3.1.97 \(\int \sqrt {a+b \cot (c+d x)} (A+B \cot (c+d x)) \, dx\) [97]

3.1.97.1 Optimal result

Integrand size = 25, antiderivative size = 122 \[ \int \sqrt {a+b \cot (c+d x)} (A+B \cot (c+d x)) \, dx=\frac {\sqrt {a-i b} (i A+B) \text {arctanh}\left (\frac {\sqrt {a+b \cot (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {\sqrt {a+i b} (i A-B) \text {arctanh}\left (\frac {\sqrt {a+b \cot (c+d x)}}{\sqrt {a+i b}}\right )}{d}-\frac {2 B \sqrt {a+b \cot (c+d x)}}{d} \]

output

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

3.1.97.2 Mathematica [A] (verified)

Time = 0.66 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.74 \[ \int \sqrt {a+b \cot (c+d x)} (A+B \cot (c+d x)) \, dx=-\frac {\frac {\left (a A b-A b \sqrt {-b^2}-b^2 B-a \sqrt {-b^2} B\right ) \text {arctanh}\left (\frac {\sqrt {a+b \cot (c+d x)}}{\sqrt {a-\sqrt {-b^2}}}\right )}{\sqrt {-b^2} \sqrt {a-\sqrt {-b^2}}}-\frac {\left (a A b+A b \sqrt {-b^2}-b^2 B+a \sqrt {-b^2} B\right ) \text {arctanh}\left (\frac {\sqrt {a+b \cot (c+d x)}}{\sqrt {a+\sqrt {-b^2}}}\right )}{\sqrt {-b^2} \sqrt {a+\sqrt {-b^2}}}+2 B \sqrt {a+b \cot (c+d x)}}{d} \]

input

Integrate[Sqrt[a + b*Cot[c + d*x]]*(A + B*Cot[c + d*x]),x]
 

output

-((((a*A*b - A*b*Sqrt[-b^2] - b^2*B - a*Sqrt[-b^2]*B)*ArcTanh[Sqrt[a + b*C 
ot[c + d*x]]/Sqrt[a - Sqrt[-b^2]]])/(Sqrt[-b^2]*Sqrt[a - Sqrt[-b^2]]) - (( 
a*A*b + A*b*Sqrt[-b^2] - b^2*B + a*Sqrt[-b^2]*B)*ArcTanh[Sqrt[a + b*Cot[c 
+ d*x]]/Sqrt[a + Sqrt[-b^2]]])/(Sqrt[-b^2]*Sqrt[a + Sqrt[-b^2]]) + 2*B*Sqr 
t[a + b*Cot[c + d*x]])/d)
 

3.1.97.3 Rubi [A] (warning: unable to verify)

Time = 0.58 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.86, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3042, 4011, 3042, 4022, 3042, 4020, 25, 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.

\(\displaystyle \int \sqrt {a+b \cot (c+d x)} (A+B \cot (c+d x)) \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4011

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4022

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4020

\(\displaystyle -\frac {i (a-i b) (A-i B) \int -\frac {1}{(1-i \cot (c+d x)) \sqrt {a+b \cot (c+d x)}}d(i \cot (c+d x))}{2 d}+\frac {i (a+i b) (A+i B) \int -\frac {1}{(i \cot (c+d x)+1) \sqrt {a+b \cot (c+d x)}}d(-i \cot (c+d x))}{2 d}-\frac {2 B \sqrt {a+b \cot (c+d x)}}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {i (a-i b) (A-i B) \int \frac {1}{(1-i \cot (c+d x)) \sqrt {a+b \cot (c+d x)}}d(i \cot (c+d x))}{2 d}-\frac {i (a+i b) (A+i B) \int \frac {1}{(i \cot (c+d x)+1) \sqrt {a+b \cot (c+d x)}}d(-i \cot (c+d x))}{2 d}-\frac {2 B \sqrt {a+b \cot (c+d x)}}{d}\)

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

\(\displaystyle -\frac {\sqrt {a-i b} (A-i B) \arctan \left (\frac {\cot (c+d x)}{\sqrt {a-i b}}\right )}{d}-\frac {\sqrt {a+i b} (A+i B) \arctan \left (\frac {\cot (c+d x)}{\sqrt {a+i b}}\right )}{d}-\frac {2 B \sqrt {a+b \cot (c+d x)}}{d}\)

input

Int[Sqrt[a + b*Cot[c + d*x]]*(A + B*Cot[c + d*x]),x]
 

output

-((Sqrt[a - I*b]*(A - I*B)*ArcTan[Cot[c + d*x]/Sqrt[a - I*b]])/d) - (Sqrt[ 
a + I*b]*(A + I*B)*ArcTan[Cot[c + d*x]/Sqrt[a + I*b]])/d - (2*B*Sqrt[a + b 
*Cot[c + d*x]])/d
 

3.1.97.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 4011

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

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]
 

3.1.97.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(813\) vs. \(2(102)=204\).

