\(\int \frac {(a+b \cot (c+d x))^3}{\sqrt {e \cot (c+d x)}} \, dx\) [64]

Optimal result

Integrand size = 25, antiderivative size = 244 \[ \int \frac {(a+b \cot (c+d x))^3}{\sqrt {e \cot (c+d x)}} \, dx=\frac {(a-b) \left (a^2+4 a b+b^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d \sqrt {e}}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d \sqrt {e}}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}+\sqrt {e} \cot (c+d x)}\right )}{\sqrt {2} d \sqrt {e}}-\frac {16 a b^2 \sqrt {e \cot (c+d x)}}{3 d e}-\frac {2 b^2 \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}{3 d e} \] Output:

1/2*(a-b)*(a^2+4*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-b)*(a^2+4*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+b)*(a^2-4*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)-16/ 
3*a*b^2*(e*cot(d*x+c))^(1/2)/d/e-2/3*b^2*(e*cot(d*x+c))^(1/2)*(a+b*cot(d*x 
+c))/d/e
 

Mathematica [C] (verified)

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

Time = 0.60 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b \cot (c+d x))^3}{\sqrt {e \cot (c+d x)}} \, dx=-\frac {\sqrt {\cot (c+d x)} \left (72 a b^2 \sqrt {\cot (c+d x)}+8 b^3 \cot ^{\frac {3}{2}}(c+d x)-8 b \left (-3 a^2+b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\cot ^2(c+d x)\right )-3 \sqrt {2} a \left (a^2-3 b^2\right ) \left (2 \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-2 \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )+\log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-\log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )\right )}{12 d \sqrt {e \cot (c+d x)}} \] Input:

Integrate[(a + b*Cot[c + d*x])^3/Sqrt[e*Cot[c + d*x]],x]
 

Output:

-1/12*(Sqrt[Cot[c + d*x]]*(72*a*b^2*Sqrt[Cot[c + d*x]] + 8*b^3*Cot[c + d*x 
]^(3/2) - 8*b*(-3*a^2 + b^2)*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 
7/4, -Cot[c + d*x]^2] - 3*Sqrt[2]*a*(a^2 - 3*b^2)*(2*ArcTan[1 - Sqrt[2]*Sq 
rt[Cot[c + d*x]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + Log[1 - Sqr 
t[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x 
]] + Cot[c + d*x]])))/(d*Sqrt[e*Cot[c + d*x]])
 

Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.17, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {3042, 4049, 27, 3042, 4113, 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.

\(\displaystyle \int \frac {(a+b \cot (c+d x))^3}{\sqrt {e \cot (c+d x)}} \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4049

\(\displaystyle -\frac {2 \int -\frac {8 a b^2 e \cot ^2(c+d x)+3 b \left (3 a^2-b^2\right ) e \cot (c+d x)+a \left (3 a^2-b^2\right ) e}{2 \sqrt {e \cot (c+d x)}}dx}{3 e}-\frac {2 b^2 \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}{3 d e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {8 a b^2 e \cot ^2(c+d x)+3 b \left (3 a^2-b^2\right ) e \cot (c+d x)+a \left (3 a^2-b^2\right ) e}{\sqrt {e \cot (c+d x)}}dx}{3 e}-\frac {2 b^2 \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}{3 d e}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {8 a b^2 e \tan \left (c+d x+\frac {\pi }{2}\right )^2-3 b \left (3 a^2-b^2\right ) e \tan \left (c+d x+\frac {\pi }{2}\right )+a \left (3 a^2-b^2\right ) e}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 e}-\frac {2 b^2 \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}{3 d e}\)

\(\Big \downarrow \) 4113

\(\displaystyle \frac {\int \frac {3 a \left (a^2-3 b^2\right ) e+3 b \left (3 a^2-b^2\right ) \cot (c+d x) e}{\sqrt {e \cot (c+d x)}}dx-\frac {16 a b^2 \sqrt {e \cot (c+d x)}}{d}}{3 e}-\frac {2 b^2 \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}{3 d e}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {3 a \left (a^2-3 b^2\right ) e-3 b \left (3 a^2-b^2\right ) e \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {16 a b^2 \sqrt {e \cot (c+d x)}}{d}}{3 e}-\frac {2 b^2 \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}{3 d e}\)

