3.88 \(\int \cos ^{14}(c+d x) (a+i a \tan (c+d x))^8 \, dx\)
Optimal. Leaf size=27 \[ -\frac{i a^{15}}{7 d (a-i a \tan (c+d x))^7} \]
[Out]
((-I/7)*a^15)/(d*(a - I*a*Tan[c + d*x])^7)
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Rubi [A] time = 0.0378439, antiderivative size = 27, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used =
{3487, 32} \[ -\frac{i a^{15}}{7 d (a-i a \tan (c+d x))^7} \]
Antiderivative was successfully verified.
[In]
Int[Cos[c + d*x]^14*(a + I*a*Tan[c + d*x])^8,x]
[Out]
((-I/7)*a^15)/(d*(a - I*a*Tan[c + d*x])^7)
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 32
Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]
Rubi steps
\begin{align*} \int \cos ^{14}(c+d x) (a+i a \tan (c+d x))^8 \, dx &=-\frac{\left (i a^{15}\right ) \operatorname{Subst}\left (\int \frac{1}{(a-x)^8} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac{i a^{15}}{7 d (a-i a \tan (c+d x))^7}\\ \end{align*}
Mathematica [B] time = 1.63517, size = 116, normalized size = 4.3 \[ \frac{a^8 (-14 i \sin (2 (c+d x))-14 i \sin (4 (c+d x))-6 i \sin (6 (c+d x))+56 \cos (2 (c+d x))+28 \cos (4 (c+d x))+8 \cos (6 (c+d x))+35) (\sin (8 (c+2 d x))-i \cos (8 (c+2 d x)))}{896 d (\cos (d x)+i \sin (d x))^8} \]
Antiderivative was successfully verified.
[In]
Integrate[Cos[c + d*x]^14*(a + I*a*Tan[c + d*x])^8,x]
[Out]
(a^8*(35 + 56*Cos[2*(c + d*x)] + 28*Cos[4*(c + d*x)] + 8*Cos[6*(c + d*x)] - (14*I)*Sin[2*(c + d*x)] - (14*I)*S
in[4*(c + d*x)] - (6*I)*Sin[6*(c + d*x)])*((-I)*Cos[8*(c + 2*d*x)] + Sin[8*(c + 2*d*x)]))/(896*d*(Cos[d*x] + I
*Sin[d*x])^8)
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Maple [B] time = 0.136, size = 689, normalized size = 25.5 \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)^14*(a+I*a*tan(d*x+c))^8,x)
[Out]
1/d*(a^8*(-1/14*sin(d*x+c)^7*cos(d*x+c)^7-1/24*sin(d*x+c)^5*cos(d*x+c)^7-1/48*sin(d*x+c)^3*cos(d*x+c)^7-1/128*
sin(d*x+c)*cos(d*x+c)^7+1/768*(cos(d*x+c)^5+5/4*cos(d*x+c)^3+15/8*cos(d*x+c))*sin(d*x+c)+5/2048*d*x+5/2048*c)-
8*I*a^8*(-1/14*sin(d*x+c)^6*cos(d*x+c)^8-1/28*sin(d*x+c)^4*cos(d*x+c)^8-1/70*sin(d*x+c)^2*cos(d*x+c)^8-1/280*c
os(d*x+c)^8)-28*a^8*(-1/14*sin(d*x+c)^5*cos(d*x+c)^9-5/168*sin(d*x+c)^3*cos(d*x+c)^9-1/112*sin(d*x+c)*cos(d*x+
c)^9+1/896*(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)+5/2048*d*x+5/2048*c)
+56*I*a^8*(-1/14*sin(d*x+c)^4*cos(d*x+c)^10-1/42*sin(d*x+c)^2*cos(d*x+c)^10-1/210*cos(d*x+c)^10)+70*a^8*(-1/14
*sin(d*x+c)^3*cos(d*x+c)^11-1/56*sin(d*x+c)*cos(d*x+c)^11+1/560*(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)+9/2048*d*x+9/2048*c)-56*I*a^8*(-1/14*sin(d*x+c)^2*cos(d
*x+c)^12-1/84*cos(d*x+c)^12)-28*a^8*(-1/14*sin(d*x+c)*cos(d*x+c)^13+1/168*(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)+33/2048*d*x+33/2048*