Time = 0.15 (sec) , antiderivative size = 814, normalized size of antiderivative = 6.67

method	result	size
parts	\(-\frac {\ln \left (b \cot \left (d x +c \right )+a +\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a}{4 d b}+\frac {\ln \left (b \cot \left (d x +c \right )+a +\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}}{4 d b}-\frac {b \arctan \left (\frac {2 \sqrt {a +b \cot \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) A}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {\ln \left (\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \cot \left (d x +c \right )-\sqrt {a^{2}+b^{2}}-a \right ) A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a}{4 d b}-\frac {\ln \left (\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \cot \left (d x +c \right )-\sqrt {a^{2}+b^{2}}-a \right ) A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}}{4 d b}+\frac {b \arctan \left (\frac {-2 \sqrt {a +b \cot \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) A}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {B \left (-2 \sqrt {a +b \cot \left (d x +c \right )}+\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \ln \left (b \cot \left (d x +c \right )+a +\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right )}{4}-\frac {\left (a -\sqrt {a^{2}+b^{2}}\right ) \arctan \left (\frac {2 \sqrt {a +b \cot \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}-\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \ln \left (\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \cot \left (d x +c \right )-\sqrt {a^{2}+b^{2}}-a \right )}{4}-\frac {\left (\sqrt {a^{2}+b^{2}}-a \right ) \arctan \left (\frac {-2 \sqrt {a +b \cot \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{d}\)	\(814\)
derivativedivides	\(-\frac {2 B \sqrt {a +b \cot \left (d x +c \right )}}{d}-\frac {\ln \left (\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \cot \left (d x +c \right )-\sqrt {a^{2}+b^{2}}-a \right ) A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}}{4 d b}+\frac {\ln \left (\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \cot \left (d x +c \right )-\sqrt {a^{2}+b^{2}}-a \right ) A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a}{4 d b}-\frac {\ln \left (\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \cot \left (d x +c \right )-\sqrt {a^{2}+b^{2}}-a \right ) B \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{4 d}-\frac {\arctan \left (\frac {-2 \sqrt {a +b \cot \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) B \sqrt {a^{2}+b^{2}}}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {b \arctan \left (\frac {-2 \sqrt {a +b \cot \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) A}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {\arctan \left (\frac {-2 \sqrt {a +b \cot \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) B a}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {\ln \left (b \cot \left (d x +c \right )+a +\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}}{4 d b}-\frac {\ln \left (b \cot \left (d x +c \right )+a +\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a}{4 d b}+\frac {\ln \left (b \cot \left (d x +c \right )+a +\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) B \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{4 d}+\frac {\arctan \left (\frac {2 \sqrt {a +b \cot \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) B \sqrt {a^{2}+b^{2}}}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}-\frac {b \arctan \left (\frac {2 \sqrt {a +b \cot \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) A}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}-\frac {\arctan \left (\frac {2 \sqrt {a +b \cot \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) B a}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\)	\(968\)
default	\(-\frac {2 B \sqrt {a +b \cot \left (d x +c \right )}}{d}-\frac {\ln \left (\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \cot \left (d x +c \right )-\sqrt {a^{2}+b^{2}}-a \right ) A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}}{4 d b}+\frac {\ln \left (\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \cot \left (d x +c \right )-\sqrt {a^{2}+b^{2}}-a \right ) A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a}{4 d b}-\frac {\ln \left (\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \cot \left (d x +c \right )-\sqrt {a^{2}+b^{2}}-a \right ) B \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{4 d}-\frac {\arctan \left (\frac {-2 \sqrt {a +b \cot \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) B \sqrt {a^{2}+b^{2}}}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {b \arctan \left (\frac {-2 \sqrt {a +b \cot \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) A}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {\arctan \left (\frac {-2 \sqrt {a +b \cot \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) B a}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {\ln \left (b \cot \left (d x +c \right )+a +\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}}{4 d b}-\frac {\ln \left (b \cot \left (d x +c \right )+a +\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a}{4 d b}+\frac {\ln \left (b \cot \left (d x +c \right )+a +\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) B \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{4 d}+\frac {\arctan \left (\frac {2 \sqrt {a +b \cot \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) B \sqrt {a^{2}+b^{2}}}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}-\frac {b \arctan \left (\frac {2 \sqrt {a +b \cot \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) A}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}-\frac {\arctan \left (\frac {2 \sqrt {a +b \cot \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) B a}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\)	\(968\)

