3.183 \(\int \frac{\cos (c+d x)}{(a+i a \tan (c+d x))^8} \, dx\)

Optimal. Leaf size=271 \[ -\frac{64 \sin ^3(c+d x)}{12155 a^8 d}+\frac{192 \sin (c+d x)}{12155 a^8 d}+\frac{128 i \cos ^3(c+d x)}{12155 d \left (a^8+i a^8 \tan (c+d x)\right )}+\frac{16 i \cos (c+d x)}{2431 a^2 d \left (a^2+i a^2 \tan (c+d x)\right )^3}+\frac{112 i \cos (c+d x)}{12155 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{168 i \cos (c+d x)}{12155 a^3 d (a+i a \tan (c+d x))^5}+\frac{24 i \cos (c+d x)}{1105 a^2 d (a+i a \tan (c+d x))^6}+\frac{3 i \cos (c+d x)}{85 a d (a+i a \tan (c+d x))^7}+\frac{i \cos (c+d x)}{17 d (a+i a \tan (c+d x))^8} \]

[Out]

(192*Sin[c + d*x])/(12155*a^8*d) - (64*Sin[c + d*x]^3)/(12155*a^8*d) + ((I/17)*Cos[c + d*x])/(d*(a + I*a*Tan[c
+ d*x])^8) + (((3*I)/85)*Cos[c + d*x])/(a*d*(a + I*a*Tan[c + d*x])^7) + (((24*I)/1105)*Cos[c + d*x])/(a^2*d*(
a + I*a*Tan[c + d*x])^6) + (((168*I)/12155)*Cos[c + d*x])/(a^3*d*(a + I*a*Tan[c + d*x])^5) + (((112*I)/12155)*
Cos[c + d*x])/(d*(a^2 + I*a^2*Tan[c + d*x])^4) + (((16*I)/2431)*Cos[c + d*x])/(a^2*d*(a^2 + I*a^2*Tan[c + d*x]
)^3) + (((128*I)/12155)*Cos[c + d*x]^3)/(d*(a^8 + I*a^8*Tan[c + d*x]))

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Rubi [A]  time = 0.312019, antiderivative size = 271, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {3502, 3500, 2633} \[ -\frac{64 \sin ^3(c+d x)}{12155 a^8 d}+\frac{192 \sin (c+d x)}{12155 a^8 d}+\frac{128 i \cos ^3(c+d x)}{12155 d \left (a^8+i a^8 \tan (c+d x)\right )}+\frac{16 i \cos (c+d x)}{2431 a^2 d \left (a^2+i a^2 \tan (c+d x)\right )^3}+\frac{112 i \cos (c+d x)}{12155 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{168 i \cos (c+d x)}{12155 a^3 d (a+i a \tan (c+d x))^5}+\frac{24 i \cos (c+d x)}{1105 a^2 d (a+i a \tan (c+d x))^6}+\frac{3 i \cos (c+d x)}{85 a d (a+i a \tan (c+d x))^7}+\frac{i \cos (c+d x)}{17 d (a+i a \tan (c+d x))^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(192*Sin[c + d*x])/(12155*a^8*d) - (64*Sin[c + d*x]^3)/(12155*a^8*d) + ((I/17)*Cos[c + d*x])/(d*(a + I*a*Tan[c
+ d*x])^8) + (((3*I)/85)*Cos[c + d*x])/(a*d*(a + I*a*Tan[c + d*x])^7) + (((24*I)/1105)*Cos[c + d*x])/(a^2*d*(
a + I*a*Tan[c + d*x])^6) + (((168*I)/12155)*Cos[c + d*x])/(a^3*d*(a + I*a*Tan[c + d*x])^5) + (((112*I)/12155)*
Cos[c + d*x])/(d*(a^2 + I*a^2*Tan[c + d*x])^4) + (((16*I)/2431)*Cos[c + d*x])/(a^2*d*(a^2 + I*a^2*Tan[c + d*x]
)^3) + (((128*I)/12155)*Cos[c + d*x]^3)/(d*(a^8 + I*a^8*Tan[c + d*x]))

