3.184 \(\int \frac{\cos ^3(c+d x)}{(a+i a \tan (c+d x))^8} \, dx\)

Optimal. Leaf size=301 \[ \frac{32 \sin ^5(c+d x)}{4199 a^8 d}-\frac{320 \sin ^3(c+d x)}{12597 a^8 d}+\frac{160 \sin (c+d x)}{4199 a^8 d}+\frac{64 i \cos ^5(c+d x)}{4199 d \left (a^8+i a^8 \tan (c+d x)\right )}+\frac{112 i \cos ^3(c+d x)}{12597 a^2 d \left (a^2+i a^2 \tan (c+d x)\right )^3}+\frac{48 i \cos ^3(c+d x)}{4199 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{66 i \cos ^3(c+d x)}{4199 a^3 d (a+i a \tan (c+d x))^5}+\frac{22 i \cos ^3(c+d x)}{969 a^2 d (a+i a \tan (c+d x))^6}+\frac{11 i \cos ^3(c+d x)}{323 a d (a+i a \tan (c+d x))^7}+\frac{i \cos ^3(c+d x)}{19 d (a+i a \tan (c+d x))^8} \]

[Out]

(160*Sin[c + d*x])/(4199*a^8*d) - (320*Sin[c + d*x]^3)/(12597*a^8*d) + (32*Sin[c + d*x]^5)/(4199*a^8*d) + ((I/
19)*Cos[c + d*x]^3)/(d*(a + I*a*Tan[c + d*x])^8) + (((11*I)/323)*Cos[c + d*x]^3)/(a*d*(a + I*a*Tan[c + d*x])^7
) + (((22*I)/969)*Cos[c + d*x]^3)/(a^2*d*(a + I*a*Tan[c + d*x])^6) + (((66*I)/4199)*Cos[c + d*x]^3)/(a^3*d*(a
+ I*a*Tan[c + d*x])^5) + (((48*I)/4199)*Cos[c + d*x]^3)/(d*(a^2 + I*a^2*Tan[c + d*x])^4) + (((112*I)/12597)*Co
s[c + d*x]^3)/(a^2*d*(a^2 + I*a^2*Tan[c + d*x])^3) + (((64*I)/4199)*Cos[c + d*x]^5)/(d*(a^8 + I*a^8*Tan[c + d*
x]))

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Rubi [A]  time = 0.384034, antiderivative size = 301, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3502, 3500, 2633} \[ \frac{32 \sin ^5(c+d x)}{4199 a^8 d}-\frac{320 \sin ^3(c+d x)}{12597 a^8 d}+\frac{160 \sin (c+d x)}{4199 a^8 d}+\frac{64 i \cos ^5(c+d x)}{4199 d \left (a^8+i a^8 \tan (c+d x)\right )}+\frac{112 i \cos ^3(c+d x)}{12597 a^2 d \left (a^2+i a^2 \tan (c+d x)\right )^3}+\frac{48 i \cos ^3(c+d x)}{4199 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{66 i \cos ^3(c+d x)}{4199 a^3 d (a+i a \tan (c+d x))^5}+\frac{22 i \cos ^3(c+d x)}{969 a^2 d (a+i a \tan (c+d x))^6}+\frac{11 i \cos ^3(c+d x)}{323 a d (a+i a \tan (c+d x))^7}+\frac{i \cos ^3(c+d x)}{19 d (a+i a \tan (c+d x))^8} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^3/(a + I*a*Tan[c + d*x])^8,x]

[Out]

