Integrand size = 24, antiderivative size = 227 \[ \int \frac {\cos (c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\frac {315 i \text {arctanh}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{2048 \sqrt {2} a^{7/2} d}+\frac {i \cos (c+d x)}{8 d (a+i a \tan (c+d x))^{7/2}}+\frac {3 i \cos (c+d x)}{32 a d (a+i a \tan (c+d x))^{5/2}}+\frac {21 i \cos (c+d x)}{256 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac {105 i \cos (c+d x)}{1024 a^3 d \sqrt {a+i a \tan (c+d x)}}-\frac {315 i \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{2048 a^4 d} \]
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)-3 15/2048*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)+21/25 6*I*cos(d*x+c)/a^2/d/(a+I*a*tan(d*x+c))^(3/2)
Time = 2.32 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.62 \[ \int \frac {\cos (c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=-\frac {\sec ^3(c+d x) \left (420+\frac {630 e^{4 i (c+d x)} \text {arctanh}\left (\sqrt {1+e^{2 i (c+d x)}}\right )}{\sqrt {1+e^{2 i (c+d x)}}}+826 \cos (2 (c+d x))-224 \cos (4 (c+d x))+474 i \sin (2 (c+d x))-288 i \sin (4 (c+d x))\right )}{4096 a^3 d (-i+\tan (c+d x))^3 \sqrt {a+i a \tan (c+d x)}} \]
-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]])
Time = 1.01 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.07, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {3042, 3983, 3042, 3983, 3042, 3983, 3042, 3983, 3042, 3971, 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.
\(\displaystyle \int \frac {\cos (c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sec (c+d x) (a+i a \tan (c+d x))^{7/2}}dx\) |
\(\Big \downarrow \) 3983 |
\(\displaystyle \frac {9 \int \frac {\cos (c+d x)}{(i \tan (c+d x) a+a)^{5/2}}dx}{16 a}+\frac {i \cos (c+d x)}{8 d (a+i a \tan (c+d x))^{7/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {9 \int \frac {1}{\sec (c+d x) (i \tan (c+d x) a+a)^{5/2}}dx}{16 a}+\frac {i \cos (c+d x)}{8 d (a+i a \tan (c+d x))^{7/2}}\) |
\(\Big \downarrow \) 3983 |
\(\displaystyle \frac {9 \left (\frac {7 \int \frac {\cos (c+d x)}{(i \tan (c+d x) a+a)^{3/2}}dx}{12 a}+\frac {i \cos (c+d x)}{6 d (a+i a \tan (c+d x))^{5/2}}\right )}{16 a}+\frac {i \cos (c+d x)}{8 d (a+i a \tan (c+d x))^{7/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {9 \left (\frac {7 \int \frac {1}{\sec (c+d x) (i \tan (c+d x) a+a)^{3/2}}dx}{12 a}+\frac {i \cos (c+d x)}{6 d (a+i a \tan (c+d x))^{5/2}}\right )}{16 a}+\frac {i \cos (c+d x)}{8 d (a+i a \tan (c+d x))^{7/2}}\) |
\(\Big \downarrow \) 3983 |
\(\displaystyle \frac {9 \left (\frac {7 \left (\frac {5 \int \frac {\cos (c+d x)}{\sqrt {i \tan (c+d x) a+a}}dx}{8 a}+\frac {i \cos (c+d x)}{4 d (a+i a \tan (c+d x))^{3/2}}\right )}{12 a}+\frac {i \cos (c+d x)}{6 d (a+i a \tan (c+d x))^{5/2}}\right )}{16 a}+\frac {i \cos (c+d x)}{8 d (a+i a \tan (c+d x))^{7/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {9 \left (\frac {7 \left (\frac {5 \int \frac {1}{\sec (c+d x) \sqrt {i \tan (c+d x) a+a}}dx}{8 a}+\frac {i \cos (c+d x)}{4 d (a+i a \tan (c+d x))^{3/2}}\right )}{12 a}+\frac {i \cos (c+d x)}{6 d (a+i a \tan (c+d x))^{5/2}}\right )}{16 a}+\frac {i \cos (c+d x)}{8 d (a+i a \tan (c+d x))^{7/2}}\) |
\(\Big \downarrow \) 3983 |
\(\displaystyle \frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \int \cos (c+d x) \sqrt {i \tan (c+d x) a+a}dx}{4 a}+\frac {i \cos (c+d x)}{2 d \sqrt {a+i a \tan (c+d x)}}\right )}{8 a}+\frac {i \cos (c+d x)}{4 d (a+i a \tan (c+d x))^{3/2}}\right )}{12 a}+\frac {i \cos (c+d x)}{6 d (a+i a \tan (c+d x))^{5/2}}\right )}{16 a}+\frac {i \cos (c+d x)}{8 d (a+i a \tan (c+d x))^{7/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \int \frac {\sqrt {i \tan (c+d x) a+a}}{\sec (c+d x)}dx}{4 a}+\frac {i \cos (c+d x)}{2 d \sqrt {a+i a \tan (c+d x)}}\right )}{8 a}+\frac {i \cos (c+d x)}{4 d (a+i a \tan (c+d x))^{3/2}}\right )}{12 a}+\frac {i \cos (c+d x)}{6 d (a+i a \tan (c+d x))^{5/2}}\right )}{16 a}+\frac {i \cos (c+d x)}{8 d (a+i a \tan (c+d x))^{7/2}}\) |
\(\Big \downarrow \) 3971 |
\(\displaystyle \frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {1}{2} a \int \frac {\sec (c+d x)}{\sqrt {i \tan (c+d x) a+a}}dx-\frac {i \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}\right )}{4 a}+\frac {i \cos (c+d x)}{2 d \sqrt {a+i a \tan (c+d x)}}\right )}{8 a}+\frac {i \cos (c+d x)}{4 d (a+i a \tan (c+d x))^{3/2}}\right )}{12 a}+\frac {i \cos (c+d x)}{6 d (a+i a \tan (c+d x))^{5/2}}\right )}{16 a}+\frac {i \cos (c+d x)}{8 d (a+i a \tan (c+d x))^{7/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {1}{2} a \int \frac {\sec (c+d x)}{\sqrt {i \tan (c+d x) a+a}}dx-\frac {i \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}\right )}{4 a}+\frac {i \cos (c+d x)}{2 d \sqrt {a+i a \tan (c+d x)}}\right )}{8 a}+\frac {i \cos (c+d x)}{4 d (a+i a \tan (c+d x))^{3/2}}\right )}{12 a}+\frac {i \cos (c+d x)}{6 d (a+i a \tan (c+d x))^{5/2}}\right )}{16 a}+\frac {i \cos (c+d x)}{8 d (a+i a \tan (c+d x))^{7/2}}\) |
\(\Big \downarrow \) 3970 |
