3.3.96 \(\int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx\) [296]

3.3.96.1 Optimal result

Integrand size = 31, antiderivative size = 399 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=-\frac {\left (a^4 A-6 a^2 A b^2+A b^4+4 a^3 b B-4 a b^3 B\right ) x}{\left (a^2+b^2\right )^4}-\frac {(4 A b-a B) \log (\sin (c+d x))}{a^5 d}+\frac {b^2 \left (20 a^6 A b+24 a^4 A b^3+16 a^2 A b^5+4 A b^7-10 a^7 B-5 a^5 b^2 B-4 a^3 b^4 B-a b^6 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^5 \left (a^2+b^2\right )^4 d}-\frac {b \left (3 a^2 A+4 A b^2-a b B\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))^3}-\frac {b \left (2 a^4 A+8 a^2 A b^2+4 A b^4-3 a^3 b B-a b^3 B\right )}{2 a^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac {b \left (a^6 A+13 a^4 A b^2+12 a^2 A b^4+4 A b^6-6 a^5 b B-3 a^3 b^3 B-a b^5 B\right )}{a^4 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))} \]

output

-(A*a^4-6*A*a^2*b^2+A*b^4+4*B*a^3*b-4*B*a*b^3)*x/(a^2+b^2)^4-(4*A*b-B*a)*l 
n(sin(d*x+c))/a^5/d+b^2*(20*A*a^6*b+24*A*a^4*b^3+16*A*a^2*b^5+4*A*b^7-10*B 
*a^7-5*B*a^5*b^2-4*B*a^3*b^4-B*a*b^6)*ln(a*cos(d*x+c)+b*sin(d*x+c))/a^5/(a 
^2+b^2)^4/d-1/3*b*(3*A*a^2+4*A*b^2-B*a*b)/a^2/(a^2+b^2)/d/(a+b*tan(d*x+c)) 
^3-A*cot(d*x+c)/a/d/(a+b*tan(d*x+c))^3-1/2*b*(2*A*a^4+8*A*a^2*b^2+4*A*b^4- 
3*B*a^3*b-B*a*b^3)/a^3/(a^2+b^2)^2/d/(a+b*tan(d*x+c))^2-b*(A*a^6+13*A*a^4* 
b^2+12*A*a^2*b^4+4*A*b^6-6*B*a^5*b-3*B*a^3*b^3-B*a*b^5)/a^4/(a^2+b^2)^3/d/ 
(a+b*tan(d*x+c))
 

3.3.96.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.58 (sec) , antiderivative size = 357, normalized size of antiderivative = 0.89 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\frac {-\frac {6 A \cot (c+d x)}{a^4}+\frac {3 i (A+i B) \log (i-\tan (c+d x))}{(a+i b)^4}+\frac {6 (-4 A b+a B) \log (\tan (c+d x))}{a^5}-\frac {3 (i A+B) \log (i+\tan (c+d x))}{(a-i b)^4}-\frac {6 b^2 \left (-20 a^6 A b-24 a^4 A b^3-16 a^2 A b^5-4 A b^7+10 a^7 B+5 a^5 b^2 B+4 a^3 b^4 B+a b^6 B\right ) \log (a+b \tan (c+d x))}{a^5 \left (a^2+b^2\right )^4}+\frac {2 b^2 (-A b+a B)}{a^2 \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac {3 b^2 \left (-4 a^2 A b-2 A b^3+3 a^3 B+a b^2 B\right )}{a^3 \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}+\frac {6 b^2 \left (-10 a^4 A b-9 a^2 A b^3-3 A b^5+6 a^5 B+3 a^3 b^2 B+a b^4 B\right )}{a^4 \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}}{6 d} \]

input

Integrate[(Cot[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^4,x]
 

output

((-6*A*Cot[c + d*x])/a^4 + ((3*I)*(A + I*B)*Log[I - Tan[c + d*x]])/(a + I* 
b)^4 + (6*(-4*A*b + a*B)*Log[Tan[c + d*x]])/a^5 - (3*(I*A + B)*Log[I + Tan 
[c + d*x]])/(a - I*b)^4 - (6*b^2*(-20*a^6*A*b - 24*a^4*A*b^3 - 16*a^2*A*b^ 
5 - 4*A*b^7 + 10*a^7*B + 5*a^5*b^2*B + 4*a^3*b^4*B + a*b^6*B)*Log[a + b*Ta 
n[c + d*x]])/(a^5*(a^2 + b^2)^4) + (2*b^2*(-(A*b) + a*B))/(a^2*(a^2 + b^2) 
*(a + b*Tan[c + d*x])^3) + (3*b^2*(-4*a^2*A*b - 2*A*b^3 + 3*a^3*B + a*b^2* 
B))/(a^3*(a^2 + b^2)^2*(a + b*Tan[c + d*x])^2) + (6*b^2*(-10*a^4*A*b - 9*a 
^2*A*b^3 - 3*A*b^5 + 6*a^5*B + 3*a^3*b^2*B + a*b^4*B))/(a^4*(a^2 + b^2)^3* 
(a + b*Tan[c + d*x])))/(6*d)
 

3.3.96.3 Rubi [A] (verified)

Time = 2.47 (sec) , antiderivative size = 459, normalized size of antiderivative = 1.15, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.516, Rules used = {3042, 4092, 3042, 4132, 27, 3042, 4132, 27, 3042, 4132, 3042, 4134, 3042, 25, 3956, 4013}

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.

