\(\int \frac {\sec (c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx\) [391]

Optimal result

Integrand size = 24, antiderivative size = 157 \[ \int \frac {\sec (c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\frac {5 i \text {arctanh}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{64 \sqrt {2} a^{7/2} d}+\frac {i \sec (c+d x)}{6 d (a+i a \tan (c+d x))^{7/2}}+\frac {5 i \sec (c+d x)}{48 a d (a+i a \tan (c+d x))^{5/2}}+\frac {5 i \sec (c+d x)}{64 a^2 d (a+i a \tan (c+d x))^{3/2}} \] Output:

5/128*I*arctanh(1/2*a^(1/2)*sec(d*x+c)*2^(1/2)/(a+I*a*tan(d*x+c))^(1/2))*2 
^(1/2)/a^(7/2)/d+1/6*I*sec(d*x+c)/d/(a+I*a*tan(d*x+c))^(7/2)+5/48*I*sec(d* 
x+c)/a/d/(a+I*a*tan(d*x+c))^(5/2)+5/64*I*sec(d*x+c)/a^2/d/(a+I*a*tan(d*x+c 
))^(3/2)
 

Mathematica [A] (verified)

Time = 1.48 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.76 \[ \int \frac {\sec (c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=-\frac {\sec ^3(c+d x) \left (52+\frac {30 e^{4 i (c+d x)} \text {arctanh}\left (\sqrt {1+e^{2 i (c+d x)}}\right )}{\sqrt {1+e^{2 i (c+d x)}}}+82 \cos (2 (c+d x))+50 i \sin (2 (c+d x))\right )}{384 a^3 d (-i+\tan (c+d x))^3 \sqrt {a+i a \tan (c+d x)}} \] Input:

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

Output:

-1/384*(Sec[c + d*x]^3*(52 + (30*E^((4*I)*(c + d*x))*ArcTanh[Sqrt[1 + E^(( 
2*I)*(c + d*x))]])/Sqrt[1 + E^((2*I)*(c + d*x))] + 82*Cos[2*(c + d*x)] + ( 
50*I)*Sin[2*(c + d*x)]))/(a^3*d*(-I + Tan[c + d*x])^3*Sqrt[a + I*a*Tan[c + 
 d*x]])
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.06, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3042, 3983, 3042, 3983, 3042, 3983, 3042, 3970, 219}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3983

\(\displaystyle \frac {5 \int \frac {\sec (c+d x)}{(i \tan (c+d x) a+a)^{5/2}}dx}{12 a}+\frac {i \sec (c+d x)}{6 d (a+i a \tan (c+d x))^{7/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {5 \int \frac {\sec (c+d x)}{(i \tan (c+d x) a+a)^{5/2}}dx}{12 a}+\frac {i \sec (c+d x)}{6 d (a+i a \tan (c+d x))^{7/2}}\)

\(\Big \downarrow \) 3983

\(\displaystyle \frac {5 \left (\frac {3 \int \frac {\sec (c+d x)}{(i \tan (c+d x) a+a)^{3/2}}dx}{8 a}+\frac {i \sec (c+d x)}{4 d (a+i a \tan (c+d x))^{5/2}}\right )}{12 a}+\frac {i \sec (c+d x)}{6 d (a+i a \tan (c+d x))^{7/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {5 \left (\frac {3 \int \frac {\sec (c+d x)}{(i \tan (c+d x) a+a)^{3/2}}dx}{8 a}+\frac {i \sec (c+d x)}{4 d (a+i a \tan (c+d x))^{5/2}}\right )}{12 a}+\frac {i \sec (c+d x)}{6 d (a+i a \tan (c+d x))^{7/2}}\)

