\(\int \frac {(a+b \cot (c+d x))^2}{(e \cot (c+d x))^{5/2}} \, dx\) [60]

Optimal result

Integrand size = 25, antiderivative size = 222 \[ \int \frac {(a+b \cot (c+d x))^2}{(e \cot (c+d x))^{5/2}} \, dx=-\frac {\left (a^2+2 a b-b^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{5/2}}+\frac {\left (a^2+2 a b-b^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{5/2}}+\frac {\left (a^2-2 a b-b^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}+\sqrt {e} \cot (c+d x)}\right )}{\sqrt {2} d e^{5/2}}+\frac {2 a^2}{3 d e (e \cot (c+d x))^{3/2}}+\frac {4 a b}{d e^2 \sqrt {e \cot (c+d x)}} \] 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^(5/2)+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^(5/2)+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^(5/2)+2/3*a^2/d/e/(e*cot(d 
*x+c))^(3/2)+4*a*b/d/e^2/(e*cot(d*x+c))^(1/2)
 

Mathematica [C] (verified)

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

Time = 0.19 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.37 \[ \int \frac {(a+b \cot (c+d x))^2}{(e \cot (c+d x))^{5/2}} \, dx=\frac {2 \left (\left (a^2-b^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},1,\frac {1}{4},-\cot ^2(c+d x)\right )+b \left (b+6 a \cot (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},1,\frac {3}{4},-\cot ^2(c+d x)\right )\right )\right )}{3 d e (e \cot (c+d x))^{3/2}} \] Input:

Integrate[(a + b*Cot[c + d*x])^2/(e*Cot[c + d*x])^(5/2),x]
 

Output:

(2*((a^2 - b^2)*Hypergeometric2F1[-3/4, 1, 1/4, -Cot[c + d*x]^2] + b*(b + 
6*a*Cot[c + d*x]*Hypergeometric2F1[-1/4, 1, 3/4, -Cot[c + d*x]^2])))/(3*d* 
e*(e*Cot[c + d*x])^(3/2))
 

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.19, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.680, Rules used = {3042, 4025, 3042, 4012, 25, 3042, 4017, 25, 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.

\(\displaystyle \int \frac {(a+b \cot (c+d x))^2}{(e \cot (c+d x))^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )^2}{\left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx\)

\(\Big \downarrow \) 4025

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4012

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4017

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1482

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

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

Input:

Int[(a + b*Cot[c + d*x])^2/(e*Cot[c + d*x])^(5/2),x]
 

Output:

(2*a^2)/(3*d*e*(e*Cot[c + d*x])^(3/2)) + ((4*a*b)/(d*Sqrt[e*Cot[c + d*x]]) 
 + (2*(((a^2 + 2*a*b - b^2)*(-(ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/S 
qrt[e]]/(Sqrt[2]*Sqrt[e])) + ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqr 
t[e]]/(Sqrt[2]*Sqrt[e])))/2 + ((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)/e^2
 

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 4012

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

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 4025

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

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.47

method	result	size
derivativedivides	\(-\frac {2 \left (-\frac {a^{2}}{3 \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {2 a b}{e \sqrt {e \cot \left (d x +c \right )}}+\frac {\frac {\left (-a^{2} e +b^{2} e \right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}-\frac {a b \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (e^{2}\right )^{\frac {1}{4}}}}{e}\right )}{d e}\)	\(326\)
default	\(-\frac {2 \left (-\frac {a^{2}}{3 \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {2 a b}{e \sqrt {e \cot \left (d x +c \right )}}+\frac {\frac {\left (-a^{2} e +b^{2} e \right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}-\frac {a b \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (e^{2}\right )^{\frac {1}{4}}}}{e}\right )}{d e}\)	\(326\)
parts	\(-\frac {2 a^{2} e \left (-\frac {1}{3 e^{2} \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}-\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 e^{4}}\right )}{d}-\frac {b^{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^{3}}+\frac {2 a b \left (\frac {2}{e^{2} \sqrt {e \cot \left (d x +c \right )}}+\frac {\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^{2} \left (e^{2}\right )^{\frac {1}{4}}}\right )}{d}\)	\(458\)

Input:

int((a+b*cot(d*x+c))^2/(e*cot(d*x+c))^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-2/d/e*(-1/3*a^2/(e*cot(d*x+c))^(3/2)-2*a*b/e/(e*cot(d*x+c))^(1/2)+1/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*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))-1/4*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*arct 
an(-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. 1318 vs. \(2 (186) = 372\).