\(\displaystyle \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {A+B \tan (c+d x)}{\tan (c+d x)^2 (a+b \tan (c+d x))^4}dx\)

\(\Big \downarrow \) 4092

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

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int \frac {4 A b \tan (c+d x)^2+a A \tan (c+d x)+4 A b-a B}{\tan (c+d x) (a+b \tan (c+d x))^4}dx}{a}-\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))^3}\)

\(\Big \downarrow \) 4132

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4132

\(\displaystyle -\frac {\frac {\frac {\int \frac {2 \cot (c+d x) \left (\left (A a^2+2 b B a-A b^2\right ) \tan (c+d x) a^3+b \left (2 A a^4-3 b B a^3+8 A b^2 a^2-b^3 B a+4 A b^4\right ) \tan ^2(c+d x)+\left (a^2+b^2\right )^2 (4 A b-a B)\right )}{(a+b \tan (c+d x))^2}dx}{2 a \left (a^2+b^2\right )}+\frac {b \left (2 a^4 A-3 a^3 b B+8 a^2 A b^2-a b^3 B+4 A b^4\right )}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a \left (a^2+b^2\right )}+\frac {b \left (3 a^2 A-a b B+4 A b^2\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}}{a}-\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))^3}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {\frac {\int \frac {\cot (c+d x) \left (\left (A a^2+2 b B a-A b^2\right ) \tan (c+d x) a^3+b \left (2 A a^4-3 b B a^3+8 A b^2 a^2-b^3 B a+4 A b^4\right ) \tan ^2(c+d x)+\left (a^2+b^2\right )^2 (4 A b-a B)\right )}{(a+b \tan (c+d x))^2}dx}{a \left (a^2+b^2\right )}+\frac {b \left (2 a^4 A-3 a^3 b B+8 a^2 A b^2-a b^3 B+4 A b^4\right )}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a \left (a^2+b^2\right )}+\frac {b \left (3 a^2 A-a b B+4 A b^2\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}}{a}-\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {\frac {\int \frac {\left (A a^2+2 b B a-A b^2\right ) \tan (c+d x) a^3+b \left (2 A a^4-3 b B a^3+8 A b^2 a^2-b^3 B a+4 A b^4\right ) \tan (c+d x)^2+\left (a^2+b^2\right )^2 (4 A b-a B)}{\tan (c+d x) (a+b \tan (c+d x))^2}dx}{a \left (a^2+b^2\right )}+\frac {b \left (2 a^4 A-3 a^3 b B+8 a^2 A b^2-a b^3 B+4 A b^4\right )}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a \left (a^2+b^2\right )}+\frac {b \left (3 a^2 A-a b B+4 A b^2\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}}{a}-\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))^3}\)

\(\Big \downarrow \) 4132

\(\displaystyle -\frac {\frac {\frac {\frac {\int \frac {\cot (c+d x) \left (\left (A a^3+3 b B a^2-3 A b^2 a-b^3 B\right ) \tan (c+d x) a^4+b \left (A a^6-6 b B a^5+13 A b^2 a^4-3 b^3 B a^3+12 A b^4 a^2-b^5 B a+4 A b^6\right ) \tan ^2(c+d x)+\left (a^2+b^2\right )^3 (4 A b-a B)\right )}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}+\frac {b \left (a^6 A-6 a^5 b B+13 a^4 A b^2-3 a^3 b^3 B+12 a^2 A b^4-a b^5 B+4 A b^6\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a \left (a^2+b^2\right )}+\frac {b \left (2 a^4 A-3 a^3 b B+8 a^2 A b^2-a b^3 B+4 A b^4\right )}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a \left (a^2+b^2\right )}+\frac {b \left (3 a^2 A-a b B+4 A b^2\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}}{a}-\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {\frac {\frac {\int \frac {\left (A a^3+3 b B a^2-3 A b^2 a-b^3 B\right ) \tan (c+d x) a^4+b \left (A a^6-6 b B a^5+13 A b^2 a^4-3 b^3 B a^3+12 A b^4 a^2-b^5 B a+4 A b^6\right ) \tan (c+d x)^2+\left (a^2+b^2\right )^3 (4 A b-a B)}{\tan (c+d x) (a+b \tan (c+d x))}dx}{a \left (a^2+b^2\right )}+\frac {b \left (a^6 A-6 a^5 b B+13 a^4 A b^2-3 a^3 b^3 B+12 a^2 A b^4-a b^5 B+4 A b^6\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a \left (a^2+b^2\right )}+\frac {b \left (2 a^4 A-3 a^3 b B+8 a^2 A b^2-a b^3 B+4 A b^4\right )}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a \left (a^2+b^2\right )}+\frac {b \left (3 a^2 A-a b B+4 A b^2\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}}{a}-\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))^3}\)

