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

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3583, 3570, 212} \[ \int \frac {\sec (c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\frac {5 i \text {arctanh}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{64 \sqrt {2} a^{7/2} d}+\frac {5 i \sec (c+d x)}{64 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac {5 i \sec (c+d x)}{48 a d (a+i a \tan (c+d x))^{5/2}}+\frac {i \sec (c+d x)}{6 d (a+i a \tan (c+d x))^{7/2}} \]

[In]

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

[Out]

(((5*I)/64)*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(Sqrt[2]*a^(7/2)*d) + ((I/6)
*Sec[c + d*x])/(d*(a + I*a*Tan[c + d*x])^(7/2)) + (((5*I)/48)*Sec[c + d*x])/(a*d*(a + I*a*Tan[c + d*x])^(5/2))
+ (((5*I)/64)*Sec[c + d*x])/(a^2*d*(a + I*a*Tan[c + d*x])^(3/2))

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 3570

Int[sec[(e_.) + (f_.)*(x_)]/Sqrt[(a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[-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 3583

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] + Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {i \sec (c+d x)}{6 d (a+i a \tan (c+d x))^{7/2}}+\frac {5 \int \frac {\sec (c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx}{12 a} \\ & = \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 \int \frac {\sec (c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx}{32 a^2} \\ & = \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}}+\frac {5 \int \frac {\sec (c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx}{128 a^3} \\ & = \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}}+\frac {(5 i) \text {Subst}\left (\int \frac {1}{2-a x^2} \, dx,x,\frac {\sec (c+d x)}{\sqrt {a+i a \tan (c+d x)}}\right )}{64 a^3 d} \\ & = \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}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.88 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.76 \[ \int \frac {\sec (c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=-\frac {\sec ^3(c+d x) \left (52+\frac {30 e^{4 i (c+d x)} \text {arctanh}\left (\sqrt {1+e^{2 i (c+d x)}}\right )}{\sqrt {1+e^{2 i (c+d x)}}}+82 \cos (2 (c+d x))+50 i \sin (2 (c+d x))\right )}{384 a^3 d (-i+\tan (c+d x))^3 \sqrt {a+i a \tan (c+d x)}} \]

[In]

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

[Out]

-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]])

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 797 vs. \(2 (126 ) = 252\).

Time = 9.87 (sec) , antiderivative size = 798, normalized size of antiderivative = 5.08

method	result	size
default	\(\frac {120 i \arctan \left (\frac {i \sin \left (d x +c \right )-\cos \left (d x +c \right )-1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \sin \left (d x +c \right )+60 i \tan \left (d x +c \right ) \arctan \left (\frac {i \sin \left (d x +c \right )-\cos \left (d x +c \right )-1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+100 i \tan \left (d x +c \right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+120 \cos \left (d x +c \right ) \arctan \left (\frac {i \sin \left (d x +c \right )-\cos \left (d x +c \right )-1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )-60 i \tan \left (d x +c \right ) \sec \left (d x +c \right ) \arctan \left (\frac {i \sin \left (d x +c \right )-\cos \left (d x +c \right )-1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+100 i \tan \left (d x +c \right ) \sec \left (d x +c \right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+60 \arctan \left (\frac {i \sin \left (d x +c \right )-\cos \left (d x +c \right )-1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+164 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-15 i \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right ) \arctan \left (\frac {i \sin \left (d x +c \right )-\cos \left (d x +c \right )-1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )-120 \sec \left (d x +c \right ) \arctan \left (\frac {i \sin \left (d x +c \right )-\cos \left (d x +c \right )-1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+164 \sec \left (d x +c \right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-45 \left (\sec ^{2}\left (d x +c \right )\right ) \arctan \left (\frac {i \sin \left (d x +c \right )-\cos \left (d x +c \right )-1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )-30 \left (\sec ^{2}\left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+15 \left (\sec ^{3}\left (d x +c \right )\right ) \arctan \left (\frac {i \sin \left (d x +c \right )-\cos \left (d x +c \right )-1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )-30 \left (\sec ^{3}\left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}{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 )}\)	\(798\)

[In]

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

[Out]

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)*(12
0*I*sin(d*x+c)*arctan(1/2*(I*sin(d*x+c)-cos(d*x+c)-1)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+60*I*
tan(d*x+c)*arctan(1/2*(I*sin(d*x+c)-cos(d*x+c)-1)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+100*I*tan
(d*x+c)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+120*cos(d*x+c)*arctan(1/2*(I*sin(d*x+c)-cos(d*x+c)-1)/(cos(d*x+c)+1
)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))-60*I*tan(d*x+c)*sec(d*x+c)*arctan(1/2*(I*sin(d*x+c)-cos(d*x+c)-1)/(cos(d
*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+100*I*tan(d*x+c)*sec(d*x+c)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+60
*arctan(1/2*(I*sin(d*x+c)-cos(d*x+c)-1)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+164*(-cos(d*x+c)/(c
os(d*x+c)+1))^(1/2)-15*I*tan(d*x+c)*sec(d*x+c)^2*arctan(1/2*(I*sin(d*x+c)-cos(d*x+c)-1)/(cos(d*x+c)+1)/(-cos(d
*x+c)/(cos(d*x+c)+1))^(1/2))-120*sec(d*x+c)*arctan(1/2*(I*sin(d*x+c)-cos(d*x+c)-1)/(cos(d*x+c)+1)/(-cos(d*x+c)
/(cos(d*x+c)+1))^(1/2))+164*sec(d*x+c)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)-45*sec(d*x+c)^2*arctan(1/2*(I*sin(d*
x+c)-cos(d*x+c)-1)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))-30*sec(d*x+c)^2*(-cos(d*x+c)/(cos(d*x+c)
+1))^(1/2)+15*sec(d*x+c)^3*arctan(1/2*(I*sin(d*x+c)-cos(d*x+c)-1)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^
(1/2))-30*sec(d*x+c)^3*(-cos(d*x+c)/(cos(d*x+c)+1))^(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. 278 vs. \(2 (118) = 236\).

Time = 0.25 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.77 \[ \int \frac {\sec (c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\frac {{\left (-15 i \, \sqrt {\frac {1}{2}} a^{4} d \sqrt {\frac {1}{a^{7} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (-\frac {5 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (i \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a^{3} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{7} d^{2}}} - i\right )} e^{\left (-i \, d x - i \, c\right )}}{32 \, a^{3} d}\right ) + 15 i \, \sqrt {\frac {1}{2}} a^{4} d \sqrt {\frac {1}{a^{7} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (-\frac {5 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (-i \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a^{3} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{7} d^{2}}} - i\right )} e^{\left (-i \, d x - i \, c\right )}}{32 \, a^{3} d}\right ) + \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (33 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 59 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 34 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 8 i\right )}\right )} e^{\left (-6 i \, d x - 6 i \, c\right )}}{384 \, a^{4} d} \]

[In]

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

[Out]

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)) +
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)) + sqrt(
2)*sqrt(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]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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