Integrand size = 24, antiderivative size = 225 \[ \int \cos ^{16}(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {a^8 x}{256}-\frac {i a^{16}}{16 d (a-i a \tan (c+d x))^8}-\frac {i a^{15}}{28 d (a-i a \tan (c+d x))^7}-\frac {i a^{14}}{48 d (a-i a \tan (c+d x))^6}-\frac {i a^{13}}{80 d (a-i a \tan (c+d x))^5}-\frac {i a^{12}}{128 d (a-i a \tan (c+d x))^4}-\frac {i a^{11}}{192 d (a-i a \tan (c+d x))^3}-\frac {i a^{10}}{256 d (a-i a \tan (c+d x))^2}-\frac {i a^9}{256 d (a-i a \tan (c+d x))} \]
1/256*a^8*x-1/16*I*a^16/d/(a-I*a*tan(d*x+c))^8-1/28*I*a^15/d/(a-I*a*tan(d* x+c))^7-1/48*I*a^14/d/(a-I*a*tan(d*x+c))^6-1/80*I*a^13/d/(a-I*a*tan(d*x+c) )^5-1/128*I*a^12/d/(a-I*a*tan(d*x+c))^4-1/192*I*a^11/d/(a-I*a*tan(d*x+c))^ 3-1/256*I*a^10/d/(a-I*a*tan(d*x+c))^2-1/256*I*a^9/d/(a-I*a*tan(d*x+c))
Time = 0.98 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.68 \[ \int \cos ^{16}(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {i a^8 \sec ^8(c+d x) (7350+12544 \cos (2 (c+d x))+7840 \cos (4 (c+d x))+3840 \cos (6 (c+d x))+1194 \cos (8 (c+d x))-3136 i \sin (2 (c+d x))-3920 i \sin (4 (c+d x))-2880 i \sin (6 (c+d x))-1089 i \sin (8 (c+d x))+840 \arctan (\tan (c+d x)) (i \cos (8 (c+d x))+\sin (8 (c+d x))))}{215040 d (i+\tan (c+d x))^8} \]
((-1/215040*I)*a^8*Sec[c + d*x]^8*(7350 + 12544*Cos[2*(c + d*x)] + 7840*Co s[4*(c + d*x)] + 3840*Cos[6*(c + d*x)] + 1194*Cos[8*(c + d*x)] - (3136*I)* Sin[2*(c + d*x)] - (3920*I)*Sin[4*(c + d*x)] - (2880*I)*Sin[6*(c + d*x)] - (1089*I)*Sin[8*(c + d*x)] + 840*ArcTan[Tan[c + d*x]]*(I*Cos[8*(c + d*x)] + Sin[8*(c + d*x)])))/(d*(I + Tan[c + d*x])^8)
Time = 0.36 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.90, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3042, 3968, 54, 2009}
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.
| \(\displaystyle \int \cos ^{16}(c+d x) (a+i a \tan (c+d x))^8 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+i a \tan (c+d x))^8}{\sec (c+d x)^{16}}dx\) |
| \(\Big \downarrow \) 3968 |
| \(\displaystyle -\frac {i a^{17} \int \frac {1}{(a-i a \tan (c+d x))^9 (i \tan (c+d x) a+a)}d(i a \tan (c+d x))}{d}\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle -\frac {i a^{17} \int \left (\frac {1}{2 (a-i a \tan (c+d x))^9 a}+\frac {1}{4 (a-i a \tan (c+d x))^8 a^2}+\frac {1}{8 (a-i a \tan (c+d x))^7 a^3}+\frac {1}{16 (a-i a \tan (c+d x))^6 a^4}+\frac {1}{32 (a-i a \tan (c+d x))^5 a^5}+\frac {1}{64 (a-i a \tan (c+d x))^4 a^6}+\frac {1}{128 (a-i a \tan (c+d x))^3 a^7}+\frac {1}{256 \left (\tan ^2(c+d x) a^2+a^2\right ) a^8}+\frac {1}{256 (a-i a \tan (c+d x))^2 a^8}\right )d(i a \tan (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {i a^{17} \left (\frac {i \arctan (\tan (c+d x))}{256 a^9}+\frac {1}{256 a^8 (a-i a \tan (c+d x))}+\frac {1}{256 a^7 (a-i a \tan (c+d x))^2}+\frac {1}{192 a^6 (a-i a \tan (c+d x))^3}+\frac {1}{128 a^5 (a-i a \tan (c+d x))^4}+\frac {1}{80 a^4 (a-i a \tan (c+d x))^5}+\frac {1}{48 a^3 (a-i a \tan (c+d x))^6}+\frac {1}{28 a^2 (a-i a \tan (c+d x))^7}+\frac {1}{16 a (a-i a \tan (c+d x))^8}\right )}{d}\) |
((-I)*a^17*(((I/256)*ArcTan[Tan[c + d*x]])/a^9 + 1/(16*a*(a - I*a*Tan[c + d*x])^8) + 1/(28*a^2*(a - I*a*Tan[c + d*x])^7) + 1/(48*a^3*(a - I*a*Tan[c + d*x])^6) + 1/(80*a^4*(a - I*a*Tan[c + d*x])^5) + 1/(128*a^5*(a - I*a*Tan [c + d*x])^4) + 1/(192*a^6*(a - I*a*Tan[c + d*x])^3) + 1/(256*a^7*(a - I*a *Tan[c + d*x])^2) + 1/(256*a^8*(a - I*a*Tan[c + d*x]))))/d
3.1.89.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_ ), x_Symbol] :> Simp[1/(a^(m - 2)*b*f) Subst[Int[(a - x)^(m/2 - 1)*(a + x )^(n + m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x] && EqQ[a^2 + b^2, 0] && IntegerQ[m/2]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 738 vs. \(2 (191 ) = 382\).
Time = 2.59 (sec) , antiderivative size = 739, normalized size of antiderivative = 3.28
\[\text {Expression too large to display}\]
1/d*(a^8*(-1/16*sin(d*x+c)^7*cos(d*x+c)^9-1/32*sin(d*x+c)^5*cos(d*x+c)^9-5 /384*sin(d*x+c)^3*cos(d*x+c)^9-1/256*sin(d*x+c)*cos(d*x+c)^9+1/2048*(cos(d *x+c)^7+7/6*cos(d*x+c)^5+35/24*cos(d*x+c)^3+35/16*cos(d*x+c))*sin(d*x+c)+3 5/32768*d*x+35/32768*c)+56*I*a^8*(-1/16*sin(d*x+c)^4*cos(d*x+c)^12-1/56*si n(d*x+c)^2*cos(d*x+c)^12-1/336*cos(d*x+c)^12)-28*a^8*(-1/16*sin(d*x+c)^5*c os(d*x+c)^11-5/224*sin(d*x+c)^3*cos(d*x+c)^11-5/896*sin(d*x+c)*cos(d*x+c)^ 11+1/1792*(cos(d*x+c)^9+9/8*cos(d*x+c)^7+21/16*cos(d*x+c)^5+105/64*cos(d*x +c)^3+315/128*cos(d*x+c))*sin(d*x+c)+45/32768*d*x+45/32768*c)-8*I*a^8*(-1/ 16*sin(d*x+c)^6*cos(d*x+c)^10-3/112*sin(d*x+c)^4*cos(d*x+c)^10-1/112*cos(d *x+c)^10*sin(d*x+c)^2-1/560*cos(d*x+c)^10)+70*a^8*(-1/16*sin(d*x+c)^3*cos( d*x+c)^13-3/224*sin(d*x+c)*cos(d*x+c)^13+1/896*(cos(d*x+c)^11+11/10*cos(d* x+c)^9+99/80*cos(d*x+c)^7+231/160*cos(d*x+c)^5+231/128*cos(d*x+c)^3+693/25 6*cos(d*x+c))*sin(d*x+c)+99/32768*d*x+99/32768*c)-1/2*I*a^8*cos(d*x+c)^16- 28*a^8*(-1/16*sin(d*x+c)*cos(d*x+c)^15+1/224*(cos(d*x+c)^13+13/12*cos(d*x+ c)^11+143/120*cos(d*x+c)^9+429/320*cos(d*x+c)^7+1001/640*cos(d*x+c)^5+1001 /512*cos(d*x+c)^3+3003/1024*cos(d*x+c))*sin(d*x+c)+429/32768*d*x+429/32768 *c)-56*I*a^8*(-1/16*cos(d*x+c)^14*sin(d*x+c)^2-1/112*cos(d*x+c)^14)+a^8*(1 /16*(cos(d*x+c)^15+15/14*cos(d*x+c)^13+65/56*cos(d*x+c)^11+143/112*cos(d*x +c)^9+1287/896*cos(d*x+c)^7+429/256*cos(d*x+c)^5+2145/1024*cos(d*x+c)^3+64 35/2048*cos(d*x+c))*sin(d*x+c)+6435/32768*d*x+6435/32768*c))
Time = 0.30 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.56 \[ \int \cos ^{16}(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {1680 \, a^{8} d x - 105 i \, a^{8} e^{\left (16 i \, d x + 16 i \, c\right )} - 960 i \, a^{8} e^{\left (14 i \, d x + 14 i \, c\right )} - 3920 i \, a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} - 9408 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} - 14700 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} - 15680 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} - 11760 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} - 6720 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )}}{430080 \, d} \]
1/430080*(1680*a^8*d*x - 105*I*a^8*e^(16*I*d*x + 16*I*c) - 960*I*a^8*e^(14 *I*d*x + 14*I*c) - 3920*I*a^8*e^(12*I*d*x + 12*I*c) - 9408*I*a^8*e^(10*I*d *x + 10*I*c) - 14700*I*a^8*e^(8*I*d*x + 8*I*c) - 15680*I*a^8*e^(6*I*d*x + 6*I*c) - 11760*I*a^8*e^(4*I*d*x + 4*I*c) - 6720*I*a^8*e^(2*I*d*x + 2*I*c)) /d
Time = 0.69 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.44 \[ \int \cos ^{16}(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {a^{8} x}{256} + \begin {cases} \frac {- 354658470655426560 i a^{8} d^{7} e^{16 i c} e^{16 i d x} - 3242591731706757120 i a^{8} d^{7} e^{14 i c} e^{14 i d x} - 13240582904469258240 i a^{8} d^{7} e^{12 i c} e^{12 i d x} - 31777398970726219776 i a^{8} d^{7} e^{10 i c} e^{10 i d x} - 49652185891759718400 i a^{8} d^{7} e^{8 i c} e^{8 i d x} - 52962331617877032960 i a^{8} d^{7} e^{6 i c} e^{6 i d x} - 39721748713407774720 i a^{8} d^{7} e^{4 i c} e^{4 i d x} - 22698142121947299840 i a^{8} d^{7} e^{2 i c} e^{2 i d x}}{1452681095804627189760 d^{8}} & \text {for}\: d^{8} \neq 0 \\x \left (\frac {a^{8} e^{16 i c}}{256} + \frac {a^{8} e^{14 i c}}{32} + \frac {7 a^{8} e^{12 i c}}{64} + \frac {7 a^{8} e^{10 i c}}{32} + \frac {35 a^{8} e^{8 i c}}{128} + \frac {7 a^{8} e^{6 i c}}{32} + \frac {7 a^{8} e^{4 i c}}{64} + \frac {a^{8} e^{2 i c}}{32}\right ) & \text {otherwise} \end {cases} \]
