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

Optimal. Leaf size=111 \[ -\frac{i a \cos ^8(c+d x)}{8 d}+\frac{a \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac{7 a \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{35 a \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{35 a \sin (c+d x) \cos (c+d x)}{128 d}+\frac{35 a x}{128} \]

[Out]

(35*a*x)/128 - ((I/8)*a*Cos[c + d*x]^8)/d + (35*a*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (35*a*Cos[c + d*x]^3*Si
n[c + d*x])/(192*d) + (7*a*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) + (a*Cos[c + d*x]^7*Sin[c + d*x])/(8*d)

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Rubi [A]  time = 0.067046, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {3486, 2635, 8} \[ -\frac{i a \cos ^8(c+d x)}{8 d}+\frac{a \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac{7 a \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{35 a \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{35 a \sin (c+d x) \cos (c+d x)}{128 d}+\frac{35 a x}{128} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(35*a*x)/128 - ((I/8)*a*Cos[c + d*x]^8)/d + (35*a*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (35*a*Cos[c + d*x]^3*Si
n[c + d*x])/(192*d) + (7*a*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) + (a*Cos[c + d*x]^7*Sin[c + d*x])/(8*d)

Rule 3486

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*(d*Sec[
e + f*x])^m)/(f*m), x] + Dist[a, Int[(d*Sec[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, m}, x] && (IntegerQ[2
*m] || NeQ[a^2 + b^2, 0])

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \cos ^8(c+d x) (a+i a \tan (c+d x)) \, dx &=-\frac{i a \cos ^8(c+d x)}{8 d}+a \int \cos ^8(c+d x) \, dx\\ &=-\frac{i a \cos ^8(c+d x)}{8 d}+\frac{a \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac{1}{8} (7 a) \int \cos ^6(c+d x) \, dx\\ &=-\frac{i a \cos ^8(c+d x)}{8 d}+\frac{7 a \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac{a \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac{1}{48} (35 a) \int \cos ^4(c+d x) \, dx\\ &=-\frac{i a \cos ^8(c+d x)}{8 d}+\frac{35 a \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac{7 a \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac{a \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac{1}{64} (35 a) \int \cos ^2(c+d x) \, dx\\ &=-\frac{i a \cos ^8(c+d x)}{8 d}+\frac{35 a \cos (c+d x) \sin (c+d x)}{128 d}+\frac{35 a \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac{7 a \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac{a \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac{1}{128} (35 a) \int 1 \, dx\\ &=\frac{35 a x}{128}-\frac{i a \cos ^8(c+d x)}{8 d}+\frac{35 a \cos (c+d x) \sin (c+d x)}{128 d}+\frac{35 a \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac{7 a \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac{a \cos ^7(c+d x) \sin (c+d x)}{8 d}\\ \end{align*}

Mathematica [A]  time = 0.111135, size = 68, normalized size = 0.61 \[ \frac{a \left (672 \sin (2 (c+d x))+168 \sin (4 (c+d x))+32 \sin (6 (c+d x))+3 \sin (8 (c+d x))-384 i \cos ^8(c+d x)+840 c+840 d x\right )}{3072 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(840*c + 840*d*x - (384*I)*Cos[c + d*x]^8 + 672*Sin[2*(c + d*x)] + 168*Sin[4*(c + d*x)] + 32*Sin[6*(c + d*x
)] + 3*Sin[8*(c + d*x)]))/(3072*d)

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Maple [A]  time = 0.094, size = 73, normalized size = 0.7 \begin{align*}{\frac{1}{d} \left ( -{\frac{i}{8}}a \left ( \cos \left ( dx+c \right ) \right ) ^{8}+a \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)),x)

[Out]

1/d*(-1/8*I*a*cos(d*x+c)^8+a*(1/8*(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)+35/128*d*x+35/128*c))

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Maxima [A]  time = 1.6621, size = 139, normalized size = 1.25 \begin{align*} \frac{105 \,{\left (d x + c\right )} a + \frac{105 \, a \tan \left (d x + c\right )^{7} + 385 \, a \tan \left (d x + c\right )^{5} + 511 \, a \tan \left (d x + c\right )^{3} + 279 \, a \tan \left (d x + c\right ) - 48 i \, a}{\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}}{384 \, 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)),x, algorithm="maxima")

[Out]

1/384*(105*(d*x + c)*a + (105*a*tan(d*x + c)^7 + 385*a*tan(d*x + c)^5 + 511*a*tan(d*x + c)^3 + 279*a*tan(d*x +
c) - 48*I*a)/(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 = 1.12426, size = 342, normalized size = 3.08 \begin{align*} \frac{{\left (840 \, a d x e^{\left (6 i \, d x + 6 i \, c\right )} - 3 i \, a e^{\left (14 i \, d x + 14 i \, c\right )} - 28 i \, a e^{\left (12 i \, d x + 12 i \, c\right )} - 126 i \, a e^{\left (10 i \, d x + 10 i \, c\right )} - 420 i \, a e^{\left (8 i \, d x + 8 i \, c\right )} + 252 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} + 42 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} + 4 i \, a\right )} e^{\left (-6 i \, d x - 6 i \, c\right )}}{3072 \, 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)),x, algorithm="fricas")