Rule 3502

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(a*(d
*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^n)/(b*f*(m + 2*n)), x] + Dist[Simplify[m + n]/(a*(m + 2*n)), Int[(d*Sec[
e + f*x])^m*(a + b*Tan[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, m}, x] && EqQ[a^2 + b^2, 0] && LtQ[n
, 0] && NeQ[m + 2*n, 0] && IntegersQ[2*m, 2*n]

Rule 3500

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(2*d^2
*(d*Sec[e + f*x])^(m - 2)*(a + b*Tan[e + f*x])^(n + 1))/(b*f*(m + 2*n)), x] - Dist[(d^2*(m - 2))/(b^2*(m + 2*n
)), Int[(d*Sec[e + f*x])^(m - 2)*(a + b*Tan[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, d, e, f, m}, x] && EqQ[a
^2 + b^2, 0] && LtQ[n, -1] && ((ILtQ[n/2, 0] && IGtQ[m - 1/2, 0]) || EqQ[n, -2] || IGtQ[m + n, 0] || (Integers
Q[n, m + 1/2] && GtQ[2*m + n + 1, 0])) && IntegerQ[2*m]

Rule 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rubi steps

\begin{align*} \int \frac{\cos (c+d x)}{(a+i a \tan (c+d x))^8} \, dx &=\frac{i \cos (c+d x)}{17 d (a+i a \tan (c+d x))^8}+\frac{9 \int \frac{\cos (c+d x)}{(a+i a \tan (c+d x))^7} \, dx}{17 a}\\ &=\frac{i \cos (c+d x)}{17 d (a+i a \tan (c+d x))^8}+\frac{3 i \cos (c+d x)}{85 a d (a+i a \tan (c+d x))^7}+\frac{24 \int \frac{\cos (c+d x)}{(a+i a \tan (c+d x))^6} \, dx}{85 a^2}\\ &=\frac{i \cos (c+d x)}{17 d (a+i a \tan (c+d x))^8}+\frac{3 i \cos (c+d x)}{85 a d (a+i a \tan (c+d x))^7}+\frac{24 i \cos (c+d x)}{1105 a^2 d (a+i a \tan (c+d x))^6}+\frac{168 \int \frac{\cos (c+d x)}{(a+i a \tan (c+d x))^5} \, dx}{1105 a^3}\\ &=\frac{i \cos (c+d x)}{17 d (a+i a \tan (c+d x))^8}+\frac{3 i \cos (c+d x)}{85 a d (a+i a \tan (c+d x))^7}+\frac{24 i \cos (c+d x)}{1105 a^2 d (a+i a \tan (c+d x))^6}+\frac{168 i \cos (c+d x)}{12155 a^3 d (a+i a \tan (c+d x))^5}+\frac{1008 \int \frac{\cos (c+d x)}{(a+i a \tan (c+d x))^4} \, dx}{12155 a^4}\\ &=\frac{i \cos (c+d x)}{17 d (a+i a \tan (c+d x))^8}+\frac{3 i \cos (c+d x)}{85 a d (a+i a \tan (c+d x))^7}+\frac{24 i \cos (c+d x)}{1105 a^2 d (a+i a \tan (c+d x))^6}+\frac{168 i \cos (c+d x)}{12155 a^3 d (a+i a \tan (c+d x))^5}+\frac{112 i \cos (c+d x)}{12155 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{112 \int \frac{\cos (c+d x)}{(a+i a \tan (c+d x))^3} \, dx}{2431 a^5}\\ &=\frac{i \cos (c+d x)}{17 d (a+i a \tan (c+d x))^8}+\frac{3 i \cos (c+d x)}{85 a d (a+i a \tan (c+d x))^7}+\frac{24 i \cos (c+d x)}{1105 a^2 d (a+i a \tan (c+d x))^6}+\frac{168 i \cos (c+d x)}{12155 a^3 d (a+i a \tan (c+d x))^5}+\frac{16 i \cos (c+d x)}{2431 