\(\int \cot (d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx\) [7]

Optimal result

Integrand size = 31, antiderivative size = 602 \[ \int \cot (d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx=-\frac {\sqrt {a^2+b^2+c \left (c+\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )} \arctan \left (\frac {b^2+(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )-b \sqrt {a^2+b^2-2 a c+c^2} \cot (d+e x)}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} \sqrt {a^2+b^2+c \left (c+\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} e}-\frac {b \text {arctanh}\left (\frac {b+2 c \cot (d+e x)}{2 \sqrt {c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{2 \sqrt {c} e}+\frac {\sqrt {a^2+b^2+c \left (c-\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )} \text {arctanh}\left (\frac {b^2+(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )+b \sqrt {a^2+b^2-2 a c+c^2} \cot (d+e x)}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} \sqrt {a^2+b^2+c \left (c-\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} e}-\frac {\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{e} \] Output:

-1/2*(a^2+b^2+c*(c+(a^2-2*a*c+b^2+c^2)^(1/2))-a*(2*c+(a^2-2*a*c+b^2+c^2)^( 
1/2)))^(1/2)*arctan(1/2*(b^2+(a-c)*(a-c-(a^2-2*a*c+b^2+c^2)^(1/2))-b*(a^2- 
2*a*c+b^2+c^2)^(1/2)*cot(e*x+d))*2^(1/2)/(a^2-2*a*c+b^2+c^2)^(1/4)/(a^2+b^ 
2+c*(c+(a^2-2*a*c+b^2+c^2)^(1/2))-a*(2*c+(a^2-2*a*c+b^2+c^2)^(1/2)))^(1/2) 
/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2))*2^(1/2)/(a^2-2*a*c+b^2+c^2)^(1/4)/ 
e-1/2*b*arctanh(1/2*(b+2*c*cot(e*x+d))/c^(1/2)/(a+b*cot(e*x+d)+c*cot(e*x+d 
)^2)^(1/2))/c^(1/2)/e+1/2*(a^2+b^2+c*(c-(a^2-2*a*c+b^2+c^2)^(1/2))-a*(2*c- 
(a^2-2*a*c+b^2+c^2)^(1/2)))^(1/2)*arctanh(1/2*(b^2+(a-c)*(a-c+(a^2-2*a*c+b 
^2+c^2)^(1/2))+b*(a^2-2*a*c+b^2+c^2)^(1/2)*cot(e*x+d))*2^(1/2)/(a^2-2*a*c+ 
b^2+c^2)^(1/4)/(a^2+b^2+c*(c-(a^2-2*a*c+b^2+c^2)^(1/2))-a*(2*c-(a^2-2*a*c+ 
b^2+c^2)^(1/2)))^(1/2)/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2))*2^(1/2)/(a^2 
-2*a*c+b^2+c^2)^(1/4)/e-(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2)/e
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.85 (sec) , antiderivative size = 324, normalized size of antiderivative = 0.54 \[ \int \cot (d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx=-\frac {\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan (d+e x) \left (b \text {arctanh}\left (\frac {2 c+b \tan (d+e x)}{2 \sqrt {c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )+\sqrt {c} \left (i \sqrt {a+i b-c} \arctan \left (\frac {i b-2 c+(2 i a-b) \tan (d+e x)}{2 \sqrt {a+i b-c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )+i \sqrt {a-i b-c} \arctan \left (\frac {i b+2 c+(2 i a+b) \tan (d+e x)}{2 \sqrt {a-i b-c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )+2 \cot (d+e x) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}\right )\right )}{2 \sqrt {c} e \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}} \] Input:

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

Output:

-1/2*(Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*Tan[d + e*x]*(b*ArcTanh[ 
(2*c + b*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x] 
^2])] + Sqrt[c]*(I*Sqrt[a + I*b - c]*ArcTan[(I*b - 2*c + ((2*I)*a - b)*Tan 
[d + e*x])/(2*Sqrt[a + I*b - c]*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2 
])] + I*Sqrt[a - I*b - c]*ArcTan[(I*b + 2*c + ((2*I)*a + b)*Tan[d + e*x])/ 
(2*Sqrt[a - I*b - c]*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2])] + 2*Cot 
[d + e*x]*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2])))/(Sqrt[c]*e*Sqrt[c 
 + b*Tan[d + e*x] + a*Tan[d + e*x]^2])
 

Rubi [A] (verified)

Time = 28.70 (sec) , antiderivative size = 666, normalized size of antiderivative = 1.11, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules used = {3042, 4184, 1354, 27, 2144, 27, 1092, 219, 1369, 25, 1363, 218, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4184

