\(\int \frac {\cos (c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx\) [392]

Optimal result

Integrand size = 24, antiderivative size = 227 \[ \int \frac {\cos (c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\frac {315 i \text {arctanh}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{2048 \sqrt {2} a^{7/2} d}+\frac {i \cos (c+d x)}{8 d (a+i a \tan (c+d x))^{7/2}}+\frac {3 i \cos (c+d x)}{32 a d (a+i a \tan (c+d x))^{5/2}}+\frac {21 i \cos (c+d x)}{256 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac {105 i \cos (c+d x)}{1024 a^3 d \sqrt {a+i a \tan (c+d x)}}-\frac {315 i \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{2048 a^4 d} \]

[Out]

315/4096*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)+105/1024*I*cos(d
*x+c)/a^3/d/(a+I*a*tan(d*x+c))^(1/2)-315/2048*I*cos(d*x+c)*(a+I*a*tan(d*x+c))^(1/2)/a^4/d+1/8*I*cos(d*x+c)/d/(
a+I*a*tan(d*x+c))^(7/2)+3/32*I*cos(d*x+c)/a/d/(a+I*a*tan(d*x+c))^(5/2)+21/256*I*cos(d*x+c)/a^2/d/(a+I*a*tan(d*
x+c))^(3/2)

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3583, 3571, 3570, 212} \[ \int \frac {\cos (c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\frac {315 i \text {arctanh}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{2048 \sqrt {2} a^{7/2} d}-\frac {315 i \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{2048 a^4 d}+\frac {105 i \cos (c+d x)}{1024 a^3 d \sqrt {a+i a \tan (c+d x)}}+\frac {21 i \cos (c+d x)}{256 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac {3 i \cos (c+d x)}{32 a d (a+i a \tan (c+d x))^{5/2}}+\frac {i \cos (c+d x)}{8 d (a+i a \tan (c+d x))^{7/2}} \]

[In]

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

[Out]

(((315*I)/2048)*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(Sqrt[2]*a^(7/2)*d) + ((
I/8)*Cos[c + d*x])/(d*(a + I*a*Tan[c + d*x])^(7/2)) + (((3*I)/32)*Cos[c + d*x])/(a*d*(a + I*a*Tan[c + d*x])^(5
/2)) + (((21*I)/256)*Cos[c + d*x])/(a^2*d*(a + I*a*Tan[c + d*x])^(3/2)) + (((105*I)/1024)*Cos[c + d*x])/(a^3*d
*Sqrt[a + I*a*Tan[c + d*x]]) - (((315*I)/2048)*Cos[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(a^4*d)

