Integrand size = 24, antiderivative size = 107 \[ \int \frac {\cos ^5(c+d x)}{(a+i a \tan (c+d x))^2} \, dx=\frac {7 \sin (c+d x)}{9 a^2 d}-\frac {7 \sin ^3(c+d x)}{9 a^2 d}+\frac {7 \sin ^5(c+d x)}{15 a^2 d}-\frac {\sin ^7(c+d x)}{9 a^2 d}+\frac {2 i \cos ^7(c+d x)}{9 d \left (a^2+i a^2 \tan (c+d x)\right )} \] Output:
7/9*sin(d*x+c)/a^2/d-7/9*sin(d*x+c)^3/a^2/d+7/15*sin(d*x+c)^5/a^2/d-1/9*si n(d*x+c)^7/a^2/d+2/9*I*cos(d*x+c)^7/d/(a^2+I*a^2*tan(d*x+c))
Time = 0.71 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.09 \[ \int \frac {\cos ^5(c+d x)}{(a+i a \tan (c+d x))^2} \, dx=\frac {i \sec ^2(c+d x) (-1050 \cos (c+d x)+378 \cos (3 (c+d x))+30 \cos (5 (c+d x))+2 \cos (7 (c+d x))-525 i \sin (c+d x)+567 i \sin (3 (c+d x))+75 i \sin (5 (c+d x))+7 i \sin (7 (c+d x)))}{2880 a^2 d (-i+\tan (c+d x))^2} \] Input:
Integrate[Cos[c + d*x]^5/(a + I*a*Tan[c + d*x])^2,x]
Output:
((I/2880)*Sec[c + d*x]^2*(-1050*Cos[c + d*x] + 378*Cos[3*(c + d*x)] + 30*C os[5*(c + d*x)] + 2*Cos[7*(c + d*x)] - (525*I)*Sin[c + d*x] + (567*I)*Sin[ 3*(c + d*x)] + (75*I)*Sin[5*(c + d*x)] + (7*I)*Sin[7*(c + d*x)]))/(a^2*d*( -I + Tan[c + d*x])^2)
Time = 0.32 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.82, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3042, 3981, 3042, 3113, 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.
| \(\displaystyle \int \frac {\cos ^5(c+d x)}{(a+i a \tan (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sec (c+d x)^5 (a+i a \tan (c+d x))^2}dx\) |
| \(\Big \downarrow \) 3981 |
| \(\displaystyle \frac {7 \int \cos ^7(c+d x)dx}{9 a^2}+\frac {2 i \cos ^7(c+d x)}{9 d \left (a^2+i a^2 \tan (c+d x)\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7 \int \sin \left (c+d x+\frac {\pi }{2}\right )^7dx}{9 a^2}+\frac {2 i \cos ^7(c+d x)}{9 d \left (a^2+i a^2 \tan (c+d x)\right )}\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle -\frac {7 \int \left (-\sin ^6(c+d x)+3 \sin ^4(c+d x)-3 \sin ^2(c+d x)+1\right )d(-\sin (c+d x))}{9 a^2 d}+\frac {2 i \cos ^7(c+d x)}{9 d \left (a^2+i a^2 \tan (c+d x)\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {7 \left (\frac {1}{7} \sin ^7(c+d x)-\frac {3}{5} \sin ^5(c+d x)+\sin ^3(c+d x)-\sin (c+d x)\right )}{9 a^2 d}+\frac {2 i \cos ^7(c+d x)}{9 d \left (a^2+i a^2 \tan (c+d x)\right )}\) |
Input:
Int[Cos[c + d*x]^5/(a + I*a*Tan[c + d*x])^2,x]
Output:
(-7*(-Sin[c + d*x] + Sin[c + d*x]^3 - (3*Sin[c + d*x]^5)/5 + Sin[c + d*x]^ 7/7))/(9*a^2*d) + (((2*I)/9)*Cos[c + d*x]^7)/(d*(a^2 + I*a^2*Tan[c + d*x]) )
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x _)])^(n_), x_Symbol] :> Simp[2*d^2*(d*Sec[e + f*x])^(m - 2)*((a + b*Tan[e + f*x])^(n + 1)/(b*f*(m + 2*n))), x] - Simp[d^2*((m - 2)/(b^2*(m + 2*n))) Int[(d*Sec[e + f*x])^(m - 2)*(a + b*Tan[e + f*x])^(n + 2), x], x] /; FreeQ[ {a, b, d, e, f, m}, x] && EqQ[a^2 + b^2, 0] && LtQ[n, -1] && ((ILtQ[n/2, 0] && IGtQ[m - 1/2, 0]) || EqQ[n, -2] || IGtQ[m + n, 0] || (IntegersQ[n, m + 1/2] && GtQ[2*m + n + 1, 0])) && IntegerQ[2*m]
Time = 1.30 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.28
| method | result | size |
| risch | \(\frac {i {\mathrm e}^{-7 i \left (d x +c \right )}}{128 a^{2} d}+\frac {i {\mathrm e}^{-9 i \left (d x +c \right )}}{1152 a^{2} d}+\frac {7 i \cos \left (d x +c \right )}{64 a^{2} d}+\frac {7 \sin \left (d x +c \right )}{16 a^{2} d}+\frac {i \cos \left (5 d x +5 c \right )}{32 a^{2} d}+\frac {11 \sin \left (5 d x +5 c \right )}{320 a^{2} d}+\frac {7 i \cos \left (3 d x +3 c \right )}{96 a^{2} d}+\frac {7 \sin \left (3 d x +3 c \right )}{64 a^{2} d}\) | \(137\) |
