\(\int \frac {1}{\sqrt {d \tan (e+f x)} (a+i a \tan (e+f x))^2} \, dx\) [176]

Optimal result

Integrand size = 28, antiderivative size = 241 \[ \int \frac {1}{\sqrt {d \tan (e+f x)} (a+i a \tan (e+f x))^2} \, dx=-\frac {\left (\frac {9}{16}-\frac {5 i}{16}\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a^2 \sqrt {d} f}+\frac {\left (\frac {9}{16}-\frac {5 i}{16}\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a^2 \sqrt {d} f}+\frac {\left (\frac {9}{16}+\frac {5 i}{16}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}+\sqrt {d} \tan (e+f x)}\right )}{\sqrt {2} a^2 \sqrt {d} f}+\frac {5 \sqrt {d \tan (e+f x)}}{8 a^2 d f (1+i \tan (e+f x))}+\frac {\sqrt {d \tan (e+f x)}}{4 d f (a+i a \tan (e+f x))^2} \] Output:

(-9/32+5/32*I)*arctan(1-2^(1/2)*(d*tan(f*x+e))^(1/2)/d^(1/2))*2^(1/2)/a^2/ 
d^(1/2)/f+(9/32-5/32*I)*arctan(1+2^(1/2)*(d*tan(f*x+e))^(1/2)/d^(1/2))*2^( 
1/2)/a^2/d^(1/2)/f+(9/32+5/32*I)*arctanh(2^(1/2)*(d*tan(f*x+e))^(1/2)/(d^( 
1/2)+d^(1/2)*tan(f*x+e)))*2^(1/2)/a^2/d^(1/2)/f+5/8*(d*tan(f*x+e))^(1/2)/a 
^2/d/f/(1+I*tan(f*x+e))+1/4*(d*tan(f*x+e))^(1/2)/d/f/(a+I*a*tan(f*x+e))^2
 

Mathematica [A] (verified)

Time = 1.83 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.51 \[ \int \frac {1}{\sqrt {d \tan (e+f x)} (a+i a \tan (e+f x))^2} \, dx=\frac {-2 \sqrt [4]{-1} \sqrt {d} \arctan \left (\frac {(-1)^{3/4} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )-7 \sqrt [4]{-1} \sqrt {d} \text {arctanh}\left (\frac {(-1)^{3/4} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )+\frac {(-7-5 i \tan (e+f x)) \sqrt {d \tan (e+f x)}}{(-i+\tan (e+f x))^2}}{8 a^2 d f} \] Input:

Integrate[1/(Sqrt[d*Tan[e + f*x]]*(a + I*a*Tan[e + f*x])^2),x]
 

Output:

(-2*(-1)^(1/4)*Sqrt[d]*ArcTan[((-1)^(3/4)*Sqrt[d*Tan[e + f*x]])/Sqrt[d]] - 
 7*(-1)^(1/4)*Sqrt[d]*ArcTanh[((-1)^(3/4)*Sqrt[d*Tan[e + f*x]])/Sqrt[d]] + 
 ((-7 - (5*I)*Tan[e + f*x])*Sqrt[d*Tan[e + f*x]])/(-I + Tan[e + f*x])^2)/( 
8*a^2*d*f)
 

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.15, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {3042, 4042, 27, 3042, 4079, 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.

\(\displaystyle \int \frac {1}{(a+i a \tan (e+f x))^2 \sqrt {d \tan (e+f x)}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{(a+i a \tan (e+f x))^2 \sqrt {d \tan (e+f x)}}dx\)

\(\Big \downarrow \) 4042

\(\displaystyle \frac {\int \frac {7 a d-3 i a d \tan (e+f x)}{2 \sqrt {d \tan (e+f x)} (i \tan (e+f x) a+a)}dx}{4 a^2 d}+\frac {\sqrt {d \tan (e+f x)}}{4 d f (a+i a \tan (e+f x))^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {7 a d-3 i a d \tan (e+f x)}{\sqrt {d \tan (e+f x)} (i \tan (e+f x) a+a)}dx}{8 a^2 d}+\frac {\sqrt {d \tan (e+f x)}}{4 d f (a+i a \tan (e+f x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {7 a d-3 i a d \tan (e+f x)}{\sqrt {d \tan (e+f x)} (i \tan (e+f x) a+a)}dx}{8 a^2 d}+\frac {\sqrt {d \tan (e+f x)}}{4 d f (a+i a \tan (e+f x))^2}\)

