\(\int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx\) [745]

Optimal result

Integrand size = 26, antiderivative size = 211 \[ \int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\frac {\left (\frac {25}{16}-\frac {21 i}{16}\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a^2 d}-\frac {\left (\frac {25}{16}-\frac {21 i}{16}\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a^2 d}+\frac {\left (\frac {25}{16}+\frac {21 i}{16}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\cot (c+d x)}}{1+\cot (c+d x)}\right )}{\sqrt {2} a^2 d}-\frac {25}{8 a^2 d \sqrt {\cot (c+d x)}}+\frac {7 i}{8 a^2 d \sqrt {\cot (c+d x)} (i+\cot (c+d x))}-\frac {1}{4 d \sqrt {\cot (c+d x)} (i a+a \cot (c+d x))^2} \] Output:

(-25/32+21/32*I)*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2))*2^(1/2)/a^2/d+(-25/32 
+21/32*I)*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))*2^(1/2)/a^2/d+(25/32+21/32*I) 
*arctanh(2^(1/2)*cot(d*x+c)^(1/2)/(1+cot(d*x+c)))*2^(1/2)/a^2/d-25/8/a^2/d 
/cot(d*x+c)^(1/2)+7/8*I/a^2/d/cot(d*x+c)^(1/2)/(I+cot(d*x+c))-1/4/d/cot(d* 
x+c)^(1/2)/(I*a+a*cot(d*x+c))^2
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

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

Time = 0.67 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.60 \[ \int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\frac {(-9+7 i \cot (c+d x)) \cot ^2(c+d x)-7 i \cot (c+d x) (i+\cot (c+d x))^2 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\tan ^2(c+d x)\right )-5 (i+\cot (c+d x))^2 \operatorname {Hypergeometric2F1}\left (1,\frac {5}{4},\frac {9}{4},-\tan ^2(c+d x)\right )}{8 a^2 d \cot ^{\frac {5}{2}}(c+d x) (i+\cot (c+d x))^2} \] Input:

Integrate[1/(Cot[c + d*x]^(7/2)*(a + I*a*Tan[c + d*x])^2),x]
 

Output:

((-9 + (7*I)*Cot[c + d*x])*Cot[c + d*x]^2 - (7*I)*Cot[c + d*x]*(I + Cot[c 
+ d*x])^2*Hypergeometric2F1[3/4, 1, 7/4, -Tan[c + d*x]^2] - 5*(I + Cot[c + 
 d*x])^2*Hypergeometric2F1[1, 5/4, 9/4, -Tan[c + d*x]^2])/(8*a^2*d*Cot[c + 
 d*x]^(5/2)*(I + Cot[c + d*x])^2)
 

Rubi [A] (verified)

Time = 0.95 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.16, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.808, Rules used = {3042, 4156, 3042, 4042, 27, 3042, 4079, 3042, 4012, 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.

\(\displaystyle \int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\cot (c+d x)^{7/2} (a+i a \tan (c+d x))^2}dx\)

\(\Big \downarrow \) 4156

\(\displaystyle \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+i a)^2}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (-a \tan \left (c+d x+\frac {\pi }{2}\right )+i a\right )^2}dx\)

\(\Big \downarrow \) 4042

\(\displaystyle \frac {\int -\frac {9 i a-5 a \cot (c+d x)}{2 \cot ^{\frac {3}{2}}(c+d x) (\cot (c+d x) a+i a)}dx}{4 a^2}-\frac {1}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {9 i a-5 a \cot (c+d x)}{\cot ^{\frac {3}{2}}(c+d x) (\cot (c+d x) a+i a)}dx}{8 a^2}-\frac {1}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int \frac {5 \tan \left (c+d x+\frac {\pi }{2}\right ) a+9 i a}{\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (i a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{8 a^2}-\frac {1}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}\)

\(\Big \downarrow \) 4079

\(\displaystyle -\frac {\frac {\int \frac {21 i \cot (c+d x) a^2+25 a^2}{\cot ^{\frac {3}{2}}(c+d x)}dx}{2 a^2}-\frac {7 i}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}-\frac {1}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {\int \frac {25 a^2-21 i a^2 \tan \left (c+d x+\frac {\pi }{2}\right )}{\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{2 a^2}-\frac {7 i}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}-\frac {1}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}\)