a**8*x/256 + Piecewise(((-354658470655426560*I*a**8*d**7*exp(16*I*c)*exp(1 6*I*d*x) - 3242591731706757120*I*a**8*d**7*exp(14*I*c)*exp(14*I*d*x) - 132 40582904469258240*I*a**8*d**7*exp(12*I*c)*exp(12*I*d*x) - 3177739897072621 9776*I*a**8*d**7*exp(10*I*c)*exp(10*I*d*x) - 49652185891759718400*I*a**8*d **7*exp(8*I*c)*exp(8*I*d*x) - 52962331617877032960*I*a**8*d**7*exp(6*I*c)* exp(6*I*d*x) - 39721748713407774720*I*a**8*d**7*exp(4*I*c)*exp(4*I*d*x) - 22698142121947299840*I*a**8*d**7*exp(2*I*c)*exp(2*I*d*x))/(145268109580462 7189760*d**8), Ne(d**8, 0)), (x*(a**8*exp(16*I*c)/256 + a**8*exp(14*I*c)/3 2 + 7*a**8*exp(12*I*c)/64 + 7*a**8*exp(10*I*c)/32 + 35*a**8*exp(8*I*c)/128 + 7*a**8*exp(6*I*c)/32 + 7*a**8*exp(4*I*c)/64 + a**8*exp(2*I*c)/32), True ))
Time = 0.37 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.09 \[ \int \cos ^{16}(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {105 \, {\left (d x + c\right )} a^{8} + \frac {105 \, a^{8} \tan \left (d x + c\right )^{15} + 805 \, a^{8} \tan \left (d x + c\right )^{13} + 2681 \, a^{8} \tan \left (d x + c\right )^{11} + 5053 \, a^{8} \tan \left (d x + c\right )^{9} + 2883 \, a^{8} \tan \left (d x + c\right )^{7} + 21504 i \, a^{8} \tan \left (d x + c\right )^{6} + 70791 \, a^{8} \tan \left (d x + c\right )^{5} - 114688 i \, a^{8} \tan \left (d x + c\right )^{4} - 117285 \, a^{8} \tan \left (d x + c\right )^{3} + 74752 i \, a^{8} \tan \left (d x + c\right )^{2} + 26775 \, a^{8} \tan \left (d x + c\right ) - 4096 i \, a^{8}}{\tan \left (d x + c\right )^{16} + 8 \, \tan \left (d x + c\right )^{14} + 28 \, \tan \left (d x + c\right )^{12} + 56 \, \tan \left (d x + c\right )^{10} + 70 \, \tan \left (d x + c\right )^{8} + 56 \, \tan \left (d x + c\right )^{6} + 28 \, \tan \left (d x + c\right )^{4} + 8 \, \tan \left (d x + c\right )^{2} + 1}}{26880 \, d} \]
1/26880*(105*(d*x + c)*a^8 + (105*a^8*tan(d*x + c)^15 + 805*a^8*tan(d*x + c)^13 + 2681*a^8*tan(d*x + c)^11 + 5053*a^8*tan(d*x + c)^9 + 2883*a^8*tan( d*x + c)^7 + 21504*I*a^8*tan(d*x + c)^6 + 70791*a^8*tan(d*x + c)^5 - 11468 8*I*a^8*tan(d*x + c)^4 - 117285*a^8*tan(d*x + c)^3 + 74752*I*a^8*tan(d*x + c)^2 + 26775*a^8*tan(d*x + c) - 4096*I*a^8)/(tan(d*x + c)^16 + 8*tan(d*x + c)^14 + 28*tan(d*x + c)^12 + 56*tan(d*x + c)^10 + 70*tan(d*x + c)^8 + 56 *tan(d*x + c)^6 + 28*tan(d*x + c)^4 + 8*tan(d*x + c)^2 + 1))/d
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1457 vs. \(2 (175) = 350\).