Time = 0.12 (sec) , antiderivative size = 1318, normalized size of antiderivative = 5.94 \[ \int \frac {(a+b \cot (c+d x))^2}{(e \cot (c+d x))^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate((a+b*cot(d*x+c))^2/(e*cot(d*x+c))^(5/2),x, algorithm="fricas")
 

Output:

1/6*(3*(d*e^3*cos(2*d*x + 2*c) + d*e^3)*sqrt(-(d^2*e^5*sqrt(-(a^8 - 12*a^6 
*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/(d^4*e^10)) + 4*a^3*b - 4*a*b^3)/(d^ 
2*e^5))*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^8*sqrt(-(a^8 - 12*a^6*b^ 
2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/(d^4*e^10)) + (a^6 - 7*a^4*b^2 + 7*a^2* 
b^4 - b^6)*d*e^3)*sqrt(-(d^2*e^5*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12 
*a^2*b^6 + b^8)/(d^4*e^10)) + 4*a^3*b - 4*a*b^3)/(d^2*e^5))) - 3*(d*e^3*co 
s(2*d*x + 2*c) + d*e^3)*sqrt(-(d^2*e^5*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^ 
4 - 12*a^2*b^6 + b^8)/(d^4*e^10)) + 4*a^3*b - 4*a*b^3)/(d^2*e^5))*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^8*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 
 12*a^2*b^6 + b^8)/(d^4*e^10)) + (a^6 - 7*a^4*b^2 + 7*a^2*b^4 - b^6)*d*e^3 
)*sqrt(-(d^2*e^5*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/ 
(d^4*e^10)) + 4*a^3*b - 4*a*b^3)/(d^2*e^5))) - 3*(d*e^3*cos(2*d*x + 2*c) + 
 d*e^3)*sqrt((d^2*e^5*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + 
b^8)/(d^4*e^10)) - 4*a^3*b + 4*a*b^3)/(d^2*e^5))*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^8*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8 
)/(d^4*e^10)) - (a^6 - 7*a^4*b^2 + 7*a^2*b^4 - b^6)*d*e^3)*sqrt((d^2*e^5*s 
qrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/(d^4*e^10)) - 4...
 

Sympy [F]

\[ \int \frac {(a+b \cot (c+d x))^2}{(e \cot (c+d x))^{5/2}} \, dx=\int \frac {\left (a + b \cot {\left (c + d x \right )}\right )^{2}}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \] Input:

integrate((a+b*cot(d*x+c))**2/(e*cot(d*x+c))**(5/2),x)
 

Output:

Integral((a + b*cot(c + d*x))**2/(e*cot(c + d*x))**(5/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b \cot (c+d x))^2}{(e \cot (c+d x))^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((a+b*cot(d*x+c))^2/(e*cot(d*x+c))^(5/2),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]

\[ \int \frac {(a+b \cot (c+d x))^2}{(e \cot (c+d x))^{5/2}} \, dx=\int { \frac {{\left (b \cot \left (d x + c\right ) + a\right )}^{2}}{\left (e \cot \left (d x + c\right )\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate((a+b*cot(d*x+c))^2/(e*cot(d*x+c))^(5/2),x, algorithm="giac")
 

Output:

integrate((b*cot(d*x + c) + a)^2/(e*cot(d*x + c))^(5/2), x)
 

Mupad [B] (verification not implemented)

Time = 10.40 (sec) , antiderivative size = 1214, normalized size of antiderivative = 5.47 \[ \int \frac {(a+b \cot (c+d x))^2}{(e \cot (c+d x))^{5/2}} \, dx=\text {Too large to display} \] Input:

int((a + b*cot(c + d*x))^2/(e*cot(c + d*x))^(5/2),x)
 

Output:

