\(\int \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2 \, dx\) [57]

Optimal result

Integrand size = 25, antiderivative size = 219 \[ \int \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2 \, dx=\frac {\left (a^2-2 a b-b^2\right ) \sqrt {e} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d}-\frac {\left (a^2-2 a b-b^2\right ) \sqrt {e} \arctan \left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d}+\frac {\left (a^2+2 a b-b^2\right ) \sqrt {e} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}+\sqrt {e} \cot (c+d x)}\right )}{\sqrt {2} d}-\frac {4 a b \sqrt {e \cot (c+d x)}}{d}-\frac {2 b^2 (e \cot (c+d x))^{3/2}}{3 d e} \] Output:

1/2*(a^2-2*a*b-b^2)*e^(1/2)*arctan(1-2^(1/2)*(e*cot(d*x+c))^(1/2)/e^(1/2)) 
*2^(1/2)/d-1/2*(a^2-2*a*b-b^2)*e^(1/2)*arctan(1+2^(1/2)*(e*cot(d*x+c))^(1/ 
2)/e^(1/2))*2^(1/2)/d+1/2*(a^2+2*a*b-b^2)*e^(1/2)*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-4*a*b*(e*cot(d*x+c))^ 
(1/2)/d-2/3*b^2*(e*cot(d*x+c))^(3/2)/d/e
 

Mathematica [C] (verified)

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

Time = 0.36 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.00 \[ \int \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2 \, dx=-\frac {\sqrt {e \cot (c+d x)} \left (4 \left (a^2-b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\cot ^2(c+d x)\right )+b \left (6 \sqrt {2} a \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-6 \sqrt {2} a \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )+24 a \sqrt {\cot (c+d x)}+4 b \cot ^{\frac {3}{2}}(c+d x)+3 \sqrt {2} a \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-3 \sqrt {2} a \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )\right )}{6 d \sqrt {\cot (c+d x)}} \] Input:

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

Output:

-1/6*(Sqrt[e*Cot[c + d*x]]*(4*(a^2 - b^2)*Cot[c + d*x]^(3/2)*Hypergeometri 
c2F1[3/4, 1, 7/4, -Cot[c + d*x]^2] + b*(6*Sqrt[2]*a*ArcTan[1 - Sqrt[2]*Sqr 
t[Cot[c + d*x]]] - 6*Sqrt[2]*a*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + 24 
*a*Sqrt[Cot[c + d*x]] + 4*b*Cot[c + d*x]^(3/2) + 3*Sqrt[2]*a*Log[1 - Sqrt[ 
2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - 3*Sqrt[2]*a*Log[1 + Sqrt[2]*Sqrt[C 
ot[c + d*x]] + Cot[c + d*x]])))/(d*Sqrt[Cot[c + d*x]])
 

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.19, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 4026, 3042, 4011, 3042, 4017, 27, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4026

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4011

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4017

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1482

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

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

Input:

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

Output:

(-4*a*b*Sqrt[e*Cot[c + d*x]])/d - (2*b^2*(e*Cot[c + d*x])^(3/2))/(3*d*e) + 
 (2*e*(-1/2*((a^2 - 2*a*b - b^2)*(-(ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x 
]])/Sqrt[e]]/(Sqrt[2]*Sqrt[e])) + ArcTan[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]*Sqrt[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 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 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 = 0.28 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.47

method	result	size
derivativedivides	\(-\frac {2 \left (\frac {b^{2} \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+2 a b e \sqrt {e \cot \left (d x +c \right )}+e^{2} \left (-\frac {a b \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 e}+\frac {\left (a^{2}-b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d e}\)	\(321\)
default	\(-\frac {2 \left (\frac {b^{2} \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+2 a b e \sqrt {e \cot \left (d x +c \right )}+e^{2} \left (-\frac {a b \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 e}+\frac {\left (a^{2}-b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d e}\)	\(321\)
parts	\(-\frac {a^{2} e \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 \left (e^{2}\right )^{\frac {1}{4}}}-\frac {2 b^{2} \left (\frac {\left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}-\frac {e^{2} \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 \left (e^{2}\right )^{\frac {1}{4}}}\right )}{d e}+\frac {2 a b \left (-2 \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 )}{4}\right )}{d}\)	\(449\)

Input:

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

Output:

-2/d/e*(1/3*b^2*(e*cot(d*x+c))^(3/2)+2*a*b*e*(e*cot(d*x+c))^(1/2)+e^2*(-1/ 
4*a/e*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))+1/8*(a^2-b^2)/(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. 1230 vs. \(2 (183) = 366\).