\(\Big \downarrow \) 4012

\(\displaystyle -\frac {\frac {\frac {50 a^2}{d \sqrt {\cot (c+d x)}}+\int \frac {21 i a^2-25 a^2 \cot (c+d x)}{\sqrt {\cot (c+d x)}}dx}{2 a^2}-\frac {7 i}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}-\frac {1}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {\frac {50 a^2}{d \sqrt {\cot (c+d x)}}+\int \frac {25 \tan \left (c+d x+\frac {\pi }{2}\right ) a^2+21 i a^2}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a^2}-\frac {7 i}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}-\frac {1}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}\)

\(\Big \downarrow \) 4017

\(\displaystyle -\frac {\frac {\frac {50 a^2}{d \sqrt {\cot (c+d x)}}+\frac {2 \int -\frac {a^2 (21 i-25 \cot (c+d x))}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}}{2 a^2}-\frac {7 i}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}-\frac {1}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {\frac {50 a^2}{d \sqrt {\cot (c+d x)}}-\frac {2 \int \frac {a^2 (21 i-25 \cot (c+d x))}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}}{2 a^2}-\frac {7 i}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}-\frac {1}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {\frac {50 a^2}{d \sqrt {\cot (c+d x)}}-\frac {2 a^2 \int \frac {21 i-25 \cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}}{2 a^2}-\frac {7 i}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}-\frac {1}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}\)

\(\Big \downarrow \) 1482

\(\displaystyle -\frac {\frac {\frac {50 a^2}{d \sqrt {\cot (c+d x)}}-\frac {2 a^2 \left (\left (\frac {25}{2}+\frac {21 i}{2}\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\left (\frac {25}{2}-\frac {21 i}{2}\right ) \int \frac {\cot (c+d x)+1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d}}{2 a^2}-\frac {7 i}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}-\frac {1}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}\)

\(\Big \downarrow \) 1476

\(\displaystyle -\frac {\frac {\frac {50 a^2}{d \sqrt {\cot (c+d x)}}-\frac {2 a^2 \left (\left (\frac {25}{2}+\frac {21 i}{2}\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\left (\frac {25}{2}-\frac {21 i}{2}\right ) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )\right )}{d}}{2 a^2}-\frac {7 i}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}-\frac {1}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}\)

\(\Big \downarrow \) 1082

\(\displaystyle -\frac {\frac {\frac {50 a^2}{d \sqrt {\cot (c+d x)}}-\frac {2 a^2 \left (\left (\frac {25}{2}+\frac {21 i}{2}\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\left (\frac {25}{2}-\frac {21 i}{2}\right ) \left (\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d}}{2 a^2}-\frac {7 i}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}-\frac {1}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}\)

\(\Big \downarrow \) 217

\(\displaystyle -\frac {\frac {\frac {50 a^2}{d \sqrt {\cot (c+d x)}}-\frac {2 a^2 \left (\left (\frac {25}{2}+\frac {21 i}{2}\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\left (\frac {25}{2}-\frac {21 i}{2}\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}}{2 a^2}-\frac {7 i}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}-\frac {1}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}\)

\(\Big \downarrow \) 1479

\(\displaystyle -\frac {\frac {\frac {50 a^2}{d \sqrt {\cot (c+d x)}}-\frac {2 a^2 \left (\left (\frac {25}{2}+\frac {21 i}{2}\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )-\left (\frac {25}{2}-\frac {21 i}{2}\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}}{2 a^2}-\frac {7 i}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}-\frac {1}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {\frac {50 a^2}{d \sqrt {\cot (c+d x)}}-\frac {2 a^2 \left (\left (\frac {25}{2}+\frac {21 i}{2}\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )-\left (\frac {25}{2}-\frac {21 i}{2}\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}}{2 a^2}-\frac {7 i}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}-\frac {1}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {\frac {50 a^2}{d \sqrt {\cot (c+d x)}}-\frac {2 a^2 \left (\left (\frac {25}{2}+\frac {21 i}{2}\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (c+d x)}+1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )-\left (\frac {25}{2}-\frac {21 i}{2}\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}}{2 a^2}-\frac {7 i}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}-\frac {1}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}\)

\(\Big \downarrow \) 1103

\(\displaystyle -\frac {\frac {\frac {50 a^2}{d \sqrt {\cot (c+d x)}}-\frac {2 a^2 \left (\left (\frac {25}{2}+\frac {21 i}{2}\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )-\left (\frac {25}{2}-\frac {21 i}{2}\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}}{2 a^2}-\frac {7 i}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}-\frac {1}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}\)

Input:

Int[1/(Cot[c + d*x]^(7/2)*(a + I*a*Tan[c + d*x])^2),x]
 

Output:

