3.69 \(\int \cos ^{12}(c+d x) (a+i a \tan (c+d x))^5 \, dx\)
Optimal. Leaf size=198 \[ -\frac{i a^{11}}{24 d (a-i a \tan (c+d x))^6}-\frac{i a^{10}}{20 d (a-i a \tan (c+d x))^5}-\frac{3 i a^9}{64 d (a-i a \tan (c+d x))^4}-\frac{i a^8}{24 d (a-i a \tan (c+d x))^3}-\frac{5 i a^7}{128 d (a-i a \tan (c+d x))^2}-\frac{3 i a^6}{64 d (a-i a \tan (c+d x))}+\frac{i a^6}{128 d (a+i a \tan (c+d x))}+\frac{7 a^5 x}{128} \]
[Out]
(7*a^5*x)/128 - ((I/24)*a^11)/(d*(a - I*a*Tan[c + d*x])^6) - ((I/20)*a^10)/(d*(a - I*a*Tan[c + d*x])^5) - (((3
*I)/64)*a^9)/(d*(a - I*a*Tan[c + d*x])^4) - ((I/24)*a^8)/(d*(a - I*a*Tan[c + d*x])^3) - (((5*I)/128)*a^7)/(d*(
a - I*a*Tan[c + d*x])^2) - (((3*I)/64)*a^6)/(d*(a - I*a*Tan[c + d*x])) + ((I/128)*a^6)/(d*(a + I*a*Tan[c + d*x
]))
________________________________________________________________________________________
Rubi [A] time = 0.11411, antiderivative size = 198, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used =
{3487, 44, 206} \[ -\frac{i a^{11}}{24 d (a-i a \tan (c+d x))^6}-\frac{i a^{10}}{20 d (a-i a \tan (c+d x))^5}-\frac{3 i a^9}{64 d (a-i a \tan (c+d x))^4}-\frac{i a^8}{24 d (a-i a \tan (c+d x))^3}-\frac{5 i a^7}{128 d (a-i a \tan (c+d x))^2}-\frac{3 i a^6}{64 d (a-i a \tan (c+d x))}+\frac{i a^6}{128 d (a+i a \tan (c+d x))}+\frac{7 a^5 x}{128} \]
Antiderivative was successfully verified.
[In]
Int[Cos[c + d*x]^12*(a + I*a*Tan[c + d*x])^5,x]
[Out]
(7*a^5*x)/128 - ((I/24)*a^11)/(d*(a - I*a*Tan[c + d*x])^6) - ((I/20)*a^10)/(d*(a - I*a*Tan[c + d*x])^5) - (((3
*I)/64)*a^9)/(d*(a - I*a*Tan[c + d*x])^4) - ((I/24)*a^8)/(d*(a - I*a*Tan[c + d*x])^3) - (((5*I)/128)*a^7)/(d*(
a - I*a*Tan[c + d*x])^2) - (((3*I)/64)*a^6)/(d*(a - I*a*Tan[c + d*x])) + ((I/128)*a^6)/(d*(a + I*a*Tan[c + d*x
]))
Rule 3487
Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[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]
Rule 44
Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rubi steps
\begin{align*} \int \cos ^{12}(c+d x) (a+i a \tan (c+d x))^5 \, dx &=-\frac{\left (i a^{13}\right ) \operatorname{Subst}\left (\int \frac{1}{(a-x)^7 (a+x)^2} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac{\left (i a^{13}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{4 a^2 (a-x)^7}+\frac{1}{4 a^3 (a-x)^6}+\frac{3}{16 a^4 (a-x)^5}+\frac{1}{8 a^5 (a-x)^4}+\frac{5}{64 a^6 (a-x)^3}+\frac{3}{64 a^7 (a-x)^2}+\frac{1}{128 a^7 (a+x)^2}+\frac{7}{128 a^7 \left (a^2-x^2\right )}\right ) \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\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{3 i a^9}{64 d (a-i a \tan (c+d x))^4}-\frac{i a^8}{24 d (a-i a \tan (c+d x))^3}-\frac{5 i a^7}{128 d (a-i a \tan (c+d x))^2}-\frac{3 i a^6}{64 d (a-i a \tan (c+d x))}+\frac{i a^6}{128 d (a+i a \tan (c+d x))}-\frac{\left (7 i a^6\right ) \operatorname{Subst}\left (\int \frac{1}{a^2-x^2} \, dx,x,i a \tan (c+d x)\right )}{128 d}\\ &=\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{3 i a^9}{64 d (a-i a \tan (c+d x))^4}-\frac{i a^8}{24 d (a-i a \tan (c+d x))^3}-\frac{5 i a^7}{128 d (a-i a \tan (c+d x))^2}-\frac{3 i a^6}{64 d (a-i a \tan (c+d x))}+\frac{i a^6}{128 d (a+i a \tan (c+d x))}\\ \end{align*}
Mathematica [A] time = 2.64195, size = 159, normalized size = 0.8 \[ \frac{a^5 (-350 \sin (c+d x)-945 \sin (3 (c+d x))-840 i d x \sin (5 (c+d x))+84 \sin (5 (c+d x))+70 \sin (7 (c+d x))-1750 i \cos (c+d x)-1575 i \cos (3 (c+d x))+840 d x \cos (5 (c+d x))-84 i \cos (5 (c+d x))+50 i \cos (7 (c+d x))) (\cos (5 (c+2 d x))+i \sin (5 (c+2 d x)))}{15360 d (\cos (d x)+i \sin (d x))^5} \]
Antiderivative was successfully verified.