\(\Big \downarrow \) 4017

\(\displaystyle \frac {\frac {2 \int -\frac {3 e \left (a \left (a^2-3 b^2\right ) e+b \left (3 a^2-b^2\right ) \cot (c+d x) e\right )}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d}-\frac {16 a b^2 \sqrt {e \cot (c+d x)}}{d}}{3 e}-\frac {2 b^2 \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}{3 d e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {6 e \int \frac {a \left (a^2-3 b^2\right ) e+b \left (3 a^2-b^2\right ) \cot (c+d x) e}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d}-\frac {16 a b^2 \sqrt {e \cot (c+d x)}}{d}}{3 e}-\frac {2 b^2 \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}{3 d e}\)

\(\Big \downarrow \) 1482

\(\displaystyle \frac {-\frac {6 e \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {\cot (c+d x) e+e}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}\right )}{d}-\frac {16 a b^2 \sqrt {e \cot (c+d x)}}{d}}{3 e}-\frac {2 b^2 \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}{3 d e}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {-\frac {6 e \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}\right )\right )}{d}-\frac {16 a b^2 \sqrt {e \cot (c+d x)}}{d}}{3 e}-\frac {2 b^2 \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}{3 d e}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {-\frac {6 e \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\int \frac {1}{-e \cot (c+d x)-1}d\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}-\frac {\int \frac {1}{-e \cot (c+d x)-1}d\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}-\frac {16 a b^2 \sqrt {e \cot (c+d x)}}{d}}{3 e}-\frac {2 b^2 \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}{3 d e}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {-\frac {6 e \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}-\frac {16 a b^2 \sqrt {e \cot (c+d x)}}{d}}{3 e}-\frac {2 b^2 \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}{3 d e}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {-\frac {6 e \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (-\frac {\int -\frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}\right )+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}-\frac {16 a b^2 \sqrt {e \cot (c+d x)}}{d}}{3 e}-\frac {2 b^2 \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}{3 d e}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {-\frac {6 e \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}\right )+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}-\frac {16 a b^2 \sqrt {e \cot (c+d x)}}{d}}{3 e}-\frac {2 b^2 \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}{3 d e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {6 e \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {e}}\right )+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}-\frac {16 a b^2 \sqrt {e \cot (c+d x)}}{d}}{3 e}-\frac {2 b^2 \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}{3 d e}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {-\frac {6 e \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\log \left (e \cot (c+d x)+\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (e \cot (c+d x)-\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d}-\frac {16 a b^2 \sqrt {e \cot (c+d x)}}{d}}{3 e}-\frac {2 b^2 \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}{3 d e}\)

Input:

Int[(a + b*Cot[c + d*x])^3/Sqrt[e*Cot[c + d*x]],x]
 

Output:

(-2*b^2*Sqrt[e*Cot[c + d*x]]*(a + b*Cot[c + d*x]))/(3*d*e) + ((-16*a*b^2*S 
qrt[e*Cot[c + d*x]])/d - (6*e*(((a - b)*(a^2 + 4*a*b + b^2)*(-(ArcTan[1 - 
(Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]]/(Sqrt[2]*Sqrt[e])) + ArcTan[1 + (S 
qrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]]/(Sqrt[2]*Sqrt[e])))/2 + ((a + b)*(a^ 
2 - 4*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]*S 
qrt[e*Cot[c + d*x]]]/(2*Sqrt[2]*Sqrt[e])))/2))/d)/(3*e)
 

Defintions of rubi rules used

rule 25

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

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 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 4049

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b^2*(a + b*Tan[e + f*x])^(m - 2)*((c 
+ d*Tan[e + f*x])^(n + 1)/(d*f*(m + n - 1))), x] + Simp[1/(d*(m + n - 1)) 
 Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^n*Simp[a^3*d*(m + n 
- 1) - b^2*(b*c*(m - 2) + a*d*(1 + n)) + b*d*(m + n - 1)*(3*a^2 - b^2)*Tan[ 
e + f*x] - b^2*(b*c*(m - 2) - a*d*(3*m + 2*n - 4))*Tan[e + f*x]^2, x], x], 
x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2 
, 0] && NeQ[c^2 + d^2, 0] && IntegerQ[2*m] && GtQ[m, 2] && (GeQ[n, -1] || I 
ntegerQ[m]) &&  !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])) 
)
 

rule 4113

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) 
+ (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*((a + 
 b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f*x])^m*Si 
mp[A - C + B*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && 
NeQ[A*b^2 - a*b*B + a^2*C, 0] &&  !LeQ[m, -1]
 