-1/4*1/(d*Sqrt[Cot[c + d*x]]*(I*a + a*Cot[c + d*x])^2) - ((-7*I)/(d*Sqrt[C 
ot[c + d*x]]*(I + Cot[c + d*x])) + ((50*a^2)/(d*Sqrt[Cot[c + d*x]]) - (2*a 
^2*((-25/2 + (21*I)/2)*(-(ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2]) 
+ ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2]) + (25/2 + (21*I)/2)*(-1/ 
2*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]/Sqrt[2] + Log[1 + Sqr 
t[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]/(2*Sqrt[2]))))/d)/(2*a^2))/(8*a^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 4042

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

rule 4079

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + 
(f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim 
p[(a*A + b*B)*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(2*f*m*( 
b*c - a*d))), x] + Simp[1/(2*a*m*(b*c - a*d))   Int[(a + b*Tan[e + f*x])^(m 
 + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(b*c*m - a*d*(2*m + n + 1)) + B*(a*c*m 
- b*d*(n + 1)) + d*(A*b - a*B)*(m + n + 1)*Tan[e + f*x], x], x], x] /; Free 
Q[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] 
 && LtQ[m, 0] &&  !GtQ[n, 0]
 

rule 4156

Int[(cot[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x 
_)]^(n_.))^(p_.), x_Symbol] :> Simp[d^(n*p)   Int[(d*Cot[e + f*x])^(m - n*p 
)*(b + a*Cot[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, d, e, f, m, n, p}, x] && 
  !IntegerQ[m] && IntegersQ[n, p]
 

Maple [A] (verified)

Time = 0.55 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.61

method	result	size
derivativedivides	\(\frac {-\frac {2}{\sqrt {\cot \left (d x +c \right )}}-\frac {\arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{\sqrt {2}-i \sqrt {2}}\right )}{2 \left (\sqrt {2}-i \sqrt {2}\right )}-\frac {\frac {9 \cot \left (d x +c \right )^{\frac {3}{2}}}{2}+\frac {11 i \sqrt {\cot \left (d x +c \right )}}{2}}{4 \left (i+\cot \left (d x +c \right )\right )^{2}}-\frac {23 \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{i \sqrt {2}+\sqrt {2}}\right )}{4 \left (i \sqrt {2}+\sqrt {2}\right )}}{a^{2} d}\)	\(128\)
default	\(\frac {-\frac {2}{\sqrt {\cot \left (d x +c \right )}}-\frac {\arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{\sqrt {2}-i \sqrt {2}}\right )}{2 \left (\sqrt {2}-i \sqrt {2}\right )}-\frac {\frac {9 \cot \left (d x +c \right )^{\frac {3}{2}}}{2}+\frac {11 i \sqrt {\cot \left (d x +c \right )}}{2}}{4 \left (i+\cot \left (d x +c \right )\right )^{2}}-\frac {23 \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{i \sqrt {2}+\sqrt {2}}\right )}{4 \left (i \sqrt {2}+\sqrt {2}\right )}}{a^{2} d}\)	\(128\)

Input:

int(1/cot(d*x+c)^(7/2)/(a+I*a*tan(d*x+c))^2,x,method=_RETURNVERBOSE)
 

Output:

1/a^2/d*(-2/cot(d*x+c)^(1/2)-1/2/(2^(1/2)-I*2^(1/2))*arctan(2*cot(d*x+c)^( 
1/2)/(2^(1/2)-I*2^(1/2)))-1/4*(9/2*cot(d*x+c)^(3/2)+11/2*I*cot(d*x+c)^(1/2 
))/(I+cot(d*x+c))^2-23/4/(I*2^(1/2)+2^(1/2))*arctan(2*cot(d*x+c)^(1/2)/(I* 
2^(1/2)+2^(1/2))))
 

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 602 vs. \(2 (156) = 312\).

Time = 0.10 (sec) , antiderivative size = 602, normalized size of antiderivative = 2.85 \[ \int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx =\text {Too large to display} \] Input:

integrate(1/cot(d*x+c)^(7/2)/(a+I*a*tan(d*x+c))^2,x, algorithm="fricas")
 

Output:

-1/16*(4*(a^2*d*e^(6*I*d*x + 6*I*c) + a^2*d*e^(4*I*d*x + 4*I*c))*sqrt(-1/1 
6*I/(a^4*d^2))*log(-2*(4*(I*a^2*d*e^(2*I*d*x + 2*I*c) - I*a^2*d)*sqrt((I*e 
^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*sqrt(-1/16*I/(a^4*d^2)) 
 - I*e^(2*I*d*x + 2*I*c))*e^(-2*I*d*x - 2*I*c)) - 4*(a^2*d*e^(6*I*d*x + 6* 
I*c) + a^2*d*e^(4*I*d*x + 4*I*c))*sqrt(-1/16*I/(a^4*d^2))*log(-2*(4*(-I*a^ 
2*d*e^(2*I*d*x + 2*I*c) + I*a^2*d)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2* 
I*d*x + 2*I*c) - 1))*sqrt(-1/16*I/(a^4*d^2)) - I*e^(2*I*d*x + 2*I*c))*e^(- 
2*I*d*x - 2*I*c)) + 4*(a^2*d*e^(6*I*d*x + 6*I*c) + a^2*d*e^(4*I*d*x + 4*I* 
c))*sqrt(529/64*I/(a^4*d^2))*log(-1/8*(8*(a^2*d*e^(2*I*d*x + 2*I*c) - a^2* 
d)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*sqrt(529/64 
*I/(a^4*d^2)) + 23)*e^(-2*I*d*x - 2*I*c)/(a^2*d)) - 4*(a^2*d*e^(6*I*d*x + 
6*I*c) + a^2*d*e^(4*I*d*x + 4*I*c))*sqrt(529/64*I/(a^4*d^2))*log(1/8*(8*(a 
^2*d*e^(2*I*d*x + 2*I*c) - a^2*d)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I 
*d*x + 2*I*c) - 1))*sqrt(529/64*I/(a^4*d^2)) - 23)*e^(-2*I*d*x - 2*I*c)/(a 
^2*d)) - sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*(42*I 
*e^(6*I*d*x + 6*I*c) - 33*I*e^(4*I*d*x + 4*I*c) - 10*I*e^(2*I*d*x + 2*I*c) 
 + I))/(a^2*d*e^(6*I*d*x + 6*I*c) + a^2*d*e^(4*I*d*x + 4*I*c))
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\text {Timed out} \] Input:

integrate(1/cot(d*x+c)**(7/2)/(a+I*a*tan(d*x+c))**2,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(1/cot(d*x+c)^(7/2)/(a+I*a*tan(d*x+c))^2,x, algorithm="maxima")
 

Output:

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati 
ve exponent.
                                                                                    
                                                                                    
 

Giac [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.43 \[ \int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\frac {-\left (23 i - 23\right ) \, \sqrt {2} \arctan \left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right ) + \left (2 i + 2\right ) \, \sqrt {2} \arctan \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right ) - 32 \, \sqrt {\tan \left (d x + c\right )} + \frac {2 \, {\left (11 i \, \tan \left (d x + c\right )^{\frac {3}{2}} + 9 \, \sqrt {\tan \left (d x + c\right )}\right )}}{{\left (\tan \left (d x + c\right ) - i\right )}^{2}}}{16 \, a^{2} d} \] Input:

integrate(1/cot(d*x+c)^(7/2)/(a+I*a*tan(d*x+c))^2,x, algorithm="giac")
 

Output:

1/16*(-(23*I - 23)*sqrt(2)*arctan((1/2*I + 1/2)*sqrt(2)*sqrt(tan(d*x + c)) 
) + (2*I + 2)*sqrt(2)*arctan(-(1/2*I - 1/2)*sqrt(2)*sqrt(tan(d*x + c))) - 
32*sqrt(tan(d*x + c)) + 2*(11*I*tan(d*x + c)^(3/2) + 9*sqrt(tan(d*x + c))) 
/(tan(d*x + c) - I)^2)/(a^2*d)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\int \frac {1}{{\mathrm {cot}\left (c+d\,x\right )}^{7/2}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \] Input:

int(1/(cot(c + d*x)^(7/2)*(a + a*tan(c + d*x)*1i)^2),x)
 

Output:

int(1/(cot(c + d*x)^(7/2)*(a + a*tan(c + d*x)*1i)^2), x)
 

Reduce [F]

\[ \int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=-\frac {\int \frac {1}{\sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{3} \tan \left (d x +c \right )^{2}-2 \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{3} \tan \left (d x +c \right ) i -\sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{3}}d x}{a^{2}} \] Input:

int(1/cot(d*x+c)^(7/2)/(a+I*a*tan(d*x+c))^2,x)
 

Output:

( - int(1/(sqrt(cot(c + d*x))*cot(c + d*x)**3*tan(c + d*x)**2 - 2*sqrt(cot 
(c + d*x))*cot(c + d*x)**3*tan(c + d*x)*i - sqrt(cot(c + d*x))*cot(c + d*x 
)**3),x))/a**2