3.83 \(\int \cos ^4(c+d x) (a+i a \tan (c+d x))^8 \, dx\)

Optimal. Leaf size=124 \[ \frac{a^8 \tan ^3(c+d x)}{3 d}-\frac{4 i a^8 \tan ^2(c+d x)}{d}-\frac{16 i a^{10}}{d (a-i a \tan (c+d x))^2}+\frac{80 i a^9}{d (a-i a \tan (c+d x))}-\frac{31 a^8 \tan (c+d x)}{d}-\frac{80 i a^8 \log (\cos (c+d x))}{d}+80 a^8 x \]

[Out]

80*a^8*x - ((80*I)*a^8*Log[Cos[c + d*x]])/d - (31*a^8*Tan[c + d*x])/d - ((4*I)*a^8*Tan[c + d*x]^2)/d + (a^8*Ta
n[c + d*x]^3)/(3*d) - ((16*I)*a^10)/(d*(a - I*a*Tan[c + d*x])^2) + ((80*I)*a^9)/(d*(a - I*a*Tan[c + d*x]))

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Rubi [A]  time = 0.077228, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {3487, 43} \[ \frac{a^8 \tan ^3(c+d x)}{3 d}-\frac{4 i a^8 \tan ^2(c+d x)}{d}-\frac{16 i a^{10}}{d (a-i a \tan (c+d x))^2}+\frac{80 i a^9}{d (a-i a \tan (c+d x))}-\frac{31 a^8 \tan (c+d x)}{d}-\frac{80 i a^8 \log (\cos (c+d x))}{d}+80 a^8 x \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^4*(a + I*a*Tan[c + d*x])^8,x]

[Out]

80*a^8*x - ((80*I)*a^8*Log[Cos[c + d*x]])/d - (31*a^8*Tan[c + d*x])/d - ((4*I)*a^8*Tan[c + d*x]^2)/d + (a^8*Ta
n[c + d*x]^3)/(3*d) - ((16*I)*a^10)/(d*(a - I*a*Tan[c + d*x])^2) + ((80*I)*a^9)/(d*(a - I*a*Tan[c + d*x]))

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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \cos ^4(c+d x) (a+i a \tan (c+d x))^8 \, dx &=-\frac{\left (i a^5\right ) \operatorname{Subst}\left (\int \frac{(a+x)^5}{(a-x)^3} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac{\left (i a^5\right ) \operatorname{Subst}\left (\int \left (-31 a^2+\frac{32 a^5}{(a-x)^3}-\frac{80 a^4}{(a-x)^2}+\frac{80 a^3}{a-x}-8 a x-x^2\right ) \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=80 a^8 x-\frac{80 i a^8 \log (\cos (c+d x))}{d}-\frac{31 a^8 \tan (c+d x)}{d}-\frac{4 i a^8 \tan ^2(c+d x)}{d}+\frac{a^8 \tan ^3(c+d x)}{3 d}-\frac{16 i a^{10}}{d (a-i a \tan (c+d x))^2}+\frac{80 i a^9}{d (a-i a \tan (c+d x))}\\ \end{align*}

Mathematica [B]  time = 2.2309, size = 566, normalized size = 4.56 \[ \frac{a^8 \sec (c) \sec ^3(c+d x) (\cos (2 (c+5 d x))+i \sin (2 (c+5 d x))) \left (-120 i d x \sin (2 c+d x)+87 \sin (2 c+d x)-180 i d x \sin (2 c+3 d x)-96 \sin (2 c+3 d x)-180 i d x \sin (4 c+3 d x)+45 \sin (4 c+3 d x)-60 i d x \sin (4 c+5 d x)-44 \sin (4 c+5 d x)-60 i d x \sin (6 c+5 d x)+3 \sin (6 c+5 d x)+180 d x \cos (2 c+3 d x)-66 i \cos (2 c+3 d x)+180 d x \cos (4 c+3 d x)+75 i \cos (4 c+3 d x)+60 d x \cos (4 c+5 d x)-50 i \cos (4 c+5 d x)+60 d x \cos (6 c+5 d x)-3 i \cos (6 c+5 d x)-90 i \cos (2 c+3 d x) \log \left (\cos ^2(c+d x)\right )+3 \cos (2 c+d x) \left (-40 i \log \left (\cos ^2(c+d x)\right )+80 d x+71 i\right )+\cos (d x) \left (-120 i \log \left (\cos ^2(c+d x)\right )+240 d x+119 i\right )-90 i \cos (4 c+3 d x) \log \left (\cos ^2(c+d x)\right )-30 i \cos (4 c+5 d x) \log \left (\cos ^2(c+d x)\right )-30 i \cos (6 c+5 d x) \log \left (\cos ^2(c+d x)\right )-60 \sin (d x) \log \left (\cos ^2(c+d x)\right )-60 \sin (2 c+d x) \log \left (\cos ^2(c+d x)\right )-90 \sin (2 c+3 d x) \log \left (\cos ^2(c+d x)\right )-90 \sin (4 c+3 d x) \log \left (\cos ^2(c+d x)\right )-30 \sin (4 c+5 d x) \log \left (\cos ^2(c+d x)\right )-30 \sin (6 c+5 d x) \log \left (\cos ^2(c+d x)\right )-120 i d x \sin (d x)-101 \sin (d x)\right )}{12 d (\cos (d x)+i \sin (d x))^8} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^4*(a + I*a*Tan[c + d*x])^8,x]