Time = 0.11 (sec) , antiderivative size = 1230, normalized size of antiderivative = 5.62 \[ \int \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2 \, dx=\text {Too large to display} \] Input:

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

Output:

-1/6*(3*d*sqrt((d^2*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^ 
8)*e^2/d^4) + 4*(a^3*b - a*b^3)*e)/d^2)*log((a^8 - 4*a^6*b^2 - 10*a^4*b^4 
- 4*a^2*b^6 + b^8)*e*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)) + ((a 
^2 - b^2)*d^3*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)*e^2 
/d^4) - 2*(a^5*b - 6*a^3*b^3 + a*b^5)*d*e)*sqrt((d^2*sqrt(-(a^8 - 12*a^6*b 
^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)*e^2/d^4) + 4*(a^3*b - a*b^3)*e)/d^2))* 
sin(2*d*x + 2*c) - 3*d*sqrt((d^2*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12 
*a^2*b^6 + b^8)*e^2/d^4) + 4*(a^3*b - a*b^3)*e)/d^2)*log((a^8 - 4*a^6*b^2 
- 10*a^4*b^4 - 4*a^2*b^6 + b^8)*e*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x 
+ 2*c)) - ((a^2 - b^2)*d^3*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b 
^6 + b^8)*e^2/d^4) - 2*(a^5*b - 6*a^3*b^3 + a*b^5)*d*e)*sqrt((d^2*sqrt(-(a 
^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)*e^2/d^4) + 4*(a^3*b - a*b 
^3)*e)/d^2))*sin(2*d*x + 2*c) - 3*d*sqrt(-(d^2*sqrt(-(a^8 - 12*a^6*b^2 + 3 
8*a^4*b^4 - 12*a^2*b^6 + b^8)*e^2/d^4) - 4*(a^3*b - a*b^3)*e)/d^2)*log((a^ 
8 - 4*a^6*b^2 - 10*a^4*b^4 - 4*a^2*b^6 + b^8)*e*sqrt((e*cos(2*d*x + 2*c) + 
 e)/sin(2*d*x + 2*c)) + ((a^2 - b^2)*d^3*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4* 
b^4 - 12*a^2*b^6 + b^8)*e^2/d^4) + 2*(a^5*b - 6*a^3*b^3 + a*b^5)*d*e)*sqrt 
(-(d^2*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)*e^2/d^4) - 
 4*(a^3*b - a*b^3)*e)/d^2))*sin(2*d*x + 2*c) + 3*d*sqrt(-(d^2*sqrt(-(a^8 - 
 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)*e^2/d^4) - 4*(a^3*b - a*b^...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

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

integrate((e*cot(d*x+c))^(1/2)*(a+b*cot(d*x+c))^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 = 10.09 (sec) , antiderivative size = 1157, normalized size of antiderivative = 5.28 \[ \int \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2 \, dx=\text {Too large to display} \] Input:

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

Output:

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

Reduce [F]

\[ \int \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2 \, dx=\frac {\sqrt {e}\, \left (-4 \sqrt {\cot \left (d x +c \right )}\, a b -2 \left (\int \frac {\sqrt {\cot \left (d x +c \right )}}{\cot \left (d x +c \right )}d x \right ) a b d +\left (\int \sqrt {\cot \left (d x +c \right )}d x \right ) a^{2} d +\left (\int \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{2}d x \right ) b^{2} d \right )}{d} \] Input:

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

Output:

(sqrt(e)*( - 4*sqrt(cot(c + d*x))*a*b - 2*int(sqrt(cot(c + d*x))/cot(c + d 
*x),x)*a*b*d + int(sqrt(cot(c + d*x)),x)*a**2*d + int(sqrt(cot(c + d*x))*c 
ot(c + d*x)**2,x)*b**2*d))/d