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

Optimal. Leaf size=417 \[ \frac{\log (\cos (e+f x)) \left (a^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+2 a b \left (-A \left (c^2-d^2\right )-2 B c d+c^2 C-C d^2\right )-b^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )\right )}{f \left (c^2+d^2\right )^2}-\frac{x \left (a^2 \left (-A \left (c^2-d^2\right )-2 B c d+c^2 C-C d^2\right )-2 a b \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )-b^2 \left (-A \left (c^2-d^2\right )-2 B c d+c^2 C-C d^2\right )\right )}{\left (c^2+d^2\right )^2}-\frac{\left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^2}{d f \left (c^2+d^2\right ) (c+d \tan (e+f x))}-\frac{(b c-a d) \left (a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (2 A d^4-B c^3 d-3 B c d^3+4 c^2 C d^2+2 c^4 C\right )\right ) \log (c+d \tan (e+f x))}{d^3 f \left (c^2+d^2\right )^2}+\frac{b^2 \tan (e+f x) \left (d^2 (A+C)-B c d+2 c^2 C\right )}{d^2 f \left (c^2+d^2\right )} \]

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

________________________________________________________________________________________

Rubi [A]  time = 1.11294, antiderivative size = 417, 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} \[ \frac{\log (\cos (e+f x)) \left (a^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+2 a b \left (-A \left (c^2-d^2\right )-2 B c d+c^2 C-C d^2\right )-b^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )\right )}{f \left (c^2+d^2\right )^2}-\frac{x \left (a^2 \left (-A \left (c^2-d^2\right )-2 B c d+c^2 C-C d^2\right )-2 a b \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )-b^2 \left (-A \left (c^2-d^2\right )-2 B c d+c^2 C-C d^2\right )\right )}{\left (c^2+d^2\right )^2}-\frac{\left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^2}{d f \left (c^2+d^2\right ) (c+d \tan (e+f x))}-\frac{(b c-a d) \left (a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (2 A d^4-B c^3 d-3 B c d^3+4 c^2 C d^2+2 c^4 C\right )\right ) \log (c+d \tan (e+f x))}{d^3 f \left (c^2+d^2\right )^2}+\frac{b^2 \tan (e+f x) \left (d^2 (A+C)-B c d+2 c^2 C\right )}{d^2 f \left (c^2+d^2\right )} \]

Antiderivative was successfully verified.