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

\(\Big \downarrow \) 1354

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2144

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 1369

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1363

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

-((((b*ArcTanh[(b + 2*c*Cot[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x] + 
 c*Cot[d + e*x]^2])])/Sqrt[c] - 2*(-(((b^2 + (a - c)*(a - c - Sqrt[a^2 + b 
^2 - 2*a*c + c^2]))*ArcTan[(b^2 + (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c 
+ c^2]) - b*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Cot[d + e*x])/(Sqrt[2]*(a^2 + b^ 
2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^ 
2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Cot[d + e*x] + c 
*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^ 
2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a* 
c + c^2])])) + ((b^2 + (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]))*Ar 
cTanh[(b^2 + (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) + b*Sqrt[a^2 
+ b^2 - 2*a*c + c^2]*Cot[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4 
)*Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a 
^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/( 
Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + 
 b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])])))/2 + Sqr 
t[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])/e)
 

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 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 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 1092

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

rule 1354

Int[((g_.) + (h_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (f 
_.)*(x_)^2)^(q_), x_Symbol] :> Simp[h*(a + b*x + c*x^2)^p*((d + f*x^2)^(q + 
 1)/(2*f*(p + q + 1))), x] - Simp[1/(2*f*(p + q + 1))   Int[(a + b*x + c*x^ 
2)^(p - 1)*(d + f*x^2)^q*Simp[h*p*(b*d) + a*(-2*g*f)*(p + q + 1) + (2*h*p*( 
c*d - a*f) + b*(-2*g*f)*(p + q + 1))*x + (h*p*((-b)*f) + c*(-2*g*f)*(p + q 
+ 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, f, g, h, q}, x] && NeQ[b^2 - 4* 
a*c, 0] && GtQ[p, 0] && NeQ[p + q + 1, 0]
 

rule 1363

Int[((g_) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f 
_.)*(x_)^2]), x_Symbol] :> Simp[-2*a*g*h   Subst[Int[1/Simp[2*a^2*g*h*c + a 
*e*x^2, x], x], x, Simp[a*h - g*c*x, x]/Sqrt[d + e*x + f*x^2]], x] /; FreeQ 
[{a, c, d, e, f, g, h}, x] && EqQ[a*h^2*e + 2*g*h*(c*d - a*f) - g^2*c*e, 0]
 

rule 1369

Int[((g_.) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + ( 
f_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[(c*d - a*f)^2 + a*c*e^2, 2]}, Simp 
[1/(2*q)   Int[Simp[(-a)*h*e - g*(c*d - a*f - q) + (h*(c*d - a*f + q) - g*c 
*e)*x, x]/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] - Simp[1/(2*q)   Int[ 
Simp[(-a)*h*e - g*(c*d - a*f + q) + (h*(c*d - a*f - q) - g*c*e)*x, x]/((a + 
 c*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, c, d, e, f, g, h}, x] 
&& NeQ[e^2 - 4*d*f, 0] && NegQ[(-a)*c]
 

rule 2144

Int[(Px_)/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), 
x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[Px, 
x, 2]}, Simp[C/c   Int[1/Sqrt[d + e*x + f*x^2], x], x] + Simp[1/c   Int[(A* 
c - a*C + B*c*x)/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, 
c, d, e, f}, x] && PolyQ[Px, x, 2]
 

rule 3042

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

rule 4184

Int[cot[(d_.) + (e_.)*(x_)]^(m_.)*((a_.) + (b_.)*(cot[(d_.) + (e_.)*(x_)]*( 
f_.))^(n_.) + (c_.)*(cot[(d_.) + (e_.)*(x_)]*(f_.))^(n2_.))^(p_), x_Symbol] 
 :> Simp[-f/e   Subst[Int[(x/f)^m*((a + b*x^n + c*x^(2*n))^p/(f^2 + x^2)), 
x], x, f*Cot[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[ 
n2, 2*n] && NeQ[b^2 - 4*a*c, 0]
 

Maple [B] (warning: unable to verify)

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

Time = 3.09 (sec) , antiderivative size = 17767874, normalized size of antiderivative = 29514.74

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

Input:

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

Output:

result too large to display
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5680 vs. \(2 (543) = 1086\).

Time = 1.87 (sec) , antiderivative size = 11381, normalized size of antiderivative = 18.91 \[ \int \cot (d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx=\text {Too large to display} \] Input:

integrate(cot(e*x+d)*(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2),x, algorithm="f 
ricas")
 

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(cot(e*x+d)*(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2),x, algorithm="m 
axima")
 

Output:

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

Giac [F]

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

integrate(cot(e*x+d)*(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2),x, algorithm="g 
iac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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