Integrand size = 24, antiderivative size = 278 \[ \int \frac {\cos ^2(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {5 x}{512 a^8}+\frac {i a}{36 d (a+i a \tan (c+d x))^9}+\frac {i}{32 d (a+i a \tan (c+d x))^8}+\frac {3 i}{112 a d (a+i a \tan (c+d x))^7}+\frac {i}{48 a^2 d (a+i a \tan (c+d x))^6}+\frac {i}{64 a^3 d (a+i a \tan (c+d x))^5}+\frac {7 i}{768 a^5 d (a+i a \tan (c+d x))^3}+\frac {3 i}{256 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac {i}{128 d \left (a^4+i a^4 \tan (c+d x)\right )^2}-\frac {i}{1024 d \left (a^8-i a^8 \tan (c+d x)\right )}+\frac {9 i}{1024 d \left (a^8+i a^8 \tan (c+d x)\right )} \]
5/512*x/a^8+1/36*I*a/d/(a+I*a*tan(d*x+c))^9+1/32*I/d/(a+I*a*tan(d*x+c))^8+ 3/112*I/a/d/(a+I*a*tan(d*x+c))^7+1/48*I/a^2/d/(a+I*a*tan(d*x+c))^6+1/64*I/ a^3/d/(a+I*a*tan(d*x+c))^5+7/768*I/a^5/d/(a+I*a*tan(d*x+c))^3+3/256*I/d/(a ^2+I*a^2*tan(d*x+c))^4+1/128*I/d/(a^4+I*a^4*tan(d*x+c))^2-1/1024*I/d/(a^8- I*a^8*tan(d*x+c))+9/1024*I/d/(a^8+I*a^8*tan(d*x+c))
Time = 0.81 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.68 \[ \int \frac {\cos ^2(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {\sec ^{10}(c+d x) (2520 \arctan (\tan (c+d x)) (\cos (8 (c+d x))+i \sin (8 (c+d x)))+i (7938+14112 \cos (2 (c+d x))+10080 \cos (4 (c+d x))+6480 \cos (6 (c+d x))+2462 \cos (8 (c+d x))-112 \cos (10 (c+d x))+3528 i \sin (2 (c+d x))+5040 i \sin (4 (c+d x))+4860 i \sin (6 (c+d x))+2147 i \sin (8 (c+d x))-140 i \sin (10 (c+d x))))}{258048 a^8 d (-i+\tan (c+d x))^9 (i+\tan (c+d x))} \]
(Sec[c + d*x]^10*(2520*ArcTan[Tan[c + d*x]]*(Cos[8*(c + d*x)] + I*Sin[8*(c + d*x)]) + I*(7938 + 14112*Cos[2*(c + d*x)] + 10080*Cos[4*(c + d*x)] + 64 80*Cos[6*(c + d*x)] + 2462*Cos[8*(c + d*x)] - 112*Cos[10*(c + d*x)] + (352 8*I)*Sin[2*(c + d*x)] + (5040*I)*Sin[4*(c + d*x)] + (4860*I)*Sin[6*(c + d* x)] + (2147*I)*Sin[8*(c + d*x)] - (140*I)*Sin[10*(c + d*x)])))/(258048*a^8 *d*(-I + Tan[c + d*x])^9*(I + Tan[c + d*x]))
Time = 0.38 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.89, 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.
