3.67 \(\int \cos ^8(c+d x) (a+i a \tan (c+d x))^5 \, dx\)

Optimal. Leaf size=27 \[ -\frac{i a^9}{4 d (a-i a \tan (c+d x))^4} \]

[Out]

((-I/4)*a^9)/(d*(a - I*a*Tan[c + d*x])^4)

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Rubi [A]  time = 0.0378692, 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} \[ -\frac{i a^9}{4 d (a-i a \tan (c+d x))^4} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^8*(a + I*a*Tan[c + d*x])^5,x]

[Out]

((-I/4)*a^9)/(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 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 ^8(c+d x) (a+i a \tan (c+d x))^5 \, dx &=-\frac{\left (i a^9\right ) \operatorname{Subst}\left (\int \frac{1}{(a-x)^5} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac{i a^9}{4 d (a-i a \tan (c+d x))^4}\\ \end{align*}

Mathematica [B]  time = 0.932438, size = 73, normalized size = 2.7 \[ \frac{a^5 (-i (2 \sin (c+d x)+3 \sin (3 (c+d x)))+10 \cos (c+d x)+5 \cos (3 (c+d x))) (\sin (5 (c+d x))-i \cos (5 (c+d x)))}{64 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^8*(a + I*a*Tan[c + d*x])^5,x]

[Out]

(a^5*(10*Cos[c + d*x] + 5*Cos[3*(c + d*x)] - I*(2*Sin[c + d*x] + 3*Sin[3*(c + d*x)]))*((-I)*Cos[5*(c + d*x)] +
Sin[5*(c + d*x)]))/(64*d)

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Maple [B]  time = 0.091, size = 301, normalized size = 11.2 \begin{align*}{\frac{1}{d} \left ( i{a}^{5} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{8}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{12}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{24}} \right ) +5\,{a}^{5} \left ( -1/8\, \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}-1/16\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}\sin \left ( dx+c \right ) +{\frac{ \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+3/2\,\cos \left ( dx+c \right ) \right ) \sin \left ( dx+c \right ) }{64}}+{\frac{3\,dx}{128}}+{\frac{3\,c}{128}} \right ) -10\,i{a}^{5} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{8}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{24}} \right ) -10\,{a}^{5} \left ( -1/8\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{7}+1/48\, \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{5}+5/4\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{15\,\cos \left ( dx+c \right ) }{8}} \right ) \sin \left ( dx+c \right ) +{\frac{5\,dx}{128}}+{\frac{5\,c}{128}} \right ) -{\frac{5\,i}{8}}{a}^{5} \left ( \cos \left ( dx+c \right ) \right ) ^{8}+{a}^{5} \left ({\frac{\sin \left ( dx+c \right ) }{8} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{7}+{\frac{7\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{6}}+{\frac{35\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{24}}+{\frac{35\,\cos \left ( dx+c \right ) }{16}} \right ) }+{\frac{35\,dx}{128}}+{\frac{35\,c}{128}} \right ) \right ) } \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))^5,x)

[Out]

1/d*(I*a^5*(-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)+5*a^5*(-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+3/12
8*c)-10*I*a^5*(-1/8*sin(d*x+c)^2*cos(d*x+c)^6-1/24*cos(d*x+c)^6)-10*a^5*(-1/8*sin(d*x+c)*cos(d*x+c)^7+1/48*(co
s(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)-5/8*I*a^5*cos(d*x+c)^8+a^5*(1/8*(co
s(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)+35/128*d*x+35/128*c))

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Maxima [B]  time = 1.64053, size = 139, normalized size = 5.15 \begin{align*} \frac{-96 i \, a^{5} \tan \left (d x + c\right )^{4} - 384 \, a^{5} \tan \left (d x + c\right )^{3} + 576 i \, a^{5} \tan \left (d x + c\right )^{2} + 384 \, a^{5} \tan \left (d x + c\right ) - 96 i \, a^{5}}{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))^5,x, algorithm="maxima")

[Out]

1/384*(-96*I*a^5*tan(d*x + c)^4 - 384*a^5*tan(d*x + c)^3 + 576*I*a^5*tan(d*x + c)^2 + 384*a^5*tan(d*x + c) - 9
6*I*a^5)/((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 [B]  time = 1.16028, size = 171, normalized size = 6.33 \begin{align*} \frac{-i \, a^{5} e^{\left (8 i \, d x + 8 i \, c\right )} - 4 i \, a^{5} e^{\left (6 i \, d x + 6 i \, c\right )} - 6 i \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} - 4 i \, a^{5} e^{\left (2 i \, d x + 2 i \, c\right )}}{64 \, 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))^5,x, algorithm="fricas")

[Out]

1/64*(-I*a^5*e^(8*I*d*x + 8*I*c) - 4*I*a^5*e^(6*I*d*x + 6*I*c) - 6*I*a^5*e^(4*I*d*x + 4*I*c) - 4*I*a^5*e^(2*I*
d*x + 2*I*c))/d

