\(\int \frac {\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx\) [486]

Optimal result

Integrand size = 31, antiderivative size = 659 \[ \int \frac {\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\frac {\left (a^4 A-6 a^2 A b^2+A b^4+4 a^3 b B-4 a b^3 B\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\tan ^2(c+d x)\right ) \tan ^{1+m}(c+d x)}{\left (a^2+b^2\right )^4 d (1+m)}-\frac {b \left (a b^6 B m \left (1-m^2\right )+3 a^2 A b^5 m \left (2-5 m+m^2\right )+A b^7 m \left (2-3 m+m^2\right )+3 a^3 b^4 B \left (2+5 m+2 m^2-m^3\right )+a^7 B \left (6-11 m+6 m^2-m^3\right )-a^6 A b \left (24-26 m+9 m^2-m^3\right )+3 a^4 A b^3 \left (8+10 m-7 m^2+m^3\right )-3 a^5 b^2 B \left (12-m-4 m^2+m^3\right )\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {b \tan (c+d x)}{a}\right ) \tan ^{1+m}(c+d x)}{6 a^4 \left (a^2+b^2\right )^4 d (1+m)}-\frac {\left (4 a^3 A b-4 a A b^3-a^4 B+6 a^2 b^2 B-b^4 B\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},-\tan ^2(c+d x)\right ) \tan ^{2+m}(c+d x)}{\left (a^2+b^2\right )^4 d (2+m)}+\frac {b (A b-a B) \tan ^{1+m}(c+d x)}{3 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {b \left (A b^3 (2-m)-a^3 B (5-m)+a^2 A b (8-m)+a b^2 B (1+m)\right ) \tan ^{1+m}(c+d x)}{6 a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {b \left (a b^4 B \left (1-m^2\right )+2 a^3 b^2 B \left (7+3 m-m^2\right )+a^4 A b \left (26-9 m+m^2\right )+2 a^2 A b^3 \left (2-6 m+m^2\right )-a^5 B \left (11-6 m+m^2\right )+A b^5 \left (2-3 m+m^2\right )\right ) \tan ^{1+m}(c+d x)}{6 a^3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))} \] Output:

(A*a^4-6*A*a^2*b^2+A*b^4+4*B*a^3*b-4*B*a*b^3)*hypergeom([1, 1/2+1/2*m],[3/ 
2+1/2*m],-tan(d*x+c)^2)*tan(d*x+c)^(1+m)/(a^2+b^2)^4/d/(1+m)-1/6*b*(a*b^6* 
B*m*(-m^2+1)+3*a^2*A*b^5*m*(m^2-5*m+2)+A*b^7*m*(m^2-3*m+2)+3*a^3*b^4*B*(-m 
^3+2*m^2+5*m+2)+a^7*B*(-m^3+6*m^2-11*m+6)-a^6*A*b*(-m^3+9*m^2-26*m+24)+3*a 
^4*A*b^3*(m^3-7*m^2+10*m+8)-3*a^5*b^2*B*(m^3-4*m^2-m+12))*hypergeom([1, 1+ 
m],[2+m],-b*tan(d*x+c)/a)*tan(d*x+c)^(1+m)/a^4/(a^2+b^2)^4/d/(1+m)-(4*A*a^ 
3*b-4*A*a*b^3-B*a^4+6*B*a^2*b^2-B*b^4)*hypergeom([1, 1+1/2*m],[2+1/2*m],-t 
an(d*x+c)^2)*tan(d*x+c)^(2+m)/(a^2+b^2)^4/d/(2+m)+1/3*b*(A*b-B*a)*tan(d*x+ 
c)^(1+m)/a/(a^2+b^2)/d/(a+b*tan(d*x+c))^3+1/6*b*(A*b^3*(2-m)-a^3*B*(5-m)+a 
^2*A*b*(8-m)+a*b^2*B*(1+m))*tan(d*x+c)^(1+m)/a^2/(a^2+b^2)^2/d/(a+b*tan(d* 
x+c))^2+1/6*b*(a*b^4*B*(-m^2+1)+2*a^3*b^2*B*(-m^2+3*m+7)+a^4*A*b*(m^2-9*m+ 
26)+2*a^2*A*b^3*(m^2-6*m+2)-a^5*B*(m^2-6*m+11)+A*b^5*(m^2-3*m+2))*tan(d*x+ 
c)^(1+m)/a^3/(a^2+b^2)^3/d/(a+b*tan(d*x+c))
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1425\) vs. \(2(659)=1318\).

