3.2.70 \(\int \frac {(a+b \tan (e+f x))^m (A+B \tan (e+f x)+C \tan ^2(e+f x))}{(c+d \tan (e+f x))^2} \, dx\) [170]

3.2.70.1 Optimal result

Integrand size = 45, antiderivative size = 403 \[ \int \frac {(a+b \tan (e+f x))^m \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^2} \, dx=\frac {(A-i B-C) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \tan (e+f x)}{a-i b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (i a+b) (c-i d)^2 f (1+m)}+\frac {(i A-B-i C) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \tan (e+f x)}{a+i b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (a+i b) (c+i d)^2 f (1+m)}-\frac {\left (a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )-b \left (A d^2 \left (c^2 (2-m)-d^2 m\right )-B c d \left (c^2 (1-m)-d^2 (1+m)\right )-c^2 C \left (c^2 m+d^2 (2+m)\right )\right )\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {d (a+b \tan (e+f x))}{b c-a d}\right ) (a+b \tan (e+f x))^{1+m}}{(b c-a d)^2 \left (c^2+d^2\right )^2 f (1+m)}+\frac {\left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^{1+m}}{(b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))} \]

output

1/2*(A-I*B-C)*hypergeom([1, 1+m],[2+m],(a+b*tan(f*x+e))/(a-I*b))*(a+b*tan( 
f*x+e))^(1+m)/(I*a+b)/(c-I*d)^2/f/(1+m)+1/2*(I*A-B-I*C)*hypergeom([1, 1+m] 
,[2+m],(a+b*tan(f*x+e))/(a+I*b))*(a+b*tan(f*x+e))^(1+m)/(a+I*b)/(c+I*d)^2/ 
f/(1+m)-(a*d^2*(2*c*(A-C)*d-B*(c^2-d^2))-b*(A*d^2*(c^2*(2-m)-d^2*m)-B*c*d* 
(c^2*(1-m)-d^2*(1+m))-c^2*C*(c^2*m+d^2*(2+m))))*hypergeom([1, 1+m],[2+m],- 
d*(a+b*tan(f*x+e))/(-a*d+b*c))*(a+b*tan(f*x+e))^(1+m)/(-a*d+b*c)^2/(c^2+d^ 
2)^2/f/(1+m)+(A*d^2-B*c*d+C*c^2)*(a+b*tan(f*x+e))^(1+m)/(-a*d+b*c)/(c^2+d^ 
2)/f/(c+d*tan(f*x+e))
 

3.2.70.2 Mathematica [A] (verified)

Time = 6.24 (sec) , antiderivative size = 563, normalized size of antiderivative = 1.40 \[ \int \frac {(a+b \tan (e+f x))^m \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^2} \, dx=-\frac {\left (A d^2-c (-c C+B d)\right ) (a+b \tan (e+f x))^{1+m}}{(-b c+a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))}-\frac {-\frac {\left (-c d (b c-a d) (B c-(A-C) d)-b c^2 \left (c^2 C-B c d+A d^2\right ) m+d^2 \left ((c C-B d) (a d-b c (1+m))-A \left (a c d-b \left (c^2-d^2 m\right )\right )\right )\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {d (a+b \tan (e+f x))}{-b c+a d}\right ) (a+b \tan (e+f x))^{1+m}}{(-b c+a d) \left (c^2+d^2\right ) f (1+m)}+\frac {\frac {i \left (-\left ((b c-a d) \left (c^2 C-2 B c d-C d^2-A \left (c^2-d^2\right )\right )\right )-i (b c-a d) \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {-i a-i b \tan (e+f x)}{-i a+b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (a+i b) f (1+m)}-\frac {i \left (-\left ((b c-a d) \left (c^2 C-2 B c d-C d^2-A \left (c^2-d^2\right )\right )\right )+i (b c-a d) \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {i a+i b \tan (e+f x)}{-i a-b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (a-i b) f (1+m)}}{c^2+d^2}}{(-b c+a d) \left (c^2+d^2\right )} \]

