\(\int \frac {(a+b \tan (e+f x))^3 (A+B \tan (e+f x)+C \tan ^2(e+f x))}{(c+d \tan (e+f x))^2} \, dx\) [77]

Optimal result

Integrand size = 45, antiderivative size = 579 \[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^2} \, dx=-\frac {\left (a^3 \left (c^2 C-2 B c d-C d^2-A \left (c^2-d^2\right )\right )-3 a b^2 \left (c^2 C-2 B c d-C d^2-A \left (c^2-d^2\right )\right )-3 a^2 b \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )+b^3 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) x}{\left (c^2+d^2\right )^2}+\frac {\left (3 a^2 b \left (c^2 C-2 B c d-C d^2-A \left (c^2-d^2\right )\right )-b^3 \left (c^2 C-2 B c d-C d^2-A \left (c^2-d^2\right )\right )+a^3 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )-3 a b^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \log (\cos (e+f x))}{\left (c^2+d^2\right )^2 f}+\frac {(b c-a d)^2 \left (b \left (3 c^4 C-2 B c^3 d+c^2 (A+5 C) d^2-4 B c d^3+3 A d^4\right )+a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \log (c+d \tan (e+f x))}{d^4 \left (c^2+d^2\right )^2 f}+\frac {b^2 \left (a d \left (3 c^2 C-B c d+(A+2 C) d^2\right )-b \left (3 c^3 C-2 B c^2 d+c (A+2 C) d^2-B d^3\right )\right ) \tan (e+f x)}{d^3 \left (c^2+d^2\right ) f}+\frac {b \left (3 c^2 C-2 B c d+(2 A+C) d^2\right ) (a+b \tan (e+f x))^2}{2 d^2 \left (c^2+d^2\right ) f}-\frac {\left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^3}{d \left (c^2+d^2\right ) f (c+d \tan (e+f x))} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.96 (sec) , antiderivative size = 1022, normalized size of antiderivative = 1.77 \[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^2} \, dx=\frac {C (a+b \tan (e+f x))^3}{2 d f (c+d \tan (e+f x))}+\frac {\frac {(-3 b c C+2 b B d+3 a C d) (a+b \tan (e+f x))^2}{d f (c+d \tan (e+f x))}+\frac {2 \left (-\frac {d^2 \left (-3 a^2 A b c^2+A b^3 c^2-a^3 B c^2+3 a b^2 B c^2+3 a^2 b c^2 C-b^3 c^2 C+2 a^3 A c d-6 a A b^2 c d-6 a^2 b B c d+2 b^3 B c d-2 a^3 c C d+6 a b^2 c C d+3 a^2 A b d^2-A b^3 d^2+a^3 B d^2-3 a b^2 B d^2-3 a^2 b C d^2+b^3 C d^2+i \left (a^3 A c^2-3 a A b^2 c^2-3 a^2 b B c^2+b^3 B c^2-a^3 c^2 C+3 a b^2 c^2 C+6 a^2 A b c d-2 A b^3 c d+2 a^3 B c d-6 a b^2 B c d-6 a^2 b c C d+2 b^3 c C d-a^3 A d^2+3 a A b^2 d^2+3 a^2 b B d^2-b^3 B d^2+a^3 C d^2-3 a b^2 C d^2\right )\right ) \log (i-\tan (e+f x))}{2 \left (c^2+d^2\right )^2 f}+\frac {d^2 \left (3 a^2 A b c^2-A b^3 c^2+a^3 B c^2-3 a b^2 B c^2-3 a^2 b c^2 C+b^3 c^2 C-2 a^3 A c d+6 a A b^2 c d+6 a^2 b B c d-2 b^3 B c d+2 a^3 c C d-6 a b^2 c C d-3 a^2 A b d^2+A b^3 d^2-a^3 B d^2+3 a b^2 B d^2+3 a^2 b C d^2-b^3 C d^2+i \left (a^3 A c^2-3 a A b^2 c^2-3 a^2 b B c^2+b^3 B c^2-a^3 c^2 C+3 a b^2 c^2 C+6 a^2 A b c d-2 A b^3 c d+2 a^3 B c d-6 a b^2 B c d-6 a^2 b c C d+2 b^3 c C d-a^3 A d^2+3 a A b^2 d^2+3 a^2 b B d^2-b^3 B d^2+a^3 C d^2-3 a b^2 C d^2\right )\right ) \log (i+\tan (e+f x))}{2 \left (c^2+d^2\right )^2 f}+\frac {(b c-a d)^2 \left (b \left (3 c^4 C-2 B c^3 d+c^2 (A+5 C) d^2-4 B c d^3+3 A d^4\right )+a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \log (c+d \tan (e+f x))}{d^2 \left (c^2+d^2\right )^2 f}-\frac {(b c-a d)^2 \left (a d \left (3 c^2 C-B c d+(A+2 C) d^2\right )-b \left (3 c^3 C-2 B c^2 d+c (A+2 C) d^2-B d^3\right )\right )}{d^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))}\right )}{d}}{2 d} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 5.36 (sec) , antiderivative size = 594, normalized size of antiderivative = 1.03, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.289, Rules used = {3042, 4128, 3042, 4130, 27, 3042, 4120, 3042, 4109, 3042, 3956, 4100, 16}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4128

