\(\int \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)} \, dx\) [509]

Optimal result

Integrand size = 23, antiderivative size = 324 \[ \int \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)} \, dx=-\frac {b \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}+\frac {b \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}}{a+\sqrt {a^2+b^2}+b \tan (c+d x)}\right )}{\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} d}-\frac {b \text {arctanh}\left (\frac {\sqrt {a+\sqrt {a^2+b^2}}-\sqrt {2} \sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} \sqrt {a-\sqrt {a^2+b^2}} d}+\frac {b \text {arctanh}\left (\frac {\sqrt {a+\sqrt {a^2+b^2}}+\sqrt {2} \sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} \sqrt {a-\sqrt {a^2+b^2}} d}-\frac {\cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.52 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.43 \[ \int \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)} \, dx=-\frac {\frac {b \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a}}-i \sqrt {a-i b} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )+i \sqrt {a+i b} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )+\cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d} \] Input:

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

Output:

-(((b*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/Sqrt[a] - I*Sqrt[a - I*b] 
*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] + I*Sqrt[a + I*b]*ArcTanh 
[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + Cot[c + d*x]*Sqrt[a + b*Tan[c + 
 d*x]])/d)
 

Rubi [A] (warning: unable to verify)

Time = 1.30 (sec) , antiderivative size = 465, normalized size of antiderivative = 1.44, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.826, Rules used = {3042, 4051, 27, 3042, 4136, 27, 3042, 3966, 483, 1449, 1142, 25, 27, 1083, 219, 1103, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4051

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4136

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3966

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

\(\Big \downarrow \) 483

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

\(\Big \downarrow \) 1449

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

\(\Big \downarrow \) 1142

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1083

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 1103

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

\(\Big \downarrow \) 4117

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

((-2*b*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d) - (4*b*((-(( 
Sqrt[a + Sqrt[a^2 + b^2]]*ArcTanh[(-(Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]) + 
2*Sqrt[a + b*Tan[c + d*x]])/(Sqrt[2]*Sqrt[a - Sqrt[a^2 + b^2]])])/Sqrt[a - 
 Sqrt[a^2 + b^2]]) + Log[Sqrt[a^2 + b^2] + b^2*Tan[c + d*x]^2 - Sqrt[2]*Sq 
rt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]]/2)/(2*Sqrt[2]*Sqrt[a + S 
qrt[a^2 + b^2]]) - ((Sqrt[a + Sqrt[a^2 + b^2]]*ArcTanh[(Sqrt[2]*Sqrt[a + S 
qrt[a^2 + b^2]] + 2*Sqrt[a + b*Tan[c + d*x]])/(Sqrt[2]*Sqrt[a - Sqrt[a^2 + 
 b^2]])])/Sqrt[a - Sqrt[a^2 + b^2]] + Log[Sqrt[a^2 + b^2] + b^2*Tan[c + d* 
x]^2 + Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]]/2)/(2*S 
qrt[2]*Sqrt[a + Sqrt[a^2 + b^2]])))/d)/2 - (Cot[c + d*x]*Sqrt[a + b*Tan[c 
+ d*x]])/d
 

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 219

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

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 483

Int[Sqrt[(c_) + (d_.)*(x_)]/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[2*d 
Subst[Int[x^2/(b*c^2 + a*d^2 - 2*b*c*x^2 + b*x^4), x], x, Sqrt[c + d*x]], x 
] /; FreeQ[{a, b, c, d}, x]
 

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{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 1142

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[(2*c*d - b*e)/(2*c)   Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) 
Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
 

rule 1449

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = 
Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*r)   Int[x^(m - 1)/(q 
 - r*x + x^2), x], x] - Simp[1/(2*c*r)   Int[x^(m - 1)/(q + r*x + x^2), x], 
 x]]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && GeQ[m, 1] && LtQ[m, 
3] && NegQ[b^2 - 4*a*c]
 

rule 3042

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

rule 3966

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b/d   Su 
bst[Int[(a + x)^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{a, b, c 
, d, n}, x] && NeQ[a^2 + b^2, 0]
 

rule 4051

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(a + b*Tan[e + f*x])^(m + 1)*((c + 
d*Tan[e + f*x])^n/(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 - 1)*Simp[a*c 
*(m + 1) - b*d*n - (b*c - a*d)*(m + 1)*Tan[e + f*x] - b*d*(m + n + 1)*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] && GtQ[n, 0] && Int 
egerQ[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]
 

Maple [B] (warning: unable to verify)

result has leaf size over 500,000. Avoiding possible recursion issues.

Time = 34.51 (sec) , antiderivative size = 1278662, normalized size of antiderivative = 3946.49

\[\text {output too large to display}\]

Input:

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

Output:

result too large to display
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 844, normalized size of antiderivative = 2.60 \[ \int \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/2*(a*d*sqrt(-(d^2*sqrt(-b^2/d^4) + a)/d^2)*log(d^3*sqrt(-(d^2*sqrt(-b^2 
/d^4) + a)/d^2)*sqrt(-b^2/d^4) + sqrt(b*tan(d*x + c) + a)*b)*tan(d*x + c) 
- a*d*sqrt(-(d^2*sqrt(-b^2/d^4) + a)/d^2)*log(-d^3*sqrt(-(d^2*sqrt(-b^2/d^ 
4) + a)/d^2)*sqrt(-b^2/d^4) + sqrt(b*tan(d*x + c) + a)*b)*tan(d*x + c) - a 
*d*sqrt((d^2*sqrt(-b^2/d^4) - a)/d^2)*log(d^3*sqrt((d^2*sqrt(-b^2/d^4) - a 
)/d^2)*sqrt(-b^2/d^4) + sqrt(b*tan(d*x + c) + a)*b)*tan(d*x + c) + a*d*sqr 
t((d^2*sqrt(-b^2/d^4) - a)/d^2)*log(-d^3*sqrt((d^2*sqrt(-b^2/d^4) - a)/d^2 
)*sqrt(-b^2/d^4) + sqrt(b*tan(d*x + c) + a)*b)*tan(d*x + c) + sqrt(a)*b*lo 
g((b*tan(d*x + c) - 2*sqrt(b*tan(d*x + c) + a)*sqrt(a) + 2*a)/tan(d*x + c) 
)*tan(d*x + c) - 2*sqrt(b*tan(d*x + c) + a)*a)/(a*d*tan(d*x + c)), 1/2*(a* 
d*sqrt(-(d^2*sqrt(-b^2/d^4) + a)/d^2)*log(d^3*sqrt(-(d^2*sqrt(-b^2/d^4) + 
a)/d^2)*sqrt(-b^2/d^4) + sqrt(b*tan(d*x + c) + a)*b)*tan(d*x + c) - a*d*sq 
rt(-(d^2*sqrt(-b^2/d^4) + a)/d^2)*log(-d^3*sqrt(-(d^2*sqrt(-b^2/d^4) + a)/ 
d^2)*sqrt(-b^2/d^4) + sqrt(b*tan(d*x + c) + a)*b)*tan(d*x + c) - a*d*sqrt( 
(d^2*sqrt(-b^2/d^4) - a)/d^2)*log(d^3*sqrt((d^2*sqrt(-b^2/d^4) - a)/d^2)*s 
qrt(-b^2/d^4) + sqrt(b*tan(d*x + c) + a)*b)*tan(d*x + c) + a*d*sqrt((d^2*s 
qrt(-b^2/d^4) - a)/d^2)*log(-d^3*sqrt((d^2*sqrt(-b^2/d^4) - a)/d^2)*sqrt(- 
b^2/d^4) + sqrt(b*tan(d*x + c) + a)*b)*tan(d*x + c) + 2*sqrt(-a)*b*arctan( 
sqrt(-a)/sqrt(b*tan(d*x + c) + a))*tan(d*x + c) - 2*sqrt(b*tan(d*x + c) + 
a)*a)/(a*d*tan(d*x + c))]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [B] (verification not implemented)

Time = 1.32 (sec) , antiderivative size = 832, normalized size of antiderivative = 2.57 \[ \int \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)} \, dx =\text {Too large to display} \] Input:

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

Output:

atan((64*a*b^11*(- a/(4*d^2) - (b*1i)/(4*d^2))^(1/2)*(a + b*tan(c + d*x))^ 
(1/2))/((16*b^13)/d + (a*b^12*16i)/d + (32*a^2*b^11)/d + (a^3*b^10*16i)/d 
+ (16*a^4*b^9)/d) - (b^12*(- a/(4*d^2) - (b*1i)/(4*d^2))^(1/2)*(a + b*tan( 
c + d*x))^(1/2)*32i)/((16*b^13)/d + (a*b^12*16i)/d + (32*a^2*b^11)/d + (a^ 
3*b^10*16i)/d + (16*a^4*b^9)/d) + (32*a^3*b^9*(- a/(4*d^2) - (b*1i)/(4*d^2 
))^(1/2)*(a + b*tan(c + d*x))^(1/2))/((16*b^13)/d + (a*b^12*16i)/d + (32*a 
^2*b^11)/d + (a^3*b^10*16i)/d + (16*a^4*b^9)/d))*(-(a + b*1i)/(4*d^2))^(1/ 
2)*2i - atan((b^12*((b*1i)/(4*d^2) - a/(4*d^2))^(1/2)*(a + b*tan(c + d*x)) 
^(1/2)*32i)/((16*b^13)/d - (a*b^12*16i)/d + (32*a^2*b^11)/d - (a^3*b^10*16 
i)/d + (16*a^4*b^9)/d) + (64*a*b^11*((b*1i)/(4*d^2) - a/(4*d^2))^(1/2)*(a 
+ b*tan(c + d*x))^(1/2))/((16*b^13)/d - (a*b^12*16i)/d + (32*a^2*b^11)/d - 
 (a^3*b^10*16i)/d + (16*a^4*b^9)/d) + (32*a^3*b^9*((b*1i)/(4*d^2) - a/(4*d 
^2))^(1/2)*(a + b*tan(c + d*x))^(1/2))/((16*b^13)/d - (a*b^12*16i)/d + (32 
*a^2*b^11)/d - (a^3*b^10*16i)/d + (16*a^4*b^9)/d))*(-(a - b*1i)/(4*d^2))^( 
1/2)*2i + (b*(a + b*tan(c + d*x))^(1/2))/(a*d - d*(a + b*tan(c + d*x))) + 
(b*atan((b^13*(a + b*tan(c + d*x))^(1/2)*128i)/(a^(1/2)*(128*b^13 + 128*a^ 
2*b^11 + 32*a^4*b^9 + (32*b^15)/a^2)) + (a^(3/2)*b^11*(a + b*tan(c + d*x)) 
^(1/2)*128i)/(128*b^13 + 128*a^2*b^11 + 32*a^4*b^9 + (32*b^15)/a^2) + (a^( 
7/2)*b^9*(a + b*tan(c + d*x))^(1/2)*32i)/(128*b^13 + 128*a^2*b^11 + 32*a^4 
*b^9 + (32*b^15)/a^2) + (b^15*(a + b*tan(c + d*x))^(1/2)*32i)/(a^(5/2)*...
 

Reduce [F]

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

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

Output:

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