\(\int \frac {1}{\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^2} \, dx\) [829]

Optimal result

Integrand size = 23, antiderivative size = 255 \[ \int \frac {1}{\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^2} \, dx=\frac {\left (a^2+2 a b-b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\left (a^2+2 a b-b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {\sqrt {b} \left (3 a^2-b^2\right ) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{\sqrt {a} \left (a^2+b^2\right )^2 d}-\frac {\left (a^2-2 a b-b^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\cot (c+d x)}}{1+\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {b \sqrt {\cot (c+d x)}}{\left (a^2+b^2\right ) d (b+a \cot (c+d x))} \] Output:

-1/2*(a^2+2*a*b-b^2)*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2))*2^(1/2)/(a^2+b^2) 
^2/d-1/2*(a^2+2*a*b-b^2)*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))*2^(1/2)/(a^2+b 
^2)^2/d+b^(1/2)*(3*a^2-b^2)*arctan(a^(1/2)*cot(d*x+c)^(1/2)/b^(1/2))/a^(1/ 
2)/(a^2+b^2)^2/d-1/2*(a^2-2*a*b-b^2)*arctanh(2^(1/2)*cot(d*x+c)^(1/2)/(1+c 
ot(d*x+c)))*2^(1/2)/(a^2+b^2)^2/d-b*cot(d*x+c)^(1/2)/(a^2+b^2)/d/(b+a*cot( 
d*x+c))
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 2.58 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.18 \[ \int \frac {1}{\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^2} \, dx=-\frac {\frac {240 b^{5/2} \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{\sqrt {a}}-240 b^2 \sqrt {\cot (c+d x)}+80 a b \cot ^{\frac {3}{2}}(c+d x)+80 a b \cot ^{\frac {3}{2}}(c+d x) \left (-1+\operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\cot ^2(c+d x)\right )\right )+\frac {24 a^2 \left (a^2+b^2\right ) \cot ^{\frac {5}{2}}(c+d x) \operatorname {Hypergeometric2F1}\left (2,\frac {5}{2},\frac {7}{2},-\frac {a \cot (c+d x)}{b}\right )}{b^2}-15 \left (a^2-b^2\right ) \left (2 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-2 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )+8 \sqrt {\cot (c+d x)}+\sqrt {2} \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-\sqrt {2} \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )}{60 \left (a^2+b^2\right )^2 d} \] Input:

Integrate[1/(Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^2),x]
 

Output:

-1/60*((240*b^(5/2)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/Sqrt[a] 
- 240*b^2*Sqrt[Cot[c + d*x]] + 80*a*b*Cot[c + d*x]^(3/2) + 80*a*b*Cot[c + 
d*x]^(3/2)*(-1 + Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2]) + (24*a^ 
2*(a^2 + b^2)*Cot[c + d*x]^(5/2)*Hypergeometric2F1[2, 5/2, 7/2, -((a*Cot[c 
 + d*x])/b)])/b^2 - 15*(a^2 - b^2)*(2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[ 
c + d*x]]] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + 8*Sqrt[Cot 
[c + d*x]] + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - 
Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/((a^2 + b^2)^ 
2*d)
 

Rubi [A] (warning: unable to verify)

Time = 1.31 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.09, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.957, Rules used = {3042, 4156, 3042, 4050, 27, 3042, 4136, 27, 3042, 4017, 25, 1482, 1476, 1082, 217, 1479, 25, 27, 1103, 4117, 73, 218}

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.

\(\displaystyle \int \frac {1}{\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^2}dx\)

\(\Big \downarrow \) 4156

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4050

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4136

\(\displaystyle -\frac {\frac {b \left (3 a^2-b^2\right ) \int \frac {\cot ^2(c+d x)+1}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}dx}{a^2+b^2}+\frac {\int -\frac {2 \left (a^2+2 b \cot (c+d x) a-b^2\right )}{\sqrt {\cot (c+d x)}}dx}{a^2+b^2}}{2 \left (a^2+b^2\right )}-\frac {b \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {b \left (3 a^2-b^2\right ) \int \frac {\cot ^2(c+d x)+1}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}dx}{a^2+b^2}-\frac {2 \int \frac {a^2+2 b \cot (c+d x) a-b^2}{\sqrt {\cot (c+d x)}}dx}{a^2+b^2}}{2 \left (a^2+b^2\right )}-\frac {b \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {b \left (3 a^2-b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {2 \int \frac {a^2-2 b \tan \left (c+d x+\frac {\pi }{2}\right ) a-b^2}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2+b^2}}{2 \left (a^2+b^2\right )}-\frac {b \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\)

