3.85 \(\int \cos ^8(c+d x) (a+i a \tan (c+d x))^8 \, dx\)

Optimal. Leaf size=43 \[ -\frac{i \left (a^3+i a^3 \tan (c+d x)\right )^4}{8 d (a-i a \tan (c+d x))^4} \]

[Out]

((-I/8)*(a^3 + I*a^3*Tan[c + d*x])^4)/(d*(a - I*a*Tan[c + d*x])^4)

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Rubi [A]  time = 0.0422143, antiderivative size = 43, 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, 37} \[ -\frac{i \left (a^3+i a^3 \tan (c+d x)\right )^4}{8 d (a-i a \tan (c+d x))^4} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^8*(a + I*a*Tan[c + d*x])^8,x]

[Out]

((-I/8)*(a^3 + I*a^3*Tan[c + d*x])^4)/(d*(a - I*a*Tan[c + d*x])^4)

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 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \cos ^8(c+d x) (a+i a \tan (c+d x))^8 \, dx &=-\frac{\left (i a^9\right ) \operatorname{Subst}\left (\int \frac{(a+x)^3}{(a-x)^5} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac{i \left (a^3+i a^3 \tan (c+d x)\right )^4}{8 d (a-i a \tan (c+d x))^4}\\ \end{align*}

Mathematica [A]  time = 0.295275, size = 31, normalized size = 0.72 \[ -\frac{i a^8 (\cos (c+d x)+i \sin (c+d x))^8}{8 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^8*(a + I*a*Tan[c + d*x])^8,x]

[Out]

((-I/8)*a^8*(Cos[c + d*x] + I*Sin[c + d*x])^8)/d

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Maple [B]  time = 0.077, size = 451, normalized size = 10.5 \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)^8*(a+I*a*tan(d*x+c))^8,x)

[Out]

1/d*(a^8*(-1/8*(sin(d*x+c)^7+7/6*sin(d*x+c)^5+35/24*sin(d*x+c)^3+35/16*sin(d*x+c))*cos(d*x+c)+35/128*d*x+35/12
8*c)-I*a^8*sin(d*x+c)^8-28*a^8*(-1/8*sin(d*x+c)^5*cos(d*x+c)^3-5/48*sin(d*x+c)^3*cos(d*x+c)^3-5/64*cos(d*x+c)^
3*sin(d*x+c)+5/128*cos(d*x+c)*sin(d*x+c)+5/128*d*x+5/128*c)+56*I*a^8*(-1/8*sin(d*x+c)^4*cos(d*x+c)^4-1/12*sin(
d*x+c)^2*cos(d*x+c)^4-1/24*cos(d*x+c)^4)+70*a^8*(-1/8*sin(d*x+c)^3*cos(d*x+c)^5-1/16*cos(d*x+c)^5*sin(d*x+c)+1
/64*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/128*d*x+3/128*c)-56*I*a^8*(-1/8*sin(d*x+c)^2*cos(d*x+c)^6-1/24*
cos(d*x+c)^6)-28*a^8*(-1/8*sin(d*x+c)*cos(d*x+c)^7+1/48*(cos(d*x+c)^5+5/4*cos(d*x+c)^3+15/8*cos(d*x+c))*sin(d*
x+c)+5/128*d*x+5/128*c)-I*a^8*cos(d*x+c)^8+a^8*(1/8*(cos(d*x+c)^7+7/6*cos(d*x+c)^5+35/24*cos(d*x+c)^3+35/16*co
s(d*x+c))*sin(d*x+c)+35/128*d*x+35/128*c))

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Maxima [B]  time = 2.41403, size = 185, normalized size = 4.3 \begin{align*} -\frac{384 \, a^{8} \tan \left (d x + c\right )^{7} - 1536 i \, a^{8} \tan \left (d x + c\right )^{6} - 2688 \, a^{8} \tan \left (d x + c\right )^{5} + 3072 i \, a^{8} \tan \left (d x + c\right )^{4} + 2688 \, a^{8} \tan \left (d x + c\right )^{3} - 1536 i \, a^{8} \tan \left (d x + c\right )^{2} - 384 \, a^{8} \tan \left (d x + c\right )}{384 \,{\left (\tan \left (d x + c\right )^{8} + 4 \, \tan \left (d x + c\right )^{6} + 6 \, \tan \left (d x + c\right )^{4} + 4 \, \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)^8*(a+I*a*tan(d*x+c))^8,x, algorithm="maxima")

[Out]

-1/384*(384*a^8*tan(d*x + c)^7 - 1536*I*a^8*tan(d*x + c)^6 - 2688*a^8*tan(d*x + c)^5 + 3072*I*a^8*tan(d*x + c)
^4 + 2688*a^8*tan(d*x + c)^3 - 1536*I*a^8*tan(d*x + c)^2 - 384*a^8*tan(d*x + c))/((tan(d*x + c)^8 + 4*tan(d*x
+ c)^6 + 6*tan(d*x + c)^4 + 4*tan(d*x + c)^2 + 1)*d)

