\(\int \frac {1}{(e \cos (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}} \, dx\) [694]

Optimal result

Integrand size = 30, antiderivative size = 350 \[ \int \frac {1}{(e \cos (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}} \, dx=-\frac {i \sqrt {2} \sqrt {a} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e}}\right ) \sec (c+d x)}{d e^{3/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i \sqrt {2} \sqrt {a} \arctan \left (1+\frac {\sqrt {2} \sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e}}\right ) \sec (c+d x)}{d e^{3/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}-\frac {i \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e} (1+\cos (c+d x)-i \sin (c+d x))}\right ) \sec (c+d x)}{d e^{3/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}} \] Output:

-I*2^(1/2)*a^(1/2)*arctan(1-2^(1/2)*(e*cos(d*x+c))^(1/2)*(a-I*a*tan(d*x+c) 
)^(1/2)/a^(1/2)/e^(1/2))*sec(d*x+c)/d/e^(3/2)/(a-I*a*tan(d*x+c))^(1/2)/(a+ 
I*a*tan(d*x+c))^(1/2)+I*2^(1/2)*a^(1/2)*arctan(1+2^(1/2)*(e*cos(d*x+c))^(1 
/2)*(a-I*a*tan(d*x+c))^(1/2)/a^(1/2)/e^(1/2))*sec(d*x+c)/d/e^(3/2)/(a-I*a* 
tan(d*x+c))^(1/2)/(a+I*a*tan(d*x+c))^(1/2)-I*2^(1/2)*a^(1/2)*arctanh(2^(1/ 
2)*(e*cos(d*x+c))^(1/2)*(a-I*a*tan(d*x+c))^(1/2)/a^(1/2)/e^(1/2)/(1+cos(d* 
x+c)-I*sin(d*x+c)))*sec(d*x+c)/d/e^(3/2)/(a-I*a*tan(d*x+c))^(1/2)/(a+I*a*t 
an(d*x+c))^(1/2)
 

Mathematica [A] (verified)

Time = 1.39 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.60 \[ \int \frac {1}{(e \cos (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}} \, dx=\frac {i e^{\frac {1}{2} i (c+d x)} \left (2 \arctan \left (1-\sqrt {2} e^{\frac {1}{2} i (c+d x)}\right )-2 \arctan \left (1+\sqrt {2} e^{\frac {1}{2} i (c+d x)}\right )+\log \left (1-\sqrt {2} e^{\frac {1}{2} i (c+d x)}+e^{i (c+d x)}\right )-\log \left (1+\sqrt {2} e^{\frac {1}{2} i (c+d x)}+e^{i (c+d x)}\right )\right )}{\sqrt {2} d e \sqrt {\frac {a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {e e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )}} \] Input:

Integrate[1/((e*Cos[c + d*x])^(3/2)*Sqrt[a + I*a*Tan[c + d*x]]),x]
 

Output:

(I*E^((I/2)*(c + d*x))*(2*ArcTan[1 - Sqrt[2]*E^((I/2)*(c + d*x))] - 2*ArcT 
an[1 + Sqrt[2]*E^((I/2)*(c + d*x))] + Log[1 - Sqrt[2]*E^((I/2)*(c + d*x)) 
+ E^(I*(c + d*x))] - Log[1 + Sqrt[2]*E^((I/2)*(c + d*x)) + E^(I*(c + d*x)) 
]))/(Sqrt[2]*d*e*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*S 
qrt[(e*(1 + E^((2*I)*(c + d*x))))/E^(I*(c + d*x))])
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.07, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3042, 3997, 3042, 3996, 826, 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.

\(\displaystyle \int \frac {1}{\sqrt {a+i a \tan (c+d x)} (e \cos (c+d x))^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\sqrt {a+i a \tan (c+d x)} (e \cos (c+d x))^{3/2}}dx\)

\(\Big \downarrow \) 3997

\(\displaystyle \frac {\sec (c+d x) \int \frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \cos (c+d x)}}dx}{e \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sec (c+d x) \int \frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \cos (c+d x)}}dx}{e \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}\)

\(\Big \downarrow \) 3996

\(\displaystyle \frac {4 i a \sec (c+d x) \int \frac {e \cos (c+d x) (a-i a \tan (c+d x))}{a^2 e^2+\cos ^2(c+d x) (a-i a \tan (c+d x))^2 e^2}d\left (\sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}\right )}{d e \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}\)