Time = 6.29 (sec) , antiderivative size = 1425, normalized size of antiderivative = 2.16 \[ \int \frac {\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx =\text {Too large to display} \] Input:

Integrate[(Tan[c + d*x]^m*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^4,x]
 

Output:

(b*(A*b - a*B)*Tan[c + d*x]^(1 + m))/(3*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x 
])^3) + (((-(a*(-3*a*b*(A*b - a*B) - a*b*(A*b - a*B)*(2 - m))) + b^2*(3*a^ 
2*A + A*b^2*(2 - m) + a*b*B*(1 + m)))*Tan[c + d*x]^(1 + m))/(2*a*(a^2 + b^ 
2)*d*(a + b*Tan[c + d*x])^2) + (((b^2*(-(a^2*b*(A*b - a*B)*(5 - m)*(1 + m) 
) + (2*a^2 + b^2*(1 - m))*(3*a^2*A + A*b^2*(2 - m) + a*b*B*(1 + m))) - a*( 
-6*a^2*b*(2*a*A*b - a^2*B + b^2*B) - a*b*(1 - m)*(A*b^3*(2 - m) - a^3*B*(5 
 - m) + a^2*A*b*(8 - m) + a*b^2*B*(1 + m))))*Tan[c + d*x]^(1 + m))/(a*(a^2 
 + b^2)*d*(a + b*Tan[c + d*x])) + (((a^2*b*(6*a^3*(2*a*A*b - a^2*B + b^2*B 
) - b^2*(1 - m)*(A*b^3*(2 - m) - a^3*B*(5 - m) + a^2*A*b*(8 - m) + a*b^2*B 
*(1 + m)) + b*(-(a^2*b*(A*b - a*B)*(5 - m)*(1 + m)) + (2*a^2 + b^2*(1 - m) 
)*(3*a^2*A + A*b^2*(2 - m) + a*b*B*(1 + m)))) - a^2*m*(b^2*(-(a^2*b*(A*b - 
 a*B)*(5 - m)*(1 + m)) + (2*a^2 + b^2*(1 - m))*(3*a^2*A + A*b^2*(2 - m) + 
a*b*B*(1 + m))) - a*(-6*a^2*b*(2*a*A*b - a^2*B + b^2*B) - a*b*(1 - m)*(A*b 
^3*(2 - m) - a^3*B*(5 - m) + a^2*A*b*(8 - m) + a*b^2*B*(1 + m)))) + b^2*(( 
a^2 - b^2*m)*(-(a^2*b*(A*b - a*B)*(5 - m)*(1 + m)) + (2*a^2 + b^2*(1 - m)) 
*(3*a^2*A + A*b^2*(2 - m) + a*b*B*(1 + m))) + a*(1 + m)*(-6*a^2*b*(2*a*A*b 
 - a^2*B + b^2*B) - a*b*(1 - m)*(A*b^3*(2 - m) - a^3*B*(5 - m) + a^2*A*b*( 
8 - m) + a*b^2*B*(1 + m)))))*Hypergeometric2F1[1, 1 + m, 2 + m, -((b*Tan[c 
 + d*x])/a)]*Tan[c + d*x]^(1 + m))/(a*(a^2 + b^2)*d*(1 + m)) + ((6*a^7*A*H 
ypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]...
 

Rubi [A] (verified)

Time = 4.08 (sec) , antiderivative size = 704, normalized size of antiderivative = 1.07, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.581, Rules used = {3042, 4092, 3042, 4132, 25, 3042, 4132, 25, 3042, 4136, 27, 3042, 4021, 3042, 3957, 278, 4117, 74}

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.

\(\displaystyle \int \frac {\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\tan (c+d x)^m (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4}dx\)

\(\Big \downarrow \) 4092

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4132

\(\displaystyle \frac {\frac {\int -\frac {\tan ^m(c+d x) \left (b (A b-a B) (5-m) (m+1) a^2+6 \left (-B a^2+2 A b a+b^2 B\right ) \tan (c+d x) a^2-b (1-m) \left (-B (5-m) a^3+A b (8-m) a^2+b^2 B (m+1) a+A b^3 (2-m)\right ) \tan ^2(c+d x)-\left (2 a^2+b^2 (1-m)\right ) \left (3 A a^2+b B (m+1) a+A b^2 (2-m)\right )\right )}{(a+b \tan (c+d x))^2}dx}{2 a \left (a^2+b^2\right )}+\frac {b \left (a^3 (-B) (5-m)+a^2 A b (8-m)+a b^2 B (m+1)+A b^3 (2-m)\right ) \tan ^{m+1}(c+d x)}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{3 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \tan ^{m+1}(c+d x)}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {b \left (a^3 (-B) (5-m)+a^2 A b (8-m)+a b^2 B (m+1)+A b^3 (2-m)\right ) \tan ^{m+1}(c+d x)}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\int \frac {\tan ^m(c+d x) \left (b (A b-a B) (5-m) (m+1) a^2+6 \left (-B a^2+2 A b a+b^2 B\right ) \tan (c+d x) a^2-b (1-m) \left (-B (5-m) a^3+A b (8-m) a^2+b^2 B (m+1) a+A b^3 (2-m)\right ) \tan ^2(c+d x)-\left (2 a^2+b^2 (1-m)\right ) \left (3 A a^2+b B (m+1) a+A b^2 (2-m)\right )\right )}{(a+b \tan (c+d x))^2}dx}{2 a \left (a^2+b^2\right )}}{3 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \tan ^{m+1}(c+d x)}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {b \left (a^3 (-B) (5-m)+a^2 A b (8-m)+a b^2 B (m+1)+A b^3 (2-m)\right ) \tan ^{m+1}(c+d x)}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\int \frac {\tan (c+d x)^m \left (b (A b-a B) (5-m) (m+1) a^2+6 \left (-B a^2+2 A b a+b^2 B\right ) \tan (c+d x) a^2-b (1-m) \left (-B (5-m) a^3+A b (8-m) a^2+b^2 B (m+1) a+A b^3 (2-m)\right ) \tan (c+d x)^2-\left (2 a^2+b^2 (1-m)\right ) \left (3 A a^2+b B (m+1) a+A b^2 (2-m)\right )\right )}{(a+b \tan (c+d x))^2}dx}{2 a \left (a^2+b^2\right )}}{3 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \tan ^{m+1}(c+d x)}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\)