\(\Big \downarrow \) 4134

\(\displaystyle -\frac {\frac {\frac {\frac {\frac {\left (a^2+b^2\right )^3 (4 A b-a B) \int \cot (c+d x)dx}{a}-\frac {b^2 \left (-10 a^7 B+20 a^6 A b-5 a^5 b^2 B+24 a^4 A b^3-4 a^3 b^4 B+16 a^2 A b^5-a b^6 B+4 A b^7\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}+\frac {a^4 x \left (a^4 A+4 a^3 b B-6 a^2 A b^2-4 a b^3 B+A b^4\right )}{a^2+b^2}}{a \left (a^2+b^2\right )}+\frac {b \left (a^6 A-6 a^5 b B+13 a^4 A b^2-3 a^3 b^3 B+12 a^2 A b^4-a b^5 B+4 A b^6\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a \left (a^2+b^2\right )}+\frac {b \left (2 a^4 A-3 a^3 b B+8 a^2 A b^2-a b^3 B+4 A b^4\right )}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a \left (a^2+b^2\right )}+\frac {b \left (3 a^2 A-a b B+4 A b^2\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}}{a}-\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {\frac {\frac {\frac {\left (a^2+b^2\right )^3 (4 A b-a B) \int -\tan \left (c+d x+\frac {\pi }{2}\right )dx}{a}-\frac {b^2 \left (-10 a^7 B+20 a^6 A b-5 a^5 b^2 B+24 a^4 A b^3-4 a^3 b^4 B+16 a^2 A b^5-a b^6 B+4 A b^7\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}+\frac {a^4 x \left (a^4 A+4 a^3 b B-6 a^2 A b^2-4 a b^3 B+A b^4\right )}{a^2+b^2}}{a \left (a^2+b^2\right )}+\frac {b \left (a^6 A-6 a^5 b B+13 a^4 A b^2-3 a^3 b^3 B+12 a^2 A b^4-a b^5 B+4 A b^6\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a \left (a^2+b^2\right )}+\frac {b \left (2 a^4 A-3 a^3 b B+8 a^2 A b^2-a b^3 B+4 A b^4\right )}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a \left (a^2+b^2\right )}+\frac {b \left (3 a^2 A-a b B+4 A b^2\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}}{a}-\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))^3}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {\frac {\frac {-\frac {\left (a^2+b^2\right )^3 (4 A b-a B) \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx}{a}-\frac {b^2 \left (-10 a^7 B+20 a^6 A b-5 a^5 b^2 B+24 a^4 A b^3-4 a^3 b^4 B+16 a^2 A b^5-a b^6 B+4 A b^7\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}+\frac {a^4 x \left (a^4 A+4 a^3 b B-6 a^2 A b^2-4 a b^3 B+A b^4\right )}{a^2+b^2}}{a \left (a^2+b^2\right )}+\frac {b \left (a^6 A-6 a^5 b B+13 a^4 A b^2-3 a^3 b^3 B+12 a^2 A b^4-a b^5 B+4 A b^6\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a \left (a^2+b^2\right )}+\frac {b \left (2 a^4 A-3 a^3 b B+8 a^2 A b^2-a b^3 B+4 A b^4\right )}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a \left (a^2+b^2\right )}+\frac {b \left (3 a^2 A-a b B+4 A b^2\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}}{a}-\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))^3}\)

\(\Big \downarrow \) 3956

\(\displaystyle -\frac {\frac {\frac {\frac {-\frac {b^2 \left (-10 a^7 B+20 a^6 A b-5 a^5 b^2 B+24 a^4 A b^3-4 a^3 b^4 B+16 a^2 A b^5-a b^6 B+4 A b^7\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}+\frac {\left (a^2+b^2\right )^3 (4 A b-a B) \log (-\sin (c+d x))}{a d}+\frac {a^4 x \left (a^4 A+4 a^3 b B-6 a^2 A b^2-4 a b^3 B+A b^4\right )}{a^2+b^2}}{a \left (a^2+b^2\right )}+\frac {b \left (a^6 A-6 a^5 b B+13 a^4 A b^2-3 a^3 b^3 B+12 a^2 A b^4-a b^5 B+4 A b^6\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a \left (a^2+b^2\right )}+\frac {b \left (2 a^4 A-3 a^3 b B+8 a^2 A b^2-a b^3 B+4 A b^4\right )}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a \left (a^2+b^2\right )}+\frac {b \left (3 a^2 A-a b B+4 A b^2\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}}{a}-\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))^3}\)

\(\Big \downarrow \) 4013

\(\displaystyle -\frac {\frac {b \left (3 a^2 A-a b B+4 A b^2\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac {\frac {b \left (2 a^4 A-3 a^3 b B+8 a^2 A b^2-a b^3 B+4 A b^4\right )}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {\frac {b \left (a^6 A-6 a^5 b B+13 a^4 A b^2-3 a^3 b^3 B+12 a^2 A b^4-a b^5 B+4 A b^6\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {\frac {\left (a^2+b^2\right )^3 (4 A b-a B) \log (-\sin (c+d x))}{a d}+\frac {a^4 x \left (a^4 A+4 a^3 b B-6 a^2 A b^2-4 a b^3 B+A b^4\right )}{a^2+b^2}-\frac {b^2 \left (-10 a^7 B+20 a^6 A b-5 a^5 b^2 B+24 a^4 A b^3-4 a^3 b^4 B+16 a^2 A b^5-a b^6 B+4 A b^7\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a d \left (a^2+b^2\right )}}{a \left (a^2+b^2\right )}}{a \left (a^2+b^2\right )}}{a \left (a^2+b^2\right )}}{a}-\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))^3}\)

input

Int[(Cot[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^4,x]
 

output

-((A*Cot[c + d*x])/(a*d*(a + b*Tan[c + d*x])^3)) - ((b*(3*a^2*A + 4*A*b^2 
- a*b*B))/(3*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) + ((b*(2*a^4*A + 8*a^ 
2*A*b^2 + 4*A*b^4 - 3*a^3*b*B - a*b^3*B))/(2*a*(a^2 + b^2)*d*(a + b*Tan[c 
+ d*x])^2) + (((a^4*(a^4*A - 6*a^2*A*b^2 + A*b^4 + 4*a^3*b*B - 4*a*b^3*B)* 
x)/(a^2 + b^2) + ((a^2 + b^2)^3*(4*A*b - a*B)*Log[-Sin[c + d*x]])/(a*d) - 
(b^2*(20*a^6*A*b + 24*a^4*A*b^3 + 16*a^2*A*b^5 + 4*A*b^7 - 10*a^7*B - 5*a^ 
5*b^2*B - 4*a^3*b^4*B - a*b^6*B)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a* 
(a^2 + b^2)*d))/(a*(a^2 + b^2)) + (b*(a^6*A + 13*a^4*A*b^2 + 12*a^2*A*b^4 
+ 4*A*b^6 - 6*a^5*b*B - 3*a^3*b^3*B - a*b^5*B))/(a*(a^2 + b^2)*d*(a + b*Ta 
n[c + d*x])))/(a*(a^2 + b^2)))/(a*(a^2 + b^2)))/a
 

