\(\int \frac {1}{(e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{3/2}} \, dx\) [428]

Optimal result

Integrand size = 30, antiderivative size = 165 \[ \int \frac {1}{(e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{3/2}} \, dx=\frac {2 i}{9 d (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{3/2}}+\frac {4 i}{15 a d (e \sec (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}}+\frac {32 i \sqrt {e \sec (c+d x)}}{45 a d e^2 \sqrt {a+i a \tan (c+d x)}}-\frac {16 i \sqrt {a+i a \tan (c+d x)}}{45 a^2 d (e \sec (c+d x))^{3/2}} \] Output:

2/9*I/d/(e*sec(d*x+c))^(3/2)/(a+I*a*tan(d*x+c))^(3/2)+4/15*I/a/d/(e*sec(d* 
x+c))^(3/2)/(a+I*a*tan(d*x+c))^(1/2)+32/45*I*(e*sec(d*x+c))^(1/2)/a/d/e^2/ 
(a+I*a*tan(d*x+c))^(1/2)-16/45*I*(a+I*a*tan(d*x+c))^(1/2)/a^2/d/(e*sec(d*x 
+c))^(3/2)
 

Mathematica [A] (verified)

Time = 1.19 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.61 \[ \int \frac {1}{(e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{3/2}} \, dx=-\frac {\sec ^3(c+d x) (-81 \cos (c+d x)+5 \cos (3 (c+d x))-54 i \sin (c+d x)+10 i \sin (3 (c+d x)))}{90 a d (e \sec (c+d x))^{3/2} (-i+\tan (c+d x)) \sqrt {a+i a \tan (c+d x)}} \] Input:

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

Output:

-1/90*(Sec[c + d*x]^3*(-81*Cos[c + d*x] + 5*Cos[3*(c + d*x)] - (54*I)*Sin[ 
c + d*x] + (10*I)*Sin[3*(c + d*x)]))/(a*d*(e*Sec[c + d*x])^(3/2)*(-I + Tan 
[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])
 

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {3042, 3983, 3042, 3983, 3042, 3978, 3042, 3969}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3983

\(\displaystyle \frac {2 \int \frac {1}{(e \sec (c+d x))^{3/2} \sqrt {i \tan (c+d x) a+a}}dx}{3 a}+\frac {2 i}{9 d (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 \int \frac {1}{(e \sec (c+d x))^{3/2} \sqrt {i \tan (c+d x) a+a}}dx}{3 a}+\frac {2 i}{9 d (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3983

\(\displaystyle \frac {2 \left (\frac {4 \int \frac {\sqrt {i \tan (c+d x) a+a}}{(e \sec (c+d x))^{3/2}}dx}{5 a}+\frac {2 i}{5 d \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}\right )}{3 a}+\frac {2 i}{9 d (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 \left (\frac {4 \int \frac {\sqrt {i \tan (c+d x) a+a}}{(e \sec (c+d x))^{3/2}}dx}{5 a}+\frac {2 i}{5 d \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}\right )}{3 a}+\frac {2 i}{9 d (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3978

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3969

\(\displaystyle \frac {2 \left (\frac {4 \left (\frac {4 i a \sqrt {e \sec (c+d x)}}{3 d e^2 \sqrt {a+i a \tan (c+d x)}}-\frac {2 i \sqrt {a+i a \tan (c+d x)}}{3 d (e \sec (c+d x))^{3/2}}\right )}{5 a}+\frac {2 i}{5 d \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}\right )}{3 a}+\frac {2 i}{9 d (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}\)

Input:

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

Output:

((2*I)/9)/(d*(e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^(3/2)) + (2*((( 
2*I)/5)/(d*(e*Sec[c + d*x])^(3/2)*Sqrt[a + I*a*Tan[c + d*x]]) + (4*((((4*I 
)/3)*a*Sqrt[e*Sec[c + d*x]])/(d*e^2*Sqrt[a + I*a*Tan[c + d*x]]) - (((2*I)/ 
3)*Sqrt[a + I*a*Tan[c + d*x]])/(d*(e*Sec[c + d*x])^(3/2))))/(5*a)))/(3*a)
 

Defintions of rubi rules used

rule 3042

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

rule 3969

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*( 
x_)])^(n_), x_Symbol] :> Simp[b*(d*Sec[e + f*x])^m*((a + b*Tan[e + f*x])^n/ 
(a*f*m)), x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0] && EqQ 
[Simplify[m + n], 0]
 

rule 3978

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

rule 3983

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

Maple [A] (verified)

Time = 6.32 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.55

method	result	size
default	\(\frac {\frac {2 i \cos \left (d x +c \right )^{3}}{9}+\frac {2 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}{9}-\frac {4 i \cos \left (d x +c \right )}{45}+\frac {16 \sin \left (d x +c \right )}{45}+\frac {32 i \sec \left (d x +c \right )}{45}}{d \sqrt {e \sec \left (d x +c \right )}\, e a \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}}\)	\(91\)