a^5 d (a+i a \tan (c+d x))^3}+\frac{112 i \cos (c+d x)}{12155 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{64 \int \frac{\cos (c+d x)}{(a+i a \tan (c+d x))^2} \, dx}{2431 a^6}\\ &=\frac{i \cos (c+d x)}{17 d (a+i a \tan (c+d x))^8}+\frac{3 i \cos (c+d x)}{85 a d (a+i a \tan (c+d x))^7}+\frac{24 i \cos (c+d x)}{1105 a^2 d (a+i a \tan (c+d x))^6}+\frac{168 i \cos (c+d x)}{12155 a^3 d (a+i a \tan (c+d x))^5}+\frac{16 i \cos (c+d x)}{2431 a^5 d (a+i a \tan (c+d x))^3}+\frac{112 i \cos (c+d x)}{12155 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{128 i \cos ^3(c+d x)}{12155 d \left (a^8+i a^8 \tan (c+d x)\right )}+\frac{192 \int \cos ^3(c+d x) \, dx}{12155 a^8}\\ &=\frac{i \cos (c+d x)}{17 d (a+i a \tan (c+d x))^8}+\frac{3 i \cos (c+d x)}{85 a d (a+i a \tan (c+d x))^7}+\frac{24 i \cos (c+d x)}{1105 a^2 d (a+i a \tan (c+d x))^6}+\frac{168 i \cos (c+d x)}{12155 a^3 d (a+i a \tan (c+d x))^5}+\frac{16 i \cos (c+d x)}{2431 a^5 d (a+i a \tan (c+d x))^3}+\frac{112 i \cos (c+d x)}{12155 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{128 i \cos ^3(c+d x)}{12155 d \left (a^8+i a^8 \tan (c+d x)\right )}-\frac{192 \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{12155 a^8 d}\\ &=\frac{192 \sin (c+d x)}{12155 a^8 d}-\frac{64 \sin ^3(c+d x)}{12155 a^8 d}+\frac{i \cos (c+d x)}{17 d (a+i a \tan (c+d x))^8}+\frac{3 i \cos (c+d x)}{85 a d (a+i a \tan (c+d x))^7}+\frac{24 i \cos (c+d x)}{1105 a^2 d (a+i a \tan (c+d x))^6}+\frac{168 i \cos (c+d x)}{12155 a^3 d (a+i a \tan (c+d x))^5}+\frac{16 i \cos (c+d x)}{2431 a^5 d (a+i a \tan (c+d x))^3}+\frac{112 i \cos (c+d x)}{12155 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{128 i \cos ^3(c+d x)}{12155 d \left (a^8+i a^8 \tan (c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 0.900893, size = 139, normalized size = 0.51 \[ -\frac{i \sec ^8(c+d x) (-24310 i \sin (c+d x)-55692 i \sin (3 (c+d x))-56100 i \sin (5 (c+d x))-51051 i \sin (7 (c+d x))+6435 i \sin (9 (c+d x))-194480 \cos (c+d x)-148512 \cos (3 (c+d x))-89760 \cos (5 (c+d x))-58344 \cos (7 (c+d x))+5720 \cos (9 (c+d x)))}{3111680 a^8 d (\tan (c+d x)-i)^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-I/3111680)*Sec[c + d*x]^8*(-194480*Cos[c + d*x] - 148512*Cos[3*(c + d*x)] - 89760*Cos[5*(c + d*x)] - 58344*
Cos[7*(c + d*x)] + 5720*Cos[9*(c + d*x)] - (24310*I)*Sin[c + d*x] - (55692*I)*Sin[3*(c + d*x)] - (56100*I)*Sin
[5*(c + d*x)] - (51051*I)*Sin[7*(c + d*x)] + (6435*I)*Sin[9*(c + d*x)]))/(a^8*d*(-I + Tan[c + d*x])^8)

