\(\int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx\) [289]

Optimal result

Integrand size = 31, antiderivative size = 352 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\frac {\left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right ) x}{\left (a^2+b^2\right )^3}-\frac {\left (a^2 A-6 A b^2+3 a b B\right ) \log (\sin (c+d x))}{a^5 d}-\frac {b^3 \left (15 a^4 A b+17 a^2 A b^3+6 A b^5-10 a^5 B-9 a^3 b^2 B-3 a b^4 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^5 \left (a^2+b^2\right )^3 d}+\frac {b \left (5 a^2 A b+6 A b^3-2 a^3 B-3 a b^2 B\right )}{2 a^3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {(2 A b-a B) \cot (c+d x)}{a^2 d (a+b \tan (c+d x))^2}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^2}+\frac {b \left (3 a^4 A b+11 a^2 A b^3+6 A b^5-a^5 B-6 a^3 b^2 B-3 a b^4 B\right )}{a^4 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \] Output:

(3*A*a^2*b-A*b^3-B*a^3+3*B*a*b^2)*x/(a^2+b^2)^3-(A*a^2-6*A*b^2+3*B*a*b)*ln 
(sin(d*x+c))/a^5/d-b^3*(15*A*a^4*b+17*A*a^2*b^3+6*A*b^5-10*B*a^5-9*B*a^3*b 
^2-3*B*a*b^4)*ln(a*cos(d*x+c)+b*sin(d*x+c))/a^5/(a^2+b^2)^3/d+1/2*b*(5*A*a 
^2*b+6*A*b^3-2*B*a^3-3*B*a*b^2)/a^3/(a^2+b^2)/d/(a+b*tan(d*x+c))^2+(2*A*b- 
B*a)*cot(d*x+c)/a^2/d/(a+b*tan(d*x+c))^2-1/2*A*cot(d*x+c)^2/a/d/(a+b*tan(d 
*x+c))^2+b*(3*A*a^4*b+11*A*a^2*b^3+6*A*b^5-B*a^5-6*B*a^3*b^2-3*B*a*b^4)/a^ 
4/(a^2+b^2)^2/d/(a+b*tan(d*x+c))
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.33 (sec) , antiderivative size = 320, normalized size of antiderivative = 0.91 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\frac {(3 A b-a B) \cot (c+d x)}{a^4 d}-\frac {A \cot ^2(c+d x)}{2 a^3 d}+\frac {(A+i B) \log (i-\tan (c+d x))}{2 (a+i b)^3 d}-\frac {\left (a^2 A-6 A b^2+3 a b B\right ) \log (\tan (c+d x))}{a^5 d}+\frac {(A-i B) \log (i+\tan (c+d x))}{2 (a-i b)^3 d}-\frac {b^3 \left (15 a^4 A b+17 a^2 A b^3+6 A b^5-10 a^5 B-9 a^3 b^2 B-3 a b^4 B\right ) \log (a+b \tan (c+d x))}{a^5 \left (a^2+b^2\right )^3 d}+\frac {b^3 (A b-a B)}{2 a^3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {b^3 \left (5 a^2 A b+3 A b^3-4 a^3 B-2 a b^2 B\right )}{a^4 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \] Input:

Integrate[(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^3,x]
 

Output:

((3*A*b - a*B)*Cot[c + d*x])/(a^4*d) - (A*Cot[c + d*x]^2)/(2*a^3*d) + ((A 
+ I*B)*Log[I - Tan[c + d*x]])/(2*(a + I*b)^3*d) - ((a^2*A - 6*A*b^2 + 3*a* 
b*B)*Log[Tan[c + d*x]])/(a^5*d) + ((A - I*B)*Log[I + Tan[c + d*x]])/(2*(a 
- I*b)^3*d) - (b^3*(15*a^4*A*b + 17*a^2*A*b^3 + 6*A*b^5 - 10*a^5*B - 9*a^3 
*b^2*B - 3*a*b^4*B)*Log[a + b*Tan[c + d*x]])/(a^5*(a^2 + b^2)^3*d) + (b^3* 
(A*b - a*B))/(2*a^3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + (b^3*(5*a^2*A* 
b + 3*A*b^3 - 4*a^3*B - 2*a*b^2*B))/(a^4*(a^2 + b^2)^2*d*(a + b*Tan[c + d* 
x]))
 

Rubi [A] (verified)

Time = 2.40 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.15, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.548, Rules used = {3042, 4092, 27, 3042, 4132, 25, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4092

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4132

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {\int \frac {A a^2+B \tan (c+d x) a^2+3 b B a-6 A b^2-3 b (2 A b-a B) \tan (c+d x)^2}{\tan (c+d x) (a+b \tan (c+d x))^3}dx}{a}-\frac {(2 A b-a B) \cot (c+d x)}{a d (a+b \tan (c+d x))^2}}{a}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^2}\)