c)-4/7*I*a^8*cos(d*x+c)^14+a^8*(1/14*(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)+429/2048*d*x+429/2048*c))
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Maxima [B] time = 1.73427, size = 232, normalized size = 8.59 \begin{align*} -\frac{30720 \, a^{8} \tan \left (d x + c\right )^{7} - 215040 i \, a^{8} \tan \left (d x + c\right )^{6} - 645120 \, a^{8} \tan \left (d x + c\right )^{5} + 1075200 i \, a^{8} \tan \left (d x + c\right )^{4} + 1075200 \, a^{8} \tan \left (d x + c\right )^{3} - 645120 i \, a^{8} \tan \left (d x + c\right )^{2} - 215040 \, a^{8} \tan \left (d x + c\right ) + 30720 i \, a^{8}}{215040 \,{\left (\tan \left (d x + c\right )^{14} + 7 \, \tan \left (d x + c\right )^{12} + 21 \, \tan \left (d x + c\right )^{10} + 35 \, \tan \left (d x + c\right )^{8} + 35 \, \tan \left (d x + c\right )^{6} + 21 \, \tan \left (d x + c\right )^{4} + 7 \, \tan \left (d x + c\right )^{2} + 1\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^14*(a+I*a*tan(d*x+c))^8,x, algorithm="maxima")
[Out]
-1/215040*(30720*a^8*tan(d*x + c)^7 - 215040*I*a^8*tan(d*x + c)^6 - 645120*a^8*tan(d*x + c)^5 + 1075200*I*a^8*
tan(d*x + c)^4 + 1075200*a^8*tan(d*x + c)^3 - 645120*I*a^8*tan(d*x + c)^2 - 215040*a^8*tan(d*x + c) + 30720*I*
a^8)/((tan(d*x + c)^14 + 7*tan(d*x + c)^12 + 21*tan(d*x + c)^10 + 35*tan(d*x + c)^8 + 35*tan(d*x + c)^6 + 21*t
an(d*x + c)^4 + 7*tan(d*x + c)^2 + 1)*d)
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Fricas [B] time = 1.91081, size = 308, normalized size = 11.41 \begin{align*} \frac{-i \, a^{8} e^{\left (14 i \, d x + 14 i \, c\right )} - 7 i \, a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} - 21 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} - 35 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} - 35 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} - 21 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} - 7 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )}}{896 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^14*(a+I*a*tan(d*x+c))^8,x, algorithm="fricas")
[Out]
1/896*(-I*a^8*e^(14*I*d*x + 14*I*c) - 7*I*a^8*e^(12*I*d*x + 12*I*c) - 21*I*a^8*e^(10*I*d*x + 10*I*c) - 35*I*a^
8*e^(8*I*d*x + 8*I*c) - 35*I*a^8*e^(6*I*d*x + 6*I*c) - 21*I*a^8*e^(4*I*d*x + 4*I*c) - 7*I*a^8*e^(2*I*d*x + 2*I
*c))/d