\(\Big \downarrow \) 826

\(\displaystyle \frac {4 i a \sec (c+d x) \left (\frac {1}{2} \int \frac {a e+\cos (c+d x) (a-i a \tan (c+d x)) e}{a^2 e^2+\cos ^2(c+d x) (a-i a \tan (c+d x))^2 e^2}d\left (\sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}\right )-\frac {1}{2} \int \frac {a e-e \cos (c+d x) (a-i a \tan (c+d x))}{a^2 e^2+\cos ^2(c+d x) (a-i a \tan (c+d x))^2 e^2}d\left (\sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}\right )\right )}{d e \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {4 i a \sec (c+d x) \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{a e+\cos (c+d x) (a-i a \tan (c+d x)) e-\sqrt {2} \sqrt {a} \sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)} \sqrt {e}}d\left (\sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}\right )+\frac {1}{2} \int \frac {1}{a e+\cos (c+d x) (a-i a \tan (c+d x)) e+\sqrt {2} \sqrt {a} \sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)} \sqrt {e}}d\left (\sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}\right )\right )-\frac {1}{2} \int \frac {a e-e \cos (c+d x) (a-i a \tan (c+d x))}{a^2 e^2+\cos ^2(c+d x) (a-i a \tan (c+d x))^2 e^2}d\left (\sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}\right )\right )}{d e \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {4 i a \sec (c+d x) \left (\frac {1}{2} \left (\frac {\int \frac {1}{-e \cos (c+d x) (a-i a \tan (c+d x))-1}d\left (1-\frac {\sqrt {2} \sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\int \frac {1}{-e \cos (c+d x) (a-i a \tan (c+d x))-1}d\left (\frac {\sqrt {2} \sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}\right )-\frac {1}{2} \int \frac {a e-e \cos (c+d x) (a-i a \tan (c+d x))}{a^2 e^2+\cos ^2(c+d x) (a-i a \tan (c+d x))^2 e^2}d\left (\sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}\right )\right )}{d e \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {4 i a \sec (c+d x) \left (\frac {1}{2} \left (\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {e \cos (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {e \cos (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}\right )-\frac {1}{2} \int \frac {a e-e \cos (c+d x) (a-i a \tan (c+d x))}{a^2 e^2+\cos ^2(c+d x) (a-i a \tan (c+d x))^2 e^2}d\left (\sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}\right )\right )}{d e \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {4 i a \sec (c+d x) \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2} \sqrt {a} \sqrt {e}-2 \sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}}{a e+\cos (c+d x) (a-i a \tan (c+d x)) e-\sqrt {2} \sqrt {a} \sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)} \sqrt {e}}d\left (\sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {e}}+\frac {\int -\frac {\sqrt {2} \left (\sqrt {a} \sqrt {e}+\sqrt {2} \sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}\right )}{a e+\cos (c+d x) (a-i a \tan (c+d x)) e+\sqrt {2} \sqrt {a} \sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)} \sqrt {e}}d\left (\sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {e}}\right )+\frac {1}{2} \left (\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {e \cos (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {e \cos (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}\right )\right )}{d e \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {4 i a \sec (c+d x) \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt {a} \sqrt {e}-2 \sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}}{a e+\cos (c+d x) (a-i a \tan (c+d x)) e-\sqrt {2} \sqrt {a} \sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)} \sqrt {e}}d\left (\sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\int \frac {\sqrt {2} \left (\sqrt {a} \sqrt {e}+\sqrt {2} \sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}\right )}{a e+\cos (c+d x) (a-i a \tan (c+d x)) e+\sqrt {2} \sqrt {a} \sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)} \sqrt {e}}d\left (\sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {e}}\right )+\frac {1}{2} \left (\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {e \cos (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {e \cos (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}\right )\right )}{d e \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {4 i a \sec (c+d x) \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt {a} \sqrt {e}-2 \sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}}{a e+\cos (c+d x) (a-i a \tan (c+d x)) e-\sqrt {2} \sqrt {a} \sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)} \sqrt {e}}d\left (\sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\int \frac {\sqrt {a} \sqrt {e}+\sqrt {2} \sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}}{a e+\cos (c+d x) (a-i a \tan (c+d x)) e+\sqrt {2} \sqrt {a} \sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)} \sqrt {e}}d\left (\sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}\right )}{2 \sqrt {a} \sqrt {e}}\right )+\frac {1}{2} \left (\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {e \cos (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {e \cos (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}\right )\right )}{d e \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {4 i a \sec (c+d x) \left (\frac {1}{2} \left (\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {e \cos (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {e \cos (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}\right )+\frac {1}{2} \left (\frac {\log \left (e \cos (c+d x) (a-i a \tan (c+d x))-\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a-i a \tan (c+d x)} \sqrt {e \cos (c+d x)}+a e\right )}{2 \sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\log \left (e \cos (c+d x) (a-i a \tan (c+d x))+\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a-i a \tan (c+d x)} \sqrt {e \cos (c+d x)}+a e\right )}{2 \sqrt {2} \sqrt {a} \sqrt {e}}\right )\right )}{d e \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}\)