3.3.96.3.1 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 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 4013

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

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 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 4134

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

3.3.96.4 Maple [A] (verified)

Time = 1.34 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.00

method	result	size
derivativedivides	\(\frac {\frac {\frac {\left (4 A \,a^{3} b -4 A a \,b^{3}-B \,a^{4}+6 B \,a^{2} b^{2}-B \,b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-A \,a^{4}+6 A \,a^{2} b^{2}-A \,b^{4}-4 B \,a^{3} b +4 B a \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}-\frac {A}{a^{4} \tan \left (d x +c \right )}+\frac {\left (-4 A b +B a \right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{5}}-\frac {b^{2} \left (4 A \,a^{2} b +2 A \,b^{3}-3 B \,a^{3}-B a \,b^{2}\right )}{2 \left (a^{2}+b^{2}\right )^{2} a^{3} \left (a +b \tan \left (d x +c \right )\right )^{2}}-\frac {b^{2} \left (10 A \,a^{4} b +9 A \,a^{2} b^{3}+3 A \,b^{5}-6 B \,a^{5}-3 B \,a^{3} b^{2}-B a \,b^{4}\right )}{\left (a^{2}+b^{2}\right )^{3} a^{4} \left (a +b \tan \left (d x +c \right )\right )}+\frac {b^{2} \left (20 A \,a^{6} b +24 A \,a^{4} b^{3}+16 A \,a^{2} b^{5}+4 A \,b^{7}-10 B \,a^{7}-5 B \,a^{5} b^{2}-4 B \,a^{3} b^{4}-B a \,b^{6}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4} a^{5}}-\frac {\left (A b -B a \right ) b^{2}}{3 \left (a^{2}+b^{2}\right ) a^{2} \left (a +b \tan \left (d x +c \right )\right )^{3}}}{d}\)	\(400\)
default	\(\frac {\frac {\frac {\left (4 A \,a^{3} b -4 A a \,b^{3}-B \,a^{4}+6 B \,a^{2} b^{2}-B \,b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-A \,a^{4}+6 A \,a^{2} b^{2}-A \,b^{4}-4 B \,a^{3} b +4 B a \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}-\frac {A}{a^{4} \tan \left (d x +c \right )}+\frac {\left (-4 A b +B a \right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{5}}-\frac {b^{2} \left (4 A \,a^{2} b +2 A \,b^{3}-3 B \,a^{3}-B a \,b^{2}\right )}{2 \left (a^{2}+b^{2}\right )^{2} a^{3} \left (a +b \tan \left (d x +c \right )\right )^{2}}-\frac {b^{2} \left (10 A \,a^{4} b +9 A \,a^{2} b^{3}+3 A \,b^{5}-6 B \,a^{5}-3 B \,a^{3} b^{2}-B a \,b^{4}\right )}{\left (a^{2}+b^{2}\right )^{3} a^{4} \left (a +b \tan \left (d x +c \right )\right )}+\frac {b^{2} \left (20 A \,a^{6} b +24 A \,a^{4} b^{3}+16 A \,a^{2} b^{5}+4 A \,b^{7}-10 B \,a^{7}-5 B \,a^{5} b^{2}-4 B \,a^{3} b^{4}-B a \,b^{6}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4} a^{5}}-\frac {\left (A b -B a \right ) b^{2}}{3 \left (a^{2}+b^{2}\right ) a^{2} \left (a +b \tan \left (d x +c \right )\right )^{3}}}{d}\)	\(400\)
norman	\(\frac {\frac {b \left (6 A \,a^{6} b +33 A \,a^{4} b^{3}+35 A \,a^{2} b^{5}+12 A \,b^{7}-10 B \,a^{5} b^{2}-9 B \,a^{3} b^{4}-3 B a \,b^{6}\right ) \left (\tan ^{2}\left (d x +c \right )\right )}{d \,a^{3} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {A}{a d}-\frac {b^{3} \left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}+4 B \,a^{3} b -4 B a \,b^{3}\right ) x \left (\tan ^{4}\left (d x +c \right )\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}+\frac {b^{2} \left (16 A \,a^{6} b +102 A \,a^{4} b^{3}+106 A \,a^{2} b^{5}+36 A \,b^{7}-35 B \,a^{5} b^{2}-28 B \,a^{3} b^{4}-9 B a \,b^{6}\right ) \left (\tan ^{3}\left (d x +c \right )\right )}{2 d \,a^{4} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {b^{3} \left (18 A \,a^{6} b +128 A \,a^{4} b^{3}+130 A \,a^{2} b^{5}+44 A \,b^{7}-47 B \,a^{5} b^{2}-34 B \,a^{3} b^{4}-11 B a \,b^{6}\right ) \left (\tan ^{4}\left (d x +c \right )\right )}{6 d \,a^{5} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {\left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}+4 B \,a^{3} b -4 B a \,b^{3}\right ) a^{3} x \tan \left (d x +c \right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}-\frac {3 b \left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}+4 B \,a^{3} b -4 B a \,b^{3}\right ) a^{2} x \left (\tan ^{2}\left (d x +c \right )\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}-\frac {3 b^{2} \left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}+4 B \,a^{3} b -4 B a \,b^{3}\right ) a x \left (\tan ^{3}\left (d x +c \right )\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}}{\tan \left (d x +c \right ) \left (a +b \tan \left (d x +c \right )\right )^{3}}+\frac {b^{2} \left (20 A \,a^{6} b +24 A \,a^{4} b^{3}+16 A \,a^{2} b^{5}+4 A \,b^{7}-10 B \,a^{7}-5 B \,a^{5} b^{2}-4 B \,a^{3} b^{4}-B a \,b^{6}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right ) a^{5} d}-\frac {\left (4 A b -B a \right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{5} d}+\frac {\left (4 A \,a^{3} b -4 A a \,b^{3}-B \,a^{4}+6 B \,a^{2} b^{2}-B \,b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}\)	\(887\)
parallelrisch	\(\text {Expression too large to display}\)	\(1195\)
risch	\(\text {Expression too large to display}\)	\(2584\)