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Sympy [B] time = 2.02483, size = 280, normalized size = 10.37 \begin{align*} \begin{cases} \frac{- 4398046511104 i a^{8} d^{6} e^{14 i c} e^{14 i d x} - 30786325577728 i a^{8} d^{6} e^{12 i c} e^{12 i d x} - 92358976733184 i a^{8} d^{6} e^{10 i c} e^{10 i d x} - 153931627888640 i a^{8} d^{6} e^{8 i c} e^{8 i d x} - 153931627888640 i a^{8} d^{6} e^{6 i c} e^{6 i d x} - 92358976733184 i a^{8} d^{6} e^{4 i c} e^{4 i d x} - 30786325577728 i a^{8} d^{6} e^{2 i c} e^{2 i d x}}{3940649673949184 d^{7}} & \text{for}\: 3940649673949184 d^{7} \neq 0 \\x \left (\frac{a^{8} e^{14 i c}}{64} + \frac{3 a^{8} e^{12 i c}}{32} + \frac{15 a^{8} e^{10 i c}}{64} + \frac{5 a^{8} e^{8 i c}}{16} + \frac{15 a^{8} e^{6 i c}}{64} + \frac{3 a^{8} e^{4 i c}}{32} + \frac{a^{8} e^{2 i c}}{64}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)**14*(a+I*a*tan(d*x+c))**8,x)
[Out]
Piecewise(((-4398046511104*I*a**8*d**6*exp(14*I*c)*exp(14*I*d*x) - 30786325577728*I*a**8*d**6*exp(12*I*c)*exp(
12*I*d*x) - 92358976733184*I*a**8*d**6*exp(10*I*c)*exp(10*I*d*x) - 153931627888640*I*a**8*d**6*exp(8*I*c)*exp(
8*I*d*x) - 153931627888640*I*a**8*d**6*exp(6*I*c)*exp(6*I*d*x) - 92358976733184*I*a**8*d**6*exp(4*I*c)*exp(4*I
*d*x) - 30786325577728*I*a**8*d**6*exp(2*I*c)*exp(2*I*d*x))/(3940649673949184*d**7), Ne(3940649673949184*d**7,
0)), (x*(a**8*exp(14*I*c)/64 + 3*a**8*exp(12*I*c)/32 + 15*a**8*exp(10*I*c)/64 + 5*a**8*exp(8*I*c)/16 + 15*a**
8*exp(6*I*c)/64 + 3*a**8*exp(4*I*c)/32 + a**8*exp(2*I*c)/64), True))
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Giac [B] time = 2.66231, size = 628, normalized size = 23.26 \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)^14*(a+I*a*tan(d*x+c))^8,x, algorithm="giac")
[Out]
1/3440640*(-3840*I*a^8*e^(42*I*d*x + 28*I*c) - 80640*I*a^8*e^(40*I*d*x + 26*I*c) - 806400*I*a^8*e^(38*I*d*x +
24*I*c) - 5107200*I*a^8*e^(36*I*d*x + 22*I*c) - 22982400*I*a^8*e^(34*I*d*x + 20*I*c) - 78140160*I*a^8*e^(32*I*
d*x + 18*I*c) - 208373760*I*a^8*e^(30*I*d*x + 16*I*c) - 446511360*I*a^8*e^(28*I*d*x + 14*I*c) - 781347840*I*a^
8*e^(26*I*d*x + 12*I*c) - 1128341760*I*a^8*e^(24*I*d*x + 10*I*c) - 1353031680*I*a^8*e^(22*I*d*x + 8*I*c) - 135
0585600*I*a^8*e^(20*I*d*x + 6*I*c) - 1121003520*I*a^8*e^(18*I*d*x + 4*I*c) - 769870080*I*a^8*e^(16*I*d*x + 2*I
*c) - 196842240*I*a^8*e^(12*I*d*x - 2*I*c) - 70452480*I*a^8*e^(10*I*d*x - 4*I*c) - 19138560*I*a^8*e^(8*I*d*x -
6*I*c) - 3709440*I*a^8*e^(6*I*d*x - 8*I*c) - 456960*I*a^8*e^(4*I*d*x - 10*I*c) - 26880*I*a^8*e^(2*I*d*x - 12*
I*c) - 433336320*I*a^8*e^(14*I*d*x))/(d*e^(28*I*d*x + 14*I*c) + 14*d*e^(26*I*d*x + 12*I*c) + 91*d*e^(24*I*d*x
+ 10*I*c) + 364*d*e^(22*I*d*x + 8*I*c) + 1001*d*e^(20*I*d*x + 6*I*c) + 2002*d*e^(18*I*d*x + 4*I*c) + 3003*d*e^
(16*I*d*x + 2*I*c) + 3003*d*e^(12*I*d*x - 2*I*c) + 2002*d*e^(10*I*d*x - 4*I*c) + 1001*d*e^(8*I*d*x - 6*I*c) +
364*d*e^(6*I*d*x - 8*I*c) + 91*d*e^(4*I*d*x - 10*I*c) + 14*d*e^(2*I*d*x - 12*I*c) + 3432*d*e^(14*I*d*x) + d*e^
(-14*I*c))