Input:

Int[1/((e*Cos[c + d*x])^(3/2)*Sqrt[a + I*a*Tan[c + d*x]]),x]
 

Output:

((4*I)*a*((-(ArcTan[1 - (Sqrt[2]*Sqrt[e*Cos[c + d*x]]*Sqrt[a - I*a*Tan[c + 
 d*x]])/(Sqrt[a]*Sqrt[e])]/(Sqrt[2]*Sqrt[a]*Sqrt[e])) + ArcTan[1 + (Sqrt[2 
]*Sqrt[e*Cos[c + d*x]]*Sqrt[a - I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e])]/(Sqr 
t[2]*Sqrt[a]*Sqrt[e]))/2 + (Log[a*e - Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[e*Cos[c 
 + d*x]]*Sqrt[a - I*a*Tan[c + d*x]] + e*Cos[c + d*x]*(a - I*a*Tan[c + d*x] 
)]/(2*Sqrt[2]*Sqrt[a]*Sqrt[e]) - Log[a*e + Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[e* 
Cos[c + d*x]]*Sqrt[a - I*a*Tan[c + d*x]] + e*Cos[c + d*x]*(a - I*a*Tan[c + 
 d*x])]/(2*Sqrt[2]*Sqrt[a]*Sqrt[e]))/2)*Sec[c + d*x])/(d*e*Sqrt[a - I*a*Ta 
n[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])
 

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 826

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 
2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s)   Int[(r + s*x^2)/(a + b*x^ 
4), x], x] - Simp[1/(2*s)   Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ 
a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] 
 && AtomQ[SplitProduct[SumBaseQ, b]]))
 

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 3042

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

rule 3996

Int[Sqrt[(a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]]/Sqrt[cos[(e_.) + (f_.)*(x_)] 
*(d_.)], x_Symbol] :> Simp[-4*(b/f)   Subst[Int[x^2/(a^2*d^2 + x^4), x], x, 
 Sqrt[d*Cos[e + f*x]]*Sqrt[a + b*Tan[e + f*x]]], x] /; FreeQ[{a, b, d, e, f 
}, x] && EqQ[a^2 + b^2, 0]
 

rule 3997

Int[1/((cos[(e_.) + (f_.)*(x_)]*(d_.))^(3/2)*Sqrt[(a_) + (b_.)*tan[(e_.) + 
(f_.)*(x_)]]), x_Symbol] :> Simp[1/(d*Cos[e + f*x]*Sqrt[a - b*Tan[e + f*x]] 
*Sqrt[a + b*Tan[e + f*x]])   Int[Sqrt[a - b*Tan[e + f*x]]/Sqrt[d*Cos[e + f* 
x]], x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0]
 

Maple [A] (warning: unable to verify)

Time = 14.54 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.66

method	result	size
default	\(\frac {\sec \left (d x +c \right ) \left (i \operatorname {arctanh}\left (\frac {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}{2 \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}}\right )-i \operatorname {arctanh}\left (\frac {\cot \left (d x +c \right )-\csc \left (d x +c \right )+1}{2 \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}}\right )+\operatorname {arctanh}\left (\frac {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}{2 \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}}\right )+\operatorname {arctanh}\left (\frac {\cot \left (d x +c \right )-\csc \left (d x +c \right )+1}{2 \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}}\right )\right ) \left (\csc \left (d x +c \right )^{2} \left (1-\cos \left (d x +c \right )\right )^{2}-1\right ) \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )+i\right ) \left (\cos \left (d x +c \right )+1\right )}{4 d \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, e \sqrt {e \cos \left (d x +c \right )}}\)	\(231\)