input

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

output

-1/4/d/b*ln(b*cot(d*x+c)+a+(a+b*cot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^ 
(1/2)+(a^2+b^2)^(1/2))*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a+1/4/d/b*ln(b*cot( 
d*x+c)+a+(a+b*cot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1 
/2))*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)-1/d*b/(2*(a^2+b^2)^(1 
/2)-2*a)^(1/2)*arctan((2*(a+b*cot(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1 
/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*A+1/4/d/b*ln((a+b*cot(d*x+c))^(1/2)*(2 
*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*cot(d*x+c)-(a^2+b^2)^(1/2)-a)*A*(2*(a^2+b^2) 
^(1/2)+2*a)^(1/2)*a-1/4/d/b*ln((a+b*cot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2 
*a)^(1/2)-b*cot(d*x+c)-(a^2+b^2)^(1/2)-a)*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)* 
(a^2+b^2)^(1/2)+1/d*b/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((-2*(a+b*cot(d* 
x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))* 
A+B/d*(-2*(a+b*cot(d*x+c))^(1/2)+1/4*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*ln(b*co 
t(d*x+c)+a+(a+b*cot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^ 
(1/2))-(a-(a^2+b^2)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*co 
t(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/ 
2))-1/4*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*ln((a+b*cot(d*x+c))^(1/2)*(2*(a^2+b^ 
2)^(1/2)+2*a)^(1/2)-b*cot(d*x+c)-(a^2+b^2)^(1/2)-a)-((a^2+b^2)^(1/2)-a)/(2 
*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((-2*(a+b*cot(d*x+c))^(1/2)+(2*(a^2+b^2) 
^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)))
 

3.1.97.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1329 vs. \(2 (96) = 192\).

Time = 0.33 (sec) , antiderivative size = 1329, normalized size of antiderivative = 10.89 \[ \int \sqrt {a+b \cot (c+d x)} (A+B \cot (c+d x)) \, dx=\text {Too large to display} \]

input

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

output

1/2*(d*sqrt((2*A*B*b + d^2*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + 
(A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) - (A^2 - B^2)*a)/d^2)*log(-(2*(A^3*B + A 
*B^3)*a + (A^4 - B^4)*b)*sqrt((b*cos(2*d*x + 2*c) + a*sin(2*d*x + 2*c) + b 
)/sin(2*d*x + 2*c)) + (A*d^3*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b 
+ (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) - (2*A*B^2*a + (A^2*B - B^3)*b)*d)*sqr 
t((2*A*B*b + d^2*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A 
^2*B^2 + B^4)*b^2)/d^4) - (A^2 - B^2)*a)/d^2)) - d*sqrt((2*A*B*b + d^2*sqr 
t(-(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d 
^4) - (A^2 - B^2)*a)/d^2)*log(-(2*(A^3*B + A*B^3)*a + (A^4 - B^4)*b)*sqrt( 
(b*cos(2*d*x + 2*c) + a*sin(2*d*x + 2*c) + b)/sin(2*d*x + 2*c)) - (A*d^3*s 
qrt(-(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2) 
/d^4) - (2*A*B^2*a + (A^2*B - B^3)*b)*d)*sqrt((2*A*B*b + d^2*sqrt(-(4*A^2* 
B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) - (A^2 
 - B^2)*a)/d^2)) - d*sqrt((2*A*B*b - d^2*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B - 
 A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) - (A^2 - B^2)*a)/d^2)*log( 
-(2*(A^3*B + A*B^3)*a + (A^4 - B^4)*b)*sqrt((b*cos(2*d*x + 2*c) + a*sin(2* 
d*x + 2*c) + b)/sin(2*d*x + 2*c)) + (A*d^3*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B 
 - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) + (2*A*B^2*a + (A^2*B - 
B^3)*b)*d)*sqrt((2*A*B*b - d^2*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a* 
b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) - (A^2 - B^2)*a)/d^2)) + d*sqrt((...
 