| \(\displaystyle \int \frac {\cos ^2(c+d x)}{(a+i a \tan (c+d x))^8} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sec (c+d x)^2 (a+i a \tan (c+d x))^8}dx\) |
| \(\Big \downarrow \) 3968 |
| \(\displaystyle -\frac {i a^3 \int \frac {1}{(a-i a \tan (c+d x))^2 (i \tan (c+d x) a+a)^{10}}d(i a \tan (c+d x))}{d}\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle -\frac {i a^3 \int \left (\frac {1}{1024 a^{10} (a-i a \tan (c+d x))^2}+\frac {9}{1024 a^{10} (i \tan (c+d x) a+a)^2}+\frac {1}{64 a^9 (i \tan (c+d x) a+a)^3}+\frac {7}{256 a^8 (i \tan (c+d x) a+a)^4}+\frac {3}{64 a^7 (i \tan (c+d x) a+a)^5}+\frac {5}{64 a^6 (i \tan (c+d x) a+a)^6}+\frac {1}{8 a^5 (i \tan (c+d x) a+a)^7}+\frac {3}{16 a^4 (i \tan (c+d x) a+a)^8}+\frac {1}{4 a^3 (i \tan (c+d x) a+a)^9}+\frac {1}{4 a^2 (i \tan (c+d x) a+a)^{10}}+\frac {5}{512 a^{10} \left (\tan ^2(c+d x) a^2+a^2\right )}\right )d(i a \tan (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {i a^3 \left (\frac {5 i \arctan (\tan (c+d x))}{512 a^{11}}+\frac {1}{1024 a^{10} (a-i a \tan (c+d x))}-\frac {9}{1024 a^{10} (a+i a \tan (c+d x))}-\frac {1}{128 a^9 (a+i a \tan (c+d x))^2}-\frac {7}{768 a^8 (a+i a \tan (c+d x))^3}-\frac {3}{256 a^7 (a+i a \tan (c+d x))^4}-\frac {1}{64 a^6 (a+i a \tan (c+d x))^5}-\frac {1}{48 a^5 (a+i a \tan (c+d x))^6}-\frac {3}{112 a^4 (a+i a \tan (c+d x))^7}-\frac {1}{32 a^3 (a+i a \tan (c+d x))^8}-\frac {1}{36 a^2 (a+i a \tan (c+d x))^9}\right )}{d}\) |
((-I)*a^3*((((5*I)/512)*ArcTan[Tan[c + d*x]])/a^11 + 1/(1024*a^10*(a - I*a *Tan[c + d*x])) - 1/(36*a^2*(a + I*a*Tan[c + d*x])^9) - 1/(32*a^3*(a + I*a *Tan[c + d*x])^8) - 3/(112*a^4*(a + I*a*Tan[c + d*x])^7) - 1/(48*a^5*(a + I*a*Tan[c + d*x])^6) - 1/(64*a^6*(a + I*a*Tan[c + d*x])^5) - 3/(256*a^7*(a + I*a*Tan[c + d*x])^4) - 7/(768*a^8*(a + I*a*Tan[c + d*x])^3) - 1/(128*a^ 9*(a + I*a*Tan[c + d*x])^2) - 9/(1024*a^10*(a + I*a*Tan[c + d*x]))))/d
3.2.74.3.1 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]
Time = 0.64 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.61
| method | result | size |
| derivativedivides | \(\frac {-\frac {5 i \ln \left (\tan \left (d x +c \right )-i\right )}{1024}+\frac {3 i}{256 \left (\tan \left (d x +c \right )-i\right )^{4}}+\frac {i}{32 \left (\tan \left (d x +c \right )-i\right )^{8}}-\frac {i}{48 \left (\tan \left (d x +c \right )-i\right )^{6}}-\frac {i}{128 \left (\tan \left (d x +c \right )-i\right )^{2}}+\frac {1}{36 \left (\tan \left (d x +c \right )-i\right )^{9}}-\frac {3}{112 \left (\tan \left (d x +c \right )-i\right )^{7}}+\frac {1}{64 \left (\tan \left (d x +c \right )-i\right )^{5}}-\frac {7}{768 \left (\tan \left (d x +c \right )-i\right )^{3}}+\frac {9}{1024 \left (\tan \left (d x +c \right )-i\right )}+\frac {5 i \ln \left (\tan \left (d x +c \right )+i\right )}{1024}+\frac {1}{1024 \tan \left (d x +c \right )+1024 i}}{d \,a^{8}}\) | \(169\) |