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.38

method	result	size
derivativedivides	\(-\frac {2 \left (\frac {b^{3} \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+3 a \,b^{2} e \sqrt {e \cot \left (d x +c \right )}+e^{2} \left (\frac {\left (a^{3} e -3 a e \,b^{2}\right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}+\frac {\left (3 a^{2} b -b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d \,e^{2}}\)	\(337\)
default	\(-\frac {2 \left (\frac {b^{3} \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+3 a \,b^{2} e \sqrt {e \cot \left (d x +c \right )}+e^{2} \left (\frac {\left (a^{3} e -3 a e \,b^{2}\right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}+\frac {\left (3 a^{2} b -b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d \,e^{2}}\)	\(337\)
parts	\(-\frac {a^{3} \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^{3} \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^{2}}-\frac {6 a \,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 {3 a^{2} 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 d \left (e^{2}\right )^{\frac {1}{4}}}\)	\(592\)

Input:

int((a+b*cot(d*x+c))^3/(e*cot(d*x+c))^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-2/d/e^2*(1/3*b^3*(e*cot(d*x+c))^(3/2)+3*a*b^2*e*(e*cot(d*x+c))^(1/2)+e^2* 
(1/8*(a^3*e-3*a*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*co 
t(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*(3*a^2*b-b^3)/(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. 1633 vs. \(2 (205) = 410\).

Time = 0.18 (sec) , antiderivative size = 1633, normalized size of antiderivative = 6.69 \[ \int \frac {(a+b \cot (c+d x))^3}{\sqrt {e \cot (c+d x)}} \, dx=\text {Too large to display} \] Input:

integrate((a+b*cot(d*x+c))^3/(e*cot(d*x+c))^(1/2),x, algorithm="fricas")
 

Output:

1/6*(3*d*e*sqrt(-(6*a^5*b - 20*a^3*b^3 + 6*a*b^5 + d^2*e*sqrt(-(a^12 - 30* 
a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/( 
d^4*e^2)))/(d^2*e))*log(-(a^12 - 12*a^10*b^2 - 27*a^8*b^4 + 27*a^4*b^8 + 1 
2*a^2*b^10 - b^12)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)) + ((3*a 
^2*b - b^3)*d^3*e^2*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 
+ 255*a^4*b^8 - 30*a^2*b^10 + b^12)/(d^4*e^2)) + (a^9 - 18*a^7*b^2 + 60*a^ 
5*b^4 - 46*a^3*b^6 + 3*a*b^8)*d*e)*sqrt(-(6*a^5*b - 20*a^3*b^3 + 6*a*b^5 + 
 d^2*e*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 
 - 30*a^2*b^10 + b^12)/(d^4*e^2)))/(d^2*e)))*sin(2*d*x + 2*c) - 3*d*e*sqrt 
(-(6*a^5*b - 20*a^3*b^3 + 6*a*b^5 + d^2*e*sqrt(-(a^12 - 30*a^10*b^2 + 255* 
a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/(d^4*e^2)))/(d^2 
*e))*log(-(a^12 - 12*a^10*b^2 - 27*a^8*b^4 + 27*a^4*b^8 + 12*a^2*b^10 - b^ 
12)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)) - ((3*a^2*b - b^3)*d^3 
*e^2*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 
 30*a^2*b^10 + b^12)/(d^4*e^2)) + (a^9 - 18*a^7*b^2 + 60*a^5*b^4 - 46*a^3* 
b^6 + 3*a*b^8)*d*e)*sqrt(-(6*a^5*b - 20*a^3*b^3 + 6*a*b^5 + d^2*e*sqrt(-(a 
^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 
+ b^12)/(d^4*e^2)))/(d^2*e)))*sin(2*d*x + 2*c) - 3*d*e*sqrt(-(6*a^5*b - 20 
*a^3*b^3 + 6*a*b^5 - d^2*e*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a 
^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/(d^4*e^2)))/(d^2*e))*log(-(a...
 

Sympy [F]

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

integrate((a+b*cot(d*x+c))**3/(e*cot(d*x+c))**(1/2),x)
 

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b \cot (c+d x))^3}{\sqrt {e \cot (c+d x)}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((a+b*cot(d*x+c))^3/(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]

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

integrate((a+b*cot(d*x+c))^3/(e*cot(d*x+c))^(1/2),x, algorithm="giac")
 

Output:

integrate((b*cot(d*x + c) + a)^3/sqrt(e*cot(d*x + c)), x)
 

Mupad [B] (verification not implemented)

Time = 9.88 (sec) , antiderivative size = 1896, normalized size of antiderivative = 7.77 \[ \int \frac {(a+b \cot (c+d x))^3}{\sqrt {e \cot (c+d x)}} \, dx=\text {Too large to display} \] Input:

int((a + b*cot(c + d*x))^3/(e*cot(c + d*x))^(1/2),x)
 

