^4*c^2*Sin[2*(e + f*x
)] + A*b^6*c^2*Sin[2*(e + f*x)] - a^3*b^3*B*c^2*Sin[2*(e + f*x)] - a*b^5*B*c^2*Sin[2*(e + f*x)] + a^4*b^2*c^2*
C*Sin[2*(e + f*x)] + a^2*b^4*c^2*C*Sin[2*(e + f*x)] - 2*a^3*A*b^3*c*d*Sin[2*(e + f*x)] - 2*a*A*b^5*c*d*Sin[2*(
e + f*x)] + 2*a^4*b^2*B*c*d*Sin[2*(e + f*x)] + 2*a^2*b^4*B*c*d*Sin[2*(e + f*x)] - 2*a^5*b*c*C*d*Sin[2*(e + f*x
)] - 2*a^3*b^3*c*C*d*Sin[2*(e + f*x)] + a^4*A*b^2*d^2*Sin[2*(e + f*x)] + a^2*A*b^4*d^2*Sin[2*(e + f*x)] - a^5*
b*B*d^2*Sin[2*(e + f*x)] - a^3*b^3*B*d^2*Sin[2*(e + f*x)] + 2*a^6*C*d^2*Sin[2*(e + f*x)] + 3*a^4*b^2*C*d^2*Sin
[2*(e + f*x)] + a^2*b^4*C*d^2*Sin[2*(e + f*x)] + a^3*A*b^3*c^2*(e + f*x)*Sin[2*(e + f*x)] - a*A*b^5*c^2*(e + f
*x)*Sin[2*(e + f*x)] + 2*a^2*b^4*B*c^2*(e + f*x)*Sin[2*(e + f*x)] - a^3*b^3*c^2*C*(e + f*x)*Sin[2*(e + f*x)] +
a*b^5*c^2*C*(e + f*x)*Sin[2*(e + f*x)] + 4*a^2*A*b^4*c*d*(e + f*x)*Sin[2*(e + f*x)] - 2*a^3*b^3*B*c*d*(e + f*
x)*Sin[2*(e + f*x)] + 2*a*b^5*B*c*d*(e + f*x)*Sin[2*(e + f*x)] - 4*a^2*b^4*c*C*d*(e + f*x)*Sin[2*(e + f*x)] -
a^3*A*b^3*d^2*(e + f*x)*Sin[2*(e + f*x)] + a*A*b^5*d^2*(e + f*x)*Sin[2*(e + f*x)] - 2*a^2*b^4*B*d^2*(e + f*x)*
Sin[2*(e + f*x)] + a^3*b^3*C*d^2*(e + f*x)*Sin[2*(e + f*x)] - a*b^5*C*d^2*(e + f*x)*Sin[2*(e + f*x)])*(c + d*T
an[e + f*x])^2)/(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])^2)

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Maple [B]  time = 0.061, size = 1554, normalized size = 3.7 \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))^2,x)

[Out]

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

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Maxima [A]  time = 1.52891, size = 670, normalized size = 1.61 \begin{align*} \frac{\frac{2 \, C d^{2} \tan \left (f x + e\right )}{b^{2}} + \frac{2 \,{\left ({\left ({\left (A - C\right )} a^{2} + 2 \, B a b -{\left (A - C\right )} b^{2}\right )} c^{2} - 2 \,{\left (B a^{2} - 2 \,{\left (A - C\right )} a b - B b^{2}\right )} c d -{\left ({\left (A - C\right )} a^{2} + 2 \, B a b -{\left (A - C\right )} b^{2}\right )} d^{2}\right )}{\left (f x + e\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac{2 \,{\left ({\left (B a^{2} b^{3} - 2 \,{\left (A - C\right )} a b^{4} - B b^{5}\right )} c^{2} - 2 \,{\left (C a^{4} b -{\left (A - 3 \, C\right )} a^{2} b^{3} - 2 \, B a b^{4} + A b^{5}\right )} c d +{\left (2 \, C a^{5} - B a^{4} b + 4 \, C a^{3} b^{2} - 3 \, B a^{2} b^{3} + 2 \, A a b^{4}\right )} d^{2}\right )} \log \left (b \tan \left (f x + e\right ) + a\right )}{a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7}} + \frac{{\left ({\left (B a^{2} - 2 \,{\left (A - C\right )} a b - B b^{2}\right )} c^{2} + 2 \,{\left ({\left (A - C\right )} a^{2} + 2 \, B a b -{\left (A - C\right )} b^{2}\right )} c d -{\left (B a^{2} - 2 \,{\left (A - C\right )} a b - B b^{2}\right )} d^{2}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac{2 \,{\left ({\left (C a^{2} b^{2} - B a b^{3} + A b^{4}\right )} c^{2} - 2 \,{\left (C a^{3} b - B a^{2} b^{2} + A a b^{3}\right )} c d +{\left (C a^{4} - B a^{3} b + A a^{2} b^{2}\right )} d^{2}\right )}}{a^{3} b^{3} + a b^{5} +{\left (a^{2} b^{4} + b^{6}\right )} \tan \left (f x + e\right )}}{2 \, f} \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))^2,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 3.56237, size = 1987, normalized size = 4.79 \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))^2,x, algorithm="fricas")

[Out]

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

[Out]

Exception raised: AttributeError

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Giac [B]  time = 1.8707, size = 1231, normalized size = 2.97 \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))^2,x, algorithm="giac")

[Out]

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