Integrand size = 24, antiderivative size = 214 \[ \int \cos ^{12}(c+d x) (a+i a \tan (c+d x))^5 \, dx=\frac {7 a^5 x}{128}-\frac {i a^{11}}{24 d (a-i a \tan (c+d x))^6}-\frac {i a^{10}}{20 d (a-i a \tan (c+d x))^5}-\frac {i a^8}{24 d (a-i a \tan (c+d x))^3}-\frac {3 i a^{13}}{64 d \left (a^2-i a^2 \tan (c+d x)\right )^4}-\frac {5 i a^{13}}{128 d \left (a^4-i a^4 \tan (c+d x)\right )^2}-\frac {3 i a^{13}}{64 d \left (a^8-i a^8 \tan (c+d x)\right )}+\frac {i a^{13}}{128 d \left (a^8+i a^8 \tan (c+d x)\right )} \] Output:
7/128*a^5*x-1/24*I*a^11/d/(a-I*a*tan(d*x+c))^6-1/20*I*a^10/d/(a-I*a*tan(d* x+c))^5-1/24*I*a^8/d/(a-I*a*tan(d*x+c))^3-3/64*I*a^13/d/(a^2-I*a^2*tan(d*x +c))^4-5/128*I*a^13/d/(a^4-I*a^4*tan(d*x+c))^2-3/64*I*a^13/d/(a^8-I*a^8*ta n(d*x+c))+1/128*I*a^13/d/(a^8+I*a^8*tan(d*x+c))
Time = 0.59 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.74 \[ \int \cos ^{12}(c+d x) (a+i a \tan (c+d x))^5 \, dx=-\frac {a^5 \sec ^7(c+d x) (-1750 \cos (c+d x)-1575 \cos (3 (c+d x))-693 \cos (5 (c+d x))+50 \cos (7 (c+d x))+350 i \sin (c+d x)+945 i \sin (3 (c+d x))-840 i \arctan (\tan (c+d x)) (\cos (5 (c+d x))-i \sin (5 (c+d x)))+525 i \sin (5 (c+d x))-70 i \sin (7 (c+d x)))}{15360 d (-i+\tan (c+d x)) (i+\tan (c+d x))^6} \] Input:
Integrate[Cos[c + d*x]^12*(a + I*a*Tan[c + d*x])^5,x]
Output:
-1/15360*(a^5*Sec[c + d*x]^7*(-1750*Cos[c + d*x] - 1575*Cos[3*(c + d*x)] - 693*Cos[5*(c + d*x)] + 50*Cos[7*(c + d*x)] + (350*I)*Sin[c + d*x] + (945* I)*Sin[3*(c + d*x)] - (840*I)*ArcTan[Tan[c + d*x]]*(Cos[5*(c + d*x)] - I*S in[5*(c + d*x)]) + (525*I)*Sin[5*(c + d*x)] - (70*I)*Sin[7*(c + d*x)]))/(d *(-I + Tan[c + d*x])*(I + Tan[c + d*x])^6)
Time = 0.36 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.85, 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.