input

int(cot(d*x+c)^2*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^4,x,method=_RETURNVERBO 
SE)
 

output

1/d*(1/(a^2+b^2)^4*(1/2*(4*A*a^3*b-4*A*a*b^3-B*a^4+6*B*a^2*b^2-B*b^4)*ln(1 
+tan(d*x+c)^2)+(-A*a^4+6*A*a^2*b^2-A*b^4-4*B*a^3*b+4*B*a*b^3)*arctan(tan(d 
*x+c)))-1/a^4*A/tan(d*x+c)+(-4*A*b+B*a)/a^5*ln(tan(d*x+c))-1/2*b^2*(4*A*a^ 
2*b+2*A*b^3-3*B*a^3-B*a*b^2)/(a^2+b^2)^2/a^3/(a+b*tan(d*x+c))^2-b^2*(10*A* 
a^4*b+9*A*a^2*b^3+3*A*b^5-6*B*a^5-3*B*a^3*b^2-B*a*b^4)/(a^2+b^2)^3/a^4/(a+ 
b*tan(d*x+c))+b^2*(20*A*a^6*b+24*A*a^4*b^3+16*A*a^2*b^5+4*A*b^7-10*B*a^7-5 
*B*a^5*b^2-4*B*a^3*b^4-B*a*b^6)/(a^2+b^2)^4/a^5*ln(a+b*tan(d*x+c))-1/3*(A* 
b-B*a)*b^2/(a^2+b^2)/a^2/(a+b*tan(d*x+c))^3)
 

3.3.96.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1510 vs. \(2 (393) = 786\).

Time = 0.44 (sec) , antiderivative size = 1510, normalized size of antiderivative = 3.78 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\text {Too large to display} \]

input

integrate(cot(d*x+c)^2*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^4,x, algorithm="f 
ricas")
 

output

-1/6*(6*A*a^12 + 24*A*a^10*b^2 + 36*A*a^8*b^4 + 24*A*a^6*b^6 + 6*A*a^4*b^8 
 + (47*B*a^7*b^5 - 74*A*a^6*b^6 + 6*B*a^5*b^7 - 42*A*a^4*b^8 + 3*B*a^3*b^9 
 - 12*A*a^2*b^10 + 6*(A*a^9*b^3 + 4*B*a^8*b^4 - 6*A*a^7*b^5 - 4*B*a^6*b^6 
+ A*a^5*b^7)*d*x)*tan(d*x + c)^4 + 3*(2*A*a^9*b^3 + 35*B*a^8*b^4 - 46*A*a^ 
7*b^5 - 12*B*a^6*b^6 + 8*A*a^5*b^7 - 5*B*a^4*b^8 + 20*A*a^3*b^9 - 2*B*a^2* 
b^10 + 8*A*a*b^11 + 6*(A*a^10*b^2 + 4*B*a^9*b^3 - 6*A*a^8*b^4 - 4*B*a^7*b^ 
5 + A*a^6*b^6)*d*x)*tan(d*x + c)^3 + 3*(6*A*a^10*b^2 + 20*B*a^9*b^3 - 6*A* 
a^8*b^4 - 37*B*a^7*b^5 + 80*A*a^6*b^6 - 18*B*a^5*b^7 + 68*A*a^4*b^8 - 5*B* 
a^3*b^9 + 20*A*a^2*b^10 + 6*(A*a^11*b + 4*B*a^10*b^2 - 6*A*a^9*b^3 - 4*B*a 
^8*b^4 + A*a^7*b^5)*d*x)*tan(d*x + c)^2 - 3*((B*a^9*b^3 - 4*A*a^8*b^4 + 4* 
B*a^7*b^5 - 16*A*a^6*b^6 + 6*B*a^5*b^7 - 24*A*a^4*b^8 + 4*B*a^3*b^9 - 16*A 
*a^2*b^10 + B*a*b^11 - 4*A*b^12)*tan(d*x + c)^4 + 3*(B*a^10*b^2 - 4*A*a^9* 
b^3 + 4*B*a^8*b^4 - 16*A*a^7*b^5 + 6*B*a^6*b^6 - 24*A*a^5*b^7 + 4*B*a^4*b^ 
8 - 16*A*a^3*b^9 + B*a^2*b^10 - 4*A*a*b^11)*tan(d*x + c)^3 + 3*(B*a^11*b - 
 4*A*a^10*b^2 + 4*B*a^9*b^3 - 16*A*a^8*b^4 + 6*B*a^7*b^5 - 24*A*a^6*b^6 + 
4*B*a^5*b^7 - 16*A*a^4*b^8 + B*a^3*b^9 - 4*A*a^2*b^10)*tan(d*x + c)^2 + (B 
*a^12 - 4*A*a^11*b + 4*B*a^10*b^2 - 16*A*a^9*b^3 + 6*B*a^8*b^4 - 24*A*a^7* 
b^5 + 4*B*a^6*b^6 - 16*A*a^5*b^7 + B*a^4*b^8 - 4*A*a^3*b^9)*tan(d*x + c))* 
log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1)) + 3*((10*B*a^7*b^5 - 20*A*a^6*b^6 
 + 5*B*a^5*b^7 - 24*A*a^4*b^8 + 4*B*a^3*b^9 - 16*A*a^2*b^10 + B*a*b^11 ...
 