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Fricas [A]  time = 1.42455, size = 46, normalized size = 1.07 \begin{align*} -\frac{i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )}}{8 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*(a+I*a*tan(d*x+c))^8,x, algorithm="fricas")

[Out]

-1/8*I*a^8*e^(8*I*d*x + 8*I*c)/d

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Sympy [A]  time = 0.948915, size = 37, normalized size = 0.86 \begin{align*} \begin{cases} - \frac{i a^{8} e^{8 i c} e^{8 i d x}}{8 d} & \text{for}\: 8 d \neq 0 \\a^{8} x e^{8 i c} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**8*(a+I*a*tan(d*x+c))**8,x)

[Out]

Piecewise((-I*a**8*exp(8*I*c)*exp(8*I*d*x)/(8*d), Ne(8*d, 0)), (a**8*x*exp(8*I*c), True))

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Giac [B]  time = 2.31477, size = 514, normalized size = 11.95 \begin{align*} \frac{-3840 i \, a^{8} e^{\left (36 i \, d x + 22 i \, c\right )} - 53760 i \, a^{8} e^{\left (34 i \, d x + 20 i \, c\right )} - 349440 i \, a^{8} e^{\left (32 i \, d x + 18 i \, c\right )} - 1397760 i \, a^{8} e^{\left (30 i \, d x + 16 i \, c\right )} - 3843840 i \, a^{8} e^{\left (28 i \, d x + 14 i \, c\right )} - 7687680 i \, a^{8} e^{\left (26 i \, d x + 12 i \, c\right )} - 11531520 i \, a^{8} e^{\left (24 i \, d x + 10 i \, c\right )} - 13178880 i \, a^{8} e^{\left (22 i \, d x + 8 i \, c\right )} - 11531520 i \, a^{8} e^{\left (20 i \, d x + 6 i \, c\right )} - 7687680 i \, a^{8} e^{\left (18 i \, d x + 4 i \, c\right )} - 3843840 i \, a^{8} e^{\left (16 i \, d x + 2 i \, c\right )} - 349440 i \, a^{8} e^{\left (12 i \, d x - 2 i \, c\right )} - 53760 i \, a^{8} e^{\left (10 i \, d x - 4 i \, c\right )} - 3840 i \, a^{8} e^{\left (8 i \, d x - 6 i \, c\right )} - 1397760 i \, a^{8} e^{\left (14 i \, d x\right )}}{30720 \,{\left (d e^{\left (28 i \, d x + 14 i \, c\right )} + 14 \, d e^{\left (26 i \, d x + 12 i \, c\right )} + 91 \, d e^{\left (24 i \, d x + 10 i \, c\right )} + 364 \, d e^{\left (22 i \, d x + 8 i \, c\right )} + 1001 \, d e^{\left (20 i \, d x + 6 i \, c\right )} + 2002 \, d e^{\left (18 i \, d x + 4 i \, c\right )} + 3003 \, d e^{\left (16 i \, d x + 2 i \, c\right )} + 3003 \, d e^{\left (12 i \, d x - 2 i \, c\right )} + 2002 \, d e^{\left (10 i \, d x - 4 i \, c\right )} + 1001 \, d e^{\left (8 i \, d x - 6 i \, c\right )} + 364 \, d e^{\left (6 i \, d x - 8 i \, c\right )} + 91 \, d e^{\left (4 i \, d x - 10 i \, c\right )} + 14 \, d e^{\left (2 i \, d x - 12 i \, c\right )} + 3432 \, d e^{\left (14 i \, d x\right )} + d e^{\left (-14 i \, c\right )}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*(a+I*a*tan(d*x+c))^8,x, algorithm="giac")

[Out]

1/30720*(-3840*I*a^8*e^(36*I*d*x + 22*I*c) - 53760*I*a^8*e^(34*I*d*x + 20*I*c) - 349440*I*a^8*e^(32*I*d*x + 18
*I*c) - 1397760*I*a^8*e^(30*I*d*x + 16*I*c) - 3843840*I*a^8*e^(28*I*d*x + 14*I*c) - 7687680*I*a^8*e^(26*I*d*x
+ 12*I*c) - 11531520*I*a^8*e^(24*I*d*x + 10*I*c) - 13178880*I*a^8*e^(22*I*d*x + 8*I*c) - 11531520*I*a^8*e^(20*
I*d*x + 6*I*c) - 7687680*I*a^8*e^(18*I*d*x + 4*I*c) - 3843840*I*a^8*e^(16*I*d*x + 2*I*c) - 349440*I*a^8*e^(12*
I*d*x - 2*I*c) - 53760*I*a^8*e^(10*I*d*x - 4*I*c) - 3840*I*a^8*e^(8*I*d*x - 6*I*c) - 1397760*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))