| derivativedivides | \(\frac {-\frac {2 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}+\frac {51 i}{16 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {49 i}{6 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {35 i}{4 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {4}{9 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}-\frac {5}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {49}{5 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {49}{8 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {99}{64 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {i}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{4}}-\frac {9 i}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{2}}+\frac {1}{20 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{5}}-\frac {13}{48 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{3}}+\frac {29}{64 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )}}{a^{2} d}\) | \(240\) |
| default | \(\frac {-\frac {2 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}+\frac {51 i}{16 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {49 i}{6 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {35 i}{4 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {4}{9 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}-\frac {5}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {49}{5 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {49}{8 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {99}{64 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {i}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{4}}-\frac {9 i}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{2}}+\frac {1}{20 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{5}}-\frac {13}{48 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{3}}+\frac {29}{64 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )}}{a^{2} d}\) | \(240\) |
Input:
int(cos(d*x+c)^5/(a+I*a*tan(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/128*I/a^2/d*exp(-7*I*(d*x+c))+1/1152*I/a^2/d*exp(-9*I*(d*x+c))+7/64*I/a^ 2/d*cos(d*x+c)+7/16*sin(d*x+c)/a^2/d+1/32*I/a^2/d*cos(5*d*x+5*c)+11/320/a^ 2/d*sin(5*d*x+5*c)+7/96*I/a^2/d*cos(3*d*x+3*c)+7/64/a^2/d*sin(3*d*x+3*c)
Time = 0.08 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.90 \[ \int \frac {\cos ^5(c+d x)}{(a+i a \tan (c+d x))^2} \, dx=\frac {{\left (-9 i \, e^{\left (14 i \, d x + 14 i \, c\right )} - 105 i \, e^{\left (12 i \, d x + 12 i \, c\right )} - 945 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 1575 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 525 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 189 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 45 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 5 i\right )} e^{\left (-9 i \, d x - 9 i \, c\right )}}{5760 \, a^{2} d} \] Input:
integrate(cos(d*x+c)^5/(a+I*a*tan(d*x+c))^2,x, algorithm="fricas")
Output:
1/5760*(-9*I*e^(14*I*d*x + 14*I*c) - 105*I*e^(12*I*d*x + 12*I*c) - 945*I*e ^(10*I*d*x + 10*I*c) + 1575*I*e^(8*I*d*x + 8*I*c) + 525*I*e^(6*I*d*x + 6*I *c) + 189*I*e^(4*I*d*x + 4*I*c) + 45*I*e^(2*I*d*x + 2*I*c) + 5*I)*e^(-9*I* d*x - 9*I*c)/(a^2*d)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 299 vs. \(2 (94) = 188\).