3.3.96.6 Sympy [F(-2)]

Exception generated. \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\text {Exception raised: AttributeError} \]

input

integrate(cot(d*x+c)**2*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))**4,x)
 

output

Exception raised: AttributeError >> 'NoneType' object has no attribute 'pr 
imitive'
 

3.3.96.7 Maxima [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 698, normalized size of antiderivative = 1.75 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=-\frac {\frac {6 \, {\left (A a^{4} + 4 \, B a^{3} b - 6 \, A a^{2} b^{2} - 4 \, B a b^{3} + A b^{4}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {6 \, {\left (10 \, B a^{7} b^{2} - 20 \, A a^{6} b^{3} + 5 \, B a^{5} b^{4} - 24 \, A a^{4} b^{5} + 4 \, B a^{3} b^{6} - 16 \, A a^{2} b^{7} + B a b^{8} - 4 \, A b^{9}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{13} + 4 \, a^{11} b^{2} + 6 \, a^{9} b^{4} + 4 \, a^{7} b^{6} + a^{5} b^{8}} + \frac {3 \, {\left (B a^{4} - 4 \, A a^{3} b - 6 \, B a^{2} b^{2} + 4 \, A a b^{3} + B b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {6 \, A a^{9} + 18 \, A a^{7} b^{2} + 18 \, A a^{5} b^{4} + 6 \, A a^{3} b^{6} + 6 \, {\left (A a^{6} b^{3} - 6 \, B a^{5} b^{4} + 13 \, A a^{4} b^{5} - 3 \, B a^{3} b^{6} + 12 \, A a^{2} b^{7} - B a b^{8} + 4 \, A b^{9}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (6 \, A a^{7} b^{2} - 27 \, B a^{6} b^{3} + 62 \, A a^{5} b^{4} - 16 \, B a^{4} b^{5} + 60 \, A a^{3} b^{6} - 5 \, B a^{2} b^{7} + 20 \, A a b^{8}\right )} \tan \left (d x + c\right )^{2} + {\left (18 \, A a^{8} b - 47 \, B a^{7} b^{2} + 128 \, A a^{6} b^{3} - 34 \, B a^{5} b^{4} + 130 \, A a^{4} b^{5} - 11 \, B a^{3} b^{6} + 44 \, A a^{2} b^{7}\right )} \tan \left (d x + c\right )}{{\left (a^{10} b^{3} + 3 \, a^{8} b^{5} + 3 \, a^{6} b^{7} + a^{4} b^{9}\right )} \tan \left (d x + c\right )^{4} + 3 \, {\left (a^{11} b^{2} + 3 \, a^{9} b^{4} + 3 \, a^{7} b^{6} + a^{5} b^{8}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{12} b + 3 \, a^{10} b^{3} + 3 \, a^{8} b^{5} + a^{6} b^{7}\right )} \tan \left (d x + c\right )^{2} + {\left (a^{13} + 3 \, a^{11} b^{2} + 3 \, a^{9} b^{4} + a^{7} b^{6}\right )} \tan \left (d x + c\right )} - \frac {6 \, {\left (B a - 4 \, A b\right )} \log \left (\tan \left (d x + c\right )\right )}{a^{5}}}{6 \, d} \]

input

integrate(cot(d*x+c)^2*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^4,x, algorithm="m 
axima")
 

output

-1/6*(6*(A*a^4 + 4*B*a^3*b - 6*A*a^2*b^2 - 4*B*a*b^3 + A*b^4)*(d*x + c)/(a 
^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) + 6*(10*B*a^7*b^2 - 20*A*a^6 
*b^3 + 5*B*a^5*b^4 - 24*A*a^4*b^5 + 4*B*a^3*b^6 - 16*A*a^2*b^7 + B*a*b^8 - 
 4*A*b^9)*log(b*tan(d*x + c) + a)/(a^13 + 4*a^11*b^2 + 6*a^9*b^4 + 4*a^7*b 
^6 + a^5*b^8) + 3*(B*a^4 - 4*A*a^3*b - 6*B*a^2*b^2 + 4*A*a*b^3 + B*b^4)*lo 
g(tan(d*x + c)^2 + 1)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) + (6 
*A*a^9 + 18*A*a^7*b^2 + 18*A*a^5*b^4 + 6*A*a^3*b^6 + 6*(A*a^6*b^3 - 6*B*a^ 
5*b^4 + 13*A*a^4*b^5 - 3*B*a^3*b^6 + 12*A*a^2*b^7 - B*a*b^8 + 4*A*b^9)*tan 
(d*x + c)^3 + 3*(6*A*a^7*b^2 - 27*B*a^6*b^3 + 62*A*a^5*b^4 - 16*B*a^4*b^5 
+ 60*A*a^3*b^6 - 5*B*a^2*b^7 + 20*A*a*b^8)*tan(d*x + c)^2 + (18*A*a^8*b - 
47*B*a^7*b^2 + 128*A*a^6*b^3 - 34*B*a^5*b^4 + 130*A*a^4*b^5 - 11*B*a^3*b^6 
 + 44*A*a^2*b^7)*tan(d*x + c))/((a^10*b^3 + 3*a^8*b^5 + 3*a^6*b^7 + a^4*b^ 
9)*tan(d*x + c)^4 + 3*(a^11*b^2 + 3*a^9*b^4 + 3*a^7*b^6 + a^5*b^8)*tan(d*x 
 + c)^3 + 3*(a^12*b + 3*a^10*b^3 + 3*a^8*b^5 + a^6*b^7)*tan(d*x + c)^2 + ( 
a^13 + 3*a^11*b^2 + 3*a^9*b^4 + a^7*b^6)*tan(d*x + c)) - 6*(B*a - 4*A*b)*l 
og(tan(d*x + c))/a^5)/d
 

3.3.96.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 846 vs. \(2 (393) = 786\).