(160*Sin[c + d*x])/(4199*a^8*d) - (320*Sin[c + d*x]^3)/(12597*a^8*d) + (32*Sin[c + d*x]^5)/(4199*a^8*d) + ((I/
19)*Cos[c + d*x]^3)/(d*(a + I*a*Tan[c + d*x])^8) + (((11*I)/323)*Cos[c + d*x]^3)/(a*d*(a + I*a*Tan[c + d*x])^7
) + (((22*I)/969)*Cos[c + d*x]^3)/(a^2*d*(a + I*a*Tan[c + d*x])^6) + (((66*I)/4199)*Cos[c + d*x]^3)/(a^3*d*(a
+ I*a*Tan[c + d*x])^5) + (((48*I)/4199)*Cos[c + d*x]^3)/(d*(a^2 + I*a^2*Tan[c + d*x])^4) + (((112*I)/12597)*Co
s[c + d*x]^3)/(a^2*d*(a^2 + I*a^2*Tan[c + d*x])^3) + (((64*I)/4199)*Cos[c + d*x]^5)/(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 ^3(c+d x)}{(a+i a \tan (c+d x))^8} \, dx &=\frac{i \cos ^3(c+d x)}{19 d (a+i a \tan (c+d x))^8}+\frac{11 \int \frac{\cos ^3(c+d x)}{(a+i a \tan (c+d x))^7} \, dx}{19 a}\\ &=\frac{i \cos ^3(c+d x)}{19 d (a+i a \tan (c+d x))^8}+\frac{11 i \cos ^3(c+d x)}{323 a d (a+i a \tan (c+d x))^7}+\frac{110 \int \frac{\cos ^3(c+d x)}{(a+i a \tan (c+d x))^6} \, dx}{323 a^2}\\ &=\frac{i \cos ^3(c+d x)}{19 d (a+i a \tan (c+d x))^8}+\frac{11 i \cos ^3(c+d x)}{323 a d (a+i a \tan (c+d x))^7}+\frac{22 i \cos ^3(c+d x)}{969 a^2 d (a+i a \tan (c+d x))^6}+\frac{66 \int \frac{\cos ^3(c+d x)}{(a+i a \tan (c+d x))^5} \, dx}{323 a^3}\\ &=\frac{i \cos ^3(c+d x)}{19 d (a+i a \tan (c+d x))^8}+\frac{11 i \cos ^3(c+d x)}{323 a d (a+i a \tan (c+d x))^7}+\frac{22 i \cos ^3(c+d x)}{969 a^2 d (a+i a \tan (c+d x))^6}+\frac{66 i \cos ^3(c+d x)}{4199 a^3 d (a+i a \tan (c+d x))^5}+\frac{528 \int \frac{\cos ^3(c+d x)}{(a+i a \tan (c+d x))^4} \, dx}{4199 a^4}\\ &=\frac{i \cos ^3(c+d x)}{19 d (a+i a \tan (c+d x))^8}+\frac{11 i \cos ^3(c+d x)}{323 a d (a+i a \tan (c+d x))^7}+\frac{22 i \cos ^3(c+d x)}{969 a^2 d (a+i a \tan (c+d x))^6}+\frac{66 i \cos ^3(c+d x)}{4199 a^3 d (a+i a \tan (c+d x))^5}+\frac{48 i \cos ^3(c+d x)}{4199 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{336 \int \frac{\cos ^3(c+d x)}{(a+i a \tan (c+d x))^3} \, dx}{4199 a^5}\\ &=\frac{i \cos ^3(c+d x)}{19 d (a+i a \tan (c+d x))^8}+\frac{11 i \cos ^3(c+d x)}{323 a d (a+i a \tan (c+d x))^7}+\frac{22 i \cos ^3(c+d x)}{969 a^2 d (a+i a \tan (c+d x))^6}+\frac{66 i \cos ^3(c+d x)}{4199 a^3 d (a+i a \tan (c+d x))^5}+\frac{112 i \cos ^3(c+d x)}{12597 a^5 d (a+i a \tan (c+d x))^3}+\frac{48 i \cos ^3(c+d x)}{4199 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{224 \int \frac{\cos ^3(c+d x)}{(a+i a \tan (c+d x))^2} \, dx}{4199 a^6}\\ &=\frac{i \cos ^3(c+d x)}{19 d (a+i a \tan (c+d x))^8}+\frac{11 i \cos ^3(c+d x)}{323 a d (a+i a \tan (c+d x))^7}+\frac{22 i \cos ^3(c+d x)}{969 a^2 d (a+i a \tan (c+d x))^6}+\frac{66 i \cos ^3(c+d x)}{4199 a^3 d (a+i a \tan (c+d x))^5}+\frac{112 i \cos ^3(c+d x)}{12597 a^5 d (a+i a \tan (c+d x))^3}+\frac{48 i \cos ^3(c+d x)}{4199 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{64 i \cos ^5(c+d x)}{4199 d \left (a^8+i a^8 \tan (c+d x)\right )}+\frac{160 \int \cos ^5(c+d x) \, dx}{4199 a^8}\\ &=\frac{i \cos ^3(c+d x)}{19 d (a+i a \tan (c+d x))^8}+\frac{11 i \cos ^3(c+d x)}{323 a d (a+i a \tan (c+d x))^7}+\frac{22 i \cos ^3(c+d x)}{969 a^2 d (a+i a \tan (c+d x))^6}+\frac{66 i \cos ^3(c+d x)}{4199 a^3 d (a+i a \tan (c+d x))^5}+\frac{112 i \cos ^3(c+d x)}{12597 a^5 d (a+i a \tan (c+d x))^3}+\frac{48 i \cos ^3(c+d x)}{4199 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{64 i \cos ^5(c+d x)}{4199 d \left (a^8+i a^8 \tan (c+d x)\right )}-\frac{160 \operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{4199 a^8 d}\\ &=\frac{160 \sin (c+d x)}{4199 a^8 d}-\frac{320 \sin ^3(c+d x)}{12597 a^8 d}+\frac{32 \sin ^5(c+d x)}{4199 a^8 d}+\frac{i \cos ^3(c+d x)}{19 d (a+i a \tan (c+d x))^8}+\frac{11 i \cos ^3(c+d x)}{323 a d (a+i a \tan (c+d x))^7}+\frac{22 i \cos ^3(c+d x)}{969 a^2 d (a+i a \tan (c+d x))^6}+\frac{66 i \cos ^3(c+d x)}{4199 a^3 d (a+i a \tan (c+d x))^5}+\frac{112 i \cos ^3(c+d x)}{12597 a^5 d (a+i a \tan (c+d x))^3}+\frac{48 i \cos ^3(c+d x)}{4199 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{64 i \cos ^5(c+d x)}{4199 d \left (a^8+i a^8 \tan (c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 1.31914, size = 161, normalized size = 0.53 \[ -\frac{i \sec ^8(c+d x) (-92378 i \sin (c+d x)-226746 i \sin (3 (c+d x))-266475 i \sin (5 (c+d x))-323323 i \sin (7 (c+d x))+73359 i \sin (9 (c+d x))+2431 i \sin (11 (c+d x))-739024 \cos (c+d x)-604656 \cos (3 (c+d x))-426360 \cos (5 (c+d x))-369512 \cos (7 (c+d x))+65208 \cos (9 (c+d x))+1768 \cos (11 (c+d x)))}{12899328 a^8 d (\tan (c+d x)-i)^8} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^3/(a + I*a*Tan[c + d*x])^8,x]