\(\Big \downarrow \) 4132

\(\displaystyle \frac {\frac {b \left (a^3 (-B) (5-m)+a^2 A b (8-m)+a b^2 B (m+1)+A b^3 (2-m)\right ) \tan ^{m+1}(c+d x)}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\int -\frac {\tan ^m(c+d x) \left (6 A a^6+b B \left (m^3-6 m^2+11 m+18\right ) a^5-A b^2 \left (m^3-9 m^2+26 m+18\right ) a^4-2 b^3 B \left (-m^3+3 m^2+7 m+3\right ) a^3-6 \left (-B a^3+3 A b a^2+3 b^2 B a-A b^3\right ) \tan (c+d x) a^3-2 A b^4 m \left (m^2-6 m+2\right ) a^2-b^5 B m \left (1-m^2\right ) a-b m \left (-B \left (m^2-6 m+11\right ) a^5+A b \left (m^2-9 m+26\right ) a^4+2 b^2 B \left (-m^2+3 m+7\right ) a^3+2 A b^3 \left (m^2-6 m+2\right ) a^2+b^4 B \left (1-m^2\right ) a+A b^5 \left (m^2-3 m+2\right )\right ) \tan ^2(c+d x)-A b^6 m \left (m^2-3 m+2\right )\right )}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}-\frac {b \left (a^5 (-B) \left (m^2-6 m+11\right )+a^4 A b \left (m^2-9 m+26\right )+2 a^3 b^2 B \left (-m^2+3 m+7\right )+2 a^2 A b^3 \left (m^2-6 m+2\right )+a b^4 B \left (1-m^2\right )+A b^5 \left (m^2-3 m+2\right )\right ) \tan ^{m+1}(c+d x)}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{2 a \left (a^2+b^2\right )}}{3 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \tan ^{m+1}(c+d x)}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {b \left (a^3 (-B) (5-m)+a^2 A b (8-m)+a b^2 B (m+1)+A b^3 (2-m)\right ) \tan ^{m+1}(c+d x)}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {-\frac {\int \frac {\tan ^m(c+d x) \left (6 A a^6+b B \left (m^3-6 m^2+11 m+18\right ) a^5-A b^2 \left (m^3-9 m^2+26 m+18\right ) a^4-2 b^3 B \left (-m^3+3 m^2+7 m+3\right ) a^3-6 \left (-B a^3+3 A b a^2+3 b^2 B a-A b^3\right ) \tan (c+d x) a^3-2 A b^4 m \left (m^2-6 m+2\right ) a^2-b^5 B m \left (1-m^2\right ) a-b m \left (-B \left (m^2-6 m+11\right ) a^5+A b \left (m^2-9 m+26\right ) a^4+2 b^2 B \left (-m^2+3 m+7\right ) a^3+2 A b^3 \left (m^2-6 m+2\right ) a^2+b^4 B \left (1-m^2\right ) a+A b^5 \left (m^2-3 m+2\right )\right ) \tan ^2(c+d x)-A b^6 m \left (m^2-3 m+2\right )\right )}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}-\frac {b \left (a^5 (-B) \left (m^2-6 m+11\right )+a^4 A b \left (m^2-9 m+26\right )+2 a^3 b^2 B \left (-m^2+3 m+7\right )+2 a^2 A b^3 \left (m^2-6 m+2\right )+a b^4 B \left (1-m^2\right )+A b^5 \left (m^2-3 m+2\right )\right ) \tan ^{m+1}(c+d x)}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{2 a \left (a^2+b^2\right )}}{3 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \tan ^{m+1}(c+d x)}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {b \left (a^3 (-B) (5-m)+a^2 A b (8-m)+a b^2 B (m+1)+A b^3 (2-m)\right ) \tan ^{m+1}(c+d x)}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {-\frac {\int \frac {\tan (c+d x)^m \left (6 A a^6+b B \left (m^3-6 m^2+11 m+18\right ) a^5-A b^2 \left (m^3-9 m^2+26 m+18\right ) a^4-2 b^3 B \left (-m^3+3 m^2+7 m+3\right ) a^3-6 \left (-B a^3+3 A b a^2+3 b^2 B a-A b^3\right ) \tan (c+d x) a^3-2 A b^4 m \left (m^2-6 m+2\right ) a^2-b^5 B m \left (1-m^2\right ) a-b m \left (-B \left (m^2-6 m+11\right ) a^5+A b \left (m^2-9 m+26\right ) a^4+2 b^2 B \left (-m^2+3 m+7\right ) a^3+2 A b^3 \left (m^2-6 m+2\right ) a^2+b^4 B \left (1-m^2\right ) a+A b^5 \left (m^2-3 m+2\right )\right ) \tan (c+d x)^2-A b^6 m \left (m^2-3 m+2\right )\right )}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}-\frac {b \left (a^5 (-B) \left (m^2-6 m+11\right )+a^4 A b \left (m^2-9 m+26\right )+2 a^3 b^2 B \left (-m^2+3 m+7\right )+2 a^2 A b^3 \left (m^2-6 m+2\right )+a b^4 B \left (1-m^2\right )+A b^5 \left (m^2-3 m+2\right )\right ) \tan ^{m+1}(c+d x)}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{2 a \left (a^2+b^2\right )}}{3 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \tan ^{m+1}(c+d x)}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\)

