\(\int \cos ^8(c+d x) (a+i a \tan (c+d x))^5 \, dx\) [67]

Optimal result

Integrand size = 24, antiderivative size = 27 \[ \int \cos ^8(c+d x) (a+i a \tan (c+d x))^5 \, dx=-\frac {i a^9}{4 d (a-i a \tan (c+d x))^4} \]

[Out]

-1/4*I*a^9/d/(a-I*a*tan(d*x+c))^4

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3568, 32} \[ \int \cos ^8(c+d x) (a+i a \tan (c+d x))^5 \, dx=-\frac {i a^9}{4 d (a-i a \tan (c+d x))^4} \]

[In]

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

[Out]

((-1/4*I)*a^9)/(d*(a - I*a*Tan[c + d*x])^4)

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]

Rule 3568

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (i a^9\right ) \text {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 [A] (verified)

Time = 0.18 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \cos ^8(c+d x) (a+i a \tan (c+d x))^5 \, dx=-\frac {i a^5}{4 d (i+\tan (c+d x))^4} \]

[In]

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

[Out]

((-1/4*I)*a^5)/(d*(I + Tan[c + d*x])^4)

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 300 vs. \(2 (23 ) = 46\).

Time = 1.38 (sec) , antiderivative size = 301, normalized size of antiderivative = 11.15

\[\frac {i a^{5} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{8}-\frac {\left (\cos ^{4}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{12}-\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{24}\right )+5 a^{5} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{8}-\frac {\left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{16}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )-10 i a^{5} \left (-\frac {\left (\cos ^{6}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{8}-\frac {\left (\cos ^{6}\left (d x +c \right )\right )}{24}\right )-10 a^{5} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{7}\left (d x +c \right )\right )}{8}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )-\frac {5 i a^{5} \left (\cos ^{8}\left (d x +c \right )\right )}{8}+a^{5} \left (\frac {\left (\cos ^{7}\left (d x +c \right )+\frac {7 \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {35 \left (\cos ^{3}\left (d x +c \right )\right )}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )}{d}\]

[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*cos(d*x+c)^4*sin(d*x+c)^2-1/24*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*cos(d*x+c)^6*sin(d*x+c)^2-1/24*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))

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (21) = 42\).

Time = 0.24 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.30 \[ \int \cos ^8(c+d x) (a+i a \tan (c+d x))^5 \, dx=\frac {-i \, a^{5} e^{\left (8 i \, d x + 8 i \, c\right )} - 4 i \, a^{5} e^{\left (6 i \, d x + 6 i \, c\right )} - 6 i \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} - 4 i \, a^{5} e^{\left (2 i \, d x + 2 i \, c\right )}}{64 \, d} \]

[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

Sympy [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 162 vs. \(2 (22) = 44\).

Time = 0.31 (sec) , antiderivative size = 162, normalized size of antiderivative = 6.00 \[ \int \cos ^8(c+d x) (a+i a \tan (c+d x))^5 \, dx=\begin {cases} \frac {- 8192 i a^{5} d^{3} e^{8 i c} e^{8 i d x} - 32768 i a^{5} d^{3} e^{6 i c} e^{6 i d x} - 49152 i a^{5} d^{3} e^{4 i c} e^{4 i d x} - 32768 i a^{5} d^{3} e^{2 i c} e^{2 i d x}}{524288 d^{4}} & \text {for}\: d^{4} \neq 0 \\x \left (\frac {a^{5} e^{8 i c}}{8} + \frac {3 a^{5} e^{6 i c}}{8} + \frac {3 a^{5} e^{4 i c}}{8} + \frac {a^{5} e^{2 i c}}{8}\right ) & \text {otherwise} \end {cases} \]

[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(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))

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (21) = 42\).

Time = 0.59 (sec) , antiderivative size = 103, normalized size of antiderivative = 3.81 \[ \int \cos ^8(c+d x) (a+i a \tan (c+d x))^5 \, dx=-\frac {i \, a^{5} \tan \left (d x + c\right )^{4} + 4 \, a^{5} \tan \left (d x + c\right )^{3} - 6 i \, a^{5} \tan \left (d x + c\right )^{2} - 4 \, a^{5} \tan \left (d x + c\right ) + i \, a^{5}}{4 \, {\left (\tan \left (d x + c\right )^{8} + 4 \, \tan \left (d x + c\right )^{6} + 6 \, \tan \left (d x + c\right )^{4} + 4 \, \tan \left (d x + c\right )^{2} + 1\right )} d} \]

[In]

integrate(cos(d*x+c)^8*(a+I*a*tan(d*x+c))^5,x, algorithm="maxima")

[Out]

-1/4*(I*a^5*tan(d*x + c)^4 + 4*a^5*tan(d*x + c)^3 - 6*I*a^5*tan(d*x + c)^2 - 4*a^5*tan(d*x + c) + 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)

Giac [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 267 vs. \(2 (21) = 42\).