[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])^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)/(c^2 + d^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]])/((c^2 + d^2)^2*f) - ((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)))*Lo
g[c + d*Tan[e + f*x]])/(d^3*(c^2 + d^2)^2*f) + (b^2*(2*c^2*C - B*c*d + (A + C)*d^2)*Tan[e + f*x])/(d^2*(c^2 +
d^2)*f) - ((c^2*C - B*c*d + A*d^2)*(a + b*Tan[e + f*x])^2)/(d*(c^2 + d^2)*f*(c + d*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{(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))^2} \, dx &=-\frac{\left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^2}{d \left (c^2+d^2\right ) f (c+d \tan (e+f x))}+\frac{\int \frac{(a+b \tan (e+f x)) \left (A d (a c+2 b d)+(2 b c-a d) (c C-B d)+d ((A-C) (b c-a d)+B (a c+b d)) \tan (e+f x)+b \left (2 c^2 C-B c d+(A+C) d^2\right ) \tan ^2(e+f x)\right )}{c+d \tan (e+f x)} \, dx}{d \left (c^2+d^2\right )}\\ &=\frac{b^2 \left (2 c^2 C-B c d+(A+C) d^2\right ) \tan (e+f x)}{d^2 \left (c^2+d^2\right ) f}-\frac{\left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^2}{d \left (c^2+d^2\right ) f (c+d \tan (e+f x))}-\frac{\int \frac{b^2 c \left (2 c^2 C-B c d+(A+C) d^2\right )-a d (A d (a c+2 b d)+(2 b c-a d) (c C-B d))-d^2 \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)+b (2 b c C-b B d-2 a C d) \left (c^2+d^2\right ) \tan ^2(e+f x)}{c+d \tan (e+f x)} \, dx}{d^2 \left (c^2+d^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 (c^2+d^2\right )^2}+\frac{b^2 \left (2 c^2 C-B c d+(A+C) d^2\right ) \tan (e+f x)}{d^2 \left (c^2+d^2\right ) f}-\frac{\left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^2}{d \left (c^2+d^2\right ) f (c+d \tan (e+f x))}-\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 (c^2+d^2\right )^2}-\frac{\left ((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 )\right ) \int \frac{1+\tan ^2(e+f x)}{c+d \tan (e+f x)} \, dx}{d^2 \left (c^2+d^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 (c^2+d^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 (c^2+d^2\right )^2 f}+\frac{b^2 \left (2 c^2 C-B c d+(A+C) d^2\right ) \tan (e+f x)}{d^2 \left (c^2+d^2\right ) f}-\frac{\left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^2}{d \left (c^2+d^2\right ) f (c+d \tan (e+f x))}-\frac{\left ((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 )\right ) \operatorname{Subst}\left (\int \frac{1}{c+x} \, dx,x,d \tan (e+f x)\right )}{d^3 \left (c^2+d^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 (c^2+d^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 (c^2+d^2\right )^2 f}-\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 ) \log (c+d \tan (e+f x))}{d^3 \left (c^2+d^2\right )^2 f}+\frac{b^2 \left (2 c^2 C-B c d+(A+C) d^2\right ) \tan (e+f x)}{d^2 \left (c^2+d^2\right ) f}-\frac{\left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^2}{d \left (c^2+d^2\right ) f (c+d \tan (e+f x))}\\ \end{align*}

Mathematica [C]  time = 7.76262, size = 2636, normalized size = 6.32 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[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])^2,x]

[Out]

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

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Maple [B]  time = 0.072, 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((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))^2,x)

[Out]

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

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Maxima [A]  time = 1.57371, size = 666, normalized size = 1.6 \begin{align*} \frac{\frac{2 \, C b^{2} \tan \left (f x + e\right )}{d^{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 )}}{c^{4} + 2 \, c^{2} d^{2} + d^{4}} - \frac{2 \,{\left (2 \, C b^{2} c^{5} + 4 \, C b^{2} c^{3} d^{2} -{\left (2 \, C a b + B b^{2}\right )} c^{4} d +{\left (B a^{2} + 2 \,{\left (A - 3 \, C\right )} a b - 3 \, B b^{2}\right )} c^{2} d^{3} - 2 \,{\left ({\left (A - C\right )} a^{2} - 2 \, B a b - A b^{2}\right )} c d^{4} -{\left (B a^{2} + 2 \, A a b\right )} d^{5}\right )} \log \left (d \tan \left (f x + e\right ) + c\right )}{c^{4} d^{3} + 2 \, c^{2} d^{5} + d^{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 )}{c^{4} + 2 \, c^{2} d^{2} + d^{4}} - \frac{2 \,{\left (C b^{2} c^{4} + A a^{2} d^{4} -{\left (2 \, C a b + B b^{2}\right )} c^{3} d +{\left (C a^{2} + 2 \, B a b + A b^{2}\right )} c^{2} d^{2} -{\left (B a^{2} + 2 \, A a b\right )} c d^{3}\right )}}{c^{3} d^{3} + c d^{5} +{\left (c^{2} d^{4} + d^{6}\right )} \tan \left (f x + e\right )}}{2 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))^2,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 3.75543, size = 1983, normalized size = 4.76 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))^2,x, algorithm="fricas")

[Out]

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

[Out]

Exception raised: AttributeError

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Giac [B]  time = 1.90879, size = 1231, normalized size = 2.95 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))^2,x, algorithm="giac")

[Out]

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