\(\int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx\) [363]

Optimal result

Integrand size = 33, antiderivative size = 289 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx=\frac {(5 A b-2 a B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{a^{7/2} d}+\frac {(i A+B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{5/2} d}-\frac {(i A-B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{5/2} d}-\frac {b \left (3 a^2 A+5 A b^2-2 a b B\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}-\frac {b \left (a^4 A+10 a^2 A b^2+5 A b^4-6 a^3 b B-2 a b^3 B\right )}{a^3 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}} \] Output:

(5*A*b-2*B*a)*arctanh((a+b*tan(d*x+c))^(1/2)/a^(1/2))/a^(7/2)/d+(I*A+B)*ar 
ctanh((a+b*tan(d*x+c))^(1/2)/(a-I*b)^(1/2))/(a-I*b)^(5/2)/d-(I*A-B)*arctan 
h((a+b*tan(d*x+c))^(1/2)/(a+I*b)^(1/2))/(a+I*b)^(5/2)/d-1/3*b*(3*A*a^2+5*A 
*b^2-2*B*a*b)/a^2/(a^2+b^2)/d/(a+b*tan(d*x+c))^(3/2)-A*cot(d*x+c)/a/d/(a+b 
*tan(d*x+c))^(3/2)-b*(A*a^4+10*A*a^2*b^2+5*A*b^4-6*B*a^3*b-2*B*a*b^3)/a^3/ 
(a^2+b^2)^2/d/(a+b*tan(d*x+c))^(1/2)
 

Mathematica [A] (verified)

Time = 3.49 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.06 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx=\frac {\frac {b \left (-3 a^2 A-5 A b^2+2 a b B\right )}{\left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac {3 a A \cot (c+d x)}{(a+b \tan (c+d x))^{3/2}}+\frac {3 \left (\frac {\left (a^2+b^2\right )^2 (5 A b-2 a B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a}}+\frac {a^3 (a+i b)^2 (i A+B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{\sqrt {a-i b}}+\frac {a^3 (a-i b)^2 (-i A+B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{\sqrt {a+i b}}-\frac {b \left (a^4 A+10 a^2 A b^2+5 A b^4-6 a^3 b B-2 a b^3 B\right )}{\sqrt {a+b \tan (c+d x)}}\right )}{a \left (a^2+b^2\right )^2}}{3 a^2 d} \] Input:

Integrate[(Cot[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(5/2) 
,x]
 

Output:

((b*(-3*a^2*A - 5*A*b^2 + 2*a*b*B))/((a^2 + b^2)*(a + b*Tan[c + d*x])^(3/2 
)) - (3*a*A*Cot[c + d*x])/(a + b*Tan[c + d*x])^(3/2) + (3*(((a^2 + b^2)^2* 
(5*A*b - 2*a*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/Sqrt[a] + (a^3* 
(a + I*b)^2*(I*A + B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/Sqr 
t[a - I*b] + (a^3*(a - I*b)^2*((-I)*A + B)*ArcTanh[Sqrt[a + b*Tan[c + d*x] 
]/Sqrt[a + I*b]])/Sqrt[a + I*b] - (b*(a^4*A + 10*a^2*A*b^2 + 5*A*b^4 - 6*a 
^3*b*B - 2*a*b^3*B))/Sqrt[a + b*Tan[c + d*x]]))/(a*(a^2 + b^2)^2))/(3*a^2* 
d)
 

Rubi [A] (warning: unable to verify)

Time = 2.49 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.19, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 4092, 27, 3042, 4132, 27, 3042, 4132, 27, 3042, 4136, 27, 3042, 4022, 3042, 4020, 25, 73, 221, 4117, 73, 221}

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.

\(\displaystyle \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {A+B \tan (c+d x)}{\tan (c+d x)^2 (a+b \tan (c+d x))^{5/2}}dx\)