\(\Big \downarrow \) 4079

\(\displaystyle \frac {\frac {\int \frac {9 a^2 d^2-5 i a^2 d^2 \tan (e+f x)}{\sqrt {d \tan (e+f x)}}dx}{2 a^2 d}+\frac {5 \sqrt {d \tan (e+f x)}}{f (1+i \tan (e+f x))}}{8 a^2 d}+\frac {\sqrt {d \tan (e+f x)}}{4 d f (a+i a \tan (e+f x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {9 a^2 d^2-5 i a^2 d^2 \tan (e+f x)}{\sqrt {d \tan (e+f x)}}dx}{2 a^2 d}+\frac {5 \sqrt {d \tan (e+f x)}}{f (1+i \tan (e+f x))}}{8 a^2 d}+\frac {\sqrt {d \tan (e+f x)}}{4 d f (a+i a \tan (e+f x))^2}\)

\(\Big \downarrow \) 4017

\(\displaystyle \frac {\frac {\int \frac {a^2 d^2 (9 d-5 i d \tan (e+f x))}{\tan ^2(e+f x) d^2+d^2}d\sqrt {d \tan (e+f x)}}{a^2 d f}+\frac {5 \sqrt {d \tan (e+f x)}}{f (1+i \tan (e+f x))}}{8 a^2 d}+\frac {\sqrt {d \tan (e+f x)}}{4 d f (a+i a \tan (e+f x))^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {d \int \frac {9 d-5 i d \tan (e+f x)}{\tan ^2(e+f x) d^2+d^2}d\sqrt {d \tan (e+f x)}}{f}+\frac {5 \sqrt {d \tan (e+f x)}}{f (1+i \tan (e+f x))}}{8 a^2 d}+\frac {\sqrt {d \tan (e+f x)}}{4 d f (a+i a \tan (e+f x))^2}\)