\(\Big \downarrow \) 4136

\(\displaystyle \frac {\frac {b \left (a^3 (-B) (5-m)+a^2 A b (8-m)+a b^2 B (m+1)+A b^3 (2-m)\right ) \tan ^{m+1}(c+d x)}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {-\frac {\frac {\int 6 \tan ^m(c+d x) \left (a^3 \left (A a^4+4 b B a^3-6 A b^2 a^2-4 b^3 B a+A b^4\right )-a^3 \left (-B a^4+4 A b a^3+6 b^2 B a^2-4 A b^3 a-b^4 B\right ) \tan (c+d x)\right )dx}{a^2+b^2}-\frac {b \left (a^7 B \left (-m^3+6 m^2-11 m+6\right )-a^6 A b \left (-m^3+9 m^2-26 m+24\right )-3 a^5 b^2 B \left (m^3-4 m^2-m+12\right )+3 a^4 A b^3 \left (m^3-7 m^2+10 m+8\right )+3 a^3 b^4 B \left (-m^3+2 m^2+5 m+2\right )+3 a^2 A b^5 m \left (m^2-5 m+2\right )+a b^6 B m \left (1-m^2\right )+A b^7 m \left (m^2-3 m+2\right )\right ) \int \frac {\tan ^m(c+d x) \left (\tan ^2(c+d x)+1\right )}{a+b \tan (c+d x)}dx}{a^2+b^2}}{a \left (a^2+b^2\right )}-\frac {b \left (a^5 (-B) \left (m^2-6 m+11\right )+a^4 A b \left (m^2-9 m+26\right )+2 a^3 b^2 B \left (-m^2+3 m+7\right )+2 a^2 A b^3 \left (m^2-6 m+2\right )+a b^4 B \left (1-m^2\right )+A b^5 \left (m^2-3 m+2\right )\right ) \tan ^{m+1}(c+d x)}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{2 a \left (a^2+b^2\right )}}{3 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \tan ^{m+1}(c+d x)}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {b \left (a^3 (-B) (5-m)+a^2 A b (8-m)+a b^2 B (m+1)+A b^3 (2-m)\right ) \tan ^{m+1}(c+d x)}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {-\frac {\frac {6 \int \tan ^m(c+d x) \left (a^3 \left (A a^4+4 b B a^3-6 A b^2 a^2-4 b^3 B a+A b^4\right )-a^3 \left (-B a^4+4 A b a^3+6 b^2 B a^2-4 A b^3 a-b^4 B\right ) \tan (c+d x)\right )dx}{a^2+b^2}-\frac {b \left (a^7 B \left (-m^3+6 m^2-11 m+6\right )-a^6 A b \left (-m^3+9 m^2-26 m+24\right )-3 a^5 b^2 B \left (m^3-4 m^2-m+12\right )+3 a^4 A b^3 \left (m^3-7 m^2+10 m+8\right )+3 a^3 b^4 B \left (-m^3+2 m^2+5 m+2\right )+3 a^2 A b^5 m \left (m^2-5 m+2\right )+a b^6 B m \left (1-m^2\right )+A b^7 m \left (m^2-3 m+2\right )\right ) \int \frac {\tan ^m(c+d x) \left (\tan ^2(c+d x)+1\right )}{a+b \tan (c+d x)}dx}{a^2+b^2}}{a \left (a^2+b^2\right )}-\frac {b \left (a^5 (-B) \left (m^2-6 m+11\right )+a^4 A b \left (m^2-9 m+26\right )+2 a^3 b^2 B \left (-m^2+3 m+7\right )+2 a^2 A b^3 \left (m^2-6 m+2\right )+a b^4 B \left (1-m^2\right )+A b^5 \left (m^2-3 m+2\right )\right ) \tan ^{m+1}(c+d x)}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{2 a \left (a^2+b^2\right )}}{3 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \tan ^{m+1}(c+d x)}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {b \left (a^3 (-B) (5-m)+a^2 A b (8-m)+a b^2 B (m+1)+A b^3 (2-m)\right ) \tan ^{m+1}(c+d x)}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {-\frac {\frac {6 \int \tan (c+d x)^m \left (a^3 \left (A a^4+4 b B a^3-6 A b^2 a^2-4 b^3 B a+A b^4\right )-a^3 \left (-B a^4+4 A b a^3+6 b^2 B a^2-4 A b^3 a-b^4 B\right ) \tan (c+d x)\right )dx}{a^2+b^2}-\frac {b \left (a^7 B \left (-m^3+6 m^2-11 m+6\right )-a^6 A b \left (-m^3+9 m^2-26 m+24\right )-3 a^5 b^2 B \left (m^3-4 m^2-m+12\right )+3 a^4 A b^3 \left (m^3-7 m^2+10 m+8\right )+3 a^3 b^4 B \left (-m^3+2 m^2+5 m+2\right )+3 a^2 A b^5 m \left (m^2-5 m+2\right )+a b^6 B m \left (1-m^2\right )+A b^7 m \left (m^2-3 m+2\right )\right ) \int \frac {\tan (c+d x)^m \left (\tan (c+d x)^2+1\right )}{a+b \tan (c+d x)}dx}{a^2+b^2}}{a \left (a^2+b^2\right )}-\frac {b \left (a^5 (-B) \left (m^2-6 m+11\right )+a^4 A b \left (m^2-9 m+26\right )+2 a^3 b^2 B \left (-m^2+3 m+7\right )+2 a^2 A b^3 \left (m^2-6 m+2\right )+a b^4 B \left (1-m^2\right )+A b^5 \left (m^2-3 m+2\right )\right ) \tan ^{m+1}(c+d x)}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{2 a \left (a^2+b^2\right )}}{3 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \tan ^{m+1}(c+d x)}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\)