\(\displaystyle \frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {i a \int \frac {1}{2-\frac {a \sec ^2(c+d x)}{i \tan (c+d x) a+a}}d\frac {\sec (c+d x)}{\sqrt {i \tan (c+d x) a+a}}}{d}-\frac {i \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}\right )}{4 a}+\frac {i \cos (c+d x)}{2 d \sqrt {a+i a \tan (c+d x)}}\right )}{8 a}+\frac {i \cos (c+d x)}{4 d (a+i a \tan (c+d x))^{3/2}}\right )}{12 a}+\frac {i \cos (c+d x)}{6 d (a+i a \tan (c+d x))^{5/2}}\right )}{16 a}+\frac {i \cos (c+d x)}{8 d (a+i a \tan (c+d x))^{7/2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {i \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{\sqrt {2} d}-\frac {i \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}\right )}{4 a}+\frac {i \cos (c+d x)}{2 d \sqrt {a+i a \tan (c+d x)}}\right )}{8 a}+\frac {i \cos (c+d x)}{4 d (a+i a \tan (c+d x))^{3/2}}\right )}{12 a}+\frac {i \cos (c+d x)}{6 d (a+i a \tan (c+d x))^{5/2}}\right )}{16 a}+\frac {i \cos (c+d x)}{8 d (a+i a \tan (c+d x))^{7/2}}\) |
((I/8)*Cos[c + d*x])/(d*(a + I*a*Tan[c + d*x])^(7/2)) + (9*(((I/6)*Cos[c + d*x])/(d*(a + I*a*Tan[c + d*x])^(5/2)) + (7*(((I/4)*Cos[c + d*x])/(d*(a + I*a*Tan[c + d*x])^(3/2)) + (5*(((I/2)*Cos[c + d*x])/(d*Sqrt[a + I*a*Tan[c + d*x]]) + (3*((I*Sqrt[a]*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(Sqrt[2]*d) - (I*Cos[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/d))/(4*a)))/(8*a)))/(12*a)))/(16*a)
3.4.92.3.1 Defintions of rubi rules used
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])
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 ]
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/(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]
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]
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
-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)-252 0*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))-2304*(-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)-3 15*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)/(-co s(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*si n(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)/(-c os(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*s ec(d*x+c)^2*arctan(1/2*(I*sin(d*x+c)-cos(d*x+c)-1)/(cos(d*x+c)+1)/(-cos...
Time = 0.26 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.32 \[ \int \frac {\cos (c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\frac {{\left (-315 i \, \sqrt {\frac {1}{2}} a^{4} d \sqrt {\frac {1}{a^{7} d^{2}}} e^{\left (8 i \, d x + 8 i \, c\right )} \log \left (-\frac {315 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (i \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a^{3} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{7} d^{2}}} - i\right )} e^{\left (-i \, d x - i \, c\right )}}{1024 \, a^{3} d}\right ) + 315 i \, \sqrt {\frac {1}{2}} a^{4} d \sqrt {\frac {1}{a^{7} d^{2}}} e^{\left (8 i \, d x + 8 i \, c\right )} \log \left (-\frac {315 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (-i \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a^{3} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{7} d^{2}}} - i\right )} e^{\left (-i \, d x - i \, c\right )}}{1024 \, a^{3} d}\right ) + \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-128 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 197 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 535 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 298 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 104 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 16 i\right )}\right )} e^{\left (-8 i \, d x - 8 i \, c\right )}}{4096 \, a^{4} d} \]
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) + 535*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 )
Timed out. \[ \int \frac {\cos (c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\text {Timed out} \]
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 \[ \int \frac {\cos (c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\text {Too large to display} \]
-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*ar ctan2(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*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))*sin(1/4*arctan2(s in(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + 2*(sqrt(2)*cos(8*d*x + 8*c) - I*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) )) + sqrt(2)*cos(8*d*x + 8*c) - I*sqrt(2)*sin(8*d*x + 8*c))*sin(7/2*arctan 2(sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))), cos(1/4*arctan2(si n(8*d*x + 8*c), cos(8*d*x + 8*c))) + 1)) + 643*(sqrt(2)*cos(8*d*x + 8*c...
\[ \int \frac {\cos (c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\int { \frac {\cos \left (d x + c\right )}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {\cos (c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\int \frac {\cos \left (c+d\,x\right )}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{7/2}} \,d x \]