Input:

int(1/(e*cos(d*x+c))^(3/2)/(a+I*a*tan(d*x+c))^(1/2),x,method=_RETURNVERBOS 
E)
 

Output:

1/4/d*sec(d*x+c)*(I*arctanh(1/2/(1/(cos(d*x+c)+1))^(1/2)*(-cot(d*x+c)+csc( 
d*x+c)+1))-I*arctanh(1/2/(1/(cos(d*x+c)+1))^(1/2)*(cot(d*x+c)-csc(d*x+c)+1 
))+arctanh(1/2/(1/(cos(d*x+c)+1))^(1/2)*(-cot(d*x+c)+csc(d*x+c)+1))+arctan 
h(1/2/(1/(cos(d*x+c)+1))^(1/2)*(cot(d*x+c)-csc(d*x+c)+1)))*(csc(d*x+c)^2*( 
1-cos(d*x+c))^2-1)*(cot(d*x+c)-csc(d*x+c)+I)/(a*(1+I*tan(d*x+c)))^(1/2)*(c 
os(d*x+c)+1)/(1/(cos(d*x+c)+1))^(1/2)/e/(e*cos(d*x+c))^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 341, normalized size of antiderivative = 0.97 \[ \int \frac {1}{(e \cos (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}} \, dx=-\frac {1}{2} \, \sqrt {\frac {4 i}{a d^{2} e^{3}}} \log \left (\frac {1}{2} \, a d e^{2} \sqrt {\frac {4 i}{a d^{2} e^{3}}} + \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}\right ) + \frac {1}{2} \, \sqrt {\frac {4 i}{a d^{2} e^{3}}} \log \left (-\frac {1}{2} \, a d e^{2} \sqrt {\frac {4 i}{a d^{2} e^{3}}} + \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}\right ) + \frac {1}{2} \, \sqrt {-\frac {4 i}{a d^{2} e^{3}}} \log \left (\frac {1}{2} \, a d e^{2} \sqrt {-\frac {4 i}{a d^{2} e^{3}}} + \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}\right ) - \frac {1}{2} \, \sqrt {-\frac {4 i}{a d^{2} e^{3}}} \log \left (-\frac {1}{2} \, a d e^{2} \sqrt {-\frac {4 i}{a d^{2} e^{3}}} + \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}\right ) \] Input:

integrate(1/(e*cos(d*x+c))^(3/2)/(a+I*a*tan(d*x+c))^(1/2),x, algorithm="fr 
icas")
 

Output:

-1/2*sqrt(4*I/(a*d^2*e^3))*log(1/2*a*d*e^2*sqrt(4*I/(a*d^2*e^3)) + sqrt(2) 
*sqrt(1/2)*sqrt(e*e^(2*I*d*x + 2*I*c) + e)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1 
))*e^(-1/2*I*d*x - 1/2*I*c)) + 1/2*sqrt(4*I/(a*d^2*e^3))*log(-1/2*a*d*e^2* 
sqrt(4*I/(a*d^2*e^3)) + sqrt(2)*sqrt(1/2)*sqrt(e*e^(2*I*d*x + 2*I*c) + e)* 
sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*e^(-1/2*I*d*x - 1/2*I*c)) + 1/2*sqrt(-4* 
I/(a*d^2*e^3))*log(1/2*a*d*e^2*sqrt(-4*I/(a*d^2*e^3)) + sqrt(2)*sqrt(1/2)* 
sqrt(e*e^(2*I*d*x + 2*I*c) + e)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*e^(-1/2* 
I*d*x - 1/2*I*c)) - 1/2*sqrt(-4*I/(a*d^2*e^3))*log(-1/2*a*d*e^2*sqrt(-4*I/ 
(a*d^2*e^3)) + sqrt(2)*sqrt(1/2)*sqrt(e*e^(2*I*d*x + 2*I*c) + e)*sqrt(a/(e 
^(2*I*d*x + 2*I*c) + 1))*e^(-1/2*I*d*x - 1/2*I*c))
 

Sympy [F]

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

integrate(1/(e*cos(d*x+c))**(3/2)/(a+I*a*tan(d*x+c))**(1/2),x)
 

Output:

Integral(1/((e*cos(c + d*x))**(3/2)*sqrt(I*a*(tan(c + d*x) - I))), x)
 

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 714 vs. \(2 (264) = 528\).