\(\displaystyle \frac {\int \frac {(a+b \tan (e+f x))^2 \left (b \left (3 C c^2-2 B d c+(2 A+C) d^2\right ) \tan ^2(e+f x)+d ((A-C) (b c-a d)+B (a c+b d)) \tan (e+f x)+A d (a c+3 b d)+(3 b c-a d) (c C-B d)\right )}{c+d \tan (e+f x)}dx}{d \left (c^2+d^2\right )}-\frac {\left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{d f \left (c^2+d^2\right ) (c+d \tan (e+f x))}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {(a+b \tan (e+f x))^2 \left (b \left (3 C c^2-2 B d c+(2 A+C) d^2\right ) \tan (e+f x)^2+d ((A-C) (b c-a d)+B (a c+b d)) \tan (e+f x)+A d (a c+3 b d)+(3 b c-a d) (c C-B d)\right )}{c+d \tan (e+f x)}dx}{d \left (c^2+d^2\right )}-\frac {\left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{d f \left (c^2+d^2\right ) (c+d \tan (e+f x))}\)

\(\Big \downarrow \) 4130

\(\displaystyle \frac {\frac {\int -\frac {2 (a+b \tan (e+f x)) \left (c \left (3 C c^2-2 B d c+(2 A+C) d^2\right ) b^2-\left (a d \left (3 C c^2-B d c+(A+2 C) d^2\right )-b \left (3 C c^3-2 B d c^2+(A+2 C) d^2 c-B d^3\right )\right ) \tan ^2(e+f x) b-a d (A d (a c+3 b d)+(3 b c-a d) (c C-B d))-d^2 \left ((B c-(A-C) d) a^2+2 b (A c-C c+B d) a-b^2 (B c-(A-C) d)\right ) \tan (e+f x)\right )}{c+d \tan (e+f x)}dx}{2 d}+\frac {b \left (d^2 (2 A+C)-2 B c d+3 c^2 C\right ) (a+b \tan (e+f x))^2}{2 d f}}{d \left (c^2+d^2\right )}-\frac {\left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{d f \left (c^2+d^2\right ) (c+d \tan (e+f x))}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {b \left (d^2 (2 A+C)-2 B c d+3 c^2 C\right ) (a+b \tan (e+f x))^2}{2 d f}-\frac {\int \frac {(a+b \tan (e+f x)) \left (c \left (3 C c^2-2 B d c+(2 A+C) d^2\right ) b^2-\left (a d \left (3 C c^2-B d c+(A+2 C) d^2\right )-b \left (3 C c^3-2 B d c^2+(A+2 C) d^2 c-B d^3\right )\right ) \tan ^2(e+f x) b-a d (A d (a c+3 b d)+(3 b c-a d) (c C-B d))-d^2 \left ((B c-(A-C) d) a^2+2 b (A c-C c+B d) a-b^2 (B c-(A-C) d)\right ) \tan (e+f x)\right )}{c+d \tan (e+f x)}dx}{d}}{d \left (c^2+d^2\right )}-\frac {\left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{d f \left (c^2+d^2\right ) (c+d \tan (e+f x))}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {b \left (d^2 (2 A+C)-2 B c d+3 c^2 C\right ) (a+b \tan (e+f x))^2}{2 d f}-\frac {\int \frac {(a+b \tan (e+f x)) \left (c \left (3 C c^2-2 B d c+(2 A+C) d^2\right ) b^2-\left (a d \left (3 C c^2-B d c+(A+2 C) d^2\right )-b \left (3 C c^3-2 B d c^2+(A+2 C) d^2 c-B d^3\right )\right ) \tan (e+f x)^2 b-a d (A d (a c+3 b d)+(3 b c-a d) (c C-B d))-d^2 \left ((B c-(A-C) d) a^2+2 b (A c-C c+B d) a-b^2 (B c-(A-C) d)\right ) \tan (e+f x)\right )}{c+d \tan (e+f x)}dx}{d}}{d \left (c^2+d^2\right )}-\frac {\left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{d f \left (c^2+d^2\right ) (c+d \tan (e+f x))}\)