\(\Big \downarrow \) 1482

\(\displaystyle \frac {\frac {d \left (\left (\frac {9}{2}+\frac {5 i}{2}\right ) \int \frac {d-d \tan (e+f x)}{\tan ^2(e+f x) d^2+d^2}d\sqrt {d \tan (e+f x)}+\left (\frac {9}{2}-\frac {5 i}{2}\right ) \int \frac {\tan (e+f x) d+d}{\tan ^2(e+f x) d^2+d^2}d\sqrt {d \tan (e+f x)}\right )}{f}+\frac {5 \sqrt {d \tan (e+f x)}}{f (1+i \tan (e+f x))}}{8 a^2 d}+\frac {\sqrt {d \tan (e+f x)}}{4 d f (a+i a \tan (e+f x))^2}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {\frac {d \left (\left (\frac {9}{2}+\frac {5 i}{2}\right ) \int \frac {d-d \tan (e+f x)}{\tan ^2(e+f x) d^2+d^2}d\sqrt {d \tan (e+f x)}+\left (\frac {9}{2}-\frac {5 i}{2}\right ) \left (\frac {1}{2} \int \frac {1}{\tan (e+f x) d+d-\sqrt {2} \sqrt {d \tan (e+f x)} \sqrt {d}}d\sqrt {d \tan (e+f x)}+\frac {1}{2} \int \frac {1}{\tan (e+f x) d+d+\sqrt {2} \sqrt {d \tan (e+f x)} \sqrt {d}}d\sqrt {d \tan (e+f x)}\right )\right )}{f}+\frac {5 \sqrt {d \tan (e+f x)}}{f (1+i \tan (e+f x))}}{8 a^2 d}+\frac {\sqrt {d \tan (e+f x)}}{4 d f (a+i a \tan (e+f x))^2}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {\frac {d \left (\left (\frac {9}{2}+\frac {5 i}{2}\right ) \int \frac {d-d \tan (e+f x)}{\tan ^2(e+f x) d^2+d^2}d\sqrt {d \tan (e+f x)}+\left (\frac {9}{2}-\frac {5 i}{2}\right ) \left (\frac {\int \frac {1}{-d \tan (e+f x)-1}d\left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d}}-\frac {\int \frac {1}{-d \tan (e+f x)-1}d\left (\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt {d}}\right )\right )}{f}+\frac {5 \sqrt {d \tan (e+f x)}}{f (1+i \tan (e+f x))}}{8 a^2 d}+\frac {\sqrt {d \tan (e+f x)}}{4 d f (a+i a \tan (e+f x))^2}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\frac {d \left (\left (\frac {9}{2}+\frac {5 i}{2}\right ) \int \frac {d-d \tan (e+f x)}{\tan ^2(e+f x) d^2+d^2}d\sqrt {d \tan (e+f x)}+\left (\frac {9}{2}-\frac {5 i}{2}\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d}}\right )\right )}{f}+\frac {5 \sqrt {d \tan (e+f x)}}{f (1+i \tan (e+f x))}}{8 a^2 d}+\frac {\sqrt {d \tan (e+f x)}}{4 d f (a+i a \tan (e+f x))^2}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {\frac {d \left (\left (\frac {9}{2}+\frac {5 i}{2}\right ) \left (-\frac {\int -\frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \tan (e+f x)}}{\tan (e+f x) d+d-\sqrt {2} \sqrt {d \tan (e+f x)} \sqrt {d}}d\sqrt {d \tan (e+f x)}}{2 \sqrt {2} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\tan (e+f x) d+d+\sqrt {2} \sqrt {d \tan (e+f x)} \sqrt {d}}d\sqrt {d \tan (e+f x)}}{2 \sqrt {2} \sqrt {d}}\right )+\left (\frac {9}{2}-\frac {5 i}{2}\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d}}\right )\right )}{f}+\frac {5 \sqrt {d \tan (e+f x)}}{f (1+i \tan (e+f x))}}{8 a^2 d}+\frac {\sqrt {d \tan (e+f x)}}{4 d f (a+i a \tan (e+f x))^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {d \left (\left (\frac {9}{2}+\frac {5 i}{2}\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \tan (e+f x)}}{\tan (e+f x) d+d-\sqrt {2} \sqrt {d \tan (e+f x)} \sqrt {d}}d\sqrt {d \tan (e+f x)}}{2 \sqrt {2} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\tan (e+f x) d+d+\sqrt {2} \sqrt {d \tan (e+f x)} \sqrt {d}}d\sqrt {d \tan (e+f x)}}{2 \sqrt {2} \sqrt {d}}\right )+\left (\frac {9}{2}-\frac {5 i}{2}\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d}}\right )\right )}{f}+\frac {5 \sqrt {d \tan (e+f x)}}{f (1+i \tan (e+f x))}}{8 a^2 d}+\frac {\sqrt {d \tan (e+f x)}}{4 d f (a+i a \tan (e+f x))^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {d \left (\left (\frac {9}{2}+\frac {5 i}{2}\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \tan (e+f x)}}{\tan (e+f x) d+d-\sqrt {2} \sqrt {d \tan (e+f x)} \sqrt {d}}d\sqrt {d \tan (e+f x)}}{2 \sqrt {2} \sqrt {d}}+\frac {\int \frac {\sqrt {d}+\sqrt {2} \sqrt {d \tan (e+f x)}}{\tan (e+f x) d+d+\sqrt {2} \sqrt {d \tan (e+f x)} \sqrt {d}}d\sqrt {d \tan (e+f x)}}{2 \sqrt {d}}\right )+\left (\frac {9}{2}-\frac {5 i}{2}\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d}}\right )\right )}{f}+\frac {5 \sqrt {d \tan (e+f x)}}{f (1+i \tan (e+f x))}}{8 a^2 d}+\frac {\sqrt {d \tan (e+f x)}}{4 d f (a+i a \tan (e+f x))^2}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {\frac {d \left (\left (\frac {9}{2}-\frac {5 i}{2}\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d}}\right )+\left (\frac {9}{2}+\frac {5 i}{2}\right ) \left (\frac {\log \left (d \tan (e+f x)+\sqrt {2} \sqrt {d} \sqrt {d \tan (e+f x)}+d\right )}{2 \sqrt {2} \sqrt {d}}-\frac {\log \left (d \tan (e+f x)-\sqrt {2} \sqrt {d} \sqrt {d \tan (e+f x)}+d\right )}{2 \sqrt {2} \sqrt {d}}\right )\right )}{f}+\frac {5 \sqrt {d \tan (e+f x)}}{f (1+i \tan (e+f x))}}{8 a^2 d}+\frac {\sqrt {d \tan (e+f x)}}{4 d f (a+i a \tan (e+f x))^2}\)