Time = 0.25 (sec) , antiderivative size = 714, normalized size of antiderivative = 2.04 \[ \int \frac {1}{(e \cos (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}} \, dx=\text {Too large to display} \] Input:

integrate(1/(e*cos(d*x+c))^(3/2)/(a+I*a*tan(d*x+c))^(1/2),x, algorithm="ma 
xima")
 

Output:

-1/4*(2*I*sqrt(2)*arctan2(sqrt(2)*cos(1/2*d*x + 1/2*c) + 1, sqrt(2)*sin(1/ 
2*d*x + 1/2*c) + 1) + 2*I*sqrt(2)*arctan2(sqrt(2)*cos(1/2*d*x + 1/2*c) + 1 
, -sqrt(2)*sin(1/2*d*x + 1/2*c) + 1) + 2*I*sqrt(2)*arctan2(sqrt(2)*cos(1/2 
*d*x + 1/2*c) - 1, sqrt(2)*sin(1/2*d*x + 1/2*c) + 1) + 2*I*sqrt(2)*arctan2 
(sqrt(2)*cos(1/2*d*x + 1/2*c) - 1, -sqrt(2)*sin(1/2*d*x + 1/2*c) + 1) - 2* 
sqrt(2)*arctan2(sqrt(2)*sin(1/2*d*x + 1/2*c) + sin(d*x + c), sqrt(2)*cos(1 
/2*d*x + 1/2*c) + cos(d*x + c) + 1) + 2*sqrt(2)*arctan2(-sqrt(2)*sin(1/2*d 
*x + 1/2*c) + sin(d*x + c), -sqrt(2)*cos(1/2*d*x + 1/2*c) + cos(d*x + c) + 
 1) + I*sqrt(2)*log(2*sqrt(2)*sin(d*x + c)*sin(1/2*d*x + 1/2*c) + 2*(sqrt( 
2)*cos(1/2*d*x + 1/2*c) + 1)*cos(d*x + c) + cos(d*x + c)^2 + 2*cos(1/2*d*x 
 + 1/2*c)^2 + sin(d*x + c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 + 2*sqrt(2)*cos(1/ 
2*d*x + 1/2*c) + 1) - I*sqrt(2)*log(-2*sqrt(2)*sin(d*x + c)*sin(1/2*d*x + 
1/2*c) - 2*(sqrt(2)*cos(1/2*d*x + 1/2*c) - 1)*cos(d*x + c) + cos(d*x + c)^ 
2 + 2*cos(1/2*d*x + 1/2*c)^2 + sin(d*x + c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 - 
 2*sqrt(2)*cos(1/2*d*x + 1/2*c) + 1) - sqrt(2)*log(2*cos(1/2*d*x + 1/2*c)^ 
2 + 2*sin(1/2*d*x + 1/2*c)^2 + 2*sqrt(2)*cos(1/2*d*x + 1/2*c) + 2*sqrt(2)* 
sin(1/2*d*x + 1/2*c) + 2) + sqrt(2)*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1 
/2*d*x + 1/2*c)^2 + 2*sqrt(2)*cos(1/2*d*x + 1/2*c) - 2*sqrt(2)*sin(1/2*d*x 
 + 1/2*c) + 2) - sqrt(2)*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/ 
2*c)^2 - 2*sqrt(2)*cos(1/2*d*x + 1/2*c) + 2*sqrt(2)*sin(1/2*d*x + 1/2*c...
 

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{(e \cos (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(1/(e*cos(d*x+c))^(3/2)/(a+I*a*tan(d*x+c))^(1/2),x, algorithm="gi 
ac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument TypeError: Bad 
Argument TypeDone
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}} \, dx=\int \frac {1}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}} \,d x \] Input:

int(1/((e*cos(c + d*x))^(3/2)*(a + a*tan(c + d*x)*1i)^(1/2)),x)
 

Output:

int(1/((e*cos(c + d*x))^(3/2)*(a + a*tan(c + d*x)*1i)^(1/2)), x)
 

Reduce [F]

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

int(1/(e*cos(d*x+c))^(3/2)/(a+I*a*tan(d*x+c))^(1/2),x)
 

Output:

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