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Sympy [B]  time = 0.830825, size = 163, normalized size = 6.04 \begin{align*} \begin{cases} \frac{- 8192 i a^{5} d^{3} e^{8 i c} e^{8 i d x} - 32768 i a^{5} d^{3} e^{6 i c} e^{6 i d x} - 49152 i a^{5} d^{3} e^{4 i c} e^{4 i d x} - 32768 i a^{5} d^{3} e^{2 i c} e^{2 i d x}}{524288 d^{4}} & \text{for}\: 524288 d^{4} \neq 0 \\x \left (\frac{a^{5} e^{8 i c}}{8} + \frac{3 a^{5} e^{6 i c}}{8} + \frac{3 a^{5} e^{4 i c}}{8} + \frac{a^{5} e^{2 i c}}{8}\right ) & \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))**5,x)

[Out]

Piecewise(((-8192*I*a**5*d**3*exp(8*I*c)*exp(8*I*d*x) - 32768*I*a**5*d**3*exp(6*I*c)*exp(6*I*d*x) - 49152*I*a*
*5*d**3*exp(4*I*c)*exp(4*I*d*x) - 32768*I*a**5*d**3*exp(2*I*c)*exp(2*I*d*x))/(524288*d**4), Ne(524288*d**4, 0)
), (x*(a**5*exp(8*I*c)/8 + 3*a**5*exp(6*I*c)/8 + 3*a**5*exp(4*I*c)/8 + a**5*exp(2*I*c)/8), True))

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Giac [B]  time = 1.61835, size = 360, normalized size = 13.33 \begin{align*} \frac{-24 i \, a^{5} e^{\left (24 i \, d x + 16 i \, c\right )} - 288 i \, a^{5} e^{\left (22 i \, d x + 14 i \, c\right )} - 1584 i \, a^{5} e^{\left (20 i \, d x + 12 i \, c\right )} - 5280 i \, a^{5} e^{\left (18 i \, d x + 10 i \, c\right )} - 11856 i \, a^{5} e^{\left (16 i \, d x + 8 i \, c\right )} - 18816 i \, a^{5} e^{\left (14 i \, d x + 6 i \, c\right )} - 21504 i \, a^{5} e^{\left (12 i \, d x + 4 i \, c\right )} - 17664 i \, a^{5} e^{\left (10 i \, d x + 2 i \, c\right )} - 3936 i \, a^{5} e^{\left (6 i \, d x - 2 i \, c\right )} - 912 i \, a^{5} e^{\left (4 i \, d x - 4 i \, c\right )} - 96 i \, a^{5} e^{\left (2 i \, d x - 6 i \, c\right )} - 10200 i \, a^{5} e^{\left (8 i \, d x\right )}}{1536 \,{\left (d e^{\left (16 i \, d x + 8 i \, c\right )} + 8 \, d e^{\left (14 i \, d x + 6 i \, c\right )} + 28 \, d e^{\left (12 i \, d x + 4 i \, c\right )} + 56 \, d e^{\left (10 i \, d x + 2 i \, c\right )} + 56 \, d e^{\left (6 i \, d x - 2 i \, c\right )} + 28 \, d e^{\left (4 i \, d x - 4 i \, c\right )} + 8 \, d e^{\left (2 i \, d x - 6 i \, c\right )} + 70 \, d e^{\left (8 i \, d x\right )} + d e^{\left (-8 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))^5,x, algorithm="giac")

[Out]

1/1536*(-24*I*a^5*e^(24*I*d*x + 16*I*c) - 288*I*a^5*e^(22*I*d*x + 14*I*c) - 1584*I*a^5*e^(20*I*d*x + 12*I*c) -
5280*I*a^5*e^(18*I*d*x + 10*I*c) - 11856*I*a^5*e^(16*I*d*x + 8*I*c) - 18816*I*a^5*e^(14*I*d*x + 6*I*c) - 2150
4*I*a^5*e^(12*I*d*x + 4*I*c) - 17664*I*a^5*e^(10*I*d*x + 2*I*c) - 3936*I*a^5*e^(6*I*d*x - 2*I*c) - 912*I*a^5*e
^(4*I*d*x - 4*I*c) - 96*I*a^5*e^(2*I*d*x - 6*I*c) - 10200*I*a^5*e^(8*I*d*x))/(d*e^(16*I*d*x + 8*I*c) + 8*d*e^(
14*I*d*x + 6*I*c) + 28*d*e^(12*I*d*x + 4*I*c) + 56*d*e^(10*I*d*x + 2*I*c) + 56*d*e^(6*I*d*x - 2*I*c) + 28*d*e^
(4*I*d*x - 4*I*c) + 8*d*e^(2*I*d*x - 6*I*c) + 70*d*e^(8*I*d*x) + d*e^(-8*I*c))