\(\Big \downarrow \) 4017

\(\displaystyle -\frac {\frac {b \left (3 a^2-b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {4 \int -\frac {a^2+2 b \cot (c+d x) a-b^2}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}}{2 \left (a^2+b^2\right )}-\frac {b \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {b \left (3 a^2-b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}+\frac {4 \int \frac {a^2+2 b \cot (c+d x) a-b^2}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}}{2 \left (a^2+b^2\right )}-\frac {b \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\)

\(\Big \downarrow \) 1482

\(\displaystyle -\frac {\frac {b \left (3 a^2-b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {4 \left (-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} \left (a^2+2 a b-b^2\right ) \int \frac {\cot (c+d x)+1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d \left (a^2+b^2\right )}}{2 \left (a^2+b^2\right )}-\frac {b \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\)

\(\Big \downarrow \) 1476

\(\displaystyle -\frac {\frac {b \left (3 a^2-b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {4 \left (-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )\right )}{d \left (a^2+b^2\right )}}{2 \left (a^2+b^2\right )}-\frac {b \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\)

\(\Big \downarrow \) 1082

\(\displaystyle -\frac {\frac {b \left (3 a^2-b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {4 \left (-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{2 \left (a^2+b^2\right )}-\frac {b \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\)

\(\Big \downarrow \) 217

\(\displaystyle -\frac {\frac {b \left (3 a^2-b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {4 \left (-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{2 \left (a^2+b^2\right )}-\frac {b \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\)

\(\Big \downarrow \) 1479

\(\displaystyle -\frac {\frac {b \left (3 a^2-b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {4 \left (-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )-\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{2 \left (a^2+b^2\right )}-\frac {b \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {b \left (3 a^2-b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {4 \left (-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )-\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{2 \left (a^2+b^2\right )}-\frac {b \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {b \left (3 a^2-b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {4 \left (-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (c+d x)}+1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )-\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{2 \left (a^2+b^2\right )}-\frac {b \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\)

\(\Big \downarrow \) 1103

\(\displaystyle -\frac {\frac {b \left (3 a^2-b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {4 \left (-\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{2 \left (a^2+b^2\right )}-\frac {b \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\)

\(\Big \downarrow \) 4117

\(\displaystyle -\frac {\frac {b \left (3 a^2-b^2\right ) \int \frac {1}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}d(-\cot (c+d x))}{d \left (a^2+b^2\right )}-\frac {4 \left (-\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{2 \left (a^2+b^2\right )}-\frac {b \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {-\frac {2 b \left (3 a^2-b^2\right ) \int \frac {1}{a \cot ^2(c+d x)+b}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}-\frac {4 \left (-\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{2 \left (a^2+b^2\right )}-\frac {b \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\)

\(\Big \downarrow \) 218

\(\displaystyle -\frac {\frac {2 \sqrt {b} \left (3 a^2-b^2\right ) \arctan \left (\frac {\sqrt {a} \cot (c+d x)}{\sqrt {b}}\right )}{\sqrt {a} d \left (a^2+b^2\right )}-\frac {4 \left (-\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{2 \left (a^2+b^2\right )}-\frac {b \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\)

Input:

Int[1/(Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^2),x]
 

Output:

-((b*Sqrt[Cot[c + d*x]])/((a^2 + b^2)*d*(b + a*Cot[c + d*x]))) - ((2*Sqrt[ 
b]*(3*a^2 - b^2)*ArcTan[(Sqrt[a]*Cot[c + d*x])/Sqrt[b]])/(Sqrt[a]*(a^2 + b 
^2)*d) - (4*(-1/2*((a^2 + 2*a*b - b^2)*(-(ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + 
d*x]]]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2])) - ((a^2 
 - 2*a*b - b^2)*(-1/2*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]/S 
qrt[2] + Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]/(2*Sqrt[2])))/ 
2))/((a^2 + b^2)*d))/(2*(a^2 + b^2))
 

Defintions of rubi rules used

rule 25

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

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 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 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 218

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

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

rule 1482

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
a*c, 2]}, Simp[(d*q + a*e)/(2*a*c)   Int[(q + c*x^2)/(a + c*x^4), x], x] + 
Simp[(d*q - a*e)/(2*a*c)   Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ[{a 
, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(- 
a)*c]
 

rule 3042

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

rule 4017

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_ 
)]], x_Symbol] :> Simp[2/f   Subst[Int[(b*c + d*x^2)/(b^2 + x^4), x], x, Sq 
rt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2, 0] & 
& NeQ[c^2 + d^2, 0]
 

rule 4050

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

rule 4117

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

rule 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]
 

rule 4156

Int[(cot[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x 
_)]^(n_.))^(p_.), x_Symbol] :> Simp[d^(n*p)   Int[(d*Cot[e + f*x])^(m - n*p 
)*(b + a*Cot[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, d, e, f, m, n, p}, x] && 
  !IntegerQ[m] && IntegersQ[n, p]
 