3.1.97.6 Sympy [F]

\[ \int \sqrt {a+b \cot (c+d x)} (A+B \cot (c+d x)) \, dx=\int \left (A + B \cot {\left (c + d x \right )}\right ) \sqrt {a + b \cot {\left (c + d x \right )}}\, dx \]

input

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

output

Integral((A + B*cot(c + d*x))*sqrt(a + b*cot(c + d*x)), x)
 

3.1.97.7 Maxima [F]

\[ \int \sqrt {a+b \cot (c+d x)} (A+B \cot (c+d x)) \, dx=\int { {\left (B \cot \left (d x + c\right ) + A\right )} \sqrt {b \cot \left (d x + c\right ) + a} \,d x } \]

input

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

output

integrate((B*cot(d*x + c) + A)*sqrt(b*cot(d*x + c) + a), x)
 

3.1.97.8 Giac [F]

\[ \int \sqrt {a+b \cot (c+d x)} (A+B \cot (c+d x)) \, dx=\int { {\left (B \cot \left (d x + c\right ) + A\right )} \sqrt {b \cot \left (d x + c\right ) + a} \,d x } \]

input

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

output

integrate((B*cot(d*x + c) + A)*sqrt(b*cot(d*x + c) + a), x)
 

3.1.97.9 Mupad [B] (verification not implemented)

Time = 15.52 (sec) , antiderivative size = 843, normalized size of antiderivative = 6.91 \[ \int \sqrt {a+b \cot (c+d x)} (A+B \cot (c+d x)) \, dx=\mathrm {atanh}\left (\frac {d^3\,\left (\frac {16\,\left (A^2\,b^4-A^2\,a^2\,b^2\right )\,\sqrt {a+b\,\mathrm {cot}\left (c+d\,x\right )}}{d^2}+\frac {16\,a\,b^2\,\left (\sqrt {-A^4\,b^2\,d^4}+A^2\,a\,d^2\right )\,\sqrt {a+b\,\mathrm {cot}\left (c+d\,x\right )}}{d^4}\right )\,\sqrt {-\frac {\sqrt {-A^4\,b^2\,d^4}+A^2\,a\,d^2}{d^4}}}{16\,\left (A^3\,a^2\,b^3+A^3\,b^5\right )}\right )\,\sqrt {-\frac {\sqrt {-A^4\,b^2\,d^4}+A^2\,a\,d^2}{d^4}}+\mathrm {atanh}\left (\frac {d^3\,\left (\frac {16\,\left (A^2\,b^4-A^2\,a^2\,b^2\right )\,\sqrt {a+b\,\mathrm {cot}\left (c+d\,x\right )}}{d^2}-\frac {16\,a\,b^2\,\left (\sqrt {-A^4\,b^2\,d^4}-A^2\,a\,d^2\right )\,\sqrt {a+b\,\mathrm {cot}\left (c+d\,x\right )}}{d^4}\right )\,\sqrt {\frac {\sqrt {-A^4\,b^2\,d^4}-A^2\,a\,d^2}{d^4}}}{16\,\left (A^3\,a^2\,b^3+A^3\,b^5\right )}\right )\,\sqrt {\frac {\sqrt {-A^4\,b^2\,d^4}-A^2\,a\,d^2}{d^4}}+2\,\mathrm {atanh}\left (\frac {32\,B^2\,b^4\,\sqrt {\frac {\sqrt {-B^4\,b^2\,d^4}}{4\,d^4}+\frac {B^2\,a}{4\,d^2}}\,\sqrt {a+b\,\mathrm {cot}\left (c+d\,x\right )}}{\frac {16\,B\,b^4\,\sqrt {-B^4\,b^2\,d^4}}{d^3}+\frac {16\,B\,a^2\,b^2\,\sqrt {-B^4\,b^2\,d^4}}{d^3}}+\frac {32\,a\,b^2\,\sqrt {\frac {\sqrt {-B^4\,b^2\,d^4}}{4\,d^4}+\frac {B^2\,a}{4\,d^2}}\,\sqrt {a+b\,\mathrm {cot}\left (c+d\,x\right )}\,\sqrt {-B^4\,b^2\,d^4}}{\frac {16\,B\,b^4\,\sqrt {-B^4\,b^2\,d^4}}{d}+\frac {16\,B\,a^2\,b^2\,\sqrt {-B^4\,b^2\,d^4}}{d}}\right )\,\sqrt {\frac {\sqrt {-B^4\,b^2\,d^4}+B^2\,a\,d^2}{4\,d^4}}-2\,\mathrm {atanh}\left (\frac {32\,B^2\,b^4\,\sqrt {\frac {B^2\,a}{4\,d^2}-\frac {\sqrt {-B^4\,b^2\,d^4}}{4\,d^4}}\,\sqrt {a+b\,\mathrm {cot}\left (c+d\,x\right )}}{\frac {16\,B\,b^4\,\sqrt {-B^4\,b^2\,d^4}}{d^3}+\frac {16\,B\,a^2\,b^2\,\sqrt {-B^4\,b^2\,d^4}}{d^3}}-\frac {32\,a\,b^2\,\sqrt {\frac {B^2\,a}{4\,d^2}-\frac {\sqrt {-B^4\,b^2\,d^4}}{4\,d^4}}\,\sqrt {a+b\,\mathrm {cot}\left (c+d\,x\right )}\,\sqrt {-B^4\,b^2\,d^4}}{\frac {16\,B\,b^4\,\sqrt {-B^4\,b^2\,d^4}}{d}+\frac {16\,B\,a^2\,b^2\,\sqrt {-B^4\,b^2\,d^4}}{d}}\right )\,\sqrt {-\frac {\sqrt {-B^4\,b^2\,d^4}-B^2\,a\,d^2}{4\,d^4}}-\frac {2\,B\,\sqrt {a+b\,\mathrm {cot}\left (c+d\,x\right )}}{d} \]