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 3571

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] + Dist[a/(2*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] && EqQ[m/2 + n, 0] && GtQ[n, 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 \cos (c+d x)}{8 d (a+i a \tan (c+d x))^{7/2}}+\frac {9 \int \frac {\cos (c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx}{16 a} \\ & = \frac {i \cos (c+d x)}{8 d (a+i a \tan (c+d x))^{7/2}}+\frac {3 i \cos (c+d x)}{32 a d (a+i a \tan (c+d x))^{5/2}}+\frac {21 \int \frac {\cos (c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx}{64 a^2} \\ & = \frac {i \cos (c+d x)}{8 d (a+i a \tan (c+d x))^{7/2}}+\frac {3 i \cos (c+d x)}{32 a d (a+i a \tan (c+d x))^{5/2}}+\frac {21 i \cos (c+d x)}{256 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac {105 \int \frac {\cos (c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx}{512 a^3} \\ & = \frac {i \cos (c+d x)}{8 d (a+i a \tan (c+d x))^{7/2}}+\frac {3 i \cos (c+d x)}{32 a d (a+i a \tan (c+d x))^{5/2}}+\frac {21 i \cos (c+d x)}{256 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac {105 i \cos (c+d x)}{1024 a^3 d \sqrt {a+i a \tan (c+d x)}}+\frac {315 \int \cos (c+d x) \sqrt {a+i a \tan (c+d x)} \, dx}{2048 a^4} \\ & = \frac {i \cos (c+d x)}{8 d (a+i a \tan (c+d x))^{7/2}}+\frac {3 i \cos (c+d x)}{32 a d (a+i a \tan (c+d x))^{5/2}}+\frac {21 i \cos (c+d x)}{256 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac {105 i \cos (c+d x)}{1024 a^3 d \sqrt {a+i a \tan (c+d x)}}-\frac {315 i \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{2048 a^4 d}+\frac {315 \int \frac {\sec (c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx}{4096 a^3} \\ & = \frac {i \cos (c+d x)}{8 d (a+i a \tan (c+d x))^{7/2}}+\frac {3 i \cos (c+d x)}{32 a d (a+i a \tan (c+d x))^{5/2}}+\frac {21 i \cos (c+d x)}{256 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac {105 i \cos (c+d x)}{1024 a^3 d \sqrt {a+i a \tan (c+d x)}}-\frac {315 i \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{2048 a^4 d}+\frac {(315 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 )}{2048 a^3 d} \\ & = \frac {315 i \text {arctanh}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{2048 \sqrt {2} a^{7/2} d}+\frac {i \cos (c+d x)}{8 d (a+i a \tan (c+d x))^{7/2}}+\frac {3 i \cos (c+d x)}{32 a d (a+i a \tan (c+d x))^{5/2}}+\frac {21 i \cos (c+d x)}{256 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac {105 i \cos (c+d x)}{1024 a^3 d \sqrt {a+i a \tan (c+d x)}}-\frac {315 i \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{2048 a^4 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.32 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.62 \[ \int \frac {\cos (c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=-\frac {\sec ^3(c+d x) \left (420+\frac {630 e^{4 i (c+d x)} \text {arctanh}\left (\sqrt {1+e^{2 i (c+d x)}}\right )}{\sqrt {1+e^{2 i (c+d x)}}}+826 \cos (2 (c+d x))-224 \cos (4 (c+d x))+474 i \sin (2 (c+d x))-288 i \sin (4 (c+d x))\right )}{4096 a^3 d (-i+\tan (c+d x))^3 \sqrt {a+i a \tan (c+d x)}} \]

[In]

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

[Out]

-1/4096*(Sec[c + d*x]^3*(420 + (630*E^((4*I)*(c + d*x))*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]])/Sqrt[1 + E^((2
*I)*(c + d*x))] + 826*Cos[2*(c + d*x)] - 224*Cos[4*(c + d*x)] + (474*I)*Sin[2*(c + d*x)] - (288*I)*Sin[4*(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. 922 vs. \(2 (184 ) = 368\).

Time = 12.80 (sec) , antiderivative size = 923, normalized size of antiderivative = 4.07

method	result	size
default	\(\text {Expression too large to display}\)	\(923\)

[In]

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

[Out]

-1/4096/d/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)/(cos(d*x+c)+1)/(a*(1+I*tan(d*x+c)))^(1/2)/(1+I*tan(d*x+c))^3/a^3*
(1792*I*cos(d*x+c)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+1792*I*cos(d*x+c)^2*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)-2
520*I*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))-230
4*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)*sin(d*x+c)+630*I*sec(d*x+c)^3*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/
2)-315*I*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)
)-2304*sin(d*x+c)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+2520*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))*sin(d*x+c)-3444*I*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+630*I*sec(d*x+c)^2*
(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+2100*tan(d*x+c)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+1260*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))-3444*I*sec(d*x+c)*(-cos(d*x+
c)/(cos(d*x+c)+1))^(1/2)-1260*I*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))+2100*tan(d*x+c)*sec(d*x+c)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)-1260*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))+2520*I*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))+945*I*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))-315*tan(d*x+c)*sec(d*x+c)^2*arct
an(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)))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.32 \[ \int \frac {\cos (c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\frac {{\left (-315 i \, \sqrt {\frac {1}{2}} a^{4} d \sqrt {\frac {1}{a^{7} d^{2}}} e^{\left (8 i \, d x + 8 i \, c\right )} \log \left (-\frac {315 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (i \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a^{3} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{7} d^{2}}} - i\right )} e^{\left (-i \, d x - i \, c\right )}}{1024 \, a^{3} d}\right ) + 315 i \, \sqrt {\frac {1}{2}} a^{4} d \sqrt {\frac {1}{a^{7} d^{2}}} e^{\left (8 i \, d x + 8 i \, c\right )} \log \left (-\frac {315 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (-i \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a^{3} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{7} d^{2}}} - i\right )} e^{\left (-i \, d x - i \, c\right )}}{1024 \, a^{3} d}\right ) + \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-128 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 197 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 535 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 298 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 104 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 16 i\right )}\right )} e^{\left (-8 i \, d x - 8 i \, c\right )}}{4096 \, a^{4} d} \]