\(\Big \downarrow \) 3983

\(\displaystyle \frac {5 \left (\frac {3 \left (\frac {\int \frac {\sec (c+d x)}{\sqrt {i \tan (c+d x) a+a}}dx}{4 a}+\frac {i \sec (c+d x)}{2 d (a+i a \tan (c+d x))^{3/2}}\right )}{8 a}+\frac {i \sec (c+d x)}{4 d (a+i a \tan (c+d x))^{5/2}}\right )}{12 a}+\frac {i \sec (c+d x)}{6 d (a+i a \tan (c+d x))^{7/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {5 \left (\frac {3 \left (\frac {\int \frac {\sec (c+d x)}{\sqrt {i \tan (c+d x) a+a}}dx}{4 a}+\frac {i \sec (c+d x)}{2 d (a+i a \tan (c+d x))^{3/2}}\right )}{8 a}+\frac {i \sec (c+d x)}{4 d (a+i a \tan (c+d x))^{5/2}}\right )}{12 a}+\frac {i \sec (c+d x)}{6 d (a+i a \tan (c+d x))^{7/2}}\)

\(\Big \downarrow \) 3970

\(\displaystyle \frac {5 \left (\frac {3 \left (\frac {i \int \frac {1}{2-\frac {a \sec ^2(c+d x)}{i \tan (c+d x) a+a}}d\frac {\sec (c+d x)}{\sqrt {i \tan (c+d x) a+a}}}{2 a d}+\frac {i \sec (c+d x)}{2 d (a+i a \tan (c+d x))^{3/2}}\right )}{8 a}+\frac {i \sec (c+d x)}{4 d (a+i a \tan (c+d x))^{5/2}}\right )}{12 a}+\frac {i \sec (c+d x)}{6 d (a+i a \tan (c+d x))^{7/2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {5 \left (\frac {3 \left (\frac {i \text {arctanh}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {i \sec (c+d x)}{2 d (a+i a \tan (c+d x))^{3/2}}\right )}{8 a}+\frac {i \sec (c+d x)}{4 d (a+i a \tan (c+d x))^{5/2}}\right )}{12 a}+\frac {i \sec (c+d x)}{6 d (a+i a \tan (c+d x))^{7/2}}\)

Input:

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

Output:

((I/6)*Sec[c + d*x])/(d*(a + I*a*Tan[c + d*x])^(7/2)) + (5*(((I/4)*Sec[c + 
 d*x])/(d*(a + I*a*Tan[c + d*x])^(5/2)) + (3*(((I/2)*ArcTanh[(Sqrt[a]*Sec[ 
c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(Sqrt[2]*a^(3/2)*d) + ((I 
/2)*Sec[c + d*x])/(d*(a + I*a*Tan[c + d*x])^(3/2))))/(8*a)))/(12*a)
 

Defintions of rubi rules used

rule 219

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

rule 3042

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

rule 3970

Int[sec[(e_.) + (f_.)*(x_)]/Sqrt[(a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]], x_S 
ymbol] :> Simp[-2*(a/(b*f))   Subst[Int[1/(2 - a*x^2), x], x, Sec[e + f*x]/ 
Sqrt[a + b*Tan[e + f*x]]], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 + b^2, 0 
]
 

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 [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 470 vs. \(2 (126 ) = 252\).

Time = 8.06 (sec) , antiderivative size = 471, normalized size of antiderivative = 3.00

method	result	size
default	\(-\frac {\tan \left (d x +c \right ) \sec \left (d x +c \right )^{2} \left (120 \cos \left (d x +c \right )^{3}+60 \cos \left (d x +c \right )^{2}-60 \cos \left (d x +c \right )-15\right ) \operatorname {arctanh}\left (\frac {\left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ) \sqrt {2}}{2 \sqrt {\cot \left (d x +c \right )^{2}-2 \cot \left (d x +c \right ) \csc \left (d x +c \right )+\csc \left (d x +c \right )^{2}-1}}\right )+i \operatorname {arctanh}\left (\frac {\left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ) \sqrt {2}}{2 \sqrt {\cot \left (d x +c \right )^{2}-2 \cot \left (d x +c \right ) \csc \left (d x +c \right )+\csc \left (d x +c \right )^{2}-1}}\right ) \left (-120 \cos \left (d x +c \right )-60+120 \sec \left (d x +c \right )+45 \sec \left (d x +c \right )^{2}-15 \sec \left (d x +c \right )^{3}\right )+i \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \left (120 \sin \left (d x +c \right )-60 \sec \left (d x +c \right ) \tan \left (d x +c \right )\right )+\sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \left (120 \cos \left (d x +c \right )-120 \sec \left (d x +c \right )+15 \sec \left (d x +c \right )^{3}\right )+i \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \left (-240 \sin \left (d x +c \right )-100 \tan \left (d x +c \right )+20 \sec \left (d x +c \right ) \tan \left (d x +c \right )\right )+\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \left (-240 \cos \left (d x +c \right )-164+76 \sec \left (d x +c \right )+30 \sec \left (d x +c \right )^{2}\right )}{384 d \left (-\tan \left (d x +c \right )+i\right )^{3} \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, a^{3} \left (\cos \left (d x +c \right )+1\right )}\)	\(471\)