\(\Big \downarrow \) 4021

\(\displaystyle \frac {\frac {b \left (a^3 (-B) (5-m)+a^2 A b (8-m)+a b^2 B (m+1)+A b^3 (2-m)\right ) \tan ^{m+1}(c+d x)}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {-\frac {\frac {6 \left (a^3 \left (a^4 A+4 a^3 b B-6 a^2 A b^2-4 a b^3 B+A b^4\right ) \int \tan ^m(c+d x)dx-a^3 \left (a^4 (-B)+4 a^3 A b+6 a^2 b^2 B-4 a A b^3-b^4 B\right ) \int \tan ^{m+1}(c+d x)dx\right )}{a^2+b^2}-\frac {b \left (a^7 B \left (-m^3+6 m^2-11 m+6\right )-a^6 A b \left (-m^3+9 m^2-26 m+24\right )-3 a^5 b^2 B \left (m^3-4 m^2-m+12\right )+3 a^4 A b^3 \left (m^3-7 m^2+10 m+8\right )+3 a^3 b^4 B \left (-m^3+2 m^2+5 m+2\right )+3 a^2 A b^5 m \left (m^2-5 m+2\right )+a b^6 B m \left (1-m^2\right )+A b^7 m \left (m^2-3 m+2\right )\right ) \int \frac {\tan (c+d x)^m \left (\tan (c+d x)^2+1\right )}{a+b \tan (c+d x)}dx}{a^2+b^2}}{a \left (a^2+b^2\right )}-\frac {b \left (a^5 (-B) \left (m^2-6 m+11\right )+a^4 A b \left (m^2-9 m+26\right )+2 a^3 b^2 B \left (-m^2+3 m+7\right )+2 a^2 A b^3 \left (m^2-6 m+2\right )+a b^4 B \left (1-m^2\right )+A b^5 \left (m^2-3 m+2\right )\right ) \tan ^{m+1}(c+d x)}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{2 a \left (a^2+b^2\right )}}{3 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \tan ^{m+1}(c+d x)}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {b \left (a^3 (-B) (5-m)+a^2 A b (8-m)+a b^2 B (m+1)+A b^3 (2-m)\right ) \tan ^{m+1}(c+d x)}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {-\frac {\frac {6 \left (a^3 \left (a^4 A+4 a^3 b B-6 a^2 A b^2-4 a b^3 B+A b^4\right ) \int \tan (c+d x)^mdx-a^3 \left (a^4 (-B)+4 a^3 A b+6 a^2 b^2 B-4 a A b^3-b^4 B\right ) \int \tan (c+d x)^{m+1}dx\right )}{a^2+b^2}-\frac {b \left (a^7 B \left (-m^3+6 m^2-11 m+6\right )-a^6 A b \left (-m^3+9 m^2-26 m+24\right )-3 a^5 b^2 B \left (m^3-4 m^2-m+12\right )+3 a^4 A b^3 \left (m^3-7 m^2+10 m+8\right )+3 a^3 b^4 B \left (-m^3+2 m^2+5 m+2\right )+3 a^2 A b^5 m \left (m^2-5 m+2\right )+a b^6 B m \left (1-m^2\right )+A b^7 m \left (m^2-3 m+2\right )\right ) \int \frac {\tan (c+d x)^m \left (\tan (c+d x)^2+1\right )}{a+b \tan (c+d x)}dx}{a^2+b^2}}{a \left (a^2+b^2\right )}-\frac {b \left (a^5 (-B) \left (m^2-6 m+11\right )+a^4 A b \left (m^2-9 m+26\right )+2 a^3 b^2 B \left (-m^2+3 m+7\right )+2 a^2 A b^3 \left (m^2-6 m+2\right )+a b^4 B \left (1-m^2\right )+A b^5 \left (m^2-3 m+2\right )\right ) \tan ^{m+1}(c+d x)}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{2 a \left (a^2+b^2\right )}}{3 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \tan ^{m+1}(c+d x)}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\)