input

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

output

-(((A*d^2 - c*(-(c*C) + B*d))*(a + b*Tan[e + f*x])^(1 + m))/((-(b*c) + a*d 
)*(c^2 + d^2)*f*(c + d*Tan[e + f*x]))) - (-(((-(c*d*(b*c - a*d)*(B*c - (A 
- C)*d)) - b*c^2*(c^2*C - B*c*d + A*d^2)*m + d^2*((c*C - B*d)*(a*d - b*c*( 
1 + m)) - A*(a*c*d - b*(c^2 - d^2*m))))*Hypergeometric2F1[1, 1 + m, 2 + m, 
 (d*(a + b*Tan[e + f*x]))/(-(b*c) + a*d)]*(a + b*Tan[e + f*x])^(1 + m))/(( 
-(b*c) + a*d)*(c^2 + d^2)*f*(1 + m))) + (((I/2)*(-((b*c - a*d)*(c^2*C - 2* 
B*c*d - C*d^2 - A*(c^2 - d^2))) - I*(b*c - a*d)*(2*c*(A - C)*d - B*(c^2 - 
d^2)))*Hypergeometric2F1[1, 1 + m, 2 + m, ((-I)*a - I*b*Tan[e + f*x])/((-I 
)*a + b)]*(a + b*Tan[e + f*x])^(1 + m))/((a + I*b)*f*(1 + m)) - ((I/2)*(-( 
(b*c - a*d)*(c^2*C - 2*B*c*d - C*d^2 - A*(c^2 - d^2))) + I*(b*c - a*d)*(2* 
c*(A - C)*d - B*(c^2 - d^2)))*Hypergeometric2F1[1, 1 + m, 2 + m, -((I*a + 
I*b*Tan[e + f*x])/((-I)*a - b))]*(a + b*Tan[e + f*x])^(1 + m))/((a - I*b)* 
f*(1 + m)))/(c^2 + d^2))/((-(b*c) + a*d)*(c^2 + d^2))
 

3.2.70.3 Rubi [A] (verified)

Time = 2.17 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.12, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.289, Rules used = {3042, 4132, 3042, 4136, 25, 3042, 4022, 3042, 4020, 25, 78, 4117, 78}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4132

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4136

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4022

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4020

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 78

\(\displaystyle \frac {\left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}+\frac {-\frac {\left (a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )-b \left (A c^2 d^2 (2-m)-A d^4 m-B \left (c^3 d (1-m)-c d^3 (m+1)\right )+c^4 (-C) m-c^2 C d^2 (m+2)\right )\right ) \int \frac {(a+b \tan (e+f x))^m \left (\tan (e+f x)^2+1\right )}{c+d \tan (e+f x)}dx}{c^2+d^2}-\frac {\frac {i (c+i d)^2 (A-i B-C) (b c-a d) (a+b \tan (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {a+b \tan (e+f x)}{a-i b}\right )}{2 f (m+1) (a-i b)}-\frac {i (c-i d)^2 (A+i B-C) (b c-a d) (a+b \tan (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {a+b \tan (e+f x)}{a+i b}\right )}{2 f (m+1) (a+i b)}}{c^2+d^2}}{\left (c^2+d^2\right ) (b c-a d)}\)

\(\Big \downarrow \) 4117

\(\displaystyle \frac {\left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}+\frac {-\frac {\left (a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )-b \left (A c^2 d^2 (2-m)-A d^4 m-B \left (c^3 d (1-m)-c d^3 (m+1)\right )+c^4 (-C) m-c^2 C d^2 (m+2)\right )\right ) \int \frac {(a+b \tan (e+f x))^m}{c+d \tan (e+f x)}d\tan (e+f x)}{f \left (c^2+d^2\right )}-\frac {\frac {i (c+i d)^2 (A-i B-C) (b c-a d) (a+b \tan (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {a+b \tan (e+f x)}{a-i b}\right )}{2 f (m+1) (a-i b)}-\frac {i (c-i d)^2 (A+i B-C) (b c-a d) (a+b \tan (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {a+b \tan (e+f x)}{a+i b}\right )}{2 f (m+1) (a+i b)}}{c^2+d^2}}{\left (c^2+d^2\right ) (b c-a d)}\)