[In]
Integrate[Cos[c + d*x]^12*(a + I*a*Tan[c + d*x])^5,x]
[Out]
(a^5*((-1750*I)*Cos[c + d*x] - (1575*I)*Cos[3*(c + d*x)] - (84*I)*Cos[5*(c + d*x)] + 840*d*x*Cos[5*(c + d*x)]
+ (50*I)*Cos[7*(c + d*x)] - 350*Sin[c + d*x] - 945*Sin[3*(c + d*x)] + 84*Sin[5*(c + d*x)] - (840*I)*d*x*Sin[5*
(c + d*x)] + 70*Sin[7*(c + d*x)])*(Cos[5*(c + 2*d*x)] + I*Sin[5*(c + 2*d*x)]))/(15360*d*(Cos[d*x] + I*Sin[d*x]
)^5)
________________________________________________________________________________________
Maple [B] time = 0.133, size = 361, normalized size = 1.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(d*x+c)^12*(a+I*a*tan(d*x+c))^5,x)
[Out]
1/d*(I*a^5*(-1/12*sin(d*x+c)^4*cos(d*x+c)^8-1/30*sin(d*x+c)^2*cos(d*x+c)^8-1/120*cos(d*x+c)^8)+5*a^5*(-1/12*si
n(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+3
5/16*cos(d*x+c))*sin(d*x+c)+7/1024*d*x+7/1024*c)-10*I*a^5*(-1/12*sin(d*x+c)^2*cos(d*x+c)^10-1/60*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))
________________________________________________________________________________________
Maxima [A] time = 1.67657, size = 252, normalized size = 1.27 \begin{align*} \frac{840 \,{\left (d x + c\right )} a^{5} + \frac{840 \, a^{5} \tan \left (d x + c\right )^{11} + 4760 \, a^{5} \tan \left (d x + c\right )^{9} + 11088 \, a^{5} \tan \left (d x + c\right )^{7} + 13488 \, a^{5} \tan \left (d x + c\right )^{5} - 1920 i \, a^{5} \tan \left (d x + c\right )^{4} + 360 \, a^{5} \tan \left (d x + c\right )^{3} + 14592 i \, a^{5} \tan \left (d x + c\right )^{2} + 14520 \, a^{5} \tan \left (d x + c\right ) - 3968 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}}{15360 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^12*(a+I*a*tan(d*x+c))^5,x, algorithm="maxima")
[Out]
1/15360*(840*(d*x + c)*a^5 + (840*a^5*tan(d*x + c)^11 + 4760*a^5*tan(d*x + c)^9 + 11088*a^5*tan(d*x + c)^7 + 1
3488*a^5*tan(d*x + c)^5 - 1920*I*a^5*tan(d*x + c)^4 + 360*a^5*tan(d*x + c)^3 + 14592*I*a^5*tan(d*x + c)^2 + 14
520*a^5*tan(d*x + c) - 3968*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
________________________________________________________________________________________
Fricas [A] time = 1.31561, size = 371, normalized size = 1.87 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^12*(a+I*a*tan(d*x+c))^5,x, algorithm="fricas")
[Out]
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) - 3
15*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
________________________________________________________________________________________
Sympy [A] time = 1.32525, size = 303, normalized size = 1.53 \begin{align*} \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}\: 51881467707308113920 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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)**12*(a+I*a*tan(d*x+c))**5,x)
[Out]
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) - 1063975411966279680*I*a**5*d**6*exp(10*I*c)*exp(8*I*d*x) - 2364389804369510