\(\Big \downarrow \) 4132

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4132

\(\displaystyle -\frac {\frac {\frac {\frac {\int \frac {\cot (c+d x) \left (-\left (\left (-B a^2+2 A b a+b^2 B\right ) \tan (c+d x) a^4\right )-b \left (-B a^5+3 A b a^4-6 b^2 B a^3+11 A b^3 a^2-3 b^4 B a+6 A b^5\right ) \tan ^2(c+d x)+\left (a^2+b^2\right )^2 \left (A a^2+3 b B a-6 A b^2\right )\right )}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}-\frac {b \left (a^5 (-B)+3 a^4 A b-6 a^3 b^2 B+11 a^2 A b^3-3 a b^4 B+6 A b^5\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a \left (a^2+b^2\right )}-\frac {b \left (-2 a^3 B+5 a^2 A b-3 a b^2 B+6 A b^3\right )}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a}-\frac {(2 A b-a B) \cot (c+d x)}{a d (a+b \tan (c+d x))^2}}{a}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {\frac {\frac {\int \frac {-\left (\left (-B a^2+2 A b a+b^2 B\right ) \tan (c+d x) a^4\right )-b \left (-B a^5+3 A b a^4-6 b^2 B a^3+11 A b^3 a^2-3 b^4 B a+6 A b^5\right ) \tan (c+d x)^2+\left (a^2+b^2\right )^2 \left (A a^2+3 b B a-6 A b^2\right )}{\tan (c+d x) (a+b \tan (c+d x))}dx}{a \left (a^2+b^2\right )}-\frac {b \left (a^5 (-B)+3 a^4 A b-6 a^3 b^2 B+11 a^2 A b^3-3 a b^4 B+6 A b^5\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a \left (a^2+b^2\right )}-\frac {b \left (-2 a^3 B+5 a^2 A b-3 a b^2 B+6 A b^3\right )}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a}-\frac {(2 A b-a B) \cot (c+d x)}{a d (a+b \tan (c+d x))^2}}{a}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^2}\)

\(\Big \downarrow \) 4134

\(\displaystyle -\frac {\frac {\frac {\frac {\frac {\left (a^2+b^2\right )^2 \left (a^2 A+3 a b B-6 A b^2\right ) \int \cot (c+d x)dx}{a}+\frac {b^3 \left (-10 a^5 B+15 a^4 A b-9 a^3 b^2 B+17 a^2 A b^3-3 a b^4 B+6 A b^5\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}-\frac {a^4 x \left (a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right )}{a^2+b^2}}{a \left (a^2+b^2\right )}-\frac {b \left (a^5 (-B)+3 a^4 A b-6 a^3 b^2 B+11 a^2 A b^3-3 a b^4 B+6 A b^5\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a \left (a^2+b^2\right )}-\frac {b \left (-2 a^3 B+5 a^2 A b-3 a b^2 B+6 A b^3\right )}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a}-\frac {(2 A b-a B) \cot (c+d x)}{a d (a+b \tan (c+d x))^2}}{a}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {\frac {\frac {\frac {\left (a^2+b^2\right )^2 \left (a^2 A+3 a b B-6 A b^2\right ) \int -\tan \left (c+d x+\frac {\pi }{2}\right )dx}{a}+\frac {b^3 \left (-10 a^5 B+15 a^4 A b-9 a^3 b^2 B+17 a^2 A b^3-3 a b^4 B+6 A b^5\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}-\frac {a^4 x \left (a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right )}{a^2+b^2}}{a \left (a^2+b^2\right )}-\frac {b \left (a^5 (-B)+3 a^4 A b-6 a^3 b^2 B+11 a^2 A b^3-3 a b^4 B+6 A b^5\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a \left (a^2+b^2\right )}-\frac {b \left (-2 a^3 B+5 a^2 A b-3 a b^2 B+6 A b^3\right )}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a}-\frac {(2 A b-a B) \cot (c+d x)}{a d (a+b \tan (c+d x))^2}}{a}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^2}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {\frac {\frac {-\frac {\left (a^2+b^2\right )^2 \left (a^2 A+3 a b B-6 A b^2\right ) \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx}{a}+\frac {b^3 \left (-10 a^5 B+15 a^4 A b-9 a^3 b^2 B+17 a^2 A b^3-3 a b^4 B+6 A b^5\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}-\frac {a^4 x \left (a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right )}{a^2+b^2}}{a \left (a^2+b^2\right )}-\frac {b \left (a^5 (-B)+3 a^4 A b-6 a^3 b^2 B+11 a^2 A b^3-3 a b^4 B+6 A b^5\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a \left (a^2+b^2\right )}-\frac {b \left (-2 a^3 B+5 a^2 A b-3 a b^2 B+6 A b^3\right )}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a}-\frac {(2 A b-a B) \cot (c+d x)}{a d (a+b \tan (c+d x))^2}}{a}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^2}\)