Input:

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

Output:

2/45/d/(e*sec(d*x+c))^(1/2)/e/a/(a*(1+I*tan(d*x+c)))^(1/2)*(5*I*cos(d*x+c) 
^3+5*sin(d*x+c)*cos(d*x+c)^2-2*I*cos(d*x+c)+8*sin(d*x+c)+16*I*sec(d*x+c))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.61 \[ \int \frac {1}{(e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{3/2}} \, dx=\frac {\sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-15 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 120 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 162 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 32 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 5 i\right )} e^{\left (-\frac {9}{2} i \, d x - \frac {9}{2} i \, c\right )}}{180 \, a^{2} d e^{2}} \] Input:

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

Output:

1/180*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(e/(e^(2*I*d*x + 2*I*c) + 1))* 
(-15*I*e^(8*I*d*x + 8*I*c) + 120*I*e^(6*I*d*x + 6*I*c) + 162*I*e^(4*I*d*x 
+ 4*I*c) + 32*I*e^(2*I*d*x + 2*I*c) + 5*I)*e^(-9/2*I*d*x - 9/2*I*c)/(a^2*d 
*e^2)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.08 \[ \int \frac {1}{(e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{3/2}} \, dx=\frac {5 i \, \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 27 i \, \cos \left (\frac {5}{9} \, \arctan \left (\sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ), \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )\right )\right ) - 15 i \, \cos \left (\frac {1}{3} \, \arctan \left (\sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ), \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )\right )\right ) + 135 i \, \cos \left (\frac {1}{9} \, \arctan \left (\sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ), \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )\right )\right ) + 5 \, \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 27 \, \sin \left (\frac {5}{9} \, \arctan \left (\sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ), \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )\right )\right ) + 15 \, \sin \left (\frac {1}{3} \, \arctan \left (\sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ), \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )\right )\right ) + 135 \, \sin \left (\frac {1}{9} \, \arctan \left (\sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ), \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )\right )\right )}{180 \, a^{\frac {3}{2}} d e^{\frac {3}{2}}} \] Input:

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

Output:

1/180*(5*I*cos(9/2*d*x + 9/2*c) + 27*I*cos(5/9*arctan2(sin(9/2*d*x + 9/2*c 
), cos(9/2*d*x + 9/2*c))) - 15*I*cos(1/3*arctan2(sin(9/2*d*x + 9/2*c), cos 
(9/2*d*x + 9/2*c))) + 135*I*cos(1/9*arctan2(sin(9/2*d*x + 9/2*c), cos(9/2* 
d*x + 9/2*c))) + 5*sin(9/2*d*x + 9/2*c) + 27*sin(5/9*arctan2(sin(9/2*d*x + 
 9/2*c), cos(9/2*d*x + 9/2*c))) + 15*sin(1/3*arctan2(sin(9/2*d*x + 9/2*c), 
 cos(9/2*d*x + 9/2*c))) + 135*sin(1/9*arctan2(sin(9/2*d*x + 9/2*c), cos(9/ 
2*d*x + 9/2*c))))/(a^(3/2)*d*e^(3/2))
 

Giac [F(-2)]

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

integrate(1/(e*sec(d*x+c))^(3/2)/(a+I*a*tan(d*x+c))^(3/2),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 [B] (verification not implemented)

Time = 1.42 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.68 \[ \int \frac {1}{(e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{3/2}} \, dx=\frac {\sqrt {\frac {e}{\cos \left (c+d\,x\right )}}\,\left (\cos \left (2\,c+2\,d\,x\right )\,12{}\mathrm {i}+\cos \left (4\,c+4\,d\,x\right )\,5{}\mathrm {i}+42\,\sin \left (2\,c+2\,d\,x\right )+5\,\sin \left (4\,c+4\,d\,x\right )+135{}\mathrm {i}\right )}{180\,a\,d\,e^2\,\sqrt {\frac {a\,\left (\cos \left (2\,c+2\,d\,x\right )+1+\sin \left (2\,c+2\,d\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,c+2\,d\,x\right )+1}}} \] Input:

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

Output:

((e/cos(c + d*x))^(1/2)*(cos(2*c + 2*d*x)*12i + cos(4*c + 4*d*x)*5i + 42*s 
in(2*c + 2*d*x) + 5*sin(4*c + 4*d*x) + 135i))/(180*a*d*e^2*((a*(cos(2*c + 
2*d*x) + sin(2*c + 2*d*x)*1i + 1))/(cos(2*c + 2*d*x) + 1))^(1/2))
 

Reduce [F]

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

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

Output:

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