| \(\displaystyle \int \cos ^{12}(c+d x) (a+i a \tan (c+d x))^5 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+i a \tan (c+d x))^5}{\sec (c+d x)^{12}}dx\) |
| \(\Big \downarrow \) 3968 |
| \(\displaystyle -\frac {i a^{13} \int \frac {1}{(a-i a \tan (c+d x))^7 (i \tan (c+d x) a+a)^2}d(i a \tan (c+d x))}{d}\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle -\frac {i a^{13} \int \left (\frac {3}{64 a^7 (a-i a \tan (c+d x))^2}+\frac {1}{128 a^7 (i \tan (c+d x) a+a)^2}+\frac {5}{64 a^6 (a-i a \tan (c+d x))^3}+\frac {1}{8 a^5 (a-i a \tan (c+d x))^4}+\frac {3}{16 a^4 (a-i a \tan (c+d x))^5}+\frac {1}{4 a^3 (a-i a \tan (c+d x))^6}+\frac {1}{4 a^2 (a-i a \tan (c+d x))^7}+\frac {7}{128 a^7 \left (\tan ^2(c+d x) a^2+a^2\right )}\right )d(i a \tan (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {i a^{13} \left (\frac {7 i \arctan (\tan (c+d x))}{128 a^8}+\frac {3}{64 a^7 (a-i a \tan (c+d x))}-\frac {1}{128 a^7 (a+i a \tan (c+d x))}+\frac {5}{128 a^6 (a-i a \tan (c+d x))^2}+\frac {1}{24 a^5 (a-i a \tan (c+d x))^3}+\frac {3}{64 a^4 (a-i a \tan (c+d x))^4}+\frac {1}{20 a^3 (a-i a \tan (c+d x))^5}+\frac {1}{24 a^2 (a-i a \tan (c+d x))^6}\right )}{d}\) |
Input:
Int[Cos[c + d*x]^12*(a + I*a*Tan[c + d*x])^5,x]
Output:
((-I)*a^13*((((7*I)/128)*ArcTan[Tan[c + d*x]])/a^8 + 1/(24*a^2*(a - I*a*Ta n[c + d*x])^6) + 1/(20*a^3*(a - I*a*Tan[c + d*x])^5) + 3/(64*a^4*(a - I*a* Tan[c + d*x])^4) + 1/(24*a^5*(a - I*a*Tan[c + d*x])^3) + 5/(128*a^6*(a - I *a*Tan[c + d*x])^2) + 3/(64*a^7*(a - I*a*Tan[c + d*x])) - 1/(128*a^7*(a + I*a*Tan[c + d*x]))))/d
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 = 0.60 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.69
\[\frac {i a^{5} \left (-\frac {\cos \left (d x +c \right )^{8} \sin \left (d x +c \right )^{4}}{12}-\frac {\cos \left (d x +c \right )^{8} \sin \left (d x +c \right )^{2}}{30}-\frac {\cos \left (d x +c \right )^{8}}{120}\right )+5 a^{5} \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{9}}{12}-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{9}}{40}+\frac {\left (\cos \left (d x +c \right )^{7}+\frac {7 \cos \left (d x +c \right )^{5}}{6}+\frac {35 \cos \left (d x +c \right )^{3}}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{320}+\frac {7 d x}{1024}+\frac {7 c}{1024}\right )-10 i a^{5} \left (-\frac {\cos \left (d x +c \right )^{10} \sin \left (d x +c \right )^{2}}{12}-\frac {\cos \left (d x +c \right )^{10}}{60}\right )-10 a^{5} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{11}}{12}+\frac {\left (\cos \left (d x +c \right )^{9}+\frac {9 \cos \left (d x +c \right )^{7}}{8}+\frac {21 \cos \left (d x +c \right )^{5}}{16}+\frac {105 \cos \left (d x +c \right )^{3}}{64}+\frac {315 \cos \left (d x +c \right )}{128}\right ) \sin \left (d x +c \right )}{120}+\frac {21 d x}{1024}+\frac {21 c}{1024}\right )-\frac {5 i a^{5} \cos \left (d x +c \right )^{12}}{12}+a^{5} \left (\frac {\left (\cos \left (d x +c \right )^{11}+\frac {11 \cos \left (d x +c \right )^{9}}{10}+\frac {99 \cos \left (d x +c \right )^{7}}{80}+\frac {231 \cos \left (d x +c \right )^{5}}{160}+\frac {231 \cos \left (d x +c \right )^{3}}{128}+\frac {693 \cos \left (d x +c \right )}{256}\right ) \sin \left (d x +c \right )}{12}+\frac {231 d x}{1024}+\frac {231 c}{1024}\right )}{d}\]