| default | \(\frac {-\frac {5 i \ln \left (\tan \left (d x +c \right )-i\right )}{1024}+\frac {3 i}{256 \left (\tan \left (d x +c \right )-i\right )^{4}}+\frac {i}{32 \left (\tan \left (d x +c \right )-i\right )^{8}}-\frac {i}{48 \left (\tan \left (d x +c \right )-i\right )^{6}}-\frac {i}{128 \left (\tan \left (d x +c \right )-i\right )^{2}}+\frac {1}{36 \left (\tan \left (d x +c \right )-i\right )^{9}}-\frac {3}{112 \left (\tan \left (d x +c \right )-i\right )^{7}}+\frac {1}{64 \left (\tan \left (d x +c \right )-i\right )^{5}}-\frac {7}{768 \left (\tan \left (d x +c \right )-i\right )^{3}}+\frac {9}{1024 \left (\tan \left (d x +c \right )-i\right )}+\frac {5 i \ln \left (\tan \left (d x +c \right )+i\right )}{1024}+\frac {1}{1024 \tan \left (d x +c \right )+1024 i}}{d \,a^{8}}\) | \(169\) |
| risch | \(\frac {5 x}{512 a^{8}}+\frac {15 i {\mathrm e}^{-4 i \left (d x +c \right )}}{512 a^{8} d}+\frac {35 i {\mathrm e}^{-6 i \left (d x +c \right )}}{1024 a^{8} d}+\frac {63 i {\mathrm e}^{-8 i \left (d x +c \right )}}{2048 a^{8} d}+\frac {21 i {\mathrm e}^{-10 i \left (d x +c \right )}}{1024 a^{8} d}+\frac {5 i {\mathrm e}^{-12 i \left (d x +c \right )}}{512 a^{8} d}+\frac {45 i {\mathrm e}^{-14 i \left (d x +c \right )}}{14336 a^{8} d}+\frac {5 i {\mathrm e}^{-16 i \left (d x +c \right )}}{8192 a^{8} d}+\frac {i {\mathrm e}^{-18 i \left (d x +c \right )}}{18432 a^{8} d}+\frac {11 i \cos \left (2 d x +2 c \right )}{512 a^{8} d}+\frac {23 \sin \left (2 d x +2 c \right )}{1024 a^{8} d}\) | \(187\) |
1/d/a^8*(-5/1024*I*ln(tan(d*x+c)-I)+3/256*I/(tan(d*x+c)-I)^4+1/32*I/(tan(d *x+c)-I)^8-1/48*I/(tan(d*x+c)-I)^6-1/128*I/(tan(d*x+c)-I)^2+1/36/(tan(d*x+ c)-I)^9-3/112/(tan(d*x+c)-I)^7+1/64/(tan(d*x+c)-I)^5-7/768/(tan(d*x+c)-I)^ 3+9/1024/(tan(d*x+c)-I)+5/1024*I*ln(tan(d*x+c)+I)+1/1024/(tan(d*x+c)+I))
Time = 0.23 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.47 \[ \int \frac {\cos ^2(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {{\left (5040 \, d x e^{\left (18 i \, d x + 18 i \, c\right )} - 252 i \, e^{\left (20 i \, d x + 20 i \, c\right )} + 11340 i \, e^{\left (16 i \, d x + 16 i \, c\right )} + 15120 i \, e^{\left (14 i \, d x + 14 i \, c\right )} + 17640 i \, e^{\left (12 i \, d x + 12 i \, c\right )} + 15876 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 10584 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 5040 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 1620 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 315 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 28 i\right )} e^{\left (-18 i \, d x - 18 i \, c\right )}}{516096 \, a^{8} d} \]
1/516096*(5040*d*x*e^(18*I*d*x + 18*I*c) - 252*I*e^(20*I*d*x + 20*I*c) + 1 1340*I*e^(16*I*d*x + 16*I*c) + 15120*I*e^(14*I*d*x + 14*I*c) + 17640*I*e^( 12*I*d*x + 12*I*c) + 15876*I*e^(10*I*d*x + 10*I*c) + 10584*I*e^(8*I*d*x + 8*I*c) + 5040*I*e^(6*I*d*x + 6*I*c) + 1620*I*e^(4*I*d*x + 4*I*c) + 315*I*e ^(2*I*d*x + 2*I*c) + 28*I)*e^(-18*I*d*x - 18*I*c)/(a^8*d)