Time = 1.31 (sec) , antiderivative size = 846, normalized size of antiderivative = 2.12 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=-\frac {\frac {6 \, {\left (A a^{4} + 4 \, B a^{3} b - 6 \, A a^{2} b^{2} - 4 \, B a b^{3} + A b^{4}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {3 \, {\left (B a^{4} - 4 \, A a^{3} b - 6 \, B a^{2} b^{2} + 4 \, A a b^{3} + B b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {6 \, {\left (10 \, B a^{7} b^{3} - 20 \, A a^{6} b^{4} + 5 \, B a^{5} b^{5} - 24 \, A a^{4} b^{6} + 4 \, B a^{3} b^{7} - 16 \, A a^{2} b^{8} + B a b^{9} - 4 \, A b^{10}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{13} b + 4 \, a^{11} b^{3} + 6 \, a^{9} b^{5} + 4 \, a^{7} b^{7} + a^{5} b^{9}} - \frac {110 \, B a^{7} b^{5} \tan \left (d x + c\right )^{3} - 220 \, A a^{6} b^{6} \tan \left (d x + c\right )^{3} + 55 \, B a^{5} b^{7} \tan \left (d x + c\right )^{3} - 264 \, A a^{4} b^{8} \tan \left (d x + c\right )^{3} + 44 \, B a^{3} b^{9} \tan \left (d x + c\right )^{3} - 176 \, A a^{2} b^{10} \tan \left (d x + c\right )^{3} + 11 \, B a b^{11} \tan \left (d x + c\right )^{3} - 44 \, A b^{12} \tan \left (d x + c\right )^{3} + 366 \, B a^{8} b^{4} \tan \left (d x + c\right )^{2} - 720 \, A a^{7} b^{5} \tan \left (d x + c\right )^{2} + 219 \, B a^{6} b^{6} \tan \left (d x + c\right )^{2} - 906 \, A a^{5} b^{7} \tan \left (d x + c\right )^{2} + 156 \, B a^{4} b^{8} \tan \left (d x + c\right )^{2} - 600 \, A a^{3} b^{9} \tan \left (d x + c\right )^{2} + 39 \, B a^{2} b^{10} \tan \left (d x + c\right )^{2} - 150 \, A a b^{11} \tan \left (d x + c\right )^{2} + 411 \, B a^{9} b^{3} \tan \left (d x + c\right ) - 792 \, A a^{8} b^{4} \tan \left (d x + c\right ) + 294 \, B a^{7} b^{5} \tan \left (d x + c\right ) - 1050 \, A a^{6} b^{6} \tan \left (d x + c\right ) + 195 \, B a^{5} b^{7} \tan \left (d x + c\right ) - 696 \, A a^{4} b^{8} \tan \left (d x + c\right ) + 48 \, B a^{3} b^{9} \tan \left (d x + c\right ) - 174 \, A a^{2} b^{10} \tan \left (d x + c\right ) + 157 \, B a^{10} b^{2} - 294 \, A a^{9} b^{3} + 136 \, B a^{8} b^{4} - 414 \, A a^{7} b^{5} + 89 \, B a^{6} b^{6} - 278 \, A a^{5} b^{7} + 22 \, B a^{4} b^{8} - 70 \, A a^{3} b^{9}}{{\left (a^{13} + 4 \, a^{11} b^{2} + 6 \, a^{9} b^{4} + 4 \, a^{7} b^{6} + a^{5} b^{8}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{3}} - \frac {6 \, {\left (B a - 4 \, A b\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{5}} + \frac {6 \, {\left (B a \tan \left (d x + c\right ) - 4 \, A b \tan \left (d x + c\right ) + A a\right )}}{a^{5} \tan \left (d x + c\right )}}{6 \, d} \]

input

integrate(cot(d*x+c)^2*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^4,x, algorithm="g 
iac")
 

output

-1/6*(6*(A*a^4 + 4*B*a^3*b - 6*A*a^2*b^2 - 4*B*a*b^3 + A*b^4)*(d*x + c)/(a 
^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) + 3*(B*a^4 - 4*A*a^3*b - 6*B 
*a^2*b^2 + 4*A*a*b^3 + B*b^4)*log(tan(d*x + c)^2 + 1)/(a^8 + 4*a^6*b^2 + 6 
*a^4*b^4 + 4*a^2*b^6 + b^8) + 6*(10*B*a^7*b^3 - 20*A*a^6*b^4 + 5*B*a^5*b^5 
 - 24*A*a^4*b^6 + 4*B*a^3*b^7 - 16*A*a^2*b^8 + B*a*b^9 - 4*A*b^10)*log(abs 
(b*tan(d*x + c) + a))/(a^13*b + 4*a^11*b^3 + 6*a^9*b^5 + 4*a^7*b^7 + a^5*b 
^9) - (110*B*a^7*b^5*tan(d*x + c)^3 - 220*A*a^6*b^6*tan(d*x + c)^3 + 55*B* 
a^5*b^7*tan(d*x + c)^3 - 264*A*a^4*b^8*tan(d*x + c)^3 + 44*B*a^3*b^9*tan(d 
*x + c)^3 - 176*A*a^2*b^10*tan(d*x + c)^3 + 11*B*a*b^11*tan(d*x + c)^3 - 4 
4*A*b^12*tan(d*x + c)^3 + 366*B*a^8*b^4*tan(d*x + c)^2 - 720*A*a^7*b^5*tan 
(d*x + c)^2 + 219*B*a^6*b^6*tan(d*x + c)^2 - 906*A*a^5*b^7*tan(d*x + c)^2 
+ 156*B*a^4*b^8*tan(d*x + c)^2 - 600*A*a^3*b^9*tan(d*x + c)^2 + 39*B*a^2*b 
^10*tan(d*x + c)^2 - 150*A*a*b^11*tan(d*x + c)^2 + 411*B*a^9*b^3*tan(d*x + 
 c) - 792*A*a^8*b^4*tan(d*x + c) + 294*B*a^7*b^5*tan(d*x + c) - 1050*A*a^6 
*b^6*tan(d*x + c) + 195*B*a^5*b^7*tan(d*x + c) - 696*A*a^4*b^8*tan(d*x + c 
) + 48*B*a^3*b^9*tan(d*x + c) - 174*A*a^2*b^10*tan(d*x + c) + 157*B*a^10*b 
^2 - 294*A*a^9*b^3 + 136*B*a^8*b^4 - 414*A*a^7*b^5 + 89*B*a^6*b^6 - 278*A* 
a^5*b^7 + 22*B*a^4*b^8 - 70*A*a^3*b^9)/((a^13 + 4*a^11*b^2 + 6*a^9*b^4 + 4 
*a^7*b^6 + a^5*b^8)*(b*tan(d*x + c) + a)^3) - 6*(B*a - 4*A*b)*log(abs(tan( 
d*x + c)))/a^5 + 6*(B*a*tan(d*x + c) - 4*A*b*tan(d*x + c) + A*a)/(a^5*t...
 