Input:
int(cos(d*x+c)^12*(a+I*a*tan(d*x+c))^5,x)
Output:
1/d*(I*a^5*(-1/12*cos(d*x+c)^8*sin(d*x+c)^4-1/30*cos(d*x+c)^8*sin(d*x+c)^2 -1/120*cos(d*x+c)^8)+5*a^5*(-1/12*sin(d*x+c)^3*cos(d*x+c)^9-1/40*sin(d*x+c )*cos(d*x+c)^9+1/320*(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)+7/1024*d*x+7/1024*c)-10*I*a^5*(-1/12*cos(d*x+c)^ 10*sin(d*x+c)^2-1/60*cos(d*x+c)^10)-10*a^5*(-1/12*sin(d*x+c)*cos(d*x+c)^11 +1/120*(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)+21/1024*d*x+21/1024*c)-5/12*I*a^5*cos(d* x+c)^12+a^5*(1/12*(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+693/256*cos(d*x+c))*sin(d*x+c)+231/ 1024*d*x+231/1024*c))
Time = 0.11 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.56 \[ \int \cos ^{12}(c+d x) (a+i a \tan (c+d x))^5 \, dx=\frac {{\left (840 \, a^{5} d x e^{\left (2 i \, d x + 2 i \, c\right )} - 10 i \, a^{5} e^{\left (14 i \, d x + 14 i \, c\right )} - 84 i \, a^{5} e^{\left (12 i \, d x + 12 i \, c\right )} - 315 i \, a^{5} e^{\left (10 i \, d x + 10 i \, c\right )} - 700 i \, a^{5} e^{\left (8 i \, d x + 8 i \, c\right )} - 1050 i \, a^{5} e^{\left (6 i \, d x + 6 i \, c\right )} - 1260 i \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} + 60 i \, a^{5}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{15360 \, d} \] Input:
integrate(cos(d*x+c)^12*(a+I*a*tan(d*x+c))^5,x, algorithm="fricas")
Output:
1/15360*(840*a^5*d*x*e^(2*I*d*x + 2*I*c) - 10*I*a^5*e^(14*I*d*x + 14*I*c) - 84*I*a^5*e^(12*I*d*x + 12*I*c) - 315*I*a^5*e^(10*I*d*x + 10*I*c) - 700*I *a^5*e^(8*I*d*x + 8*I*c) - 1050*I*a^5*e^(6*I*d*x + 6*I*c) - 1260*I*a^5*e^( 4*I*d*x + 4*I*c) + 60*I*a^5)*e^(-2*I*d*x - 2*I*c)/d
Time = 0.48 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.41 \[ \int \cos ^{12}(c+d x) (a+i a \tan (c+d x))^5 \, dx=\frac {7 a^{5} x}{128} + \begin {cases} \frac {\left (- 33776997205278720 i a^{5} d^{6} e^{14 i c} e^{12 i d x} - 283726776524341248 i a^{5} d^{6} e^{12 i c} e^{10 i d x} - 1063975411966279680 i a^{5} d^{6} e^{10 i c} e^{8 i d x} - 2364389804369510400 i a^{5} d^{6} e^{8 i c} e^{6 i d x} - 3546584706554265600 i a^{5} d^{6} e^{6 i c} e^{4 i d x} - 4255901647865118720 i a^{5} d^{6} e^{4 i c} e^{2 i d x} + 202661983231672320 i a^{5} d^{6} e^{- 2 i d x}\right ) e^{- 2 i c}}{51881467707308113920 d^{7}} & \text {for}\: d^{7} e^{2 i c} \neq 0 \\x \left (- \frac {7 a^{5}}{128} + \frac {\left (a^{5} e^{14 i c} + 7 a^{5} e^{12 i c} + 21 a^{5} e^{10 i c} + 35 a^{5} e^{8 i c} + 35 a^{5} e^{6 i c} + 21 a^{5} e^{4 i c} + 7 a^{5} e^{2 i c} + a^{5}\right ) e^{- 2 i c}}{128}\right ) & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)**12*(a+I*a*tan(d*x+c))**5,x)