Time = 0.47 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.42 \[ \int \frac {\cos ^2(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\begin {cases} \frac {\left (- 2495687119199326634196634435584 i a^{72} d^{9} e^{92 i c} e^{2 i d x} + 112305920363969698538848549601280 i a^{72} d^{9} e^{88 i c} e^{- 2 i d x} + 149741227151959598051798066135040 i a^{72} d^{9} e^{86 i c} e^{- 4 i d x} + 174698098343952864393764410490880 i a^{72} d^{9} e^{84 i c} e^{- 6 i d x} + 157228288509557577954387969441792 i a^{72} d^{9} e^{82 i c} e^{- 8 i d x} + 104818859006371718636258646294528 i a^{72} d^{9} e^{80 i c} e^{- 10 i d x} + 49913742383986532683932688711680 i a^{72} d^{9} e^{78 i c} e^{- 12 i d x} + 16043702909138528362692649943040 i a^{72} d^{9} e^{76 i c} e^{- 14 i d x} + 3119608898999158292745793044480 i a^{72} d^{9} e^{74 i c} e^{- 16 i d x} + 277298568799925181577403826176 i a^{72} d^{9} e^{72 i c} e^{- 18 i d x}\right ) e^{- 90 i c}}{5111167220120220946834707324076032 a^{80} d^{10}} & \text {for}\: a^{80} d^{10} e^{90 i c} \neq 0 \\x \left (\frac {\left (e^{20 i c} + 10 e^{18 i c} + 45 e^{16 i c} + 120 e^{14 i c} + 210 e^{12 i c} + 252 e^{10 i c} + 210 e^{8 i c} + 120 e^{6 i c} + 45 e^{4 i c} + 10 e^{2 i c} + 1\right ) e^{- 18 i c}}{1024 a^{8}} - \frac {5}{512 a^{8}}\right ) & \text {otherwise} \end {cases} + \frac {5 x}{512 a^{8}} \]
Piecewise(((-2495687119199326634196634435584*I*a**72*d**9*exp(92*I*c)*exp( 2*I*d*x) + 112305920363969698538848549601280*I*a**72*d**9*exp(88*I*c)*exp( -2*I*d*x) + 149741227151959598051798066135040*I*a**72*d**9*exp(86*I*c)*exp (-4*I*d*x) + 174698098343952864393764410490880*I*a**72*d**9*exp(84*I*c)*ex p(-6*I*d*x) + 157228288509557577954387969441792*I*a**72*d**9*exp(82*I*c)*e xp(-8*I*d*x) + 104818859006371718636258646294528*I*a**72*d**9*exp(80*I*c)* exp(-10*I*d*x) + 49913742383986532683932688711680*I*a**72*d**9*exp(78*I*c) *exp(-12*I*d*x) + 16043702909138528362692649943040*I*a**72*d**9*exp(76*I*c )*exp(-14*I*d*x) + 3119608898999158292745793044480*I*a**72*d**9*exp(74*I*c )*exp(-16*I*d*x) + 277298568799925181577403826176*I*a**72*d**9*exp(72*I*c) *exp(-18*I*d*x))*exp(-90*I*c)/(5111167220120220946834707324076032*a**80*d* *10), Ne(a**80*d**10*exp(90*I*c), 0)), (x*((exp(20*I*c) + 10*exp(18*I*c) + 45*exp(16*I*c) + 120*exp(14*I*c) + 210*exp(12*I*c) + 252*exp(10*I*c) + 21 0*exp(8*I*c) + 120*exp(6*I*c) + 45*exp(4*I*c) + 10*exp(2*I*c) + 1)*exp(-18 *I*c)/(1024*a**8) - 5/(512*a**8)), True)) + 5*x/(512*a**8)
Exception generated. \[ \int \frac {\cos ^2(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\text {Exception raised: RuntimeError} \]