Input:

Int[1/(Sqrt[d*Tan[e + f*x]]*(a + I*a*Tan[e + f*x])^2),x]
 

Output:

Sqrt[d*Tan[e + f*x]]/(4*d*f*(a + I*a*Tan[e + f*x])^2) + ((d*((9/2 - (5*I)/ 
2)*(-(ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]]/(Sqrt[2]*Sqrt[d]) 
) + ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]]/(Sqrt[2]*Sqrt[d])) 
+ (9/2 + (5*I)/2)*(-1/2*Log[d + d*Tan[e + f*x] - Sqrt[2]*Sqrt[d]*Sqrt[d*Ta 
n[e + f*x]]]/(Sqrt[2]*Sqrt[d]) + Log[d + d*Tan[e + f*x] + Sqrt[2]*Sqrt[d]* 
Sqrt[d*Tan[e + f*x]]]/(2*Sqrt[2]*Sqrt[d]))))/f + (5*Sqrt[d*Tan[e + f*x]])/ 
(f*(1 + I*Tan[e + f*x])))/(8*a^2*d)
 

Defintions of rubi rules used

rule 25

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

rule 27

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

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

rule 1482

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
a*c, 2]}, Simp[(d*q + a*e)/(2*a*c)   Int[(q + c*x^2)/(a + c*x^4), x], x] + 
Simp[(d*q - a*e)/(2*a*c)   Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ[{a 
, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(- 
a)*c]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4017

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_ 
)]], x_Symbol] :> Simp[2/f   Subst[Int[(b*c + d*x^2)/(b^2 + x^4), x], x, Sq 
rt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2, 0] & 
& NeQ[c^2 + d^2, 0]
 

rule 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]
 

Maple [A] (verified)

Time = 1.53 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.49

method	result	size
derivativedivides	\(\frac {2 d^{3} \left (-\frac {\frac {-\frac {5 i \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{2}-\frac {7 d \sqrt {d \tan \left (f x +e \right )}}{2}}{\left (i d \tan \left (f x +e \right )+d \right )^{2}}+\frac {7 i \arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {-i d}}\right )}{2 \sqrt {-i d}}}{8 d^{3}}+\frac {i \arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {i d}}\right )}{8 d^{3} \sqrt {i d}}\right )}{f \,a^{2}}\)	\(118\)
default	\(\frac {2 d^{3} \left (-\frac {\frac {-\frac {5 i \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{2}-\frac {7 d \sqrt {d \tan \left (f x +e \right )}}{2}}{\left (i d \tan \left (f x +e \right )+d \right )^{2}}+\frac {7 i \arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {-i d}}\right )}{2 \sqrt {-i d}}}{8 d^{3}}+\frac {i \arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {i d}}\right )}{8 d^{3} \sqrt {i d}}\right )}{f \,a^{2}}\)	\(118\)

Input:

int(1/(d*tan(f*x+e))^(1/2)/(a+I*a*tan(f*x+e))^2,x,method=_RETURNVERBOSE)
 

Output:

2/f/a^2*d^3*(-1/8/d^3*((-5/2*I*(d*tan(f*x+e))^(3/2)-7/2*d*(d*tan(f*x+e))^( 
1/2))/(I*d*tan(f*x+e)+d)^2+7/2*I/(-I*d)^(1/2)*arctan((d*tan(f*x+e))^(1/2)/ 
(-I*d)^(1/2)))+1/8*I/d^3/(I*d)^(1/2)*arctan((d*tan(f*x+e))^(1/2)/(I*d)^(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. 557 vs. \(2 (179) = 358\).