[In]

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

[Out]

1/4096*(-315*I*sqrt(1/2)*a^4*d*sqrt(1/(a^7*d^2))*e^(8*I*d*x + 8*I*c)*log(-315/1024*(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)) + 315*I*sqrt(1/2)*a^4*d*sqrt(1/(a^7*d^2))*e^(8*I*d*x + 8*I*c)*log(-315/1024*(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))*(-128*I*e^(10*I*d*x + 10*I*c) + 197*I*e^(8*I*d*x + 8*I*c) + 53
5*I*e^(6*I*d*x + 6*I*c) + 298*I*e^(4*I*d*x + 4*I*c) + 104*I*e^(2*I*d*x + 2*I*c) + 16*I))*e^(-8*I*d*x - 8*I*c)/
(a^4*d)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

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. 2779 vs. \(2 (172) = 344\).

Time = 0.49 (sec) , antiderivative size = 2779, normalized size of antiderivative = 12.24 \[ \int \frac {\cos (c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/16384*(4*(cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*
d*x + 8*c)))^2 + 2*cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 1)^(3/4)*(325*((-I*sqrt(2)*cos(8*d*x
+ 8*c) - sqrt(2)*sin(8*d*x + 8*c))*cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + (-I*sqrt(2)*cos(8
*d*x + 8*c) - sqrt(2)*sin(8*d*x + 8*c))*sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + 2*(-I*sqrt(2)
*cos(8*d*x + 8*c) - sqrt(2)*sin(8*d*x + 8*c))*cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) - I*sqrt(2)
*cos(8*d*x + 8*c) - sqrt(2)*sin(8*d*x + 8*c))*cos(7/2*arctan2(sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*
c))), cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 1)) + 643*(-I*sqrt(2)*cos(8*d*x + 8*c) - sqrt(2)*
sin(8*d*x + 8*c))*cos(3/2*arctan2(sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))), cos(1/4*arctan2(sin(8*
d*x + 8*c), cos(8*d*x + 8*c))) + 1)) + 325*((sqrt(2)*cos(8*d*x + 8*c) - I*sqrt(2)*sin(8*d*x + 8*c))*cos(1/4*ar
ctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + (sqrt(2)*cos(8*d*x + 8*c) - I*sqrt(2)*sin(8*d*x + 8*c))*sin(1/4
*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + 2*(sqrt(2)*cos(8*d*x + 8*c) - I*sqrt(2)*sin(8*d*x + 8*c))*co
s(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + sqrt(2)*cos(8*d*x + 8*c) - I*sqrt(2)*sin(8*d*x + 8*c))*si
n(7/2*arctan2(sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))), cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*
x + 8*c))) + 1)) + 643*(sqrt(2)*cos(8*d*x + 8*c) - I*sqrt(2)*sin(8*d*x + 8*c))*sin(3/2*arctan2(sin(1/4*arctan2
(sin(8*d*x + 8*c), cos(8*d*x + 8*c))), cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 1)))*sqrt(a) + 4
*(cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))
)^2 + 2*cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 1)^(1/4)*(765*((I*sqrt(2)*cos(8*d*x + 8*c) + sq
rt(2)*sin(8*d*x + 8*c))*cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + (I*sqrt(2)*cos(8*d*x + 8*c) +
sqrt(2)*sin(8*d*x + 8*c))*sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + 2*(I*sqrt(2)*cos(8*d*x + 8
*c) + sqrt(2)*sin(8*d*x + 8*c))*cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + I*sqrt(2)*cos(8*d*x + 8
*c) + sqrt(2)*sin(8*d*x + 8*c))*cos(5/2*arctan2(sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))), cos(1/4*
arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 1)) + (187*I*sqrt(2)*cos(8*d*x + 8*c) + 187*sqrt(2)*sin(8*d*x +
8*c) + 128*I*sqrt(2))*cos(1/2*arctan2(sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))), cos(1/4*arctan2(s
in(8*d*x + 8*c), cos(8*d*x + 8*c))) + 1)) - 765*((sqrt(2)*cos(8*d*x + 8*c) - I*sqrt(2)*sin(8*d*x + 8*c))*cos(1
/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + (sqrt(2)*cos(8*d*x + 8*c) - I*sqrt(2)*sin(8*d*x + 8*c))*si
n(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + 2*(sqrt(2)*cos(8*d*x + 8*c) - I*sqrt(2)*sin(8*d*x + 8*c
))*cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + sqrt(2)*cos(8*d*x + 8*c) - I*sqrt(2)*sin(8*d*x + 8*c