((2*a^2)/3 + 4*a*b*cot(c + d*x))/(d*e*(e*cot(c + d*x))^(3/2)) - 2*atanh((3 
2*a^4*d^3*e^8*(e*cot(c + d*x))^(1/2)*((a^4*1i)/(4*d^2*e^5) + (b^4*1i)/(4*d 
^2*e^5) + (a*b^3)/(d^2*e^5) - (a^3*b)/(d^2*e^5) - (a^2*b^2*3i)/(2*d^2*e^5) 
)^(1/2))/(b^6*d^2*e^6*16i - a^6*d^2*e^6*16i + 32*a*b^5*d^2*e^6 + 32*a^5*b* 
d^2*e^6 - a^2*b^4*d^2*e^6*112i - 192*a^3*b^3*d^2*e^6 + a^4*b^2*d^2*e^6*112 
i) + (32*b^4*d^3*e^8*(e*cot(c + d*x))^(1/2)*((a^4*1i)/(4*d^2*e^5) + (b^4*1 
i)/(4*d^2*e^5) + (a*b^3)/(d^2*e^5) - (a^3*b)/(d^2*e^5) - (a^2*b^2*3i)/(2*d 
^2*e^5))^(1/2))/(b^6*d^2*e^6*16i - a^6*d^2*e^6*16i + 32*a*b^5*d^2*e^6 + 32 
*a^5*b*d^2*e^6 - a^2*b^4*d^2*e^6*112i - 192*a^3*b^3*d^2*e^6 + a^4*b^2*d^2* 
e^6*112i) - (192*a^2*b^2*d^3*e^8*(e*cot(c + d*x))^(1/2)*((a^4*1i)/(4*d^2*e 
^5) + (b^4*1i)/(4*d^2*e^5) + (a*b^3)/(d^2*e^5) - (a^3*b)/(d^2*e^5) - (a^2* 
b^2*3i)/(2*d^2*e^5))^(1/2))/(b^6*d^2*e^6*16i - a^6*d^2*e^6*16i + 32*a*b^5* 
d^2*e^6 + 32*a^5*b*d^2*e^6 - a^2*b^4*d^2*e^6*112i - 192*a^3*b^3*d^2*e^6 + 
a^4*b^2*d^2*e^6*112i))*(((a^3*b*4i - a*b^3*4i + a^4 + b^4 - 6*a^2*b^2)*1i) 
/(4*d^2*e^5))^(1/2) - 2*atanh((32*a^4*d^3*e^8*(e*cot(c + d*x))^(1/2)*((a*b 
^3)/(d^2*e^5) - (b^4*1i)/(4*d^2*e^5) - (a^4*1i)/(4*d^2*e^5) - (a^3*b)/(d^2 
*e^5) + (a^2*b^2*3i)/(2*d^2*e^5))^(1/2))/(a^6*d^2*e^6*16i - b^6*d^2*e^6*16 
i + 32*a*b^5*d^2*e^6 + 32*a^5*b*d^2*e^6 + a^2*b^4*d^2*e^6*112i - 192*a^3*b 
^3*d^2*e^6 - a^4*b^2*d^2*e^6*112i) + (32*b^4*d^3*e^8*(e*cot(c + d*x))^(1/2 
)*((a*b^3)/(d^2*e^5) - (b^4*1i)/(4*d^2*e^5) - (a^4*1i)/(4*d^2*e^5) - (a...
 

Reduce [F]

\[ \int \frac {(a+b \cot (c+d x))^2}{(e \cot (c+d x))^{5/2}} \, dx=\frac {\sqrt {e}\, \left (\left (\int \frac {\sqrt {\cot \left (d x +c \right )}}{\cot \left (d x +c \right )}d x \right ) b^{2}+\left (\int \frac {\sqrt {\cot \left (d x +c \right )}}{\cot \left (d x +c \right )^{3}}d x \right ) a^{2}+2 \left (\int \frac {\sqrt {\cot \left (d x +c \right )}}{\cot \left (d x +c \right )^{2}}d x \right ) a b \right )}{e^{3}} \] Input:

int((a+b*cot(d*x+c))^2/(e*cot(d*x+c))^(5/2),x)
 

Output:

(sqrt(e)*(int(sqrt(cot(c + d*x))/cot(c + d*x),x)*b**2 + int(sqrt(cot(c + d 
*x))/cot(c + d*x)**3,x)*a**2 + 2*int(sqrt(cot(c + d*x))/cot(c + d*x)**2,x) 
*a*b))/e**3