Time = 1.28 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.59 \[ \int \frac {\cos ^2(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=-\frac {-\frac {2520 i \, \log \left (\tan \left (d x + c\right ) + i\right )}{a^{8}} + \frac {2520 i \, \log \left (\tan \left (d x + c\right ) - i\right )}{a^{8}} + \frac {504 \, {\left (5 i \, \tan \left (d x + c\right ) - 6\right )}}{a^{8} {\left (\tan \left (d x + c\right ) + i\right )}} + \frac {-7129 i \, \tan \left (d x + c\right )^{9} - 68697 \, \tan \left (d x + c\right )^{8} + 296964 i \, \tan \left (d x + c\right )^{7} + 758772 \, \tan \left (d x + c\right )^{6} - 1271214 i \, \tan \left (d x + c\right )^{5} - 1465758 \, \tan \left (d x + c\right )^{4} + 1191540 i \, \tan \left (d x + c\right )^{3} + 693828 \, \tan \left (d x + c\right )^{2} - 295425 i \, \tan \left (d x + c\right ) - 89553}{a^{8} {\left (\tan \left (d x + c\right ) - i\right )}^{9}}}{516096 \, d} \]
-1/516096*(-2520*I*log(tan(d*x + c) + I)/a^8 + 2520*I*log(tan(d*x + c) - I )/a^8 + 504*(5*I*tan(d*x + c) - 6)/(a^8*(tan(d*x + c) + I)) + (-7129*I*tan (d*x + c)^9 - 68697*tan(d*x + c)^8 + 296964*I*tan(d*x + c)^7 + 758772*tan( d*x + c)^6 - 1271214*I*tan(d*x + c)^5 - 1465758*tan(d*x + c)^4 + 1191540*I *tan(d*x + c)^3 + 693828*tan(d*x + c)^2 - 295425*I*tan(d*x + c) - 89553)/( a^8*(tan(d*x + c) - I)^9))/d
Time = 6.25 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.85 \[ \int \frac {\cos ^2(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {5\,x}{512\,a^8}+\frac {\frac {163\,{\mathrm {tan}\left (c+d\,x\right )}^2}{448\,a^8}-\frac {10}{63\,a^8}-\frac {\mathrm {tan}\left (c+d\,x\right )\,9019{}\mathrm {i}}{32256\,a^8}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,393{}\mathrm {i}}{1792\,a^8}+\frac {11\,{\mathrm {tan}\left (c+d\,x\right )}^4}{64\,a^8}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^5\,1{}\mathrm {i}}{2\,a^8}-\frac {95\,{\mathrm {tan}\left (c+d\,x\right )}^6}{192\,a^8}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^7\,205{}\mathrm {i}}{768\,a^8}+\frac {5\,{\mathrm {tan}\left (c+d\,x\right )}^8}{64\,a^8}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^9\,5{}\mathrm {i}}{512\,a^8}}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^{10}\,1{}\mathrm {i}+8\,{\mathrm {tan}\left (c+d\,x\right )}^9-{\mathrm {tan}\left (c+d\,x\right )}^8\,27{}\mathrm {i}-48\,{\mathrm {tan}\left (c+d\,x\right )}^7+{\mathrm {tan}\left (c+d\,x\right )}^6\,42{}\mathrm {i}+{\mathrm {tan}\left (c+d\,x\right )}^4\,42{}\mathrm {i}+48\,{\mathrm {tan}\left (c+d\,x\right )}^3-{\mathrm {tan}\left (c+d\,x\right )}^2\,27{}\mathrm {i}-8\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )} \]
(5*x)/(512*a^8) + ((163*tan(c + d*x)^2)/(448*a^8) - 10/(63*a^8) - (tan(c + d*x)*9019i)/(32256*a^8) + (tan(c + d*x)^3*393i)/(1792*a^8) + (11*tan(c + d*x)^4)/(64*a^8) + (tan(c + d*x)^5*1i)/(2*a^8) - (95*tan(c + d*x)^6)/(192* a^8) - (tan(c + d*x)^7*205i)/(768*a^8) + (5*tan(c + d*x)^8)/(64*a^8) + (ta n(c + d*x)^9*5i)/(512*a^8))/(d*(48*tan(c + d*x)^3 - tan(c + d*x)^2*27i - 8 *tan(c + d*x) + tan(c + d*x)^4*42i + tan(c + d*x)^6*42i - 48*tan(c + d*x)^ 7 - tan(c + d*x)^8*27i + 8*tan(c + d*x)^9 + tan(c + d*x)^10*1i + 1i))