[Out]

(a^8*Sec[c]*Sec[c + d*x]^3*(Cos[2*(c + 5*d*x)] + I*Sin[2*(c + 5*d*x)])*((-66*I)*Cos[2*c + 3*d*x] + 180*d*x*Cos
[2*c + 3*d*x] + (75*I)*Cos[4*c + 3*d*x] + 180*d*x*Cos[4*c + 3*d*x] - (50*I)*Cos[4*c + 5*d*x] + 60*d*x*Cos[4*c
+ 5*d*x] - (3*I)*Cos[6*c + 5*d*x] + 60*d*x*Cos[6*c + 5*d*x] + 3*Cos[2*c + d*x]*(71*I + 80*d*x - (40*I)*Log[Cos
[c + d*x]^2]) + Cos[d*x]*(119*I + 240*d*x - (120*I)*Log[Cos[c + d*x]^2]) - (90*I)*Cos[2*c + 3*d*x]*Log[Cos[c +
d*x]^2] - (90*I)*Cos[4*c + 3*d*x]*Log[Cos[c + d*x]^2] - (30*I)*Cos[4*c + 5*d*x]*Log[Cos[c + d*x]^2] - (30*I)*
Cos[6*c + 5*d*x]*Log[Cos[c + d*x]^2] - 101*Sin[d*x] - (120*I)*d*x*Sin[d*x] - 60*Log[Cos[c + d*x]^2]*Sin[d*x] +
87*Sin[2*c + d*x] - (120*I)*d*x*Sin[2*c + d*x] - 60*Log[Cos[c + d*x]^2]*Sin[2*c + d*x] - 96*Sin[2*c + 3*d*x]
- (180*I)*d*x*Sin[2*c + 3*d*x] - 90*Log[Cos[c + d*x]^2]*Sin[2*c + 3*d*x] + 45*Sin[4*c + 3*d*x] - (180*I)*d*x*S
in[4*c + 3*d*x] - 90*Log[Cos[c + d*x]^2]*Sin[4*c + 3*d*x] - 44*Sin[4*c + 5*d*x] - (60*I)*d*x*Sin[4*c + 5*d*x]
- 30*Log[Cos[c + d*x]^2]*Sin[4*c + 5*d*x] + 3*Sin[6*c + 5*d*x] - (60*I)*d*x*Sin[6*c + 5*d*x] - 30*Log[Cos[c +
d*x]^2]*Sin[6*c + 5*d*x]))/(12*d*(Cos[d*x] + I*Sin[d*x])^8)

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Maple [B]  time = 0.075, size = 306, normalized size = 2.5 \begin{align*} -2\,{\frac{{a}^{8}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{d}}-{\frac{40\,i{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{d}}-{\frac{4\,i{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{d}}-{\frac{4\,i{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{29\,{a}^{8} \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) }{4\,d}}-28\,{\frac{{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{d\cos \left ( dx+c \right ) }}-{\frac{2\,i{a}^{8} \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{d}}-{\frac{34\,i{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{d}}+80\,{a}^{8}x+{\frac{{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{9}}{3\,d \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}-2\,{\frac{{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{9}}{d\cos \left ( dx+c \right ) }}-{\frac{91\,{a}^{8}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{3\,d}}-{\frac{665\,{a}^{8}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{12\,d}}-{\frac{345\,{a}^{8}\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) }{4\,d}}+80\,{\frac{{a}^{8}c}{d}}-{\frac{80\,i{a}^{8}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4*(a+I*a*tan(d*x+c))^8,x)