3.3.96.9 Mupad [B] (verification not implemented)

Time = 12.85 (sec) , antiderivative size = 576, normalized size of antiderivative = 1.44 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\frac {b^2\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (-10\,B\,a^7+20\,A\,a^6\,b-5\,B\,a^5\,b^2+24\,A\,a^4\,b^3-4\,B\,a^3\,b^4+16\,A\,a^2\,b^5-B\,a\,b^6+4\,A\,b^7\right )}{a^5\,d\,{\left (a^2+b^2\right )}^4}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (4\,A\,b-B\,a\right )}{a^5\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (A+B\,1{}\mathrm {i}\right )}{2\,d\,\left (a^4\,1{}\mathrm {i}-4\,a^3\,b-a^2\,b^2\,6{}\mathrm {i}+4\,a\,b^3+b^4\,1{}\mathrm {i}\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B+A\,1{}\mathrm {i}\right )}{2\,d\,\left (a^4-a^3\,b\,4{}\mathrm {i}-6\,a^2\,b^2+a\,b^3\,4{}\mathrm {i}+b^4\right )}-\frac {\frac {A}{a}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (A\,a^6\,b^3-6\,B\,a^5\,b^4+13\,A\,a^4\,b^5-3\,B\,a^3\,b^6+12\,A\,a^2\,b^7-B\,a\,b^8+4\,A\,b^9\right )}{a^4\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (6\,A\,a^6\,b^2-27\,B\,a^5\,b^3+62\,A\,a^4\,b^4-16\,B\,a^3\,b^5+60\,A\,a^2\,b^6-5\,B\,a\,b^7+20\,A\,b^8\right )}{2\,a^3\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (18\,A\,a^6\,b-47\,B\,a^5\,b^2+128\,A\,a^4\,b^3-34\,B\,a^3\,b^4+130\,A\,a^2\,b^5-11\,B\,a\,b^6+44\,A\,b^7\right )}{6\,a^2\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}}{d\,\left (a^3\,\mathrm {tan}\left (c+d\,x\right )+3\,a^2\,b\,{\mathrm {tan}\left (c+d\,x\right )}^2+3\,a\,b^2\,{\mathrm {tan}\left (c+d\,x\right )}^3+b^3\,{\mathrm {tan}\left (c+d\,x\right )}^4\right )} \]

input

int((cot(c + d*x)^2*(A + B*tan(c + d*x)))/(a + b*tan(c + d*x))^4,x)
 

output

(b^2*log(a + b*tan(c + d*x))*(4*A*b^7 - 10*B*a^7 + 16*A*a^2*b^5 + 24*A*a^4 
*b^3 - 4*B*a^3*b^4 - 5*B*a^5*b^2 + 20*A*a^6*b - B*a*b^6))/(a^5*d*(a^2 + b^ 
2)^4) - (log(tan(c + d*x))*(4*A*b - B*a))/(a^5*d) - (log(tan(c + d*x) - 1i 
)*(A + B*1i))/(2*d*(4*a*b^3 - 4*a^3*b + a^4*1i + b^4*1i - a^2*b^2*6i)) - ( 
log(tan(c + d*x) + 1i)*(A*1i + B))/(2*d*(a*b^3*4i - a^3*b*4i + a^4 + b^4 - 
 6*a^2*b^2)) - (A/a + (tan(c + d*x)^3*(4*A*b^9 + 12*A*a^2*b^7 + 13*A*a^4*b 
^5 + A*a^6*b^3 - 3*B*a^3*b^6 - 6*B*a^5*b^4 - B*a*b^8))/(a^4*(a^6 + b^6 + 3 
*a^2*b^4 + 3*a^4*b^2)) + (tan(c + d*x)^2*(20*A*b^8 + 60*A*a^2*b^6 + 62*A*a 
^4*b^4 + 6*A*a^6*b^2 - 16*B*a^3*b^5 - 27*B*a^5*b^3 - 5*B*a*b^7))/(2*a^3*(a 
^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (tan(c + d*x)*(44*A*b^7 + 130*A*a^2*b 
^5 + 128*A*a^4*b^3 - 34*B*a^3*b^4 - 47*B*a^5*b^2 + 18*A*a^6*b - 11*B*a*b^6 
))/(6*a^2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)))/(d*(a^3*tan(c + d*x) + b^3 
*tan(c + d*x)^4 + 3*a^2*b*tan(c + d*x)^2 + 3*a*b^2*tan(c + d*x)^3))