Time = 0.41 (sec) , antiderivative size = 299, normalized size of antiderivative = 2.79 \[ \int \frac {\cos ^5(c+d x)}{(a+i a \tan (c+d x))^2} \, dx=\begin {cases} \frac {\left (- 227994731135631360 i a^{14} d^{7} e^{30 i c} e^{5 i d x} - 2659938529915699200 i a^{14} d^{7} e^{28 i c} e^{3 i d x} - 23939446769241292800 i a^{14} d^{7} e^{26 i c} e^{i d x} + 39899077948735488000 i a^{14} d^{7} e^{24 i c} e^{- i d x} + 13299692649578496000 i a^{14} d^{7} e^{22 i c} e^{- 3 i d x} + 4787889353848258560 i a^{14} d^{7} e^{20 i c} e^{- 5 i d x} + 1139973655678156800 i a^{14} d^{7} e^{18 i c} e^{- 7 i d x} + 126663739519795200 i a^{14} d^{7} e^{16 i c} e^{- 9 i d x}\right ) e^{- 25 i c}}{145916627926804070400 a^{16} d^{8}} & \text {for}\: a^{16} d^{8} e^{25 i c} \neq 0 \\\frac {x \left (e^{14 i c} + 7 e^{12 i c} + 21 e^{10 i c} + 35 e^{8 i c} + 35 e^{6 i c} + 21 e^{4 i c} + 7 e^{2 i c} + 1\right ) e^{- 9 i c}}{128 a^{2}} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)**5/(a+I*a*tan(d*x+c))**2,x)
Output:
Piecewise(((-227994731135631360*I*a**14*d**7*exp(30*I*c)*exp(5*I*d*x) - 26 59938529915699200*I*a**14*d**7*exp(28*I*c)*exp(3*I*d*x) - 2393944676924129 2800*I*a**14*d**7*exp(26*I*c)*exp(I*d*x) + 39899077948735488000*I*a**14*d* *7*exp(24*I*c)*exp(-I*d*x) + 13299692649578496000*I*a**14*d**7*exp(22*I*c) *exp(-3*I*d*x) + 4787889353848258560*I*a**14*d**7*exp(20*I*c)*exp(-5*I*d*x ) + 1139973655678156800*I*a**14*d**7*exp(18*I*c)*exp(-7*I*d*x) + 126663739 519795200*I*a**14*d**7*exp(16*I*c)*exp(-9*I*d*x))*exp(-25*I*c)/(1459166279 26804070400*a**16*d**8), Ne(a**16*d**8*exp(25*I*c), 0)), (x*(exp(14*I*c) + 7*exp(12*I*c) + 21*exp(10*I*c) + 35*exp(8*I*c) + 35*exp(6*I*c) + 21*exp(4 *I*c) + 7*exp(2*I*c) + 1)*exp(-9*I*c)/(128*a**2), True))
Exception generated. \[ \int \frac {\cos ^5(c+d x)}{(a+i a \tan (c+d x))^2} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(cos(d*x+c)^5/(a+I*a*tan(d*x+c))^2,x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (93) = 186\).
