\(\int \frac {(a+b \cot (c+d x))^2}{\sqrt {e \cot (c+d x)}} \, dx\) [58]

Optimal result

Integrand size = 25, antiderivative size = 199 \[ \int \frac {(a+b \cot (c+d x))^2}{\sqrt {e \cot (c+d x)}} \, dx=\frac {\left (a^2+2 a b-b^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d \sqrt {e}}-\frac {\left (a^2+2 a b-b^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d \sqrt {e}}-\frac {\left (a^2-2 a b-b^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}+\sqrt {e} \cot (c+d x)}\right )}{\sqrt {2} d \sqrt {e}}-\frac {2 b^2 \sqrt {e \cot (c+d x)}}{d e} \] Output:

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

Mathematica [C] (verified)

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

Time = 0.56 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b \cot (c+d x))^2}{\sqrt {e \cot (c+d x)}} \, dx=-\frac {\sqrt {\cot (c+d x)} \left (2 b^2 \sqrt {\cot (c+d x)}+\frac {4}{3} a b \cot ^{\frac {3}{2}}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\cot ^2(c+d x)\right )-\frac {\left (a^2-b^2\right ) \left (2 \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-2 \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )+\log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-\log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )}{2 \sqrt {2}}\right )}{d \sqrt {e \cot (c+d x)}} \] Input:

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

Output:

-((Sqrt[Cot[c + d*x]]*(2*b^2*Sqrt[Cot[c + d*x]] + (4*a*b*Cot[c + d*x]^(3/2 
)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2])/3 - ((a^2 - b^2)*(2*Arc 
Tan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d* 
x]]] + Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Log[1 + Sqrt[2 
]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/(2*Sqrt[2])))/(d*Sqrt[e*Cot[c + d*x 
]]))
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.20, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {3042, 4026, 3042, 4017, 25, 1482, 1476, 1082, 217, 1479, 25, 27, 1103}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4026

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4017

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1482

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

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

Input:

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

Output:

(-2*b^2*Sqrt[e*Cot[c + d*x]])/(d*e) + (2*(-1/2*((a^2 + 2*a*b - b^2)*(-(Arc 
Tan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]]/(Sqrt[2]*Sqrt[e])) + ArcTa 
n[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]]/(Sqrt[2]*Sqrt[e]))) - ((a^2 
- 2*a*b - b^2)*(-1/2*Log[e + e*Cot[c + d*x] - Sqrt[2]*Sqrt[e]*Sqrt[e*Cot[c 
 + d*x]]]/(Sqrt[2]*Sqrt[e]) + Log[e + e*Cot[c + d*x] + Sqrt[2]*Sqrt[e]*Sqr 
t[e*Cot[c + d*x]]]/(2*Sqrt[2]*Sqrt[e])))/2))/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 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 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 4026

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

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.54

method	result	size
derivativedivides	\(-\frac {2 \left (b^{2} \sqrt {e \cot \left (d x +c \right )}+e \left (\frac {\left (a^{2} e -b^{2} e \right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}+\frac {a b \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d e}\)	\(306\)
default	\(-\frac {2 \left (b^{2} \sqrt {e \cot \left (d x +c \right )}+e \left (\frac {\left (a^{2} e -b^{2} e \right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}+\frac {a b \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d e}\)	\(306\)
parts	\(-\frac {a^{2} \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d e}-\frac {2 b^{2} \left (\sqrt {e \cot \left (d x +c \right )}-\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8}\right )}{d e}-\frac {a b \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{2 d \left (e^{2}\right )^{\frac {1}{4}}}\)	\(431\)

Input:

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

Output:

-2/d/e*(b^2*(e*cot(d*x+c))^(1/2)+e*(1/8*(a^2*e-b^2*e)*(e^2)^(1/4)/e^2*2^(1 
/2)*(ln((e*cot(d*x+c)+(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2) 
)/(e*cot(d*x+c)-(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2)))+2*a 
rctan(2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)-2*arctan(-2^(1/2)/(e^2)^ 
(1/4)*(e*cot(d*x+c))^(1/2)+1))+1/4*a*b/(e^2)^(1/4)*2^(1/2)*(ln((e*cot(d*x+ 
c)-(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2))/(e*cot(d*x+c)+(e^ 
2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2)))+2*arctan(2^(1/2)/(e^2) 
^(1/4)*(e*cot(d*x+c))^(1/2)+1)-2*arctan(-2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c) 
)^(1/2)+1))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1183 vs. \(2 (166) = 332\).