input

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

output

atanh((d^3*((16*(A^2*b^4 - A^2*a^2*b^2)*(a + b*cot(c + d*x))^(1/2))/d^2 + 
(16*a*b^2*((-A^4*b^2*d^4)^(1/2) + A^2*a*d^2)*(a + b*cot(c + d*x))^(1/2))/d 
^4)*(-((-A^4*b^2*d^4)^(1/2) + A^2*a*d^2)/d^4)^(1/2))/(16*(A^3*b^5 + A^3*a^ 
2*b^3)))*(-((-A^4*b^2*d^4)^(1/2) + A^2*a*d^2)/d^4)^(1/2) + atanh((d^3*((16 
*(A^2*b^4 - A^2*a^2*b^2)*(a + b*cot(c + d*x))^(1/2))/d^2 - (16*a*b^2*((-A^ 
4*b^2*d^4)^(1/2) - A^2*a*d^2)*(a + b*cot(c + d*x))^(1/2))/d^4)*(((-A^4*b^2 
*d^4)^(1/2) - A^2*a*d^2)/d^4)^(1/2))/(16*(A^3*b^5 + A^3*a^2*b^3)))*(((-A^4 
*b^2*d^4)^(1/2) - A^2*a*d^2)/d^4)^(1/2) + 2*atanh((32*B^2*b^4*((-B^4*b^2*d 
^4)^(1/2)/(4*d^4) + (B^2*a)/(4*d^2))^(1/2)*(a + b*cot(c + d*x))^(1/2))/((1 
6*B*b^4*(-B^4*b^2*d^4)^(1/2))/d^3 + (16*B*a^2*b^2*(-B^4*b^2*d^4)^(1/2))/d^ 
3) + (32*a*b^2*((-B^4*b^2*d^4)^(1/2)/(4*d^4) + (B^2*a)/(4*d^2))^(1/2)*(a + 
 b*cot(c + d*x))^(1/2)*(-B^4*b^2*d^4)^(1/2))/((16*B*b^4*(-B^4*b^2*d^4)^(1/ 
2))/d + (16*B*a^2*b^2*(-B^4*b^2*d^4)^(1/2))/d))*(((-B^4*b^2*d^4)^(1/2) + B 
^2*a*d^2)/(4*d^4))^(1/2) - 2*atanh((32*B^2*b^4*((B^2*a)/(4*d^2) - (-B^4*b^ 
2*d^4)^(1/2)/(4*d^4))^(1/2)*(a + b*cot(c + d*x))^(1/2))/((16*B*b^4*(-B^4*b 
^2*d^4)^(1/2))/d^3 + (16*B*a^2*b^2*(-B^4*b^2*d^4)^(1/2))/d^3) - (32*a*b^2* 
((B^2*a)/(4*d^2) - (-B^4*b^2*d^4)^(1/2)/(4*d^4))^(1/2)*(a + b*cot(c + d*x) 
)^(1/2)*(-B^4*b^2*d^4)^(1/2))/((16*B*b^4*(-B^4*b^2*d^4)^(1/2))/d + (16*B*a 
^2*b^2*(-B^4*b^2*d^4)^(1/2))/d))*(-((-B^4*b^2*d^4)^(1/2) - B^2*a*d^2)/(4*d 
^4))^(1/2) - (2*B*(a + b*cot(c + d*x))^(1/2))/d