\(\Big \downarrow \) 4092

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4132

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4132

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4136

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4022

\(\displaystyle -\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}-\frac {\frac {2 b \left (3 a^2 A-2 a b B+5 A b^2\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac {\frac {2 b \left (a^4 A-6 a^3 b B+10 a^2 A b^2-2 a b^3 B+5 A b^4\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\left (a^2+b^2\right )^2 (5 A b-2 a B) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx+2 \left (\frac {1}{2} a^3 (a-i b)^2 (A+i B) \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx+\frac {1}{2} a^3 (a+i b)^2 (A-i B) \int \frac {i \tan (c+d x)+1}{\sqrt {a+b \tan (c+d x)}}dx\right )}{a \left (a^2+b^2\right )}}{a \left (a^2+b^2\right )}}{2 a}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}-\frac {\frac {2 b \left (3 a^2 A-2 a b B+5 A b^2\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac {\frac {2 b \left (a^4 A-6 a^3 b B+10 a^2 A b^2-2 a b^3 B+5 A b^4\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\left (a^2+b^2\right )^2 (5 A b-2 a B) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx+2 \left (\frac {1}{2} a^3 (a-i b)^2 (A+i B) \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx+\frac {1}{2} a^3 (a+i b)^2 (A-i B) \int \frac {i \tan (c+d x)+1}{\sqrt {a+b \tan (c+d x)}}dx\right )}{a \left (a^2+b^2\right )}}{a \left (a^2+b^2\right )}}{2 a}\)

\(\Big \downarrow \) 4020

\(\displaystyle -\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}-\frac {\frac {2 b \left (3 a^2 A-2 a b B+5 A b^2\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac {\frac {2 b \left (a^4 A-6 a^3 b B+10 a^2 A b^2-2 a b^3 B+5 A b^4\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\left (a^2+b^2\right )^2 (5 A b-2 a B) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx+2 \left (\frac {i a^3 (a+i b)^2 (A-i B) \int -\frac {1}{(1-i \tan (c+d x)) \sqrt {a+b \tan (c+d x)}}d(i \tan (c+d x))}{2 d}-\frac {i a^3 (a-i b)^2 (A+i B) \int -\frac {1}{(i \tan (c+d x)+1) \sqrt {a+b \tan (c+d x)}}d(-i \tan (c+d x))}{2 d}\right )}{a \left (a^2+b^2\right )}}{a \left (a^2+b^2\right )}}{2 a}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}-\frac {\frac {2 b \left (3 a^2 A-2 a b B+5 A b^2\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac {\frac {2 b \left (a^4 A-6 a^3 b B+10 a^2 A b^2-2 a b^3 B+5 A b^4\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\left (a^2+b^2\right )^2 (5 A b-2 a B) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx+2 \left (\frac {i a^3 (a-i b)^2 (A+i B) \int \frac {1}{(i \tan (c+d x)+1) \sqrt {a+b \tan (c+d x)}}d(-i \tan (c+d x))}{2 d}-\frac {i a^3 (a+i b)^2 (A-i B) \int \frac {1}{(1-i \tan (c+d x)) \sqrt {a+b \tan (c+d x)}}d(i \tan (c+d x))}{2 d}\right )}{a \left (a^2+b^2\right )}}{a \left (a^2+b^2\right )}}{2 a}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}-\frac {\frac {2 b \left (3 a^2 A-2 a b B+5 A b^2\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac {\frac {2 b \left (a^4 A-6 a^3 b B+10 a^2 A b^2-2 a b^3 B+5 A b^4\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\left (a^2+b^2\right )^2 (5 A b-2 a B) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx+2 \left (\frac {a^3 (a-i b)^2 (A+i B) \int \frac {1}{-\frac {i \tan ^2(c+d x)}{b}-\frac {i a}{b}+1}d\sqrt {a+b \tan (c+d x)}}{b d}+\frac {a^3 (a+i b)^2 (A-i B) \int \frac {1}{\frac {i \tan ^2(c+d x)}{b}+\frac {i a}{b}+1}d\sqrt {a+b \tan (c+d x)}}{b d}\right )}{a \left (a^2+b^2\right )}}{a \left (a^2+b^2\right )}}{2 a}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}-\frac {\frac {2 b \left (3 a^2 A-2 a b B+5 A b^2\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac {\frac {2 b \left (a^4 A-6 a^3 b B+10 a^2 A b^2-2 a b^3 B+5 A b^4\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\left (a^2+b^2\right )^2 (5 A b-2 a B) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx+2 \left (\frac {a^3 (a+i b)^2 (A-i B) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}}+\frac {a^3 (a-i b)^2 (A+i B) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d \sqrt {a+i b}}\right )}{a \left (a^2+b^2\right )}}{a \left (a^2+b^2\right )}}{2 a}\)