\(\Big \downarrow \) 3956

\(\displaystyle -\frac {\frac {\frac {\frac {\frac {b^3 \left (-10 a^5 B+15 a^4 A b-9 a^3 b^2 B+17 a^2 A b^3-3 a b^4 B+6 A b^5\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}+\frac {\left (a^2+b^2\right )^2 \left (a^2 A+3 a b B-6 A b^2\right ) \log (-\sin (c+d x))}{a d}-\frac {a^4 x \left (a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right )}{a^2+b^2}}{a \left (a^2+b^2\right )}-\frac {b \left (a^5 (-B)+3 a^4 A b-6 a^3 b^2 B+11 a^2 A b^3-3 a b^4 B+6 A b^5\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a \left (a^2+b^2\right )}-\frac {b \left (-2 a^3 B+5 a^2 A b-3 a b^2 B+6 A b^3\right )}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a}-\frac {(2 A b-a B) \cot (c+d x)}{a d (a+b \tan (c+d x))^2}}{a}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^2}\)

\(\Big \downarrow \) 4013

\(\displaystyle -\frac {\frac {\frac {\frac {\frac {\left (a^2+b^2\right )^2 \left (a^2 A+3 a b B-6 A b^2\right ) \log (-\sin (c+d x))}{a d}-\frac {a^4 x \left (a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right )}{a^2+b^2}+\frac {b^3 \left (-10 a^5 B+15 a^4 A b-9 a^3 b^2 B+17 a^2 A b^3-3 a b^4 B+6 A b^5\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a d \left (a^2+b^2\right )}}{a \left (a^2+b^2\right )}-\frac {b \left (a^5 (-B)+3 a^4 A b-6 a^3 b^2 B+11 a^2 A b^3-3 a b^4 B+6 A b^5\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a \left (a^2+b^2\right )}-\frac {b \left (-2 a^3 B+5 a^2 A b-3 a b^2 B+6 A b^3\right )}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a}-\frac {(2 A b-a B) \cot (c+d x)}{a d (a+b \tan (c+d x))^2}}{a}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^2}\)

Input:

Int[(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^3,x]
 

Output:

-1/2*(A*Cot[c + d*x]^2)/(a*d*(a + b*Tan[c + d*x])^2) - (-(((2*A*b - a*B)*C 
ot[c + d*x])/(a*d*(a + b*Tan[c + d*x])^2)) + (-1/2*(b*(5*a^2*A*b + 6*A*b^3 
 - 2*a^3*B - 3*a*b^2*B))/(a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + ((-((a 
^4*(3*a^2*A*b - A*b^3 - a^3*B + 3*a*b^2*B)*x)/(a^2 + b^2)) + ((a^2 + b^2)^ 
2*(a^2*A - 6*A*b^2 + 3*a*b*B)*Log[-Sin[c + d*x]])/(a*d) + (b^3*(15*a^4*A*b 
 + 17*a^2*A*b^3 + 6*A*b^5 - 10*a^5*B - 9*a^3*b^2*B - 3*a*b^4*B)*Log[a*Cos[ 
c + d*x] + b*Sin[c + d*x]])/(a*(a^2 + b^2)*d))/(a*(a^2 + b^2)) - (b*(3*a^4 
*A*b + 11*a^2*A*b^3 + 6*A*b^5 - a^5*B - 6*a^3*b^2*B - 3*a*b^4*B))/(a*(a^2 
+ b^2)*d*(a + b*Tan[c + d*x])))/(a*(a^2 + b^2)))/a)/a
 

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]
 

Maple [A] (verified)

Time = 1.16 (sec) , antiderivative size = 320, normalized size of antiderivative = 0.91