Time = 0.09 (sec) , antiderivative size = 557, normalized size of antiderivative = 2.31 \[ \int \frac {1}{\sqrt {d \tan (e+f x)} (a+i a \tan (e+f x))^2} \, dx =\text {Too large to display} \] Input:

integrate(1/(d*tan(f*x+e))^(1/2)/(a+I*a*tan(f*x+e))^2,x, algorithm="fricas 
")
 

Output:

-1/16*(4*a^2*d*f*sqrt(-1/16*I/(a^4*d*f^2))*e^(4*I*f*x + 4*I*e)*log(-2*(4*( 
a^2*d*f*e^(2*I*f*x + 2*I*e) + a^2*d*f)*sqrt((-I*d*e^(2*I*f*x + 2*I*e) + I* 
d)/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(-1/16*I/(a^4*d*f^2)) + I*d*e^(2*I*f*x + 
 2*I*e))*e^(-2*I*f*x - 2*I*e)) - 4*a^2*d*f*sqrt(-1/16*I/(a^4*d*f^2))*e^(4* 
I*f*x + 4*I*e)*log(2*(4*(a^2*d*f*e^(2*I*f*x + 2*I*e) + a^2*d*f)*sqrt((-I*d 
*e^(2*I*f*x + 2*I*e) + I*d)/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(-1/16*I/(a^4*d 
*f^2)) - I*d*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e)) - 4*a^2*d*f*sqrt(4 
9/64*I/(a^4*d*f^2))*e^(4*I*f*x + 4*I*e)*log(1/8*(8*(a^2*f*e^(2*I*f*x + 2*I 
*e) + a^2*f)*sqrt((-I*d*e^(2*I*f*x + 2*I*e) + I*d)/(e^(2*I*f*x + 2*I*e) + 
1))*sqrt(49/64*I/(a^4*d*f^2)) + 7*I)*e^(-2*I*f*x - 2*I*e)/(a^2*f)) + 4*a^2 
*d*f*sqrt(49/64*I/(a^4*d*f^2))*e^(4*I*f*x + 4*I*e)*log(-1/8*(8*(a^2*f*e^(2 
*I*f*x + 2*I*e) + a^2*f)*sqrt((-I*d*e^(2*I*f*x + 2*I*e) + I*d)/(e^(2*I*f*x 
 + 2*I*e) + 1))*sqrt(49/64*I/(a^4*d*f^2)) - 7*I)*e^(-2*I*f*x - 2*I*e)/(a^2 
*f)) - sqrt((-I*d*e^(2*I*f*x + 2*I*e) + I*d)/(e^(2*I*f*x + 2*I*e) + 1))*(6 
*e^(4*I*f*x + 4*I*e) + 7*e^(2*I*f*x + 2*I*e) + 1))*e^(-4*I*f*x - 4*I*e)/(a 
^2*d*f)
 

Sympy [F]

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

integrate(1/(d*tan(f*x+e))**(1/2)/(a+I*a*tan(f*x+e))**2,x)
 

Output:

-Integral(1/(sqrt(d*tan(e + f*x))*tan(e + f*x)**2 - 2*I*sqrt(d*tan(e + f*x 
))*tan(e + f*x) - sqrt(d*tan(e + f*x))), x)/a**2
 

Maxima [F(-2)]