[Out]

1/3072*(840*a*d*x*e^(6*I*d*x + 6*I*c) - 3*I*a*e^(14*I*d*x + 14*I*c) - 28*I*a*e^(12*I*d*x + 12*I*c) - 126*I*a*e
^(10*I*d*x + 10*I*c) - 420*I*a*e^(8*I*d*x + 8*I*c) + 252*I*a*e^(4*I*d*x + 4*I*c) + 42*I*a*e^(2*I*d*x + 2*I*c)
+ 4*I*a)*e^(-6*I*d*x - 6*I*c)/d

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Sympy [A]  time = 1.04046, size = 280, normalized size = 2.52 \begin{align*} \frac{35 a x}{128} + \begin{cases} \frac{\left (- 10133099161583616 i a d^{6} e^{20 i c} e^{8 i d x} - 94575592174780416 i a d^{6} e^{18 i c} e^{6 i d x} - 425590164786511872 i a d^{6} e^{16 i c} e^{4 i d x} - 1418633882621706240 i a d^{6} e^{14 i c} e^{2 i d x} + 851180329573023744 i a d^{6} e^{10 i c} e^{- 2 i d x} + 141863388262170624 i a d^{6} e^{8 i c} e^{- 4 i d x} + 13510798882111488 i a d^{6} e^{6 i c} e^{- 6 i d x}\right ) e^{- 12 i c}}{10376293541461622784 d^{7}} & \text{for}\: 10376293541461622784 d^{7} e^{12 i c} \neq 0 \\x \left (- \frac{35 a}{128} + \frac{\left (a e^{14 i c} + 7 a e^{12 i c} + 21 a e^{10 i c} + 35 a e^{8 i c} + 35 a e^{6 i c} + 21 a e^{4 i c} + 7 a e^{2 i c} + a\right ) e^{- 6 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)**8*(a+I*a*tan(d*x+c)),x)

[Out]

35*a*x/128 + Piecewise(((-10133099161583616*I*a*d**6*exp(20*I*c)*exp(8*I*d*x) - 94575592174780416*I*a*d**6*exp
(18*I*c)*exp(6*I*d*x) - 425590164786511872*I*a*d**6*exp(16*I*c)*exp(4*I*d*x) - 1418633882621706240*I*a*d**6*ex
p(14*I*c)*exp(2*I*d*x) + 851180329573023744*I*a*d**6*exp(10*I*c)*exp(-2*I*d*x) + 141863388262170624*I*a*d**6*e
xp(8*I*c)*exp(-4*I*d*x) + 13510798882111488*I*a*d**6*exp(6*I*c)*exp(-6*I*d*x))*exp(-12*I*c)/(10376293541461622
784*d**7), Ne(10376293541461622784*d**7*exp(12*I*c), 0)), (x*(-35*a/128 + (a*exp(14*I*c) + 7*a*exp(12*I*c) + 2
1*a*exp(10*I*c) + 35*a*exp(8*I*c) + 35*a*exp(6*I*c) + 21*a*exp(4*I*c) + 7*a*exp(2*I*c) + a)*exp(-6*I*c)/128),
True))

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Giac [A]  time = 1.16079, size = 204, normalized size = 1.84 \begin{align*} \frac{{\left (840 \, a d x e^{\left (6 i \, d x + 2 i \, c\right )} + 84 i \, a e^{\left (6 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 84 i \, a e^{\left (6 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x\right )} + e^{\left (-2 i \, c\right )}\right ) - 3 i \, a e^{\left (14 i \, d x + 10 i \, c\right )} - 28 i \, a e^{\left (12 i \, d x + 8 i \, c\right )} - 126 i \, a e^{\left (10 i \, d x + 6 i \, c\right )} - 420 i \, a e^{\left (8 i \, d x + 4 i \, c\right )} + 42 i \, a e^{\left (2 i \, d x - 2 i \, c\right )} + 252 i \, a e^{\left (4 i \, d x\right )} + 4 i \, a e^{\left (-4 i \, c\right )}\right )} e^{\left (-6 i \, d x - 2 i \, c\right )}}{3072 \, 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)),x, algorithm="giac")

[Out]

1/3072*(840*a*d*x*e^(6*I*d*x + 2*I*c) + 84*I*a*e^(6*I*d*x + 2*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 84*I*a*e^(6*
I*d*x + 2*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) - 3*I*a*e^(14*I*d*x + 10*I*c) - 28*I*a*e^(12*I*d*x + 8*I*c) - 126
*I*a*e^(10*I*d*x + 6*I*c) - 420*I*a*e^(8*I*d*x + 4*I*c) + 42*I*a*e^(2*I*d*x - 2*I*c) + 252*I*a*e^(4*I*d*x) + 4
*I*a*e^(-4*I*c))*e^(-6*I*d*x - 2*I*c)/d