))*sin(5/2*arctan2(sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))), cos(1/4*arctan2(sin(8*d*x + 8*c), cos
(8*d*x + 8*c))) + 1)) - (187*sqrt(2)*cos(8*d*x + 8*c) - 187*I*sqrt(2)*sin(8*d*x + 8*c) + 128*sqrt(2))*sin(1/2*
arctan2(sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))), cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*
c))) + 1)))*sqrt(a) + 315*(2*sqrt(2)*arctan2((cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + sin(1/4
*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + 2*cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 1)^
(1/4)*sin(1/2*arctan2(sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))), cos(1/4*arctan2(sin(8*d*x + 8*c),
cos(8*d*x + 8*c))) + 1)), (cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + sin(1/4*arctan2(sin(8*d*x
+ 8*c), cos(8*d*x + 8*c)))^2 + 2*cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 1)^(1/4)*cos(1/2*arcta
n2(sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))), cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))
+ 1)) + 1) - 2*sqrt(2)*arctan2((cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + sin(1/4*arctan2(sin(8
*d*x + 8*c), cos(8*d*x + 8*c)))^2 + 2*cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 1)^(1/4)*sin(1/2*
arctan2(sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))), cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*
c))) + 1)), (cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*
d*x + 8*c)))^2 + 2*cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 1)^(1/4)*cos(1/2*arctan2(sin(1/4*arc
tan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))), cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 1)) - 1) - I
*sqrt(2)*log(sqrt(cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + sin(1/4*arctan2(sin(8*d*x + 8*c), c
os(8*d*x + 8*c)))^2 + 2*cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 1)*cos(1/2*arctan2(sin(1/4*arct
an2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))), cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 1))^2 + sqrt(
cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^
2 + 2*cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 1)*sin(1/2*arctan2(sin(1/4*arctan2(sin(8*d*x + 8*
c), cos(8*d*x + 8*c))), cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 1))^2 + 2*(cos(1/4*arctan2(sin(
8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + 2*cos(1/4*arctan
2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 1)^(1/4)*cos(1/2*arctan2(sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x
+ 8*c))), cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 1)) + 1) + I*sqrt(2)*log(sqrt(cos(1/4*arctan2
(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + 2*cos(1/4*a
rctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 1)*cos(1/2*arctan2(sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x +
8*c))), cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 1))^2 + sqrt(cos(1/4*arctan2(sin(8*d*x + 8*c),
cos(8*d*x + 8*c)))^2 + sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + 2*cos(1/4*arctan2(sin(8*d*x +
8*c), cos(8*d*x + 8*c))) + 1)*sin(1/2*arctan2(sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))), cos(1/4*a
rctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 1))^2 - 2*(cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^
2 + sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + 2*cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8
*c))) + 1)^(1/4)*cos(1/2*arctan2(sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))), cos(1/4*arctan2(sin(8*d
*x + 8*c), cos(8*d*x + 8*c))) + 1)) + 1))*sqrt(a))/(a^4*d)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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