method	result	size
derivativedivides	\(\frac {\frac {\frac {\left (A \,a^{3}-3 A a \,b^{2}+3 B \,a^{2} b -B \,b^{3}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (3 A \,a^{2} b -A \,b^{3}-B \,a^{3}+3 B a \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}-\frac {A}{2 a^{3} \tan \left (d x +c \right )^{2}}-\frac {-3 A b +B a}{a^{4} \tan \left (d x +c \right )}+\frac {\left (-A \,a^{2}+6 A \,b^{2}-3 B a b \right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{5}}+\frac {b^{3} \left (5 A \,a^{2} b +3 A \,b^{3}-4 B \,a^{3}-2 B a \,b^{2}\right )}{\left (a^{2}+b^{2}\right )^{2} a^{4} \left (a +b \tan \left (d x +c \right )\right )}-\frac {b^{3} \left (15 A \,a^{4} b +17 A \,a^{2} b^{3}+6 A \,b^{5}-10 B \,a^{5}-9 B \,a^{3} b^{2}-3 B a \,b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3} a^{5}}+\frac {\left (A b -B a \right ) b^{3}}{2 \left (a^{2}+b^{2}\right ) a^{3} \left (a +b \tan \left (d x +c \right )\right )^{2}}}{d}\)	\(320\)
default	\(\frac {\frac {\frac {\left (A \,a^{3}-3 A a \,b^{2}+3 B \,a^{2} b -B \,b^{3}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (3 A \,a^{2} b -A \,b^{3}-B \,a^{3}+3 B a \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}-\frac {A}{2 a^{3} \tan \left (d x +c \right )^{2}}-\frac {-3 A b +B a}{a^{4} \tan \left (d x +c \right )}+\frac {\left (-A \,a^{2}+6 A \,b^{2}-3 B a b \right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{5}}+\frac {b^{3} \left (5 A \,a^{2} b +3 A \,b^{3}-4 B \,a^{3}-2 B a \,b^{2}\right )}{\left (a^{2}+b^{2}\right )^{2} a^{4} \left (a +b \tan \left (d x +c \right )\right )}-\frac {b^{3} \left (15 A \,a^{4} b +17 A \,a^{2} b^{3}+6 A \,b^{5}-10 B \,a^{5}-9 B \,a^{3} b^{2}-3 B a \,b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3} a^{5}}+\frac {\left (A b -B a \right ) b^{3}}{2 \left (a^{2}+b^{2}\right ) a^{3} \left (a +b \tan \left (d x +c \right )\right )^{2}}}{d}\)	\(320\)
parallelrisch	\(\frac {-30 b^{3} \left (A \,a^{4} b +\frac {17}{15} A \,a^{2} b^{3}+\frac {2}{5} A \,b^{5}-\frac {2}{3} B \,a^{5}-\frac {3}{5} B \,a^{3} b^{2}-\frac {1}{5} B a \,b^{4}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2} \ln \left (a +b \tan \left (d x +c \right )\right )+a^{5} \left (a +b \tan \left (d x +c \right )\right )^{2} \left (A \,a^{3}-3 A a \,b^{2}+3 B \,a^{2} b -B \,b^{3}\right ) \ln \left (\sec \left (d x +c \right )^{2}\right )-2 \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )^{2} \left (A \,a^{2}-6 A \,b^{2}+3 B a b \right ) \ln \left (\tan \left (d x +c \right )\right )+6 \left (-\frac {B x \,a^{8} d}{3}+b \left (A d x +\frac {2 B}{3}\right ) a^{7}-\frac {11 b^{2} \left (-\frac {6 B d x}{11}+A \right ) a^{6}}{6}-\frac {b^{3} \left (A d x -\frac {21 B}{2}\right ) a^{5}}{3}-\frac {22 A \,a^{4} b^{4}}{3}+\frac {13 B \,a^{3} b^{5}}{3}-\frac {17 A \,a^{2} b^{6}}{2}+\frac {3 B a \,b^{7}}{2}-3 A \,b^{8}\right ) b^{2} \tan \left (d x +c \right )^{2}+12 a b \left (-\frac {B x \,a^{8} d}{3}+b \left (A d x +\frac {B}{2}\right ) a^{7}-\frac {4 b^{2} \left (-\frac {3 B d x}{4}+A \right ) a^{6}}{3}-\frac {b^{3} \left (A d x -7 B \right ) a^{5}}{3}-5 A \,a^{4} b^{4}+\frac {17 B \,a^{3} b^{5}}{6}-\frac {17 A \,a^{2} b^{6}}{3}+B a \,b^{7}-2 A \,b^{8}\right ) \tan \left (d x +c \right )-\left (A a \left (a^{2}+b^{2}\right )^{3} \cot \left (d x +c \right )^{2}-4 \left (A b -\frac {B a}{2}\right ) \left (a^{2}+b^{2}\right )^{3} \cot \left (d x +c \right )-6 a^{4} x d \left (A \,a^{2} b -\frac {1}{3} A \,b^{3}-\frac {1}{3} B \,a^{3}+B a \,b^{2}\right )\right ) a^{3}}{2 \left (a^{2}+b^{2}\right )^{3} a^{5} d \left (a +b \tan \left (d x +c \right )\right )^{2}}\)	\(492\)
norman	\(\frac {\frac {\left (2 A b -B a \right ) \tan \left (d x +c \right )}{a^{2} d}+\frac {b^{2} \left (3 A \,a^{2} b -A \,b^{3}-B \,a^{3}+3 B a \,b^{2}\right ) x \tan \left (d x +c \right )^{4}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}+\frac {\left (3 A \,a^{2} b -A \,b^{3}-B \,a^{3}+3 B a \,b^{2}\right ) a^{2} x \tan \left (d x +c \right )^{2}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}-\frac {A}{2 a d}-\frac {b^{2} \left (11 A \,a^{4} b^{2}+33 A \,a^{2} b^{4}+18 A \,b^{6}-4 B \,a^{5} b -17 B \,a^{3} b^{3}-9 B a \,b^{5}\right ) \tan \left (d x +c \right )^{4}}{2 a^{5} d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {b \left (8 A \,a^{4} b^{2}+22 A \,a^{2} b^{4}+12 A \,b^{6}-3 B \,a^{5} b -11 B \,a^{3} b^{3}-6 B a \,b^{5}\right ) \tan \left (d x +c \right )^{3}}{d \,a^{4} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {2 b \left (3 A \,a^{2} b -A \,b^{3}-B \,a^{3}+3 B a \,b^{2}\right ) a x \tan \left (d x +c \right )^{3}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}}{\tan \left (d x +c \right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}-\frac {\left (A \,a^{2}-6 A \,b^{2}+3 B a b \right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{5} d}+\frac {\left (A \,a^{3}-3 A a \,b^{2}+3 B \,a^{2} b -B \,b^{3}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d \left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right )}-\frac {b^{3} \left (15 A \,a^{4} b +17 A \,a^{2} b^{3}+6 A \,b^{5}-10 B \,a^{5}-9 B \,a^{3} b^{2}-3 B a \,b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{5} d \left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right )}\)	\(610\)
risch	\(\text {Expression too large to display}\)	\(2061\)