\(\Big \downarrow \) 3957

\(\displaystyle \frac {\frac {b \left (a^3 (-B) (5-m)+a^2 A b (8-m)+a b^2 B (m+1)+A b^3 (2-m)\right ) \tan ^{m+1}(c+d x)}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {-\frac {\frac {6 \left (\frac {a^3 \left (a^4 A+4 a^3 b B-6 a^2 A b^2-4 a b^3 B+A b^4\right ) \int \frac {\tan ^m(c+d x)}{\tan ^2(c+d x)+1}d\tan (c+d x)}{d}-\frac {a^3 \left (a^4 (-B)+4 a^3 A b+6 a^2 b^2 B-4 a A b^3-b^4 B\right ) \int \frac {\tan ^{m+1}(c+d x)}{\tan ^2(c+d x)+1}d\tan (c+d x)}{d}\right )}{a^2+b^2}-\frac {b \left (a^7 B \left (-m^3+6 m^2-11 m+6\right )-a^6 A b \left (-m^3+9 m^2-26 m+24\right )-3 a^5 b^2 B \left (m^3-4 m^2-m+12\right )+3 a^4 A b^3 \left (m^3-7 m^2+10 m+8\right )+3 a^3 b^4 B \left (-m^3+2 m^2+5 m+2\right )+3 a^2 A b^5 m \left (m^2-5 m+2\right )+a b^6 B m \left (1-m^2\right )+A b^7 m \left (m^2-3 m+2\right )\right ) \int \frac {\tan (c+d x)^m \left (\tan (c+d x)^2+1\right )}{a+b \tan (c+d x)}dx}{a^2+b^2}}{a \left (a^2+b^2\right )}-\frac {b \left (a^5 (-B) \left (m^2-6 m+11\right )+a^4 A b \left (m^2-9 m+26\right )+2 a^3 b^2 B \left (-m^2+3 m+7\right )+2 a^2 A b^3 \left (m^2-6 m+2\right )+a b^4 B \left (1-m^2\right )+A b^5 \left (m^2-3 m+2\right )\right ) \tan ^{m+1}(c+d x)}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{2 a \left (a^2+b^2\right )}}{3 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \tan ^{m+1}(c+d x)}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\)

\(\Big \downarrow \) 278

\(\displaystyle \frac {\frac {b \left (a^3 (-B) (5-m)+a^2 A b (8-m)+a b^2 B (m+1)+A b^3 (2-m)\right ) \tan ^{m+1}(c+d x)}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {-\frac {\frac {6 \left (\frac {a^3 \left (a^4 A+4 a^3 b B-6 a^2 A b^2-4 a b^3 B+A b^4\right ) \tan ^{m+1}(c+d x) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\tan ^2(c+d x)\right )}{d (m+1)}-\frac {a^3 \left (a^4 (-B)+4 a^3 A b+6 a^2 b^2 B-4 a A b^3-b^4 B\right ) \tan ^{m+2}(c+d x) \operatorname {Hypergeometric2F1}\left (1,\frac {m+2}{2},\frac {m+4}{2},-\tan ^2(c+d x)\right )}{d (m+2)}\right )}{a^2+b^2}-\frac {b \left (a^7 B \left (-m^3+6 m^2-11 m+6\right )-a^6 A b \left (-m^3+9 m^2-26 m+24\right )-3 a^5 b^2 B \left (m^3-4 m^2-m+12\right )+3 a^4 A b^3 \left (m^3-7 m^2+10 m+8\right )+3 a^3 b^4 B \left (-m^3+2 m^2+5 m+2\right )+3 a^2 A b^5 m \left (m^2-5 m+2\right )+a b^6 B m \left (1-m^2\right )+A b^7 m \left (m^2-3 m+2\right )\right ) \int \frac {\tan (c+d x)^m \left (\tan (c+d x)^2+1\right )}{a+b \tan (c+d x)}dx}{a^2+b^2}}{a \left (a^2+b^2\right )}-\frac {b \left (a^5 (-B) \left (m^2-6 m+11\right )+a^4 A b \left (m^2-9 m+26\right )+2 a^3 b^2 B \left (-m^2+3 m+7\right )+2 a^2 A b^3 \left (m^2-6 m+2\right )+a b^4 B \left (1-m^2\right )+A b^5 \left (m^2-3 m+2\right )\right ) \tan ^{m+1}(c+d x)}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{2 a \left (a^2+b^2\right )}}{3 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \tan ^{m+1}(c+d x)}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\)