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

integrate(1/(d*tan(f*x+e))^(1/2)/(a+I*a*tan(f*x+e))^2,x, algorithm="maxima 
")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.68 \[ \int \frac {1}{\sqrt {d \tan (e+f x)} (a+i a \tan (e+f x))^2} \, dx=\frac {\frac {2 i \, \sqrt {2} \arctan \left (\frac {2 \, \sqrt {d \tan \left (f x + e\right )} {\left | d \right |}}{i \, \sqrt {2} d^{\frac {3}{2}} + \sqrt {2} \sqrt {d} {\left | d \right |}}\right )}{\sqrt {d} {\left (\frac {i \, d}{{\left | d \right |}} + 1\right )}} - \frac {7 i \, \sqrt {2} \arctan \left (\frac {2 \, \sqrt {d \tan \left (f x + e\right )} {\left | d \right |}}{-i \, \sqrt {2} d^{\frac {3}{2}} + \sqrt {2} \sqrt {d} {\left | d \right |}}\right )}{\sqrt {d} {\left (-\frac {i \, d}{{\left | d \right |}} + 1\right )}} + \frac {-5 i \, \sqrt {d \tan \left (f x + e\right )} d \tan \left (f x + e\right ) - 7 \, \sqrt {d \tan \left (f x + e\right )} d}{{\left (d \tan \left (f x + e\right ) - i \, d\right )}^{2}}}{8 \, a^{2} f} \] Input:

integrate(1/(d*tan(f*x+e))^(1/2)/(a+I*a*tan(f*x+e))^2,x, algorithm="giac")
 

Output:

1/8*(2*I*sqrt(2)*arctan(2*sqrt(d*tan(f*x + e))*abs(d)/(I*sqrt(2)*d^(3/2) + 
 sqrt(2)*sqrt(d)*abs(d)))/(sqrt(d)*(I*d/abs(d) + 1)) - 7*I*sqrt(2)*arctan( 
2*sqrt(d*tan(f*x + e))*abs(d)/(-I*sqrt(2)*d^(3/2) + sqrt(2)*sqrt(d)*abs(d) 
))/(sqrt(d)*(-I*d/abs(d) + 1)) + (-5*I*sqrt(d*tan(f*x + e))*d*tan(f*x + e) 
 - 7*sqrt(d*tan(f*x + e))*d)/(d*tan(f*x + e) - I*d)^2)/(a^2*f)
 

Mupad [B] (verification not implemented)

Time = 2.73 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.70 \[ \int \frac {1}{\sqrt {d \tan (e+f x)} (a+i a \tan (e+f x))^2} \, dx=\frac {\frac {7\,d\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{8\,a^2\,f}+\frac {{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}\,5{}\mathrm {i}}{8\,a^2\,f}}{-d^2\,{\mathrm {tan}\left (e+f\,x\right )}^2+d^2\,\mathrm {tan}\left (e+f\,x\right )\,2{}\mathrm {i}+d^2}+\mathrm {atan}\left (8\,a^2\,f\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,\sqrt {-\frac {1{}\mathrm {i}}{64\,a^4\,d\,f^2}}\right )\,\sqrt {-\frac {1{}\mathrm {i}}{64\,a^4\,d\,f^2}}\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {16\,a^2\,f\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,\sqrt {\frac {49{}\mathrm {i}}{256\,a^4\,d\,f^2}}}{7}\right )\,\sqrt {\frac {49{}\mathrm {i}}{256\,a^4\,d\,f^2}}\,2{}\mathrm {i} \] Input:

int(1/((d*tan(e + f*x))^(1/2)*(a + a*tan(e + f*x)*1i)^2),x)
 

Output:

(((d*tan(e + f*x))^(3/2)*5i)/(8*a^2*f) + (7*d*(d*tan(e + f*x))^(1/2))/(8*a 
^2*f))/(d^2*tan(e + f*x)*2i + d^2 - d^2*tan(e + f*x)^2) + atan(8*a^2*f*(d* 
tan(e + f*x))^(1/2)*(-1i/(64*a^4*d*f^2))^(1/2))*(-1i/(64*a^4*d*f^2))^(1/2) 
*2i - atan((16*a^2*f*(d*tan(e + f*x))^(1/2)*(49i/(256*a^4*d*f^2))^(1/2))/7 
)*(49i/(256*a^4*d*f^2))^(1/2)*2i
 

Reduce [F]

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

int(1/(d*tan(f*x+e))^(1/2)/(a+I*a*tan(f*x+e))^2,x)
 

Output:

( - int(1/(sqrt(tan(e + f*x))*tan(e + f*x)**2 - 2*sqrt(tan(e + f*x))*tan(e 
 + f*x)*i - sqrt(tan(e + f*x))),x))/(sqrt(d)*a**2)