Time = 0.78 (sec) , antiderivative size = 267, normalized size of antiderivative = 9.89 \[ \int \cos ^8(c+d x) (a+i a \tan (c+d x))^5 \, dx=-\frac {i \, a^{5} e^{\left (24 i \, d x + 16 i \, c\right )} + 12 i \, a^{5} e^{\left (22 i \, d x + 14 i \, c\right )} + 66 i \, a^{5} e^{\left (20 i \, d x + 12 i \, c\right )} + 220 i \, a^{5} e^{\left (18 i \, d x + 10 i \, c\right )} + 494 i \, a^{5} e^{\left (16 i \, d x + 8 i \, c\right )} + 784 i \, a^{5} e^{\left (14 i \, d x + 6 i \, c\right )} + 896 i \, a^{5} e^{\left (12 i \, d x + 4 i \, c\right )} + 736 i \, a^{5} e^{\left (10 i \, d x + 2 i \, c\right )} + 164 i \, a^{5} e^{\left (6 i \, d x - 2 i \, c\right )} + 38 i \, a^{5} e^{\left (4 i \, d x - 4 i \, c\right )} + 4 i \, a^{5} e^{\left (2 i \, d x - 6 i \, c\right )} + 425 i \, a^{5} e^{\left (8 i \, d x\right )}}{64 \, {\left (d e^{\left (16 i \, d x + 8 i \, c\right )} + 8 \, d e^{\left (14 i \, d x + 6 i \, c\right )} + 28 \, d e^{\left (12 i \, d x + 4 i \, c\right )} + 56 \, d e^{\left (10 i \, d x + 2 i \, c\right )} + 56 \, d e^{\left (6 i \, d x - 2 i \, c\right )} + 28 \, d e^{\left (4 i \, d x - 4 i \, c\right )} + 8 \, d e^{\left (2 i \, d x - 6 i \, c\right )} + 70 \, d e^{\left (8 i \, d x\right )} + d e^{\left (-8 i \, c\right )}\right )}} \]

[In]

integrate(cos(d*x+c)^8*(a+I*a*tan(d*x+c))^5,x, algorithm="giac")

[Out]

-1/64*(I*a^5*e^(24*I*d*x + 16*I*c) + 12*I*a^5*e^(22*I*d*x + 14*I*c) + 66*I*a^5*e^(20*I*d*x + 12*I*c) + 220*I*a
^5*e^(18*I*d*x + 10*I*c) + 494*I*a^5*e^(16*I*d*x + 8*I*c) + 784*I*a^5*e^(14*I*d*x + 6*I*c) + 896*I*a^5*e^(12*I
*d*x + 4*I*c) + 736*I*a^5*e^(10*I*d*x + 2*I*c) + 164*I*a^5*e^(6*I*d*x - 2*I*c) + 38*I*a^5*e^(4*I*d*x - 4*I*c)
+ 4*I*a^5*e^(2*I*d*x - 6*I*c) + 425*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))

Mupad [B] (verification not implemented)

Time = 3.74 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.33 \[ \int \cos ^8(c+d x) (a+i a \tan (c+d x))^5 \, dx=-\frac {\frac {a^5\,{\cos \left (c+d\,x\right )}^4\,1{}\mathrm {i}}{4}+a^5\,{\cos \left (c+d\,x\right )}^6\,\left (\mathrm {tan}\left (c+d\,x\right )-2{}\mathrm {i}\right )-2\,a^5\,{\cos \left (c+d\,x\right )}^8\,\left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{d} \]

[In]

int(cos(c + d*x)^8*(a + a*tan(c + d*x)*1i)^5,x)

[Out]

-((a^5*cos(c + d*x)^4*1i)/4 + a^5*cos(c + d*x)^6*(tan(c + d*x) - 2i) - 2*a^5*cos(c + d*x)^8*(tan(c + d*x) - 1i
))/d