[Out]

-2/d*a^8*cos(d*x+c)*sin(d*x+c)^7-40*I/d*a^8*sin(d*x+c)^2-4*I/d*a^8*sin(d*x+c)^6-4*I/d*a^8*sin(d*x+c)^8/cos(d*x
+c)^2+29/4/d*a^8*cos(d*x+c)^3*sin(d*x+c)-28/d*a^8*sin(d*x+c)^7/cos(d*x+c)-2*I/d*a^8*cos(d*x+c)^4-34*I/d*a^8*si
n(d*x+c)^4+80*a^8*x+1/3/d*a^8*sin(d*x+c)^9/cos(d*x+c)^3-2/d*a^8*sin(d*x+c)^9/cos(d*x+c)-91/3/d*a^8*cos(d*x+c)*
sin(d*x+c)^5-665/12/d*a^8*cos(d*x+c)*sin(d*x+c)^3-345/4/d*a^8*sin(d*x+c)*cos(d*x+c)+80/d*a^8*c-80*I*a^8*ln(cos
(d*x+c))/d

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Maxima [A]  time = 1.57149, size = 184, normalized size = 1.48 \begin{align*} \frac{8 \, a^{8} \tan \left (d x + c\right )^{3} - 96 i \, a^{8} \tan \left (d x + c\right )^{2} + 1920 \,{\left (d x + c\right )} a^{8} + 960 i \, a^{8} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 744 \, a^{8} \tan \left (d x + c\right ) - \frac{3 \,{\left (640 \, a^{8} \tan \left (d x + c\right )^{3} - 768 i \, a^{8} \tan \left (d x + c\right )^{2} + 384 \, a^{8} \tan \left (d x + c\right ) - 512 i \, a^{8}\right )}}{\tan \left (d x + c\right )^{4} + 2 \, \tan \left (d x + c\right )^{2} + 1}}{24 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*(a+I*a*tan(d*x+c))^8,x, algorithm="maxima")

[Out]

1/24*(8*a^8*tan(d*x + c)^3 - 96*I*a^8*tan(d*x + c)^2 + 1920*(d*x + c)*a^8 + 960*I*a^8*log(tan(d*x + c)^2 + 1)
- 744*a^8*tan(d*x + c) - 3*(640*a^8*tan(d*x + c)^3 - 768*I*a^8*tan(d*x + c)^2 + 384*a^8*tan(d*x + c) - 512*I*a
^8)/(tan(d*x + c)^4 + 2*tan(d*x + c)^2 + 1))/d

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Fricas [A]  time = 1.53678, size = 536, normalized size = 4.32 \begin{align*} \frac{-12 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} + 60 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} + 252 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} + 36 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} - 324 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} - 188 i \, a^{8} +{\left (-240 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} - 720 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} - 720 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} - 240 i \, a^{8}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )}{3 \,{\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*(a+I*a*tan(d*x+c))^8,x, algorithm="fricas")

[Out]

1/3*(-12*I*a^8*e^(10*I*d*x + 10*I*c) + 60*I*a^8*e^(8*I*d*x + 8*I*c) + 252*I*a^8*e^(6*I*d*x + 6*I*c) + 36*I*a^8
*e^(4*I*d*x + 4*I*c) - 324*I*a^8*e^(2*I*d*x + 2*I*c) - 188*I*a^8 + (-240*I*a^8*e^(6*I*d*x + 6*I*c) - 720*I*a^8
*e^(4*I*d*x + 4*I*c) - 720*I*a^8*e^(2*I*d*x + 2*I*c) - 240*I*a^8)*log(e^(2*I*d*x + 2*I*c) + 1))/(d*e^(6*I*d*x
+ 6*I*c) + 3*d*e^(4*I*d*x + 4*I*c) + 3*d*e^(2*I*d*x + 2*I*c) + d)