Input:

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

Output:

-1/384/d/(-tan(d*x+c)+I)^3/(a*(1+I*tan(d*x+c)))^(1/2)/(-cos(d*x+c)/(cos(d* 
x+c)+1))^(1/2)/a^3/(cos(d*x+c)+1)*(tan(d*x+c)*sec(d*x+c)^2*(120*cos(d*x+c) 
^3+60*cos(d*x+c)^2-60*cos(d*x+c)-15)*arctanh(1/2/(cot(d*x+c)^2-2*cot(d*x+c 
)*csc(d*x+c)+csc(d*x+c)^2-1)^(1/2)*(I-cot(d*x+c)+csc(d*x+c))*2^(1/2))+I*ar 
ctanh(1/2/(cot(d*x+c)^2-2*cot(d*x+c)*csc(d*x+c)+csc(d*x+c)^2-1)^(1/2)*(I-c 
ot(d*x+c)+csc(d*x+c))*2^(1/2))*(-120*cos(d*x+c)-60+120*sec(d*x+c)+45*sec(d 
*x+c)^2-15*sec(d*x+c)^3)+I*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1 
20*sin(d*x+c)-60*sec(d*x+c)*tan(d*x+c))+2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c) 
+1))^(1/2)*(120*cos(d*x+c)-120*sec(d*x+c)+15*sec(d*x+c)^3)+I*(-cos(d*x+c)/ 
(cos(d*x+c)+1))^(1/2)*(-240*sin(d*x+c)-100*tan(d*x+c)+20*sec(d*x+c)*tan(d* 
x+c))+(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(-240*cos(d*x+c)-164+76*sec(d*x+c 
)+30*sec(d*x+c)^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. 278 vs. \(2 (118) = 236\).

Time = 0.11 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.77 \[ \int \frac {\sec (c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\frac {{\left (-15 i \, \sqrt {\frac {1}{2}} a^{4} d \sqrt {\frac {1}{a^{7} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (-\frac {5 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (i \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a^{3} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{7} d^{2}}} - i\right )} e^{\left (-i \, d x - i \, c\right )}}{32 \, a^{3} d}\right ) + 15 i \, \sqrt {\frac {1}{2}} a^{4} d \sqrt {\frac {1}{a^{7} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (-\frac {5 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (-i \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a^{3} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{7} d^{2}}} - i\right )} e^{\left (-i \, d x - i \, c\right )}}{32 \, a^{3} d}\right ) + \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (33 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 59 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 34 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 8 i\right )}\right )} e^{\left (-6 i \, d x - 6 i \, c\right )}}{384 \, a^{4} d} \] Input:

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

Output:

1/384*(-15*I*sqrt(1/2)*a^4*d*sqrt(1/(a^7*d^2))*e^(6*I*d*x + 6*I*c)*log(-5/ 
32*(sqrt(2)*sqrt(1/2)*(I*a^3*d*e^(2*I*d*x + 2*I*c) + I*a^3*d)*sqrt(a/(e^(2 
*I*d*x + 2*I*c) + 1))*sqrt(1/(a^7*d^2)) - I)*e^(-I*d*x - I*c)/(a^3*d)) + 1 
5*I*sqrt(1/2)*a^4*d*sqrt(1/(a^7*d^2))*e^(6*I*d*x + 6*I*c)*log(-5/32*(sqrt( 
2)*sqrt(1/2)*(-I*a^3*d*e^(2*I*d*x + 2*I*c) - I*a^3*d)*sqrt(a/(e^(2*I*d*x + 
 2*I*c) + 1))*sqrt(1/(a^7*d^2)) - I)*e^(-I*d*x - I*c)/(a^3*d)) + sqrt(2)*s 
qrt(a/(e^(2*I*d*x + 2*I*c) + 1))*(33*I*e^(6*I*d*x + 6*I*c) + 59*I*e^(4*I*d 
*x + 4*I*c) + 34*I*e^(2*I*d*x + 2*I*c) + 8*I))*e^(-6*I*d*x - 6*I*c)/(a^4*d 
)
 