Output:

atan((((16*(e*cot(c + d*x))^(1/2)*(a^6*e^2 - b^6*e^2 + 15*a^2*b^4*e^2 - 15 
*a^4*b^2*e^2))/d^2 - (8*(4*a^3*d^2*e^3 - 12*a*b^2*d^2*e^3)*(((a*b^5*6i + a 
^5*b*6i - a^6 + b^6 - 15*a^2*b^4 - a^3*b^3*20i + 15*a^4*b^2)*1i)/(4*d^2*e) 
)^(1/2))/d^3)*(((a*b^5*6i + a^5*b*6i - a^6 + b^6 - 15*a^2*b^4 - a^3*b^3*20 
i + 15*a^4*b^2)*1i)/(4*d^2*e))^(1/2)*1i + ((16*(e*cot(c + d*x))^(1/2)*(a^6 
*e^2 - b^6*e^2 + 15*a^2*b^4*e^2 - 15*a^4*b^2*e^2))/d^2 + (8*(4*a^3*d^2*e^3 
 - 12*a*b^2*d^2*e^3)*(((a*b^5*6i + a^5*b*6i - a^6 + b^6 - 15*a^2*b^4 - a^3 
*b^3*20i + 15*a^4*b^2)*1i)/(4*d^2*e))^(1/2))/d^3)*(((a*b^5*6i + a^5*b*6i - 
 a^6 + b^6 - 15*a^2*b^4 - a^3*b^3*20i + 15*a^4*b^2)*1i)/(4*d^2*e))^(1/2)*1 
i)/(((16*(e*cot(c + d*x))^(1/2)*(a^6*e^2 - b^6*e^2 + 15*a^2*b^4*e^2 - 15*a 
^4*b^2*e^2))/d^2 + (8*(4*a^3*d^2*e^3 - 12*a*b^2*d^2*e^3)*(((a*b^5*6i + a^5 
*b*6i - a^6 + b^6 - 15*a^2*b^4 - a^3*b^3*20i + 15*a^4*b^2)*1i)/(4*d^2*e))^ 
(1/2))/d^3)*(((a*b^5*6i + a^5*b*6i - a^6 + b^6 - 15*a^2*b^4 - a^3*b^3*20i 
+ 15*a^4*b^2)*1i)/(4*d^2*e))^(1/2) - ((16*(e*cot(c + d*x))^(1/2)*(a^6*e^2 
- b^6*e^2 + 15*a^2*b^4*e^2 - 15*a^4*b^2*e^2))/d^2 - (8*(4*a^3*d^2*e^3 - 12 
*a*b^2*d^2*e^3)*(((a*b^5*6i + a^5*b*6i - a^6 + b^6 - 15*a^2*b^4 - a^3*b^3* 
20i + 15*a^4*b^2)*1i)/(4*d^2*e))^(1/2))/d^3)*(((a*b^5*6i + a^5*b*6i - a^6 
+ b^6 - 15*a^2*b^4 - a^3*b^3*20i + 15*a^4*b^2)*1i)/(4*d^2*e))^(1/2) + (16* 
(3*a^8*b*e^2 - b^9*e^2 + 6*a^4*b^5*e^2 + 8*a^6*b^3*e^2))/d^3))*(((a*b^5*6i 
 + a^5*b*6i - a^6 + b^6 - 15*a^2*b^4 - a^3*b^3*20i + 15*a^4*b^2)*1i)/(4...
 

Reduce [F]

\[ \int \frac {(a+b \cot (c+d x))^3}{\sqrt {e \cot (c+d x)}} \, dx=\frac {\sqrt {e}\, \left (-6 \sqrt {\cot \left (d x +c \right )}\, a \,b^{2}+\left (\int \frac {\sqrt {\cot \left (d x +c \right )}}{\cot \left (d x +c \right )}d x \right ) a^{3} d -3 \left (\int \frac {\sqrt {\cot \left (d x +c \right )}}{\cot \left (d x +c \right )}d x \right ) a \,b^{2} d +3 \left (\int \sqrt {\cot \left (d x +c \right )}d x \right ) a^{2} b d +\left (\int \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{2}d x \right ) b^{3} d \right )}{d e} \] Input:

int((a+b*cot(d*x+c))^3/(e*cot(d*x+c))^(1/2),x)
 

Output:

(sqrt(e)*( - 6*sqrt(cot(c + d*x))*a*b**2 + int(sqrt(cot(c + d*x))/cot(c + 
d*x),x)*a**3*d - 3*int(sqrt(cot(c + d*x))/cot(c + d*x),x)*a*b**2*d + 3*int 
(sqrt(cot(c + d*x)),x)*a**2*b*d + int(sqrt(cot(c + d*x))*cot(c + d*x)**2,x 
)*b**3*d))/(d*e)