\(\Big \downarrow \) 4120

\(\displaystyle \frac {\frac {b \left (d^2 (2 A+C)-2 B c d+3 c^2 C\right ) (a+b \tan (e+f x))^2}{2 d f}-\frac {-\frac {\int \frac {c \left (3 C c^3-2 B d c^2+(A+2 C) d^2 c-B d^3\right ) b^3-3 a c d \left (2 C c^2-B d c+(A+C) d^2\right ) b^2+\left (c^2+d^2\right ) \left (\left (3 C c^2-2 B d c+(A-C) d^2\right ) b^2-3 a d (2 c C-B d) b+3 a^2 C d^2\right ) \tan ^2(e+f x) b+3 a^2 d^2 \left (C c^2-B d c+A d^2\right ) b+a^3 d^3 (A c-C c+B d)+d^3 \left ((B c-(A-C) d) a^3+3 b (A c-C c+B d) a^2-3 b^2 (B c-(A-C) d) a-b^3 (A c-C c+B d)\right ) \tan (e+f x)}{c+d \tan (e+f x)}dx}{d}-\frac {b^2 \tan (e+f x) \left (a d \left (d^2 (A+2 C)-B c d+3 c^2 C\right )-b \left (c d^2 (A+2 C)-2 B c^2 d-B d^3+3 c^3 C\right )\right )}{d f}}{d}}{d \left (c^2+d^2\right )}-\frac {\left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{d f \left (c^2+d^2\right ) (c+d \tan (e+f x))}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {b \left (d^2 (2 A+C)-2 B c d+3 c^2 C\right ) (a+b \tan (e+f x))^2}{2 d f}-\frac {-\frac {\int \frac {c \left (3 C c^3-2 B d c^2+(A+2 C) d^2 c-B d^3\right ) b^3-3 a c d \left (2 C c^2-B d c+(A+C) d^2\right ) b^2+\left (c^2+d^2\right ) \left (\left (3 C c^2-2 B d c+(A-C) d^2\right ) b^2-3 a d (2 c C-B d) b+3 a^2 C d^2\right ) \tan (e+f x)^2 b+3 a^2 d^2 \left (C c^2-B d c+A d^2\right ) b+a^3 d^3 (A c-C c+B d)+d^3 \left ((B c-(A-C) d) a^3+3 b (A c-C c+B d) a^2-3 b^2 (B c-(A-C) d) a-b^3 (A c-C c+B d)\right ) \tan (e+f x)}{c+d \tan (e+f x)}dx}{d}-\frac {b^2 \tan (e+f x) \left (a d \left (d^2 (A+2 C)-B c d+3 c^2 C\right )-b \left (c d^2 (A+2 C)-2 B c^2 d-B d^3+3 c^3 C\right )\right )}{d f}}{d}}{d \left (c^2+d^2\right )}-\frac {\left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{d f \left (c^2+d^2\right ) (c+d \tan (e+f x))}\)

\(\Big \downarrow \) 4109

\(\displaystyle \frac {\frac {b \left (d^2 (2 A+C)-2 B c d+3 c^2 C\right ) (a+b \tan (e+f x))^2}{2 d f}-\frac {-\frac {-\frac {d^3 \left (a^3 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+3 a^2 b \left (-A \left (c^2-d^2\right )-2 B c d+c^2 C-C d^2\right )-3 a b^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )-b^3 \left (-A \left (c^2-d^2\right )-2 B c d+c^2 C-C d^2\right )\right ) \int \tan (e+f x)dx}{c^2+d^2}+\frac {(b c-a d)^2 \left (a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (c^2 d^2 (A+5 C)+3 A d^4-2 B c^3 d-4 B c d^3+3 c^4 C\right )\right ) \int \frac {\tan ^2(e+f x)+1}{c+d \tan (e+f x)}dx}{c^2+d^2}-\frac {d^3 x \left (a^3 \left (-A \left (c^2-d^2\right )-2 B c d+c^2 C-C d^2\right )-3 a^2 b \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )-3 a b^2 \left (-A \left (c^2-d^2\right )-2 B c d+c^2 C-C d^2\right )+b^3 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )\right )}{c^2+d^2}}{d}-\frac {b^2 \tan (e+f x) \left (a d \left (d^2 (A+2 C)-B c d+3 c^2 C\right )-b \left (c d^2 (A+2 C)-2 B c^2 d-B d^3+3 c^3 C\right )\right )}{d f}}{d}}{d \left (c^2+d^2\right )}-\frac {\left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{d f \left (c^2+d^2\right ) (c+d \tan (e+f x))}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {b \left (d^2 (2 A+C)-2 B c d+3 c^2 C\right ) (a+b \tan (e+f x))^2}{2 d f}-\frac {-\frac {-\frac {d^3 \left (a^3 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+3 a^2 b \left (-A \left (c^2-d^2\right )-2 B c d+c^2 C-C d^2\right )-3 a b^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )-b^3 \left (-A \left (c^2-d^2\right )-2 B c d+c^2 C-C d^2\right )\right ) \int \tan (e+f x)dx}{c^2+d^2}+\frac {(b c-a d)^2 \left (a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (c^2 d^2 (A+5 C)+3 A d^4-2 B c^3 d-4 B c d^3+3 c^4 C\right )\right ) \int \frac {\tan (e+f x)^2+1}{c+d \tan (e+f x)}dx}{c^2+d^2}-\frac {d^3 x \left (a^3 \left (-A \left (c^2-d^2\right )-2 B c d+c^2 C-C d^2\right )-3 a^2 b \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )-3 a b^2 \left (-A \left (c^2-d^2\right )-2 B c d+c^2 C-C d^2\right )+b^3 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )\right )}{c^2+d^2}}{d}-\frac {b^2 \tan (e+f x) \left (a d \left (d^2 (A+2 C)-B c d+3 c^2 C\right )-b \left (c d^2 (A+2 C)-2 B c^2 d-B d^3+3 c^3 C\right )\right )}{d f}}{d}}{d \left (c^2+d^2\right )}-\frac {\left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{d f \left (c^2+d^2\right ) (c+d \tan (e+f x))}\)