Sympy [F]

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

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

Output:

Integral(sec(c + d*x)/(I*a*(tan(c + d*x) - I))**(7/2), x)
 

Maxima [F]

\[ \int \frac {\sec (c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\int { \frac {\sec \left (d x + c\right )}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \] Input:

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

Output:

integrate(sec(d*x + c)/(I*a*tan(d*x + c) + a)^(7/2), x)
 

Giac [F(-2)]

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

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

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

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

Output:

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

Reduce [F]

\[ \int \frac {\sec (c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(sqrt(a)*( - 10*sqrt(tan(c + d*x)*i + 1)*sec(c + d*x)*i - int(( - sqrt(tan 
(c + d*x)*i + 1)*sec(c + d*x)*tan(c + d*x)**4)/(tan(c + d*x)**5*i + 3*tan( 
c + d*x)**4 - 2*tan(c + d*x)**3*i + 2*tan(c + d*x)**2 - 3*tan(c + d*x)*i - 
 1),x)*tan(c + d*x)**2*d - int(( - sqrt(tan(c + d*x)*i + 1)*sec(c + d*x)*t 
an(c + d*x)**4)/(tan(c + d*x)**5*i + 3*tan(c + d*x)**4 - 2*tan(c + d*x)**3 
*i + 2*tan(c + d*x)**2 - 3*tan(c + d*x)*i - 1),x)*d + 6*int(( - sqrt(tan(c 
 + d*x)*i + 1)*sec(c + d*x)*tan(c + d*x)**2)/(tan(c + d*x)**5*i + 3*tan(c 
+ d*x)**4 - 2*tan(c + d*x)**3*i + 2*tan(c + d*x)**2 - 3*tan(c + d*x)*i - 1 
),x)*tan(c + d*x)**2*d + 6*int(( - sqrt(tan(c + d*x)*i + 1)*sec(c + d*x)*t 
an(c + d*x)**2)/(tan(c + d*x)**5*i + 3*tan(c + d*x)**4 - 2*tan(c + d*x)**3 
*i + 2*tan(c + d*x)**2 - 3*tan(c + d*x)*i - 1),x)*d - 4*int(( - sqrt(tan(c 
 + d*x)*i + 1)*sec(c + d*x)*tan(c + d*x))/(tan(c + d*x)**5*i + 3*tan(c + d 
*x)**4 - 2*tan(c + d*x)**3*i + 2*tan(c + d*x)**2 - 3*tan(c + d*x)*i - 1),x 
)*tan(c + d*x)**2*d*i - 4*int(( - sqrt(tan(c + d*x)*i + 1)*sec(c + d*x)*ta 
n(c + d*x))/(tan(c + d*x)**5*i + 3*tan(c + d*x)**4 - 2*tan(c + d*x)**3*i + 
 2*tan(c + d*x)**2 - 3*tan(c + d*x)*i - 1),x)*d*i - int(( - sqrt(tan(c + d 
*x)*i + 1)*sec(c + d*x))/(tan(c + d*x)**5*i + 3*tan(c + d*x)**4 - 2*tan(c 
+ d*x)**3*i + 2*tan(c + d*x)**2 - 3*tan(c + d*x)*i - 1),x)*tan(c + d*x)**2 
*d - int(( - sqrt(tan(c + d*x)*i + 1)*sec(c + d*x))/(tan(c + d*x)**5*i + 3 
*tan(c + d*x)**4 - 2*tan(c + d*x)**3*i + 2*tan(c + d*x)**2 - 3*tan(c + ...