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Sympy [A]  time = 2.8681, size = 201, normalized size = 1.62 \begin{align*} - 64 a^{8} \left (\begin{cases} - \frac{i e^{2 i d x}}{2 d} & \text{for}\: d \neq 0 \\x & \text{otherwise} \end{cases}\right ) e^{2 i c} + 16 a^{8} \left (\begin{cases} - \frac{i e^{4 i d x}}{4 d} & \text{for}\: d \neq 0 \\x & \text{otherwise} \end{cases}\right ) e^{4 i c} - \frac{80 i a^{8} \log{\left (e^{2 i d x} + e^{- 2 i c} \right )}}{d} + \frac{- \frac{80 i a^{8} e^{- 2 i c} e^{4 i d x}}{d} - \frac{140 i a^{8} e^{- 4 i c} e^{2 i d x}}{d} - \frac{188 i a^{8} e^{- 6 i c}}{3 d}}{e^{6 i d x} + 3 e^{- 2 i c} e^{4 i d x} + 3 e^{- 4 i c} e^{2 i d x} + e^{- 6 i c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**4*(a+I*a*tan(d*x+c))**8,x)

[Out]

-64*a**8*Piecewise((-I*exp(2*I*d*x)/(2*d), Ne(d, 0)), (x, True))*exp(2*I*c) + 16*a**8*Piecewise((-I*exp(4*I*d*
x)/(4*d), Ne(d, 0)), (x, True))*exp(4*I*c) - 80*I*a**8*log(exp(2*I*d*x) + exp(-2*I*c))/d + (-80*I*a**8*exp(-2*
I*c)*exp(4*I*d*x)/d - 140*I*a**8*exp(-4*I*c)*exp(2*I*d*x)/d - 188*I*a**8*exp(-6*I*c)/(3*d))/(exp(6*I*d*x) + 3*
exp(-2*I*c)*exp(4*I*d*x) + 3*exp(-4*I*c)*exp(2*I*d*x) + exp(-6*I*c))

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Giac [B]  time = 2.19591, size = 1060, normalized size = 8.55 \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)^4*(a+I*a*tan(d*x+c))^8,x, algorithm="giac")

[Out]

1/1344*(-107520*I*a^8*e^(28*I*d*x + 14*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 1505280*I*a^8*e^(26*I*d*x + 12*I*c)
*log(e^(2*I*d*x + 2*I*c) + 1) - 9784320*I*a^8*e^(24*I*d*x + 10*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 39137280*I*
a^8*e^(22*I*d*x + 8*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 107627520*I*a^8*e^(20*I*d*x + 6*I*c)*log(e^(2*I*d*x +
2*I*c) + 1) - 215255040*I*a^8*e^(18*I*d*x + 4*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 322882560*I*a^8*e^(16*I*d*x
+ 2*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 322882560*I*a^8*e^(12*I*d*x - 2*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 21
5255040*I*a^8*e^(10*I*d*x - 4*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 107627520*I*a^8*e^(8*I*d*x - 6*I*c)*log(e^(2
*I*d*x + 2*I*c) + 1) - 39137280*I*a^8*e^(6*I*d*x - 8*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 9784320*I*a^8*e^(4*I*
d*x - 10*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 1505280*I*a^8*e^(2*I*d*x - 12*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) -
369008640*I*a^8*e^(14*I*d*x)*log(e^(2*I*d*x + 2*I*c) + 1) - 107520*I*a^8*e^(-14*I*c)*log(e^(2*I*d*x + 2*I*c)
+ 1) - 5376*I*a^8*e^(32*I*d*x + 18*I*c) - 32256*I*a^8*e^(30*I*d*x + 16*I*c) + 112896*I*a^8*e^(28*I*d*x + 14*I*
c) + 1849344*I*a^8*e^(26*I*d*x + 12*I*c) + 8902656*I*a^8*e^(24*I*d*x + 10*I*c) + 24220672*I*a^8*e^(22*I*d*x +
8*I*c) + 40941824*I*a^8*e^(20*I*d*x + 6*I*c) + 39542272*I*a^8*e^(18*I*d*x + 4*I*c) + 5795328*I*a^8*e^(16*I*d*x
+ 2*I*c) - 80602368*I*a^8*e^(12*I*d*x - 2*I*c) - 77650944*I*a^8*e^(10*I*d*x - 4*I*c) - 49588224*I*a^8*e^(8*I*
d*x - 6*I*c) - 21590016*I*a^8*e^(6*I*d*x - 8*I*c) - 6212864*I*a^8*e^(4*I*d*x - 10*I*c) - 1071616*I*a^8*e^(2*I*
d*x - 12*I*c) - 46007808*I*a^8*e^(14*I*d*x) - 84224*I*a^8*e^(-14*I*c))/(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))