\(\Big \downarrow \) 3956

\(\displaystyle \frac {\frac {b \left (d^2 (2 A+C)-2 B c d+3 c^2 C\right ) (a+b \tan (e+f x))^2}{2 d f}-\frac {-\frac {\frac {(b c-a d)^2 \left (a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (c^2 d^2 (A+5 C)+3 A d^4-2 B c^3 d-4 B c d^3+3 c^4 C\right )\right ) \int \frac {\tan (e+f x)^2+1}{c+d \tan (e+f x)}dx}{c^2+d^2}+\frac {d^3 \log (\cos (e+f x)) \left (a^3 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+3 a^2 b \left (-A \left (c^2-d^2\right )-2 B c d+c^2 C-C d^2\right )-3 a b^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )-b^3 \left (-A \left (c^2-d^2\right )-2 B c d+c^2 C-C d^2\right )\right )}{f \left (c^2+d^2\right )}-\frac {d^3 x \left (a^3 \left (-A \left (c^2-d^2\right )-2 B c d+c^2 C-C d^2\right )-3 a^2 b \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )-3 a b^2 \left (-A \left (c^2-d^2\right )-2 B c d+c^2 C-C d^2\right )+b^3 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )\right )}{c^2+d^2}}{d}-\frac {b^2 \tan (e+f x) \left (a d \left (d^2 (A+2 C)-B c d+3 c^2 C\right )-b \left (c d^2 (A+2 C)-2 B c^2 d-B d^3+3 c^3 C\right )\right )}{d f}}{d}}{d \left (c^2+d^2\right )}-\frac {\left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{d f \left (c^2+d^2\right ) (c+d \tan (e+f x))}\)

\(\Big \downarrow \) 4100

\(\displaystyle \frac {\frac {b \left (d^2 (2 A+C)-2 B c d+3 c^2 C\right ) (a+b \tan (e+f x))^2}{2 d f}-\frac {-\frac {\frac {(b c-a d)^2 \left (a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (c^2 d^2 (A+5 C)+3 A d^4-2 B c^3 d-4 B c d^3+3 c^4 C\right )\right ) \int \frac {1}{c+d \tan (e+f x)}d(d \tan (e+f x))}{d f \left (c^2+d^2\right )}+\frac {d^3 \log (\cos (e+f x)) \left (a^3 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+3 a^2 b \left (-A \left (c^2-d^2\right )-2 B c d+c^2 C-C d^2\right )-3 a b^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )-b^3 \left (-A \left (c^2-d^2\right )-2 B c d+c^2 C-C d^2\right )\right )}{f \left (c^2+d^2\right )}-\frac {d^3 x \left (a^3 \left (-A \left (c^2-d^2\right )-2 B c d+c^2 C-C d^2\right )-3 a^2 b \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )-3 a b^2 \left (-A \left (c^2-d^2\right )-2 B c d+c^2 C-C d^2\right )+b^3 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )\right )}{c^2+d^2}}{d}-\frac {b^2 \tan (e+f x) \left (a d \left (d^2 (A+2 C)-B c d+3 c^2 C\right )-b \left (c d^2 (A+2 C)-2 B c^2 d-B d^3+3 c^3 C\right )\right )}{d f}}{d}}{d \left (c^2+d^2\right )}-\frac {\left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{d f \left (c^2+d^2\right ) (c+d \tan (e+f x))}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {\frac {b \left (d^2 (2 A+C)-2 B c d+3 c^2 C\right ) (a+b \tan (e+f x))^2}{2 d f}-\frac {-\frac {\frac {d^3 \log (\cos (e+f x)) \left (a^3 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+3 a^2 b \left (-A \left (c^2-d^2\right )-2 B c d+c^2 C-C d^2\right )-3 a b^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )-b^3 \left (-A \left (c^2-d^2\right )-2 B c d+c^2 C-C d^2\right )\right )}{f \left (c^2+d^2\right )}-\frac {d^3 x \left (a^3 \left (-A \left (c^2-d^2\right )-2 B c d+c^2 C-C d^2\right )-3 a^2 b \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )-3 a b^2 \left (-A \left (c^2-d^2\right )-2 B c d+c^2 C-C d^2\right )+b^3 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )\right )}{c^2+d^2}+\frac {(b c-a d)^2 \left (a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (c^2 d^2 (A+5 C)+3 A d^4-2 B c^3 d-4 B c d^3+3 c^4 C\right )\right ) \log (c+d \tan (e+f x))}{d f \left (c^2+d^2\right )}}{d}-\frac {b^2 \tan (e+f x) \left (a d \left (d^2 (A+2 C)-B c d+3 c^2 C\right )-b \left (c d^2 (A+2 C)-2 B c^2 d-B d^3+3 c^3 C\right )\right )}{d f}}{d}}{d \left (c^2+d^2\right )}-\frac {\left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{d f \left (c^2+d^2\right ) (c+d \tan (e+f x))}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 16

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

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3956

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

rule 4100

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_) + (C_.)*tan[(e_.) + 
 (f_.)*(x_)]^2), x_Symbol] :> Simp[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 4109

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] + (Simp[(A*b^2 - a*b*B + a^2*C)/(a^2 + b^2)   Int[( 
1 + Tan[e + f*x]^2)/(a + b*Tan[e + f*x]), x], x] - Simp[(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 4120

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] - Simp[1/(d*(n + 2))   Int[(c + d*Tan[e + f*x])^n*Si 
mp[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] /; FreeQ[{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 4128