\(\Big \downarrow \) 4117

\(\displaystyle -\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}-\frac {\frac {2 b \left (3 a^2 A-2 a b B+5 A b^2\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac {\frac {2 b \left (a^4 A-6 a^3 b B+10 a^2 A b^2-2 a b^3 B+5 A b^4\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\frac {\left (a^2+b^2\right )^2 (5 A b-2 a B) \int \frac {\cot (c+d x)}{\sqrt {a+b \tan (c+d x)}}d\tan (c+d x)}{d}+2 \left (\frac {a^3 (a+i b)^2 (A-i B) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}}+\frac {a^3 (a-i b)^2 (A+i B) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d \sqrt {a+i b}}\right )}{a \left (a^2+b^2\right )}}{a \left (a^2+b^2\right )}}{2 a}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}-\frac {\frac {2 b \left (3 a^2 A-2 a b B+5 A b^2\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac {\frac {2 b \left (a^4 A-6 a^3 b B+10 a^2 A b^2-2 a b^3 B+5 A b^4\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\frac {2 \left (a^2+b^2\right )^2 (5 A b-2 a B) \int \frac {1}{\frac {a+b \tan (c+d x)}{b}-\frac {a}{b}}d\sqrt {a+b \tan (c+d x)}}{b d}+2 \left (\frac {a^3 (a+i b)^2 (A-i B) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}}+\frac {a^3 (a-i b)^2 (A+i B) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d \sqrt {a+i b}}\right )}{a \left (a^2+b^2\right )}}{a \left (a^2+b^2\right )}}{2 a}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}-\frac {\frac {2 b \left (3 a^2 A-2 a b B+5 A b^2\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac {\frac {2 b \left (a^4 A-6 a^3 b B+10 a^2 A b^2-2 a b^3 B+5 A b^4\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {-\frac {2 \left (a^2+b^2\right )^2 (5 A b-2 a B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}+2 \left (\frac {a^3 (a+i b)^2 (A-i B) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}}+\frac {a^3 (a-i b)^2 (A+i B) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d \sqrt {a+i b}}\right )}{a \left (a^2+b^2\right )}}{a \left (a^2+b^2\right )}}{2 a}\)

Input:

Int[(Cot[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(5/2),x]
 

Output:

-((A*Cot[c + d*x])/(a*d*(a + b*Tan[c + d*x])^(3/2))) - ((2*b*(3*a^2*A + 5* 
A*b^2 - 2*a*b*B))/(3*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) + ((2*((a 
^3*(a + I*b)^2*(A - I*B)*ArcTan[Tan[c + d*x]/Sqrt[a - I*b]])/(Sqrt[a - I*b 
]*d) + (a^3*(a - I*b)^2*(A + I*B)*ArcTan[Tan[c + d*x]/Sqrt[a + I*b]])/(Sqr 
t[a + I*b]*d)) - (2*(a^2 + b^2)^2*(5*A*b - 2*a*B)*ArcTanh[Sqrt[a + b*Tan[c 
 + d*x]]/Sqrt[a]])/(Sqrt[a]*d))/(a*(a^2 + b^2)) + (2*b*(a^4*A + 10*a^2*A*b 
^2 + 5*A*b^4 - 6*a^3*b*B - 2*a*b^3*B))/(a*(a^2 + b^2)*d*Sqrt[a + b*Tan[c + 
 d*x]]))/(a*(a^2 + b^2)))/(2*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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 3042

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

rule 4020

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

rule 4022

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

rule 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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(12967\) vs. \(2(257)=514\).

Time = 0.18 (sec) , antiderivative size = 12968, normalized size of antiderivative = 44.87

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(12968\)
default	\(\text {Expression too large to display}\)	\(12968\)

Input:

int(cot(d*x+c)^2*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^(5/2),x,method=_RETURNV 
ERBOSE)
 

Output:

result too large to display
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 7530 vs. \(2 (252) = 504\).