Time = 0.24 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.84 \[ \int \frac {\cos ^5(c+d x)}{(a+i a \tan (c+d x))^2} \, dx=\frac {\frac {3 \, {\left (435 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1470 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2060 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1330 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 353\right )}}{a^{2} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right )}^{5}} + \frac {4455 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 26460 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 78120 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 137340 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 157374 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 118356 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 57744 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 16596 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2339}{a^{2} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{9}}}{2880 \, d} \] Input:
integrate(cos(d*x+c)^5/(a+I*a*tan(d*x+c))^2,x, algorithm="giac")
Output:
1/2880*(3*(435*tan(1/2*d*x + 1/2*c)^4 + 1470*I*tan(1/2*d*x + 1/2*c)^3 - 20 60*tan(1/2*d*x + 1/2*c)^2 - 1330*I*tan(1/2*d*x + 1/2*c) + 353)/(a^2*(tan(1 /2*d*x + 1/2*c) + I)^5) + (4455*tan(1/2*d*x + 1/2*c)^8 - 26460*I*tan(1/2*d *x + 1/2*c)^7 - 78120*tan(1/2*d*x + 1/2*c)^6 + 137340*I*tan(1/2*d*x + 1/2* c)^5 + 157374*tan(1/2*d*x + 1/2*c)^4 - 118356*I*tan(1/2*d*x + 1/2*c)^3 - 5 7744*tan(1/2*d*x + 1/2*c)^2 + 16596*I*tan(1/2*d*x + 1/2*c) + 2339)/(a^2*(t an(1/2*d*x + 1/2*c) - I)^9))/d
Time = 2.86 (sec) , antiderivative size = 216, normalized size of antiderivative = 2.02 \[ \int \frac {\cos ^5(c+d x)}{(a+i a \tan (c+d x))^2} \, dx=\frac {\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {191\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}-\frac {1289\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{64}+\frac {649\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{64}-\frac {41\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{32}+\frac {41\,\sin \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{32}-\frac {7\,\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{64}+\frac {7\,\sin \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )}{64}+\frac {\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )\,525{}\mathrm {i}}{32}-\frac {\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )\,205{}\mathrm {i}}{32}+\frac {\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )\,1{}\mathrm {i}}{2}-\frac {\cos \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )\,1{}\mathrm {i}}{2}+\frac {\cos \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )\,1{}\mathrm {i}}{32}-\frac {\cos \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )\,1{}\mathrm {i}}{32}\right )\,2{}\mathrm {i}}{45\,a^2\,d\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}\right )}^9\,{\left (\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}\right )}^5} \] Input:
int(cos(c + d*x)^5/(a + a*tan(c + d*x)*1i)^2,x)
Output:
(cos(c/2 + (d*x)/2)*((cos((3*c)/2 + (3*d*x)/2)*525i)/32 - (cos((5*c)/2 + ( 5*d*x)/2)*205i)/32 + (cos((7*c)/2 + (7*d*x)/2)*1i)/2 - (cos((9*c)/2 + (9*d *x)/2)*1i)/2 + (cos((11*c)/2 + (11*d*x)/2)*1i)/32 - (cos((13*c)/2 + (13*d* x)/2)*1i)/32 + (191*sin(c/2 + (d*x)/2))/16 - (1289*sin((3*c)/2 + (3*d*x)/2 ))/64 + (649*sin((5*c)/2 + (5*d*x)/2))/64 - (41*sin((7*c)/2 + (7*d*x)/2))/ 32 + (41*sin((9*c)/2 + (9*d*x)/2))/32 - (7*sin((11*c)/2 + (11*d*x)/2))/64 + (7*sin((13*c)/2 + (13*d*x)/2))/64)*2i)/(45*a^2*d*(cos(c/2 + (d*x)/2) + s in(c/2 + (d*x)/2)*1i)^9*(cos(c/2 + (d*x)/2)*1i + sin(c/2 + (d*x)/2))^5)
\[ \int \frac {\cos ^5(c+d x)}{(a+i a \tan (c+d x))^2} \, dx=-\frac {\int \frac {\cos \left (d x +c \right )^{5}}{\tan \left (d x +c \right )^{2}-2 \tan \left (d x +c \right ) i -1}d x}{a^{2}} \] Input:
int(cos(d*x+c)^5/(a+I*a*tan(d*x+c))^2,x)
Output:
( - int(cos(c + d*x)**5/(tan(c + d*x)**2 - 2*tan(c + d*x)*i - 1),x))/a**2