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) 
 + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*d^2 + c*(c*C - B*d))*(a + b*Tan[e + 
 f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Sim 
p[1/(d*(n + 1)*(c^2 + d^2))   Int[(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 4130

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

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 829, normalized size of antiderivative = 1.43

method	result	size
derivativedivides	\(\frac {\frac {b^{2} \left (\frac {\tan \left (f x +e \right )^{2} C b d}{2}+B \tan \left (f x +e \right ) b d +3 d \tan \left (f x +e \right ) C a -2 \tan \left (f x +e \right ) C b c \right )}{d^{3}}+\frac {\frac {\left (-2 A \,a^{3} c d +3 A \,a^{2} b \,c^{2}-3 A \,a^{2} b \,d^{2}+6 A a \,b^{2} c d -A \,b^{3} c^{2}+b^{3} A \,d^{2}+B \,a^{3} c^{2}-B \,a^{3} d^{2}+6 B \,a^{2} b c d -3 B a \,b^{2} c^{2}+3 B a \,b^{2} d^{2}-2 B \,b^{3} c d +2 C \,a^{3} c d -3 C \,a^{2} b \,c^{2}+3 C \,a^{2} b \,d^{2}-6 C a \,b^{2} c d +C \,b^{3} c^{2}-C \,b^{3} d^{2}\right ) \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}+\left (A \,a^{3} c^{2}-A \,a^{3} d^{2}+6 A \,a^{2} b c d -3 A a \,b^{2} c^{2}+3 A a \,b^{2} d^{2}-2 A \,b^{3} c d +2 B \,a^{3} c d -3 B \,a^{2} b \,c^{2}+3 B \,a^{2} b \,d^{2}-6 B a \,b^{2} c d +B \,b^{3} c^{2}-B \,b^{3} d^{2}-C \,a^{3} c^{2}+C \,a^{3} d^{2}-6 C \,a^{2} b c d +3 C a \,b^{2} c^{2}-3 C a \,b^{2} d^{2}+2 C \,b^{3} c d \right ) \arctan \left (\tan \left (f x +e \right )\right )}{\left (c^{2}+d^{2}\right )^{2}}-\frac {A \,a^{3} d^{5}-3 A b c \,d^{4} a^{2}+3 A a \,b^{2} d^{3} c^{2}-A \,b^{3} c^{3} d^{2}-B c \,d^{4} a^{3}+3 B b \,c^{2} d^{3} a^{2}-3 B \,b^{2} c^{3} a \,d^{2}+B \,b^{3} c^{4} d +C \,c^{2} d^{3} a^{3}-3 C b \,c^{3} a^{2} d^{2}+3 C a \,b^{2} c^{4} d -C \,b^{3} c^{5}}{d^{4} \left (c^{2}+d^{2}\right ) \left (c +d \tan \left (f x +e \right )\right )}+\frac {\left (2 A \,a^{3} c \,d^{5}-3 A \,a^{2} b \,c^{2} d^{4}+3 A \,a^{2} b \,d^{6}-6 A a \,b^{2} c \,d^{5}+A \,b^{3} c^{4} d^{2}+3 A \,b^{3} c^{2} d^{4}-B \,a^{3} c^{2} d^{4}+B \,a^{3} d^{6}-6 B \,a^{2} b c \,d^{5}+3 B a \,b^{2} c^{4} d^{2}+9 B a \,b^{2} c^{2} d^{4}-2 B \,b^{3} c^{5} d -4 B \,b^{3} c^{3} d^{3}-2 C \,a^{3} c \,d^{5}+3 C \,a^{2} b \,c^{4} d^{2}+9 C \,a^{2} b \,c^{2} d^{4}-6 C a \,b^{2} c^{5} d -12 C a \,b^{2} c^{3} d^{3}+3 C \,b^{3} c^{6}+5 C \,b^{3} c^{4} d^{2}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{d^{4} \left (c^{2}+d^{2}\right )^{2}}}{f}\)	\(829\)
default	\(\frac {\frac {b^{2} \left (\frac {\tan \left (f x +e \right )^{2} C b d}{2}+B \tan \left (f x +e \right ) b d +3 d \tan \left (f x +e \right ) C a -2 \tan \left (f x +e \right ) C b c \right )}{d^{3}}+\frac {\frac {\left (-2 A \,a^{3} c d +3 A \,a^{2} b \,c^{2}-3 A \,a^{2} b \,d^{2}+6 A a \,b^{2} c d -A \,b^{3} c^{2}+b^{3} A \,d^{2}+B \,a^{3} c^{2}-B \,a^{3} d^{2}+6 B \,a^{2} b c d -3 B a \,b^{2} c^{2}+3 B a \,b^{2} d^{2}-2 B \,b^{3} c d +2 C \,a^{3} c d -3 C \,a^{2} b \,c^{2}+3 C \,a^{2} b \,d^{2}-6 C a \,b^{2} c d +C \,b^{3} c^{2}-C \,b^{3} d^{2}\right ) \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}+\left (A \,a^{3} c^{2}-A \,a^{3} d^{2}+6 A \,a^{2} b c d -3 A a \,b^{2} c^{2}+3 A a \,b^{2} d^{2}-2 A \,b^{3} c d +2 B \,a^{3} c d -3 B \,a^{2} b \,c^{2}+3 B \,a^{2} b \,d^{2}-6 B a \,b^{2} c d +B \,b^{3} c^{2}-B \,b^{3} d^{2}-C \,a^{3} c^{2}+C \,a^{3} d^{2}-6 C \,a^{2} b c d +3 C a \,b^{2} c^{2}-3 C a \,b^{2} d^{2}+2 C \,b^{3} c d \right ) \arctan \left (\tan \left (f x +e \right )\right )}{\left (c^{2}+d^{2}\right )^{2}}-\frac {A \,a^{3} d^{5}-3 A b c \,d^{4} a^{2}+3 A a \,b^{2} d^{3} c^{2}-A \,b^{3} c^{3} d^{2}-B c \,d^{4} a^{3}+3 B b \,c^{2} d^{3} a^{2}-3 B \,b^{2} c^{3} a \,d^{2}+B \,b^{3} c^{4} d +C \,c^{2} d^{3} a^{3}-3 C b \,c^{3} a^{2} d^{2}+3 C a \,b^{2} c^{4} d -C \,b^{3} c^{5}}{d^{4} \left (c^{2}+d^{2}\right ) \left (c +d \tan \left (f x +e \right )\right )}+\frac {\left (2 A \,a^{3} c \,d^{5}-3 A \,a^{2} b \,c^{2} d^{4}+3 A \,a^{2} b \,d^{6}-6 A a \,b^{2} c \,d^{5}+A \,b^{3} c^{4} d^{2}+3 A \,b^{3} c^{2} d^{4}-B \,a^{3} c^{2} d^{4}+B \,a^{3} d^{6}-6 B \,a^{2} b c \,d^{5}+3 B a \,b^{2} c^{4} d^{2}+9 B a \,b^{2} c^{2} d^{4}-2 B \,b^{3} c^{5} d -4 B \,b^{3} c^{3} d^{3}-2 C \,a^{3} c \,d^{5}+3 C \,a^{2} b \,c^{4} d^{2}+9 C \,a^{2} b \,c^{2} d^{4}-6 C a \,b^{2} c^{5} d -12 C a \,b^{2} c^{3} d^{3}+3 C \,b^{3} c^{6}+5 C \,b^{3} c^{4} d^{2}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{d^{4} \left (c^{2}+d^{2}\right )^{2}}}{f}\)	\(829\)
norman	\(\text {Expression too large to display}\)	\(1074\)
parallelrisch	\(\text {Expression too large to display}\)	\(3121\)
risch	\(\text {Expression too large to display}\)	\(4264\)