Input:

int(cot(d*x+c)^3*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^3,x,method=_RETURNVERBO 
SE)
 

Output:

1/d*(1/(a^2+b^2)^3*(1/2*(A*a^3-3*A*a*b^2+3*B*a^2*b-B*b^3)*ln(1+tan(d*x+c)^ 
2)+(3*A*a^2*b-A*b^3-B*a^3+3*B*a*b^2)*arctan(tan(d*x+c)))-1/2/a^3*A/tan(d*x 
+c)^2-(-3*A*b+B*a)/a^4/tan(d*x+c)+(-A*a^2+6*A*b^2-3*B*a*b)/a^5*ln(tan(d*x+ 
c))+b^3*(5*A*a^2*b+3*A*b^3-4*B*a^3-2*B*a*b^2)/(a^2+b^2)^2/a^4/(a+b*tan(d*x 
+c))-b^3*(15*A*a^4*b+17*A*a^2*b^3+6*A*b^5-10*B*a^5-9*B*a^3*b^2-3*B*a*b^4)/ 
(a^2+b^2)^3/a^5*ln(a+b*tan(d*x+c))+1/2*(A*b-B*a)*b^3/(a^2+b^2)/a^3/(a+b*ta 
n(d*x+c))^2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1065 vs. \(2 (346) = 692\).

Time = 0.25 (sec) , antiderivative size = 1065, normalized size of antiderivative = 3.03 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx =\text {Too large to display} \] Input:

integrate(cot(d*x+c)^3*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^3,x, algorithm="f 
ricas")
 

Output:

-1/2*(A*a^10 + 3*A*a^8*b^2 + 3*A*a^6*b^4 + A*a^4*b^6 + (A*a^8*b^2 + 3*A*a^ 
6*b^4 - 9*B*a^5*b^5 + 14*A*a^4*b^6 - 3*B*a^3*b^7 + 6*A*a^2*b^8 + 2*(B*a^8* 
b^2 - 3*A*a^7*b^3 - 3*B*a^6*b^4 + A*a^5*b^5)*d*x)*tan(d*x + c)^4 + 2*(A*a^ 
9*b + B*a^8*b^2 - 2*B*a^6*b^4 + 6*B*a^4*b^6 - 11*A*a^3*b^7 + 3*B*a^2*b^8 - 
 6*A*a*b^9 + 2*(B*a^9*b - 3*A*a^8*b^2 - 3*B*a^7*b^3 + A*a^6*b^4)*d*x)*tan( 
d*x + c)^3 + (A*a^10 + 4*B*a^9*b - 8*A*a^8*b^2 + 12*B*a^7*b^3 - 30*A*a^6*b 
^4 + 23*B*a^5*b^5 - 45*A*a^4*b^6 + 9*B*a^3*b^7 - 18*A*a^2*b^8 + 2*(B*a^10 
- 3*A*a^9*b - 3*B*a^8*b^2 + A*a^7*b^3)*d*x)*tan(d*x + c)^2 + ((A*a^8*b^2 + 
 3*B*a^7*b^3 - 3*A*a^6*b^4 + 9*B*a^5*b^5 - 15*A*a^4*b^6 + 9*B*a^3*b^7 - 17 
*A*a^2*b^8 + 3*B*a*b^9 - 6*A*b^10)*tan(d*x + c)^4 + 2*(A*a^9*b + 3*B*a^8*b 
^2 - 3*A*a^7*b^3 + 9*B*a^6*b^4 - 15*A*a^5*b^5 + 9*B*a^4*b^6 - 17*A*a^3*b^7 
 + 3*B*a^2*b^8 - 6*A*a*b^9)*tan(d*x + c)^3 + (A*a^10 + 3*B*a^9*b - 3*A*a^8 
*b^2 + 9*B*a^7*b^3 - 15*A*a^6*b^4 + 9*B*a^5*b^5 - 17*A*a^4*b^6 + 3*B*a^3*b 
^7 - 6*A*a^2*b^8)*tan(d*x + c)^2)*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1)) 
 - ((10*B*a^5*b^5 - 15*A*a^4*b^6 + 9*B*a^3*b^7 - 17*A*a^2*b^8 + 3*B*a*b^9 
- 6*A*b^10)*tan(d*x + c)^4 + 2*(10*B*a^6*b^4 - 15*A*a^5*b^5 + 9*B*a^4*b^6 
- 17*A*a^3*b^7 + 3*B*a^2*b^8 - 6*A*a*b^9)*tan(d*x + c)^3 + (10*B*a^7*b^3 - 
 15*A*a^6*b^4 + 9*B*a^5*b^5 - 17*A*a^4*b^6 + 3*B*a^3*b^7 - 6*A*a^2*b^8)*ta 
n(d*x + c)^2)*log((b^2*tan(d*x + c)^2 + 2*a*b*tan(d*x + c) + a^2)/(tan(d*x 
 + c)^2 + 1)) + 2*(B*a^10 - 2*A*a^9*b + 3*B*a^8*b^2 - 6*A*a^7*b^3 + 3*B...
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\text {Exception raised: AttributeError} \] Input:

integrate(cot(d*x+c)**3*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))**3,x)
 

Output:

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

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 541, normalized size of antiderivative = 1.54 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=-\frac {\frac {2 \, {\left (B a^{3} - 3 \, A a^{2} b - 3 \, B a b^{2} + A b^{3}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {2 \, {\left (10 \, B a^{5} b^{3} - 15 \, A a^{4} b^{4} + 9 \, B a^{3} b^{5} - 17 \, A a^{2} b^{6} + 3 \, B a b^{7} - 6 \, A b^{8}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{11} + 3 \, a^{9} b^{2} + 3 \, a^{7} b^{4} + a^{5} b^{6}} - \frac {{\left (A a^{3} + 3 \, B a^{2} b - 3 \, A a b^{2} - B b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {A a^{7} + 2 \, A a^{5} b^{2} + A a^{3} b^{4} + 2 \, {\left (B a^{5} b^{2} - 3 \, A a^{4} b^{3} + 6 \, B a^{3} b^{4} - 11 \, A a^{2} b^{5} + 3 \, B a b^{6} - 6 \, A b^{7}\right )} \tan \left (d x + c\right )^{3} + {\left (4 \, B a^{6} b - 11 \, A a^{5} b^{2} + 17 \, B a^{4} b^{3} - 33 \, A a^{3} b^{4} + 9 \, B a^{2} b^{5} - 18 \, A a b^{6}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (B a^{7} - 2 \, A a^{6} b + 2 \, B a^{5} b^{2} - 4 \, A a^{4} b^{3} + B a^{3} b^{4} - 2 \, A a^{2} b^{5}\right )} \tan \left (d x + c\right )}{{\left (a^{8} b^{2} + 2 \, a^{6} b^{4} + a^{4} b^{6}\right )} \tan \left (d x + c\right )^{4} + 2 \, {\left (a^{9} b + 2 \, a^{7} b^{3} + a^{5} b^{5}\right )} \tan \left (d x + c\right )^{3} + {\left (a^{10} + 2 \, a^{8} b^{2} + a^{6} b^{4}\right )} \tan \left (d x + c\right )^{2}} + \frac {2 \, {\left (A a^{2} + 3 \, B a b - 6 \, A b^{2}\right )} \log \left (\tan \left (d x + c\right )\right )}{a^{5}}}{2 \, d} \] Input:

integrate(cot(d*x+c)^3*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^3,x, algorithm="m 
axima")
 

Output:

-1/2*(2*(B*a^3 - 3*A*a^2*b - 3*B*a*b^2 + A*b^3)*(d*x + c)/(a^6 + 3*a^4*b^2 
 + 3*a^2*b^4 + b^6) - 2*(10*B*a^5*b^3 - 15*A*a^4*b^4 + 9*B*a^3*b^5 - 17*A* 
a^2*b^6 + 3*B*a*b^7 - 6*A*b^8)*log(b*tan(d*x + c) + a)/(a^11 + 3*a^9*b^2 + 
 3*a^7*b^4 + a^5*b^6) - (A*a^3 + 3*B*a^2*b - 3*A*a*b^2 - B*b^3)*log(tan(d* 
x + c)^2 + 1)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + (A*a^7 + 2*A*a^5*b^2 + 
 A*a^3*b^4 + 2*(B*a^5*b^2 - 3*A*a^4*b^3 + 6*B*a^3*b^4 - 11*A*a^2*b^5 + 3*B 
*a*b^6 - 6*A*b^7)*tan(d*x + c)^3 + (4*B*a^6*b - 11*A*a^5*b^2 + 17*B*a^4*b^ 
3 - 33*A*a^3*b^4 + 9*B*a^2*b^5 - 18*A*a*b^6)*tan(d*x + c)^2 + 2*(B*a^7 - 2 
*A*a^6*b + 2*B*a^5*b^2 - 4*A*a^4*b^3 + B*a^3*b^4 - 2*A*a^2*b^5)*tan(d*x + 
c))/((a^8*b^2 + 2*a^6*b^4 + a^4*b^6)*tan(d*x + c)^4 + 2*(a^9*b + 2*a^7*b^3 
 + a^5*b^5)*tan(d*x + c)^3 + (a^10 + 2*a^8*b^2 + a^6*b^4)*tan(d*x + c)^2) 
+ 2*(A*a^2 + 3*B*a*b - 6*A*b^2)*log(tan(d*x + c))/a^5)/d
 

Giac [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 563, normalized size of antiderivative = 1.60 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=-\frac {{\left (B a^{3} - 3 \, A a^{2} b - 3 \, B a b^{2} + A b^{3}\right )} {\left (d x + c\right )}}{a^{6} d + 3 \, a^{4} b^{2} d + 3 \, a^{2} b^{4} d + b^{6} d} + \frac {{\left (A a^{3} + 3 \, B a^{2} b - 3 \, A a b^{2} - B b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, {\left (a^{6} d + 3 \, a^{4} b^{2} d + 3 \, a^{2} b^{4} d + b^{6} d\right )}} + \frac {{\left (10 \, B a^{5} b^{4} - 15 \, A a^{4} b^{5} + 9 \, B a^{3} b^{6} - 17 \, A a^{2} b^{7} + 3 \, B a b^{8} - 6 \, A b^{9}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{11} b d + 3 \, a^{9} b^{3} d + 3 \, a^{7} b^{5} d + a^{5} b^{7} d} - \frac {{\left (A a^{2} + 3 \, B a b - 6 \, A b^{2}\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{5} d} - \frac {A a^{9} + 3 \, A a^{7} b^{2} + 3 \, A a^{5} b^{4} + A a^{3} b^{6} + 2 \, {\left (B a^{7} b^{2} - 3 \, A a^{6} b^{3} + 7 \, B a^{5} b^{4} - 14 \, A a^{4} b^{5} + 9 \, B a^{3} b^{6} - 17 \, A a^{2} b^{7} + 3 \, B a b^{8} - 6 \, A b^{9}\right )} \tan \left (d x + c\right )^{3} + {\left (4 \, B a^{8} b - 11 \, A a^{7} b^{2} + 21 \, B a^{6} b^{3} - 44 \, A a^{5} b^{4} + 26 \, B a^{4} b^{5} - 51 \, A a^{3} b^{6} + 9 \, B a^{2} b^{7} - 18 \, A a b^{8}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (B a^{9} - 2 \, A a^{8} b + 3 \, B a^{7} b^{2} - 6 \, A a^{6} b^{3} + 3 \, B a^{5} b^{4} - 6 \, A a^{4} b^{5} + B a^{3} b^{6} - 2 \, A a^{2} b^{7}\right )} \tan \left (d x + c\right )}{2 \, {\left (a^{2} + b^{2}\right )}^{3} {\left (b \tan \left (d x + c\right ) + a\right )}^{2} a^{4} d \tan \left (d x + c\right )^{2}} \] Input:

integrate(cot(d*x+c)^3*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^3,x, algorithm="g 
iac")
 

Output:

-(B*a^3 - 3*A*a^2*b - 3*B*a*b^2 + A*b^3)*(d*x + c)/(a^6*d + 3*a^4*b^2*d + 
3*a^2*b^4*d + b^6*d) + 1/2*(A*a^3 + 3*B*a^2*b - 3*A*a*b^2 - B*b^3)*log(tan 
(d*x + c)^2 + 1)/(a^6*d + 3*a^4*b^2*d + 3*a^2*b^4*d + b^6*d) + (10*B*a^5*b 
^4 - 15*A*a^4*b^5 + 9*B*a^3*b^6 - 17*A*a^2*b^7 + 3*B*a*b^8 - 6*A*b^9)*log( 
abs(b*tan(d*x + c) + a))/(a^11*b*d + 3*a^9*b^3*d + 3*a^7*b^5*d + a^5*b^7*d 
) - (A*a^2 + 3*B*a*b - 6*A*b^2)*log(abs(tan(d*x + c)))/(a^5*d) - 1/2*(A*a^ 
9 + 3*A*a^7*b^2 + 3*A*a^5*b^4 + A*a^3*b^6 + 2*(B*a^7*b^2 - 3*A*a^6*b^3 + 7 
*B*a^5*b^4 - 14*A*a^4*b^5 + 9*B*a^3*b^6 - 17*A*a^2*b^7 + 3*B*a*b^8 - 6*A*b 
^9)*tan(d*x + c)^3 + (4*B*a^8*b - 11*A*a^7*b^2 + 21*B*a^6*b^3 - 44*A*a^5*b 
^4 + 26*B*a^4*b^5 - 51*A*a^3*b^6 + 9*B*a^2*b^7 - 18*A*a*b^8)*tan(d*x + c)^ 
2 + 2*(B*a^9 - 2*A*a^8*b + 3*B*a^7*b^2 - 6*A*a^6*b^3 + 3*B*a^5*b^4 - 6*A*a 
^4*b^5 + B*a^3*b^6 - 2*A*a^2*b^7)*tan(d*x + c))/((a^2 + b^2)^3*(b*tan(d*x 
+ c) + a)^2*a^4*d*tan(d*x + c)^2)
 

Mupad [B] (verification not implemented)

Time = 10.33 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.23 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (2\,A\,b-B\,a\right )}{a^2}-\frac {A}{2\,a}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (-B\,a^5\,b^2+3\,A\,a^4\,b^3-6\,B\,a^3\,b^4+11\,A\,a^2\,b^5-3\,B\,a\,b^6+6\,A\,b^7\right )}{a^4\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (-4\,B\,a^5\,b+11\,A\,a^4\,b^2-17\,B\,a^3\,b^3+33\,A\,a^2\,b^4-9\,B\,a\,b^5+18\,A\,b^6\right )}{2\,a^3\,\left (a^4+2\,a^2\,b^2+b^4\right )}}{d\,\left (a^2\,{\mathrm {tan}\left (c+d\,x\right )}^2+2\,a\,b\,{\mathrm {tan}\left (c+d\,x\right )}^3+b^2\,{\mathrm {tan}\left (c+d\,x\right )}^4\right )}+\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (\frac {A}{a^3}-\frac {A\,a+3\,B\,b}{{\left (a^2+b^2\right )}^2}-\frac {6\,A\,b^2}{a^5}+\frac {3\,B\,b}{a^4}+\frac {4\,b^2\,\left (A\,a+B\,b\right )}{{\left (a^2+b^2\right )}^3}\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-B+A\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^3\,1{}\mathrm {i}+3\,a^2\,b+a\,b^2\,3{}\mathrm {i}-b^3\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (A\,a^2+3\,B\,a\,b-6\,A\,b^2\right )}{a^5\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (A-B\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^3+a^2\,b\,3{}\mathrm {i}+3\,a\,b^2-b^3\,1{}\mathrm {i}\right )} \] Input:

int((cot(c + d*x)^3*(A + B*tan(c + d*x)))/(a + b*tan(c + d*x))^3,x)
 

Output:

((tan(c + d*x)*(2*A*b - B*a))/a^2 - A/(2*a) + (tan(c + d*x)^3*(6*A*b^7 + 1 
1*A*a^2*b^5 + 3*A*a^4*b^3 - 6*B*a^3*b^4 - B*a^5*b^2 - 3*B*a*b^6))/(a^4*(a^ 
4 + b^4 + 2*a^2*b^2)) + (tan(c + d*x)^2*(18*A*b^6 + 33*A*a^2*b^4 + 11*A*a^ 
4*b^2 - 17*B*a^3*b^3 - 9*B*a*b^5 - 4*B*a^5*b))/(2*a^3*(a^4 + b^4 + 2*a^2*b 
^2)))/(d*(a^2*tan(c + d*x)^2 + b^2*tan(c + d*x)^4 + 2*a*b*tan(c + d*x)^3)) 
 + (log(a + b*tan(c + d*x))*(A/a^3 - (A*a + 3*B*b)/(a^2 + b^2)^2 - (6*A*b^ 
2)/a^5 + (3*B*b)/a^4 + (4*b^2*(A*a + B*b))/(a^2 + b^2)^3))/d - (log(tan(c 
+ d*x) - 1i)*(A*1i - B))/(2*d*(a*b^2*3i + 3*a^2*b - a^3*1i - b^3)) - (log( 
tan(c + d*x))*(A*a^2 - 6*A*b^2 + 3*B*a*b))/(a^5*d) - (log(tan(c + d*x) + 1 
i)*(A - B*1i))/(2*d*(3*a*b^2 + a^2*b*3i - a^3 - b^3*1i))
 

Reduce [F]

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

int(cot(d*x+c)^3*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^3,x)
 

Output:

int(cot(d*x+c)^3*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^3,x)