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.09

method	result	size
derivativedivides	\(-\frac {\frac {\frac {\left (a^{2}-b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}+\frac {a b \sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{2}}{\left (a^{2}+b^{2}\right )^{2}}-\frac {2 b \left (\frac {\left (-\frac {a^{2}}{2}-\frac {b^{2}}{2}\right ) \sqrt {\cot \left (d x +c \right )}}{b +a \cot \left (d x +c \right )}+\frac {\left (3 a^{2}-b^{2}\right ) \arctan \left (\frac {a \sqrt {\cot \left (d x +c \right )}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{\left (a^{2}+b^{2}\right )^{2}}}{d}\)	\(279\)
default	\(-\frac {\frac {\frac {\left (a^{2}-b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}+\frac {a b \sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{2}}{\left (a^{2}+b^{2}\right )^{2}}-\frac {2 b \left (\frac {\left (-\frac {a^{2}}{2}-\frac {b^{2}}{2}\right ) \sqrt {\cot \left (d x +c \right )}}{b +a \cot \left (d x +c \right )}+\frac {\left (3 a^{2}-b^{2}\right ) \arctan \left (\frac {a \sqrt {\cot \left (d x +c \right )}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{\left (a^{2}+b^{2}\right )^{2}}}{d}\)	\(279\)

Input:

int(1/cot(d*x+c)^(1/2)/(a+b*tan(d*x+c))^2,x,method=_RETURNVERBOSE)
 

Output:

-1/d*(2/(a^2+b^2)^2*(1/8*(a^2-b^2)*2^(1/2)*(ln((cot(d*x+c)+2^(1/2)*cot(d*x 
+c)^(1/2)+1)/(cot(d*x+c)-2^(1/2)*cot(d*x+c)^(1/2)+1))+2*arctan(1+2^(1/2)*c 
ot(d*x+c)^(1/2))+2*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2)))+1/4*a*b*2^(1/2)*(l 
n((cot(d*x+c)-2^(1/2)*cot(d*x+c)^(1/2)+1)/(cot(d*x+c)+2^(1/2)*cot(d*x+c)^( 
1/2)+1))+2*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))+2*arctan(-1+2^(1/2)*cot(d*x+ 
c)^(1/2))))-2*b/(a^2+b^2)^2*((-1/2*a^2-1/2*b^2)*cot(d*x+c)^(1/2)/(b+a*cot( 
d*x+c))+1/2*(3*a^2-b^2)/(a*b)^(1/2)*arctan(a*cot(d*x+c)^(1/2)/(a*b)^(1/2)) 
))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1472 vs. \(2 (225) = 450\).

Time = 0.22 (sec) , antiderivative size = 2973, normalized size of antiderivative = 11.66 \[ \int \frac {1}{\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^2} \, dx=\text {Too large to display} \] Input:

integrate(1/cot(d*x+c)^(1/2)/(a+b*tan(d*x+c))^2,x, algorithm="fricas")
 

Output:

[-1/2*(2*sqrt(1/2)*((a^4*b + 2*a^2*b^3 + b^5)*d*tan(d*x + c) + (a^5 + 2*a^ 
3*b^2 + a*b^4)*d)*sqrt((a^4 + 4*a^3*b + 2*a^2*b^2 - 4*a*b^3 + b^4)/((a^8 + 
 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^2))*arctan(-((a^8 + 4*a^6*b^2 
+ 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^2*sqrt((a^4 + 4*a^3*b + 2*a^2*b^2 - 4*a*b 
^3 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^2))*sqrt((a^4 
 - 4*a^3*b + 2*a^2*b^2 + 4*a*b^3 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4* 
a^2*b^6 + b^8)*d^2)) + 2*sqrt(1/2)*(a^6 - 2*a^5*b + a^4*b^2 - 4*a^3*b^3 - 
a^2*b^4 - 2*a*b^5 - b^6)*d*sqrt((a^4 + 4*a^3*b + 2*a^2*b^2 - 4*a*b^3 + b^4 
)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^2))*sqrt(tan(d*x + c) 
))/(a^4 - 6*a^2*b^2 + b^4)) + 2*sqrt(1/2)*((a^4*b + 2*a^2*b^3 + b^5)*d*tan 
(d*x + c) + (a^5 + 2*a^3*b^2 + a*b^4)*d)*sqrt((a^4 + 4*a^3*b + 2*a^2*b^2 - 
 4*a*b^3 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^2))*arc 
tan(((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^2*sqrt((a^4 + 4*a^3 
*b + 2*a^2*b^2 - 4*a*b^3 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 
+ b^8)*d^2))*sqrt((a^4 - 4*a^3*b + 2*a^2*b^2 + 4*a*b^3 + b^4)/((a^8 + 4*a^ 
6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^2)) - 2*sqrt(1/2)*(a^6 - 2*a^5*b + 
a^4*b^2 - 4*a^3*b^3 - a^2*b^4 - 2*a*b^5 - b^6)*d*sqrt((a^4 + 4*a^3*b + 2*a 
^2*b^2 - 4*a*b^3 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d 
^2))*sqrt(tan(d*x + c)))/(a^4 - 6*a^2*b^2 + b^4)) - sqrt(1/2)*((a^4*b + 2* 
a^2*b^3 + b^5)*d*tan(d*x + c) + (a^5 + 2*a^3*b^2 + a*b^4)*d)*sqrt((a^4 ...
 