Input:

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

Output:

1/f*(b^2/d^3*(1/2*tan(f*x+e)^2*C*b*d+B*tan(f*x+e)*b*d+3*d*tan(f*x+e)*C*a-2 
*tan(f*x+e)*C*b*c)+1/(c^2+d^2)^2*(1/2*(-2*A*a^3*c*d+3*A*a^2*b*c^2-3*A*a^2* 
b*d^2+6*A*a*b^2*c*d-A*b^3*c^2+A*b^3*d^2+B*a^3*c^2-B*a^3*d^2+6*B*a^2*b*c*d- 
3*B*a*b^2*c^2+3*B*a*b^2*d^2-2*B*b^3*c*d+2*C*a^3*c*d-3*C*a^2*b*c^2+3*C*a^2* 
b*d^2-6*C*a*b^2*c*d+C*b^3*c^2-C*b^3*d^2)*ln(1+tan(f*x+e)^2)+(A*a^3*c^2-A*a 
^3*d^2+6*A*a^2*b*c*d-3*A*a*b^2*c^2+3*A*a*b^2*d^2-2*A*b^3*c*d+2*B*a^3*c*d-3 
*B*a^2*b*c^2+3*B*a^2*b*d^2-6*B*a*b^2*c*d+B*b^3*c^2-B*b^3*d^2-C*a^3*c^2+C*a 
^3*d^2-6*C*a^2*b*c*d+3*C*a*b^2*c^2-3*C*a*b^2*d^2+2*C*b^3*c*d)*arctan(tan(f 
*x+e)))-1/d^4*(A*a^3*d^5-3*A*a^2*b*c*d^4+3*A*a*b^2*c^2*d^3-A*b^3*c^3*d^2-B 
*a^3*c*d^4+3*B*a^2*b*c^2*d^3-3*B*a*b^2*c^3*d^2+B*b^3*c^4*d+C*a^3*c^2*d^3-3 
*C*a^2*b*c^3*d^2+3*C*a*b^2*c^4*d-C*b^3*c^5)/(c^2+d^2)/(c+d*tan(f*x+e))+1/d 
^4*(2*A*a^3*c*d^5-3*A*a^2*b*c^2*d^4+3*A*a^2*b*d^6-6*A*a*b^2*c*d^5+A*b^3*c^ 
4*d^2+3*A*b^3*c^2*d^4-B*a^3*c^2*d^4+B*a^3*d^6-6*B*a^2*b*c*d^5+3*B*a*b^2*c^ 
4*d^2+9*B*a*b^2*c^2*d^4-2*B*b^3*c^5*d-4*B*b^3*c^3*d^3-2*C*a^3*c*d^5+3*C*a^ 
2*b*c^4*d^2+9*C*a^2*b*c^2*d^4-6*C*a*b^2*c^5*d-12*C*a*b^2*c^3*d^3+3*C*b^3*c 
^6+5*C*b^3*c^4*d^2)/(c^2+d^2)^2*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. 1477 vs. \(2 (577) = 1154\).