Time = 54.90 (sec) , antiderivative size = 15079, normalized size of antiderivative = 52.18 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(cot(d*x+c)^2*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^(5/2),x, algorith 
m="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx=\int \frac {\left (A + B \tan {\left (c + d x \right )}\right ) \cot ^{2}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(cot(d*x+c)**2*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))**(5/2),x)
 

Output:

Integral((A + B*tan(c + d*x))*cot(c + d*x)**2/(a + b*tan(c + d*x))**(5/2), 
 x)
 

Maxima [F(-1)]

Timed out. \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(cot(d*x+c)^2*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^(5/2),x, algorith 
m="maxima")
 

Output:

Timed out
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(cot(d*x+c)^2*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^(5/2),x, algorith 
m="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{%%%{1,[9,22,7]%%%}+%%%{10,[9,20,7]%%%}+%%%{45,[9,18,7]%%%} 
+%%%{120,
 

Mupad [B] (verification not implemented)

Time = 9.73 (sec) , antiderivative size = 67465, normalized size of antiderivative = 233.44 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx=\text {Too large to display} \] Input:

int((cot(c + d*x)^2*(A + B*tan(c + d*x)))/(a + b*tan(c + d*x))^(5/2),x)
 

Output:

((2*(a + b*tan(c + d*x))*(5*A*b^5 + 11*A*a^2*b^3 - 8*B*a^3*b^2 - 2*B*a*b^4 
))/(3*(a*b^2 + a^3)^2) + (2*(A*b^3 - B*a*b^2))/(3*a*(a^2 + b^2)) - ((a + b 
*tan(c + d*x))^2*(5*A*b^5 + 10*A*a^2*b^3 - 6*B*a^3*b^2 + A*a^4*b - 2*B*a*b 
^4))/(a^3*(a^2 + b^2)^2))/(d*(a + b*tan(c + d*x))^(5/2) - a*d*(a + b*tan(c 
 + d*x))^(3/2)) + atan(((((((8*A^2*a^5*d^2 - 8*B^2*a^5*d^2 - 80*A^2*a^3*b^ 
2*d^2 + 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 + 40*A^2*a*b^4*d^2 - 40*B^2*a* 
b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/4 - (A^4 + 2*A^2*B^2 + 
 B^4)*(16*a^10*d^4 + 16*b^10*d^4 + 80*a^2*b^8*d^4 + 160*a^4*b^6*d^4 + 160* 
a^6*b^4*d^4 + 80*a^8*b^2*d^4))^(1/2) - 4*A^2*a^5*d^2 + 4*B^2*a^5*d^2 + 40* 
A^2*a^3*b^2*d^2 - 40*B^2*a^3*b^2*d^2 - 8*A*B*b^5*d^2 - 20*A^2*a*b^4*d^2 + 
20*B^2*a*b^4*d^2 + 80*A*B*a^2*b^3*d^2 - 40*A*B*a^4*b*d^2)/(16*(a^10*d^4 + 
b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5*a^8*b^2*d^4 
)))^(1/2)*((((((8*A^2*a^5*d^2 - 8*B^2*a^5*d^2 - 80*A^2*a^3*b^2*d^2 + 80*B^ 
2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 + 40*A^2*a*b^4*d^2 - 40*B^2*a*b^4*d^2 - 160 
*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/4 - (A^4 + 2*A^2*B^2 + B^4)*(16*a^1 
0*d^4 + 16*b^10*d^4 + 80*a^2*b^8*d^4 + 160*a^4*b^6*d^4 + 160*a^6*b^4*d^4 + 
 80*a^8*b^2*d^4))^(1/2) - 4*A^2*a^5*d^2 + 4*B^2*a^5*d^2 + 40*A^2*a^3*b^2*d 
^2 - 40*B^2*a^3*b^2*d^2 - 8*A*B*b^5*d^2 - 20*A^2*a*b^4*d^2 + 20*B^2*a*b^4* 
d^2 + 80*A*B*a^2*b^3*d^2 - 40*A*B*a^4*b*d^2)/(16*(a^10*d^4 + b^10*d^4 + 5* 
a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5*a^8*b^2*d^4)))^(1/2)*...
 

Reduce [F]

\[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx=\int \frac {\sqrt {a +\tan \left (d x +c \right ) b}\, \cot \left (d x +c \right )^{2}}{\tan \left (d x +c \right )^{2} b^{2}+2 \tan \left (d x +c \right ) a b +a^{2}}d x \] Input:

int(cot(d*x+c)^2*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^(5/2),x)
 

Output:

int((sqrt(tan(c + d*x)*b + a)*cot(c + d*x)**2)/(tan(c + d*x)**2*b**2 + 2*t 
an(c + d*x)*a*b + a**2),x)