400*I*a**5*d**6*exp(8*I*c)*exp(6*I*d*x) - 3546584706554265600*I*a**5*d**6*exp(6*I*c)*exp(4*I*d*x) - 4255901647
865118720*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)/(518
81467707308113920*d**7), Ne(51881467707308113920*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))
________________________________________________________________________________________
Giac [B] time = 1.86856, size = 1234, normalized size = 6.23 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^12*(a+I*a*tan(d*x+c))^5,x, algorithm="giac")
[Out]
1/245760*(13440*a^5*d*x*e^(18*I*d*x + 10*I*c) + 107520*a^5*d*x*e^(16*I*d*x + 8*I*c) + 376320*a^5*d*x*e^(14*I*d
*x + 6*I*c) + 752640*a^5*d*x*e^(12*I*d*x + 4*I*c) + 940800*a^5*d*x*e^(10*I*d*x + 2*I*c) + 376320*a^5*d*x*e^(6*
I*d*x - 2*I*c) + 107520*a^5*d*x*e^(4*I*d*x - 4*I*c) + 13440*a^5*d*x*e^(2*I*d*x - 6*I*c) + 752640*a^5*d*x*e^(8*
I*d*x) - 4710*I*a^5*e^(18*I*d*x + 10*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 37680*I*a^5*e^(16*I*d*x + 8*I*c)*log(
e^(2*I*d*x + 2*I*c) + 1) - 131880*I*a^5*e^(14*I*d*x + 6*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 263760*I*a^5*e^(12
*I*d*x + 4*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 329700*I*a^5*e^(10*I*d*x + 2*I*c)*log(e^(2*I*d*x + 2*I*c) + 1)
- 131880*I*a^5*e^(6*I*d*x - 2*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 37680*I*a^5*e^(4*I*d*x - 4*I*c)*log(e^(2*I*d
*x + 2*I*c) + 1) - 4710*I*a^5*e^(2*I*d*x - 6*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 263760*I*a^5*e^(8*I*d*x)*log(
e^(2*I*d*x + 2*I*c) + 1) + 4710*I*a^5*e^(18*I*d*x + 10*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 37680*I*a^5*e^(16*
I*d*x + 8*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 131880*I*a^5*e^(14*I*d*x + 6*I*c)*log(e^(2*I*d*x) + e^(-2*I*c))
+ 263760*I*a^5*e^(12*I*d*x + 4*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 329700*I*a^5*e^(10*I*d*x + 2*I*c)*log(e^(
2*I*d*x) + e^(-2*I*c)) + 131880*I*a^5*e^(6*I*d*x - 2*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 37680*I*a^5*e^(4*I*d
*x - 4*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 4710*I*a^5*e^(2*I*d*x - 6*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 263
760*I*a^5*e^(8*I*d*x)*log(e^(2*I*d*x) + e^(-2*I*c)) - 160*I*a^5*e^(30*I*d*x + 22*I*c) - 2624*I*a^5*e^(28*I*d*x
+ 20*I*c) - 20272*I*a^5*e^(26*I*d*x + 18*I*c) - 98112*I*a^5*e^(24*I*d*x + 16*I*c) - 333984*I*a^5*e^(22*I*d*x
+ 14*I*c) - 853440*I*a^5*e^(20*I*d*x + 12*I*c) - 1691424*I*a^5*e^(18*I*d*x + 10*I*c) - 2609472*I*a^5*e^(16*I*d
*x + 8*I*c) - 3076512*I*a^5*e^(14*I*d*x + 6*I*c) - 2680384*I*a^5*e^(12*I*d*x + 4*I*c) - 1640240*I*a^5*e^(10*I*
d*x + 2*I*c) - 124320*I*a^5*e^(6*I*d*x - 2*I*c) + 6720*I*a^5*e^(4*I*d*x - 4*I*c) + 7680*I*a^5*e^(2*I*d*x - 6*I
*c) - 642880*I*a^5*e^(8*I*d*x) + 960*I*a^5*e^(-8*I*c))/(d*e^(18*I*d*x + 10*I*c) + 8*d*e^(16*I*d*x + 8*I*c) + 2
8*d*e^(14*I*d*x + 6*I*c) + 56*d*e^(12*I*d*x + 4*I*c) + 70*d*e^(10*I*d*x + 2*I*c) + 28*d*e^(6*I*d*x - 2*I*c) +
8*d*e^(4*I*d*x - 4*I*c) + d*e^(2*I*d*x - 6*I*c) + 56*d*e^(8*I*d*x))