Output:
7*a**5*x/128 + Piecewise(((-33776997205278720*I*a**5*d**6*exp(14*I*c)*exp( 12*I*d*x) - 283726776524341248*I*a**5*d**6*exp(12*I*c)*exp(10*I*d*x) - 106 3975411966279680*I*a**5*d**6*exp(10*I*c)*exp(8*I*d*x) - 236438980436951040 0*I*a**5*d**6*exp(8*I*c)*exp(6*I*d*x) - 3546584706554265600*I*a**5*d**6*ex p(6*I*c)*exp(4*I*d*x) - 4255901647865118720*I*a**5*d**6*exp(4*I*c)*exp(2*I *d*x) + 202661983231672320*I*a**5*d**6*exp(-2*I*d*x))*exp(-2*I*c)/(5188146 7707308113920*d**7), Ne(d**7*exp(2*I*c), 0)), (x*(-7*a**5/128 + (a**5*exp( 14*I*c) + 7*a**5*exp(12*I*c) + 21*a**5*exp(10*I*c) + 35*a**5*exp(8*I*c) + 35*a**5*exp(6*I*c) + 21*a**5*exp(4*I*c) + 7*a**5*exp(2*I*c) + a**5)*exp(-2 *I*c)/128), True))
Time = 0.11 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.87 \[ \int \cos ^{12}(c+d x) (a+i a \tan (c+d x))^5 \, dx=\frac {105 \, {\left (d x + c\right )} a^{5} + \frac {105 \, a^{5} \tan \left (d x + c\right )^{11} + 595 \, a^{5} \tan \left (d x + c\right )^{9} + 1386 \, a^{5} \tan \left (d x + c\right )^{7} + 1686 \, a^{5} \tan \left (d x + c\right )^{5} - 240 i \, a^{5} \tan \left (d x + c\right )^{4} + 45 \, a^{5} \tan \left (d x + c\right )^{3} + 1824 i \, a^{5} \tan \left (d x + c\right )^{2} + 1815 \, a^{5} \tan \left (d x + c\right ) - 496 i \, a^{5}}{\tan \left (d x + c\right )^{12} + 6 \, \tan \left (d x + c\right )^{10} + 15 \, \tan \left (d x + c\right )^{8} + 20 \, \tan \left (d x + c\right )^{6} + 15 \, \tan \left (d x + c\right )^{4} + 6 \, \tan \left (d x + c\right )^{2} + 1}}{1920 \, d} \] Input:
integrate(cos(d*x+c)^12*(a+I*a*tan(d*x+c))^5,x, algorithm="maxima")
Output:
1/1920*(105*(d*x + c)*a^5 + (105*a^5*tan(d*x + c)^11 + 595*a^5*tan(d*x + c )^9 + 1386*a^5*tan(d*x + c)^7 + 1686*a^5*tan(d*x + c)^5 - 240*I*a^5*tan(d* x + c)^4 + 45*a^5*tan(d*x + c)^3 + 1824*I*a^5*tan(d*x + c)^2 + 1815*a^5*ta n(d*x + c) - 496*I*a^5)/(tan(d*x + c)^12 + 6*tan(d*x + c)^10 + 15*tan(d*x + c)^8 + 20*tan(d*x + c)^6 + 15*tan(d*x + c)^4 + 6*tan(d*x + c)^2 + 1))/d
Time = 0.29 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.56 \[ \int \cos ^{12}(c+d x) (a+i a \tan (c+d x))^5 \, dx=\frac {1}{3840} i \, a^{5} {\left (\frac {105 \, \log \left (\tan \left (d x + c\right ) + i\right )}{d} - \frac {105 \, \log \left (\tan \left (d x + c\right ) - i\right )}{d} - \frac {2 \, {\left (105 i \, \tan \left (d x + c\right )^{6} - 525 \, \tan \left (d x + c\right )^{5} - 980 i \, \tan \left (d x + c\right )^{4} + 700 \, \tan \left (d x + c\right )^{3} - 189 i \, \tan \left (d x + c\right )^{2} + 665 \, \tan \left (d x + c\right ) + 496 i\right )}}{d {\left (\tan \left (d x + c\right ) + i\right )}^{6} {\left (\tan \left (d x + c\right ) - i\right )}}\right )} \] Input:
integrate(cos(d*x+c)^12*(a+I*a*tan(d*x+c))^5,x, algorithm="giac")