Time = 1.53 (sec) , antiderivative size = 1457, normalized size of antiderivative = 6.48 \[ \int \cos ^{16}(c+d x) (a+i a \tan (c+d x))^8 \, dx=\text {Too large to display} \]
1/13762560*(53760*a^8*d*x*e^(28*I*d*x + 14*I*c) + 752640*a^8*d*x*e^(26*I*d *x + 12*I*c) + 4892160*a^8*d*x*e^(24*I*d*x + 10*I*c) + 19568640*a^8*d*x*e^ (22*I*d*x + 8*I*c) + 53813760*a^8*d*x*e^(20*I*d*x + 6*I*c) + 107627520*a^8 *d*x*e^(18*I*d*x + 4*I*c) + 161441280*a^8*d*x*e^(16*I*d*x + 2*I*c) + 16144 1280*a^8*d*x*e^(12*I*d*x - 2*I*c) + 107627520*a^8*d*x*e^(10*I*d*x - 4*I*c) + 53813760*a^8*d*x*e^(8*I*d*x - 6*I*c) + 19568640*a^8*d*x*e^(6*I*d*x - 8* I*c) + 4892160*a^8*d*x*e^(4*I*d*x - 10*I*c) + 752640*a^8*d*x*e^(2*I*d*x - 12*I*c) + 184504320*a^8*d*x*e^(14*I*d*x) + 53760*a^8*d*x*e^(-14*I*c) - 259 35*I*a^8*e^(28*I*d*x + 14*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 363090*I*a^8 *e^(26*I*d*x + 12*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 2360085*I*a^8*e^(24* I*d*x + 10*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 9440340*I*a^8*e^(22*I*d*x + 8*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 25960935*I*a^8*e^(20*I*d*x + 6*I*c) *log(e^(2*I*d*x + 2*I*c) + 1) - 51921870*I*a^8*e^(18*I*d*x + 4*I*c)*log(e^ (2*I*d*x + 2*I*c) + 1) - 77882805*I*a^8*e^(16*I*d*x + 2*I*c)*log(e^(2*I*d* x + 2*I*c) + 1) - 77882805*I*a^8*e^(12*I*d*x - 2*I*c)*log(e^(2*I*d*x + 2*I *c) + 1) - 51921870*I*a^8*e^(10*I*d*x - 4*I*c)*log(e^(2*I*d*x + 2*I*c) + 1 ) - 25960935*I*a^8*e^(8*I*d*x - 6*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 9440 340*I*a^8*e^(6*I*d*x - 8*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 2360085*I*a^8 *e^(4*I*d*x - 10*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 363090*I*a^8*e^(2*I*d *x - 12*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 89008920*I*a^8*e^(14*I*d*x)...
Time = 5.67 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.87 \[ \int \cos ^{16}(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {a^8\,x}{256}-\frac {-\frac {a^8\,{\mathrm {tan}\left (c+d\,x\right )}^7}{256}-\frac {a^8\,{\mathrm {tan}\left (c+d\,x\right )}^6\,1{}\mathrm {i}}{32}+\frac {85\,a^8\,{\mathrm {tan}\left (c+d\,x\right )}^5}{768}+\frac {a^8\,{\mathrm {tan}\left (c+d\,x\right )}^4\,11{}\mathrm {i}}{48}-\frac {1193\,a^8\,{\mathrm {tan}\left (c+d\,x\right )}^3}{3840}-\frac {a^8\,{\mathrm {tan}\left (c+d\,x\right )}^2\,143{}\mathrm {i}}{480}+\frac {5993\,a^8\,\mathrm {tan}\left (c+d\,x\right )}{26880}+\frac {a^8\,16{}\mathrm {i}}{105}}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^8+{\mathrm {tan}\left (c+d\,x\right )}^7\,8{}\mathrm {i}-28\,{\mathrm {tan}\left (c+d\,x\right )}^6-{\mathrm {tan}\left (c+d\,x\right )}^5\,56{}\mathrm {i}+70\,{\mathrm {tan}\left (c+d\,x\right )}^4+{\mathrm {tan}\left (c+d\,x\right )}^3\,56{}\mathrm {i}-28\,{\mathrm {tan}\left (c+d\,x\right )}^2-\mathrm {tan}\left (c+d\,x\right )\,8{}\mathrm {i}+1\right )} \]
(a^8*x)/256 - ((5993*a^8*tan(c + d*x))/26880 + (a^8*16i)/105 - (a^8*tan(c + d*x)^2*143i)/480 - (1193*a^8*tan(c + d*x)^3)/3840 + (a^8*tan(c + d*x)^4* 11i)/48 + (85*a^8*tan(c + d*x)^5)/768 - (a^8*tan(c + d*x)^6*1i)/32 - (a^8* tan(c + d*x)^7)/256)/(d*(tan(c + d*x)^3*56i - 28*tan(c + d*x)^2 - tan(c + d*x)*8i + 70*tan(c + d*x)^4 - tan(c + d*x)^5*56i - 28*tan(c + d*x)^6 + tan (c + d*x)^7*8i + tan(c + d*x)^8 + 1))