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Maple [A]  time = 0.118, size = 306, normalized size = 1.1 \begin{align*} 2\,{\frac{1}{d{a}^{8}} \left ({\frac{{\frac{19109\,i}{5}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{10}}}+{\frac{784\,i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{14}}}+{\frac{{\frac{1793\,i}{256}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{2}}}-{\frac{2692\,i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{12}}}-{\frac{{\frac{7937\,i}{64}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{4}}}-{\frac{64\,i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{16}}}-{\frac{{\frac{10241\,i}{4}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{8}}}+{\frac{{\frac{13313\,i}{16}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{6}}}+{\frac{128}{17\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{17}}}-{\frac{1376}{5\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{15}}}+{\frac{21400}{13\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{13}}}-{\frac{38954}{11\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{11}}}+{\frac{6847}{2\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{9}}}-{\frac{12799}{8\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{7}}}+{\frac{57083}{160\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{5}}}-{\frac{4351}{128\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{3}}}+{\frac{511}{512\,\tan \left ( 1/2\,dx+c/2 \right ) -512\,i}}+{\frac{1}{512\,\tan \left ( 1/2\,dx+c/2 \right ) +512\,i}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d/a^8*(19109/5*I/(tan(1/2*d*x+1/2*c)-I)^10+784*I/(tan(1/2*d*x+1/2*c)-I)^14+1793/256*I/(tan(1/2*d*x+1/2*c)-I)
^2-2692*I/(tan(1/2*d*x+1/2*c)-I)^12-7937/64*I/(tan(1/2*d*x+1/2*c)-I)^4-64*I/(tan(1/2*d*x+1/2*c)-I)^16-10241/4*
I/(tan(1/2*d*x+1/2*c)-I)^8+13313/16*I/(tan(1/2*d*x+1/2*c)-I)^6+128/17/(tan(1/2*d*x+1/2*c)-I)^17-1376/5/(tan(1/
2*d*x+1/2*c)-I)^15+21400/13/(tan(1/2*d*x+1/2*c)-I)^13-38954/11/(tan(1/2*d*x+1/2*c)-I)^11+6847/2/(tan(1/2*d*x+1
/2*c)-I)^9-12799/8/(tan(1/2*d*x+1/2*c)-I)^7+57083/160/(tan(1/2*d*x+1/2*c)-I)^5-4351/128/(tan(1/2*d*x+1/2*c)-I)
^3+511/512/(tan(1/2*d*x+1/2*c)-I)+1/512/(tan(1/2*d*x+1/2*c)+I))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 2.66626, size = 452, normalized size = 1.67 \begin{align*} \frac{{\left (-12155 i \, e^{\left (18 i \, d x + 18 i \, c\right )} + 109395 i \, e^{\left (16 i \, d x + 16 i \, c\right )} + 145860 i \, e^{\left (14 i \, d x + 14 i \, c\right )} + 204204 i \, e^{\left (12 i \, d x + 12 i \, c\right )} + 218790 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 170170 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 92820 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 33660 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 7293 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 715 i\right )} e^{\left (-17 i \, d x - 17 i \, c\right )}}{6223360 \, a^{8} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6223360*(-12155*I*e^(18*I*d*x + 18*I*c) + 109395*I*e^(16*I*d*x + 16*I*c) + 145860*I*e^(14*I*d*x + 14*I*c) +
204204*I*e^(12*I*d*x + 12*I*c) + 218790*I*e^(10*I*d*x + 10*I*c) + 170170*I*e^(8*I*d*x + 8*I*c) + 92820*I*e^(6*
I*d*x + 6*I*c) + 33660*I*e^(4*I*d*x + 4*I*c) + 7293*I*e^(2*I*d*x + 2*I*c) + 715*I)*e^(-17*I*d*x - 17*I*c)/(a^8
*d)