Time = 0.90 (sec) , antiderivative size = 1477, normalized size of antiderivative = 2.55 \[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^2} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 28.69 (sec) , antiderivative size = 24300, normalized size of antiderivative = 41.97 \[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((zoo*x*(a + b*tan(e))**3*(A + B*tan(e) + C*tan(e)**2)/tan(e)**2, 
 Eq(c, 0) & Eq(d, 0) & Eq(f, 0)), ((A*a**3*x + 3*A*a**2*b*log(tan(e + f*x) 
**2 + 1)/(2*f) - 3*A*a*b**2*x + 3*A*a*b**2*tan(e + f*x)/f - A*b**3*log(tan 
(e + f*x)**2 + 1)/(2*f) + A*b**3*tan(e + f*x)**2/(2*f) + B*a**3*log(tan(e 
+ f*x)**2 + 1)/(2*f) - 3*B*a**2*b*x + 3*B*a**2*b*tan(e + f*x)/f - 3*B*a*b* 
*2*log(tan(e + f*x)**2 + 1)/(2*f) + 3*B*a*b**2*tan(e + f*x)**2/(2*f) + B*b 
**3*x + B*b**3*tan(e + f*x)**3/(3*f) - B*b**3*tan(e + f*x)/f - C*a**3*x + 
C*a**3*tan(e + f*x)/f - 3*C*a**2*b*log(tan(e + f*x)**2 + 1)/(2*f) + 3*C*a* 
*2*b*tan(e + f*x)**2/(2*f) + 3*C*a*b**2*x + C*a*b**2*tan(e + f*x)**3/f - 3 
*C*a*b**2*tan(e + f*x)/f + C*b**3*log(tan(e + f*x)**2 + 1)/(2*f) + C*b**3* 
tan(e + f*x)**4/(4*f) - C*b**3*tan(e + f*x)**2/(2*f))/c**2, Eq(d, 0)), (-A 
*a**3*f*x*tan(e + f*x)**2/(4*d**2*f*tan(e + f*x)**2 - 8*I*d**2*f*tan(e + f 
*x) - 4*d**2*f) + 2*I*A*a**3*f*x*tan(e + f*x)/(4*d**2*f*tan(e + f*x)**2 - 
8*I*d**2*f*tan(e + f*x) - 4*d**2*f) + A*a**3*f*x/(4*d**2*f*tan(e + f*x)**2 
 - 8*I*d**2*f*tan(e + f*x) - 4*d**2*f) - A*a**3*tan(e + f*x)/(4*d**2*f*tan 
(e + f*x)**2 - 8*I*d**2*f*tan(e + f*x) - 4*d**2*f) + 2*I*A*a**3/(4*d**2*f* 
tan(e + f*x)**2 - 8*I*d**2*f*tan(e + f*x) - 4*d**2*f) + 3*I*A*a**2*b*f*x*t 
an(e + f*x)**2/(4*d**2*f*tan(e + f*x)**2 - 8*I*d**2*f*tan(e + f*x) - 4*d** 
2*f) + 6*A*a**2*b*f*x*tan(e + f*x)/(4*d**2*f*tan(e + f*x)**2 - 8*I*d**2*f* 
tan(e + f*x) - 4*d**2*f) - 3*I*A*a**2*b*f*x/(4*d**2*f*tan(e + f*x)**2 -...
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 684, normalized size of antiderivative = 1.18 \[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^2} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.72 (sec) , antiderivative size = 1039, normalized size of antiderivative = 1.79 \[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^2} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 12.48 (sec) , antiderivative size = 701, normalized size of antiderivative = 1.21 \[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^2} \, dx=\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (\frac {B\,b^3+3\,C\,a\,b^2}{d^2}-\frac {2\,C\,b^3\,c}{d^3}\right )}{f}-\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,\left (B\,a^3-A\,b^3+C\,b^3+3\,A\,a^2\,b-3\,B\,a\,b^2-3\,C\,a^2\,b+A\,a^3\,1{}\mathrm {i}+B\,b^3\,1{}\mathrm {i}-C\,a^3\,1{}\mathrm {i}-A\,a\,b^2\,3{}\mathrm {i}-B\,a^2\,b\,3{}\mathrm {i}+C\,a\,b^2\,3{}\mathrm {i}\right )}{2\,f\,\left (-c^2+c\,d\,2{}\mathrm {i}+d^2\right )}+\frac {\ln \left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )\,\left (d^4\,\left (3\,A\,b^3\,c^2-B\,a^3\,c^2-3\,A\,a^2\,b\,c^2+9\,B\,a\,b^2\,c^2+9\,C\,a^2\,b\,c^2\right )-d^5\,\left (2\,C\,a^3\,c-2\,A\,a^3\,c+6\,A\,a\,b^2\,c+6\,B\,a^2\,b\,c\right )-d^3\,\left (4\,B\,b^3\,c^3+12\,C\,a\,b^2\,c^3\right )+d^6\,\left (B\,a^3+3\,A\,b\,a^2\right )-d\,\left (2\,B\,b^3\,c^5+6\,C\,a\,b^2\,c^5\right )+d^2\,\left (A\,b^3\,c^4+5\,C\,b^3\,c^4+3\,B\,a\,b^2\,c^4+3\,C\,a^2\,b\,c^4\right )+3\,C\,b^3\,c^6\right )}{f\,\left (c^4\,d^4+2\,c^2\,d^6+d^8\right )}-\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )-\mathrm {i}\right )\,\left (A\,a^3-A\,b^3\,1{}\mathrm {i}+B\,a^3\,1{}\mathrm {i}+B\,b^3-C\,a^3+C\,b^3\,1{}\mathrm {i}-3\,A\,a\,b^2+A\,a^2\,b\,3{}\mathrm {i}-B\,a\,b^2\,3{}\mathrm {i}-3\,B\,a^2\,b+3\,C\,a\,b^2-C\,a^2\,b\,3{}\mathrm {i}\right )}{2\,f\,\left (-c^2\,1{}\mathrm {i}+2\,c\,d+d^2\,1{}\mathrm {i}\right )}-\frac {C\,a^3\,c^2\,d^3-B\,a^3\,c\,d^4+A\,a^3\,d^5-3\,C\,a^2\,b\,c^3\,d^2+3\,B\,a^2\,b\,c^2\,d^3-3\,A\,a^2\,b\,c\,d^4+3\,C\,a\,b^2\,c^4\,d-3\,B\,a\,b^2\,c^3\,d^2+3\,A\,a\,b^2\,c^2\,d^3-C\,b^3\,c^5+B\,b^3\,c^4\,d-A\,b^3\,c^3\,d^2}{d\,f\,\left (\mathrm {tan}\left (e+f\,x\right )\,d^4+c\,d^3\right )\,\left (c^2+d^2\right )}+\frac {C\,b^3\,{\mathrm {tan}\left (e+f\,x\right )}^2}{2\,d^2\,f} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 2547, normalized size of antiderivative = 4.40 \[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 2*log(tan(e + f*x)**2 + 1)*tan(e + f*x)*a**4*c**2*d**6 + 4*log(tan(e + 
 f*x)**2 + 1)*tan(e + f*x)*a**3*b*c**3*d**5 - 4*log(tan(e + f*x)**2 + 1)*t 
an(e + f*x)*a**3*b*c*d**7 + 2*log(tan(e + f*x)**2 + 1)*tan(e + f*x)*a**3*c 
**3*d**6 + 12*log(tan(e + f*x)**2 + 1)*tan(e + f*x)*a**2*b**2*c**2*d**6 - 
3*log(tan(e + f*x)**2 + 1)*tan(e + f*x)*a**2*b*c**4*d**5 + 3*log(tan(e + f 
*x)**2 + 1)*tan(e + f*x)*a**2*b*c**2*d**7 - 4*log(tan(e + f*x)**2 + 1)*tan 
(e + f*x)*a*b**3*c**3*d**5 + 4*log(tan(e + f*x)**2 + 1)*tan(e + f*x)*a*b** 
3*c*d**7 - 6*log(tan(e + f*x)**2 + 1)*tan(e + f*x)*a*b**2*c**3*d**6 - 2*lo 
g(tan(e + f*x)**2 + 1)*tan(e + f*x)*b**4*c**2*d**6 + log(tan(e + f*x)**2 + 
 1)*tan(e + f*x)*b**3*c**4*d**5 - log(tan(e + f*x)**2 + 1)*tan(e + f*x)*b* 
*3*c**2*d**7 - 2*log(tan(e + f*x)**2 + 1)*a**4*c**3*d**5 + 4*log(tan(e + f 
*x)**2 + 1)*a**3*b*c**4*d**4 - 4*log(tan(e + f*x)**2 + 1)*a**3*b*c**2*d**6 
 + 2*log(tan(e + f*x)**2 + 1)*a**3*c**4*d**5 + 12*log(tan(e + f*x)**2 + 1) 
*a**2*b**2*c**3*d**5 - 3*log(tan(e + f*x)**2 + 1)*a**2*b*c**5*d**4 + 3*log 
(tan(e + f*x)**2 + 1)*a**2*b*c**3*d**6 - 4*log(tan(e + f*x)**2 + 1)*a*b**3 
*c**4*d**4 + 4*log(tan(e + f*x)**2 + 1)*a*b**3*c**2*d**6 - 6*log(tan(e + f 
*x)**2 + 1)*a*b**2*c**4*d**5 - 2*log(tan(e + f*x)**2 + 1)*b**4*c**3*d**5 + 
 log(tan(e + f*x)**2 + 1)*b**3*c**5*d**4 - log(tan(e + f*x)**2 + 1)*b**3*c 
**3*d**6 + 4*log(tan(e + f*x)*d + c)*tan(e + f*x)*a**4*c**2*d**6 - 8*log(t 
an(e + f*x)*d + c)*tan(e + f*x)*a**3*b*c**3*d**5 + 8*log(tan(e + f*x)*d...