\(\Big \downarrow \) 4117

\(\displaystyle \frac {\frac {b \left (a^3 (-B) (5-m)+a^2 A b (8-m)+a b^2 B (m+1)+A b^3 (2-m)\right ) \tan ^{m+1}(c+d x)}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {-\frac {\frac {6 \left (\frac {a^3 \left (a^4 A+4 a^3 b B-6 a^2 A b^2-4 a b^3 B+A b^4\right ) \tan ^{m+1}(c+d x) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\tan ^2(c+d x)\right )}{d (m+1)}-\frac {a^3 \left (a^4 (-B)+4 a^3 A b+6 a^2 b^2 B-4 a A b^3-b^4 B\right ) \tan ^{m+2}(c+d x) \operatorname {Hypergeometric2F1}\left (1,\frac {m+2}{2},\frac {m+4}{2},-\tan ^2(c+d x)\right )}{d (m+2)}\right )}{a^2+b^2}-\frac {b \left (a^7 B \left (-m^3+6 m^2-11 m+6\right )-a^6 A b \left (-m^3+9 m^2-26 m+24\right )-3 a^5 b^2 B \left (m^3-4 m^2-m+12\right )+3 a^4 A b^3 \left (m^3-7 m^2+10 m+8\right )+3 a^3 b^4 B \left (-m^3+2 m^2+5 m+2\right )+3 a^2 A b^5 m \left (m^2-5 m+2\right )+a b^6 B m \left (1-m^2\right )+A b^7 m \left (m^2-3 m+2\right )\right ) \int \frac {\tan ^m(c+d x)}{a+b \tan (c+d x)}d\tan (c+d x)}{d \left (a^2+b^2\right )}}{a \left (a^2+b^2\right )}-\frac {b \left (a^5 (-B) \left (m^2-6 m+11\right )+a^4 A b \left (m^2-9 m+26\right )+2 a^3 b^2 B \left (-m^2+3 m+7\right )+2 a^2 A b^3 \left (m^2-6 m+2\right )+a b^4 B \left (1-m^2\right )+A b^5 \left (m^2-3 m+2\right )\right ) \tan ^{m+1}(c+d x)}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{2 a \left (a^2+b^2\right )}}{3 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \tan ^{m+1}(c+d x)}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\)

\(\Big \downarrow \) 74

\(\displaystyle \frac {b (A b-a B) \tan ^{m+1}(c+d x)}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac {\frac {b \left (a^3 (-B) (5-m)+a^2 A b (8-m)+a b^2 B (m+1)+A b^3 (2-m)\right ) \tan ^{m+1}(c+d x)}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {-\frac {b \left (a^5 (-B) \left (m^2-6 m+11\right )+a^4 A b \left (m^2-9 m+26\right )+2 a^3 b^2 B \left (-m^2+3 m+7\right )+2 a^2 A b^3 \left (m^2-6 m+2\right )+a b^4 B \left (1-m^2\right )+A b^5 \left (m^2-3 m+2\right )\right ) \tan ^{m+1}(c+d x)}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}-\frac {\frac {6 \left (\frac {a^3 \left (a^4 A+4 a^3 b B-6 a^2 A b^2-4 a b^3 B+A b^4\right ) \tan ^{m+1}(c+d x) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\tan ^2(c+d x)\right )}{d (m+1)}-\frac {a^3 \left (a^4 (-B)+4 a^3 A b+6 a^2 b^2 B-4 a A b^3-b^4 B\right ) \tan ^{m+2}(c+d x) \operatorname {Hypergeometric2F1}\left (1,\frac {m+2}{2},\frac {m+4}{2},-\tan ^2(c+d x)\right )}{d (m+2)}\right )}{a^2+b^2}-\frac {b \left (a^7 B \left (-m^3+6 m^2-11 m+6\right )-a^6 A b \left (-m^3+9 m^2-26 m+24\right )-3 a^5 b^2 B \left (m^3-4 m^2-m+12\right )+3 a^4 A b^3 \left (m^3-7 m^2+10 m+8\right )+3 a^3 b^4 B \left (-m^3+2 m^2+5 m+2\right )+3 a^2 A b^5 m \left (m^2-5 m+2\right )+a b^6 B m \left (1-m^2\right )+A b^7 m \left (m^2-3 m+2\right )\right ) \tan ^{m+1}(c+d x) \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,-\frac {b \tan (c+d x)}{a}\right )}{a d (m+1) \left (a^2+b^2\right )}}{a \left (a^2+b^2\right )}}{2 a \left (a^2+b^2\right )}}{3 a \left (a^2+b^2\right )}\)

Input:

Int[(Tan[c + d*x]^m*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^4,x]
 

Output:

(b*(A*b - a*B)*Tan[c + d*x]^(1 + m))/(3*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x 
])^3) + ((b*(A*b^3*(2 - m) - a^3*B*(5 - m) + a^2*A*b*(8 - m) + a*b^2*B*(1 
+ m))*Tan[c + d*x]^(1 + m))/(2*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) - ( 
-((b*(a*b^4*B*(1 - m^2) + 2*a^3*b^2*B*(7 + 3*m - m^2) + a^4*A*b*(26 - 9*m 
+ m^2) + 2*a^2*A*b^3*(2 - 6*m + m^2) - a^5*B*(11 - 6*m + m^2) + A*b^5*(2 - 
 3*m + m^2))*Tan[c + d*x]^(1 + m))/(a*(a^2 + b^2)*d*(a + b*Tan[c + d*x]))) 
 - (-((b*(a*b^6*B*m*(1 - m^2) + 3*a^2*A*b^5*m*(2 - 5*m + m^2) + A*b^7*m*(2 
 - 3*m + m^2) + 3*a^3*b^4*B*(2 + 5*m + 2*m^2 - m^3) + a^7*B*(6 - 11*m + 6* 
m^2 - m^3) - a^6*A*b*(24 - 26*m + 9*m^2 - m^3) + 3*a^4*A*b^3*(8 + 10*m - 7 
*m^2 + m^3) - 3*a^5*b^2*B*(12 - m - 4*m^2 + m^3))*Hypergeometric2F1[1, 1 + 
 m, 2 + m, -((b*Tan[c + d*x])/a)]*Tan[c + d*x]^(1 + m))/(a*(a^2 + b^2)*d*( 
1 + m))) + (6*((a^3*(a^4*A - 6*a^2*A*b^2 + A*b^4 + 4*a^3*b*B - 4*a*b^3*B)* 
Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^( 
1 + m))/(d*(1 + m)) - (a^3*(4*a^3*A*b - 4*a*A*b^3 - a^4*B + 6*a^2*b^2*B - 
b^4*B)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, -Tan[c + d*x]^2]*Tan[c + 
 d*x]^(2 + m))/(d*(2 + m))))/(a^2 + b^2))/(a*(a^2 + b^2)))/(2*a*(a^2 + b^2 
)))/(3*a*(a^2 + b^2))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 74

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x 
)^(m + 1)/(b*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] 
/; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[c, 0] 
 &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))
 

rule 278

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( 
c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( 
-b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && (ILtQ[p, 0 
] || GtQ[a, 0])
 

rule 3042

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

rule 3957

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b/d   Subst[Int 
[x^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{b, c, d, n}, x] && 
!IntegerQ[n]
 

rule 4021

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

rule 4092

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

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]
 

Maple [F]

\[\int \frac {\tan \left (d x +c \right )^{m} \left (A +B \tan \left (d x +c \right )\right )}{\left (a +b \tan \left (d x +c \right )\right )^{4}}d x\]

Input:

int(tan(d*x+c)^m*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^4,x)
 

Output:

int(tan(d*x+c)^m*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^4,x)
 

Fricas [F]

\[ \int \frac {\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\int { \frac {{\left (B \tan \left (d x + c\right ) + A\right )} \tan \left (d x + c\right )^{m}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{4}} \,d x } \] Input:

integrate(tan(d*x+c)^m*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^4,x, algorithm="f 
ricas")
 

Output:

integral((B*tan(d*x + c) + A)*tan(d*x + c)^m/(b^4*tan(d*x + c)^4 + 4*a*b^3 
*tan(d*x + c)^3 + 6*a^2*b^2*tan(d*x + c)^2 + 4*a^3*b*tan(d*x + c) + a^4), 
x)
 

Sympy [F]

\[ \int \frac {\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\int \frac {\left (A + B \tan {\left (c + d x \right )}\right ) \tan ^{m}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{4}}\, dx \] Input:

integrate(tan(d*x+c)**m*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))**4,x)
 

Output:

Integral((A + B*tan(c + d*x))*tan(c + d*x)**m/(a + b*tan(c + d*x))**4, x)
 

Maxima [F]

\[ \int \frac {\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\int { \frac {{\left (B \tan \left (d x + c\right ) + A\right )} \tan \left (d x + c\right )^{m}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{4}} \,d x } \] Input:

integrate(tan(d*x+c)^m*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^4,x, algorithm="m 
axima")
 

Output:

integrate((B*tan(d*x + c) + A)*tan(d*x + c)^m/(b*tan(d*x + c) + a)^4, x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(tan(d*x+c)^m*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^4,x, algorithm="g 
iac")
 

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:Unable to divide, perhaps due to rounding error%%%{1,[0 
,1,0]%%%} / %%%{1,[0,0,4]%%%} Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\int \frac {{\mathrm {tan}\left (c+d\,x\right )}^m\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )}{{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^4} \,d x \] Input:

int((tan(c + d*x)^m*(A + B*tan(c + d*x)))/(a + b*tan(c + d*x))^4,x)
 

Output:

int((tan(c + d*x)^m*(A + B*tan(c + d*x)))/(a + b*tan(c + d*x))^4, x)
 

Reduce [F]

\[ \int \frac {\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\left (\int \frac {\tan \left (d x +c \right )^{m}}{\tan \left (d x +c \right )^{3} b^{3} m -2 \tan \left (d x +c \right )^{3} b^{3}+3 \tan \left (d x +c \right )^{2} a \,b^{2} m -6 \tan \left (d x +c \right )^{2} a \,b^{2}+3 \tan \left (d x +c \right ) a^{2} b m -6 \tan \left (d x +c \right ) a^{2} b +a^{3} m -2 a^{3}}d x \right ) \left (m -2\right ) \] Input:

int(tan(d*x+c)^m*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^4,x)
 

Output:

int(tan(c + d*x)**m/(tan(c + d*x)**3*b**3*m - 2*tan(c + d*x)**3*b**3 + 3*t 
an(c + d*x)**2*a*b**2*m - 6*tan(c + d*x)**2*a*b**2 + 3*tan(c + d*x)*a**2*b 
*m - 6*tan(c + d*x)*a**2*b + a**3*m - 2*a**3),x)*(m - 2)