Sympy [F]

\[ \int \frac {1}{\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^2} \, dx=\int \frac {1}{\left (a + b \tan {\left (c + d x \right )}\right )^{2} \sqrt {\cot {\left (c + d x \right )}}}\, dx \] Input:

integrate(1/cot(d*x+c)**(1/2)/(a+b*tan(d*x+c))**2,x)
 

Output:

Integral(1/((a + b*tan(c + d*x))**2*sqrt(cot(c + d*x))), x)
 

Maxima [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^2} \, dx=\frac {\frac {4 \, {\left (3 \, a^{2} b - b^{3}\right )} \arctan \left (\frac {a}{\sqrt {a b} \sqrt {\tan \left (d x + c\right )}}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a b}} - \frac {2 \, \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {4 \, b}{{\left (a^{2} b + b^{3} + \frac {a^{3} + a b^{2}}{\tan \left (d x + c\right )}\right )} \sqrt {\tan \left (d x + c\right )}}}{4 \, d} \] Input:

integrate(1/cot(d*x+c)^(1/2)/(a+b*tan(d*x+c))^2,x, algorithm="maxima")
 

Output:

1/4*(4*(3*a^2*b - b^3)*arctan(a/(sqrt(a*b)*sqrt(tan(d*x + c))))/((a^4 + 2* 
a^2*b^2 + b^4)*sqrt(a*b)) - (2*sqrt(2)*(a^2 + 2*a*b - b^2)*arctan(1/2*sqrt 
(2)*(sqrt(2) + 2/sqrt(tan(d*x + c)))) + 2*sqrt(2)*(a^2 + 2*a*b - b^2)*arct 
an(-1/2*sqrt(2)*(sqrt(2) - 2/sqrt(tan(d*x + c)))) + sqrt(2)*(a^2 - 2*a*b - 
 b^2)*log(sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) - sqrt(2)*(a^2 
- 2*a*b - b^2)*log(-sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1))/(a^4 
 + 2*a^2*b^2 + b^4) - 4*b/((a^2*b + b^3 + (a^3 + a*b^2)/tan(d*x + c))*sqrt 
(tan(d*x + c))))/d
 

Giac [F]

\[ \int \frac {1}{\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^2} \, dx=\int { \frac {1}{{\left (b \tan \left (d x + c\right ) + a\right )}^{2} \sqrt {\cot \left (d x + c\right )}} \,d x } \] Input:

integrate(1/cot(d*x+c)^(1/2)/(a+b*tan(d*x+c))^2,x, algorithm="giac")
 

Output:

integrate(1/((b*tan(d*x + c) + a)^2*sqrt(cot(d*x + c))), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^2} \, dx=\int \frac {1}{\sqrt {\mathrm {cot}\left (c+d\,x\right )}\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^2} \,d x \] Input:

int(1/(cot(c + d*x)^(1/2)*(a + b*tan(c + d*x))^2),x)
 

Output:

int(1/(cot(c + d*x)^(1/2)*(a + b*tan(c + d*x))^2), x)
 

Reduce [F]

\[ \int \frac {1}{\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^2} \, dx=\int \frac {\sqrt {\cot \left (d x +c \right )}}{\cot \left (d x +c \right ) \tan \left (d x +c \right )^{2} b^{2}+2 \cot \left (d x +c \right ) \tan \left (d x +c \right ) a b +\cot \left (d x +c \right ) a^{2}}d x \] Input:

int(1/cot(d*x+c)^(1/2)/(a+b*tan(d*x+c))^2,x)
 

Output:

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