Output:
1/3840*I*a^5*(105*log(tan(d*x + c) + I)/d - 105*log(tan(d*x + c) - I)/d - 2*(105*I*tan(d*x + c)^6 - 525*tan(d*x + c)^5 - 980*I*tan(d*x + c)^4 + 700* tan(d*x + c)^3 - 189*I*tan(d*x + c)^2 + 665*tan(d*x + c) + 496*I)/(d*(tan( d*x + c) + I)^6*(tan(d*x + c) - I)))
Time = 2.18 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.80 \[ \int \cos ^{12}(c+d x) (a+i a \tan (c+d x))^5 \, dx=\frac {7\,a^5\,x}{128}-\frac {-\frac {7\,a^5\,{\mathrm {tan}\left (c+d\,x\right )}^6}{128}-\frac {a^5\,{\mathrm {tan}\left (c+d\,x\right )}^5\,35{}\mathrm {i}}{128}+\frac {49\,a^5\,{\mathrm {tan}\left (c+d\,x\right )}^4}{96}+\frac {a^5\,{\mathrm {tan}\left (c+d\,x\right )}^3\,35{}\mathrm {i}}{96}+\frac {63\,a^5\,{\mathrm {tan}\left (c+d\,x\right )}^2}{640}+\frac {a^5\,\mathrm {tan}\left (c+d\,x\right )\,133{}\mathrm {i}}{384}-\frac {31\,a^5}{120}}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^7+{\mathrm {tan}\left (c+d\,x\right )}^6\,5{}\mathrm {i}-9\,{\mathrm {tan}\left (c+d\,x\right )}^5-{\mathrm {tan}\left (c+d\,x\right )}^4\,5{}\mathrm {i}-5\,{\mathrm {tan}\left (c+d\,x\right )}^3-{\mathrm {tan}\left (c+d\,x\right )}^2\,9{}\mathrm {i}+5\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )} \] Input:
int(cos(c + d*x)^12*(a + a*tan(c + d*x)*1i)^5,x)
Output:
(7*a^5*x)/128 - ((a^5*tan(c + d*x)*133i)/384 - (31*a^5)/120 + (63*a^5*tan( c + d*x)^2)/640 + (a^5*tan(c + d*x)^3*35i)/96 + (49*a^5*tan(c + d*x)^4)/96 - (a^5*tan(c + d*x)^5*35i)/128 - (7*a^5*tan(c + d*x)^6)/128)/(d*(5*tan(c + d*x) - tan(c + d*x)^2*9i - 5*tan(c + d*x)^3 - tan(c + d*x)^4*5i - 9*tan( c + d*x)^5 + tan(c + d*x)^6*5i + tan(c + d*x)^7 + 1i))
Time = 0.16 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.81 \[ \int \cos ^{12}(c+d x) (a+i a \tan (c+d x))^5 \, dx=\frac {a^{5} \left (-2560 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{11}+11776 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{9}-21552 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7}+19656 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}-9030 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+1815 \cos \left (d x +c \right ) \sin \left (d x +c \right )-2560 \sin \left (d x +c \right )^{12} i +13056 \sin \left (d x +c \right )^{10} i -27120 \sin \left (d x +c \right )^{8} i +29120 \sin \left (d x +c \right )^{6} i -16800 \sin \left (d x +c \right )^{4} i +4800 \sin \left (d x +c \right )^{2} i +105 d x \right )}{1920 d} \] Input:
int(cos(d*x+c)^12*(a+I*a*tan(d*x+c))^5,x)
Output:
(a**5*( - 2560*cos(c + d*x)*sin(c + d*x)**11 + 11776*cos(c + d*x)*sin(c + d*x)**9 - 21552*cos(c + d*x)*sin(c + d*x)**7 + 19656*cos(c + d*x)*sin(c + d*x)**5 - 9030*cos(c + d*x)*sin(c + d*x)**3 + 1815*cos(c + d*x)*sin(c + d* x) - 2560*sin(c + d*x)**12*i + 13056*sin(c + d*x)**10*i - 27120*sin(c + d* x)**8*i + 29120*sin(c + d*x)**6*i - 16800*sin(c + d*x)**4*i + 4800*sin(c + d*x)**2*i + 105*d*x))/(1920*d)