\(\Big \downarrow \) 78

\(\displaystyle \frac {\left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}+\frac {-\frac {(a+b \tan (e+f x))^{m+1} \left (a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )-b \left (A c^2 d^2 (2-m)-A d^4 m-B \left (c^3 d (1-m)-c d^3 (m+1)\right )+c^4 (-C) m-c^2 C d^2 (m+2)\right )\right ) \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,-\frac {d (a+b \tan (e+f x))}{b c-a d}\right )}{f (m+1) \left (c^2+d^2\right ) (b c-a d)}-\frac {\frac {i (c+i d)^2 (A-i B-C) (b c-a d) (a+b \tan (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {a+b \tan (e+f x)}{a-i b}\right )}{2 f (m+1) (a-i b)}-\frac {i (c-i d)^2 (A+i B-C) (b c-a d) (a+b \tan (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {a+b \tan (e+f x)}{a+i b}\right )}{2 f (m+1) (a+i b)}}{c^2+d^2}}{\left (c^2+d^2\right ) (b c-a d)}\)

input

Int[((a + b*Tan[e + f*x])^m*(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])^(1 + m))/((b*c - a*d)*(c^2 + 
 d^2)*f*(c + d*Tan[e + f*x])) + (-(((a*d^2*(2*c*(A - C)*d - B*(c^2 - d^2)) 
 - b*(A*c^2*d^2*(2 - m) - c^4*C*m - A*d^4*m - c^2*C*d^2*(2 + m) - B*(c^3*d 
*(1 - m) - c*d^3*(1 + m))))*Hypergeometric2F1[1, 1 + m, 2 + m, -((d*(a + b 
*Tan[e + f*x]))/(b*c - a*d))]*(a + b*Tan[e + f*x])^(1 + m))/((b*c - a*d)*( 
c^2 + d^2)*f*(1 + m))) - (((I/2)*(A - I*B - C)*(c + I*d)^2*(b*c - a*d)*Hyp 
ergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a - I*b)]*(a + b*Tan 
[e + f*x])^(1 + m))/((a - I*b)*f*(1 + m)) - ((I/2)*(A + I*B - C)*(c - I*d) 
^2*(b*c - a*d)*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a 
+ I*b)]*(a + b*Tan[e + f*x])^(1 + m))/((a + I*b)*f*(1 + m)))/(c^2 + d^2))/ 
((b*c - a*d)*(c^2 + d^2))
 

3.2.70.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 78

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(b 
*c - a*d)^n*((a + b*x)^(m + 1)/(b^(n + 1)*(m + 1)))*Hypergeometric2F1[-n, m 
 + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m}, x] 
 &&  !IntegerQ[m] && IntegerQ[n]
 

rule 3042

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

rule 4020

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + 
(f_.)*(x_)]), x_Symbol] :> Simp[c*(d/f)   Subst[Int[(a + (b/d)*x)^m/(d^2 + 
c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[ 
b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]
 

rule 4022

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)]), x_Symbol] :> Simp[(c + I*d)/2   Int[(a + b*Tan[e + f*x])^m*( 
1 - I*Tan[e + f*x]), x], x] + Simp[(c - I*d)/2   Int[(a + b*Tan[e + f*x])^m 
*(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c 
 - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] &&  !IntegerQ[m]
 

rule 4117

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

rule 4132

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*b^2 - a*(b*B - a*C))*(a + b*Tan[e + 
 f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + 
b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 + b^2))   Int[(a + b*Tan[e + 
f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1) - b^2*d* 
(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d 
)*(A*b - a*B - b*C)*Tan[e + f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Ta 
n[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] && LtQ[m, -1] && 
!(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
 

rule 4136

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

3.2.70.4 Maple [F]

\[\int \frac {\left (a +b \tan \left (f x +e \right )\right )^{m} \left (A +B \tan \left (f x +e \right )+C \tan \left (f x +e \right )^{2}\right )}{\left (c +d \tan \left (f x +e \right )\right )^{2}}d x\]

input

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

output

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

3.2.70.5 Fricas [F]

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

input

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

output

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

3.2.70.6 Sympy [F(-2)]

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

input

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

output

Exception raised: HeuristicGCDFailed >> no luck
 

3.2.70.7 Maxima [F]

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

input

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

output

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

3.2.70.8 Giac [F]

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

input

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

output

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

3.2.70.9 Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \tan (e+f x))^m \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^2} \, dx=\int \frac {{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^m\,\left (C\,{\mathrm {tan}\left (e+f\,x\right )}^2+B\,\mathrm {tan}\left (e+f\,x\right )+A\right )}{{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^2} \,d x \]

input

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

output

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