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Sympy [A]  time = 3.49903, size = 369, normalized size = 1.36 \begin{align*} \begin{cases} \frac{\left (- 143500911498201343931187200 i a^{72} d^{9} e^{82 i c} e^{i d x} + 1291508203483812095380684800 i a^{72} d^{9} e^{80 i c} e^{- i d x} + 1722010937978416127174246400 i a^{72} d^{9} e^{78 i c} e^{- 3 i d x} + 2410815313169782578043944960 i a^{72} d^{9} e^{76 i c} e^{- 5 i d x} + 2583016406967624190761369600 i a^{72} d^{9} e^{74 i c} e^{- 7 i d x} + 2009012760974818815036620800 i a^{72} d^{9} e^{72 i c} e^{- 9 i d x} + 1095825142349901171838156800 i a^{72} d^{9} e^{70 i c} e^{- 11 i d x} + 397387139533480644732518400 i a^{72} d^{9} e^{68 i c} e^{- 13 i d x} + 86100546898920806358712320 i a^{72} d^{9} e^{66 i c} e^{- 15 i d x} + 8441230088129490819481600 i a^{72} d^{9} e^{64 i c} e^{- 17 i d x}\right ) e^{- 81 i c}}{73472466687079088092767846400 a^{80} d^{10}} & \text{for}\: 73472466687079088092767846400 a^{80} d^{10} e^{81 i c} \neq 0 \\\frac{x \left (e^{18 i c} + 9 e^{16 i c} + 36 e^{14 i c} + 84 e^{12 i c} + 126 e^{10 i c} + 126 e^{8 i c} + 84 e^{6 i c} + 36 e^{4 i c} + 9 e^{2 i c} + 1\right ) e^{- 17 i c}}{512 a^{8}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise(((-143500911498201343931187200*I*a**72*d**9*exp(82*I*c)*exp(I*d*x) + 1291508203483812095380684800*I*
a**72*d**9*exp(80*I*c)*exp(-I*d*x) + 1722010937978416127174246400*I*a**72*d**9*exp(78*I*c)*exp(-3*I*d*x) + 241
0815313169782578043944960*I*a**72*d**9*exp(76*I*c)*exp(-5*I*d*x) + 2583016406967624190761369600*I*a**72*d**9*e
xp(74*I*c)*exp(-7*I*d*x) + 2009012760974818815036620800*I*a**72*d**9*exp(72*I*c)*exp(-9*I*d*x) + 1095825142349
901171838156800*I*a**72*d**9*exp(70*I*c)*exp(-11*I*d*x) + 397387139533480644732518400*I*a**72*d**9*exp(68*I*c)
*exp(-13*I*d*x) + 86100546898920806358712320*I*a**72*d**9*exp(66*I*c)*exp(-15*I*d*x) + 84412300881294908194816
00*I*a**72*d**9*exp(64*I*c)*exp(-17*I*d*x))*exp(-81*I*c)/(73472466687079088092767846400*a**80*d**10), Ne(73472
466687079088092767846400*a**80*d**10*exp(81*I*c), 0)), (x*(exp(18*I*c) + 9*exp(16*I*c) + 36*exp(14*I*c) + 84*e
xp(12*I*c) + 126*exp(10*I*c) + 126*exp(8*I*c) + 84*exp(6*I*c) + 36*exp(4*I*c) + 9*exp(2*I*c) + 1)*exp(-17*I*c)
/(512*a**8), True))

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Giac [A]  time = 1.17698, size = 336, normalized size = 1.24 \begin{align*} \frac{\frac{12155}{a^{8}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + i\right )}} + \frac{6211205 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{16} - 55791450 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{15} - 303072770 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{14} + 1091397450 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} + 2909561798 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{12} - 5901218466 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} - 9405145178 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} + 11877161010 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 12017308160 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 9710430158 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 6263238566 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 3172666718 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 1247921210 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 365303990 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 77883902 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 10498214 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 982907}{a^{8}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - i\right )}^{17}}}{3111680 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3111680*(12155/(a^8*(tan(1/2*d*x + 1/2*c) + I)) + (6211205*tan(1/2*d*x + 1/2*c)^16 - 55791450*I*tan(1/2*d*x
+ 1/2*c)^15 - 303072770*tan(1/2*d*x + 1/2*c)^14 + 1091397450*I*tan(1/2*d*x + 1/2*c)^13 + 2909561798*tan(1/2*d*
x + 1/2*c)^12 - 5901218466*I*tan(1/2*d*x + 1/2*c)^11 - 9405145178*tan(1/2*d*x + 1/2*c)^10 + 11877161010*I*tan(
1/2*d*x + 1/2*c)^9 + 12017308160*tan(1/2*d*x + 1/2*c)^8 - 9710430158*I*tan(1/2*d*x + 1/2*c)^7 - 6263238566*tan
(1/2*d*x + 1/2*c)^6 + 3172666718*I*tan(1/2*d*x + 1/2*c)^5 + 1247921210*tan(1/2*d*x + 1/2*c)^4 - 365303990*I*ta
n(1/2*d*x + 1/2*c)^3 - 77883902*tan(1/2*d*x + 1/2*c)^2 + 10498214*I*tan(1/2*d*x + 1/2*c) + 982907)/(a^8*(tan(1
/2*d*x + 1/2*c) - I)^17))/d