Time = 0.10 (sec) , antiderivative size = 1183, normalized size of antiderivative = 5.94 \[ \int \frac {(a+b \cot (c+d x))^2}{\sqrt {e \cot (c+d x)}} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/2*(d*e*sqrt(-(4*a^3*b - 4*a*b^3 + d^2*e*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^ 
4*b^4 - 12*a^2*b^6 + b^8)/(d^4*e^2)))/(d^2*e))*log((a^8 - 4*a^6*b^2 - 10*a 
^4*b^4 - 4*a^2*b^6 + b^8)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)) 
+ (2*a*b*d^3*e^2*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/ 
(d^4*e^2)) + (a^6 - 7*a^4*b^2 + 7*a^2*b^4 - b^6)*d*e)*sqrt(-(4*a^3*b - 4*a 
*b^3 + d^2*e*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/(d^4 
*e^2)))/(d^2*e))) - d*e*sqrt(-(4*a^3*b - 4*a*b^3 + d^2*e*sqrt(-(a^8 - 12*a 
^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/(d^4*e^2)))/(d^2*e))*log((a^8 - 4* 
a^6*b^2 - 10*a^4*b^4 - 4*a^2*b^6 + b^8)*sqrt((e*cos(2*d*x + 2*c) + e)/sin( 
2*d*x + 2*c)) - (2*a*b*d^3*e^2*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a 
^2*b^6 + b^8)/(d^4*e^2)) + (a^6 - 7*a^4*b^2 + 7*a^2*b^4 - b^6)*d*e)*sqrt(- 
(4*a^3*b - 4*a*b^3 + d^2*e*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b 
^6 + b^8)/(d^4*e^2)))/(d^2*e))) - d*e*sqrt(-(4*a^3*b - 4*a*b^3 - d^2*e*sqr 
t(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/(d^4*e^2)))/(d^2*e)) 
*log((a^8 - 4*a^6*b^2 - 10*a^4*b^4 - 4*a^2*b^6 + b^8)*sqrt((e*cos(2*d*x + 
2*c) + e)/sin(2*d*x + 2*c)) + (2*a*b*d^3*e^2*sqrt(-(a^8 - 12*a^6*b^2 + 38* 
a^4*b^4 - 12*a^2*b^6 + b^8)/(d^4*e^2)) - (a^6 - 7*a^4*b^2 + 7*a^2*b^4 - b^ 
6)*d*e)*sqrt(-(4*a^3*b - 4*a*b^3 - d^2*e*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4* 
b^4 - 12*a^2*b^6 + b^8)/(d^4*e^2)))/(d^2*e))) + d*e*sqrt(-(4*a^3*b - 4*a*b 
^3 - d^2*e*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/(d^...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b \cot (c+d x))^2}{\sqrt {e \cot (c+d x)}} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 10.20 (sec) , antiderivative size = 1234, normalized size of antiderivative = 6.20 \[ \int \frac {(a+b \cot (c+d x))^2}{\sqrt {e \cot (c+d x)}} \, dx=\text {Too large to display} \] Input:

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

Output:

2*atanh((32*a^4*e^2*(e*cot(c + d*x))^(1/2)*((a*b^3)/(d^2*e) - (b^4*1i)/(4* 
d^2*e) - (a^4*1i)/(4*d^2*e) - (a^3*b)/(d^2*e) + (a^2*b^2*3i)/(2*d^2*e))^(1 
/2))/((a^6*e^2*16i)/d - (b^6*e^2*16i)/d + (32*a*b^5*e^2)/d + (32*a^5*b*e^2 
)/d + (a^2*b^4*e^2*112i)/d - (192*a^3*b^3*e^2)/d - (a^4*b^2*e^2*112i)/d) + 
 (32*b^4*e^2*(e*cot(c + d*x))^(1/2)*((a*b^3)/(d^2*e) - (b^4*1i)/(4*d^2*e) 
- (a^4*1i)/(4*d^2*e) - (a^3*b)/(d^2*e) + (a^2*b^2*3i)/(2*d^2*e))^(1/2))/(( 
a^6*e^2*16i)/d - (b^6*e^2*16i)/d + (32*a*b^5*e^2)/d + (32*a^5*b*e^2)/d + ( 
a^2*b^4*e^2*112i)/d - (192*a^3*b^3*e^2)/d - (a^4*b^2*e^2*112i)/d) - (192*a 
^2*b^2*e^2*(e*cot(c + d*x))^(1/2)*((a*b^3)/(d^2*e) - (b^4*1i)/(4*d^2*e) - 
(a^4*1i)/(4*d^2*e) - (a^3*b)/(d^2*e) + (a^2*b^2*3i)/(2*d^2*e))^(1/2))/((a^ 
6*e^2*16i)/d - (b^6*e^2*16i)/d + (32*a*b^5*e^2)/d + (32*a^5*b*e^2)/d + (a^ 
2*b^4*e^2*112i)/d - (192*a^3*b^3*e^2)/d - (a^4*b^2*e^2*112i)/d))*((a*b^3)/ 
(d^2*e) - (b^4*1i)/(4*d^2*e) - (a^4*1i)/(4*d^2*e) - (a^3*b)/(d^2*e) + (a^2 
*b^2*3i)/(2*d^2*e))^(1/2) + 2*atanh((32*a^4*e^2*(e*cot(c + d*x))^(1/2)*((a 
^4*1i)/(4*d^2*e) + (b^4*1i)/(4*d^2*e) + (a*b^3)/(d^2*e) - (a^3*b)/(d^2*e) 
- (a^2*b^2*3i)/(2*d^2*e))^(1/2))/((b^6*e^2*16i)/d - (a^6*e^2*16i)/d + (32* 
a*b^5*e^2)/d + (32*a^5*b*e^2)/d - (a^2*b^4*e^2*112i)/d - (192*a^3*b^3*e^2) 
/d + (a^4*b^2*e^2*112i)/d) + (32*b^4*e^2*(e*cot(c + d*x))^(1/2)*((a^4*1i)/ 
(4*d^2*e) + (b^4*1i)/(4*d^2*e) + (a*b^3)/(d^2*e) - (a^3*b)/(d^2*e) - (a^2* 
b^2*3i)/(2*d^2*e))^(1/2))/((b^6*e^2*16i)/d - (a^6*e^2*16i)/d + (32*a*b^...
 

Reduce [F]

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

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

Output:

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