[Out]

((-I/12899328)*Sec[c + d*x]^8*(-739024*Cos[c + d*x] - 604656*Cos[3*(c + d*x)] - 426360*Cos[5*(c + d*x)] - 3695
12*Cos[7*(c + d*x)] + 65208*Cos[9*(c + d*x)] + 1768*Cos[11*(c + d*x)] - (92378*I)*Sin[c + d*x] - (226746*I)*Si
n[3*(c + d*x)] - (266475*I)*Sin[5*(c + d*x)] - (323323*I)*Sin[7*(c + d*x)] + (73359*I)*Sin[9*(c + d*x)] + (243
1*I)*Sin[11*(c + d*x)]))/(a^8*d*(-I + Tan[c + d*x])^8)

________________________________________________________________________________________

Maple [A]  time = 0.125, size = 372, normalized size = 1.2 \begin{align*} 2\,{\frac{1}{d{a}^{8}} \left ({\frac{-992\,i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{16}}}-{\frac{{\frac{32525\,i}{8}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{8}}}+{\frac{{\frac{7181\,i}{1024}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{2}}}+{\frac{{\frac{32417\,i}{4}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{10}}}-{\frac{{\frac{i}{1024}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) +i \right ) ^{2}}}-{\frac{{\frac{25468\,i}{3}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{12}}}+{\frac{4428\,i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{14}}}-{\frac{{\frac{2177\,i}{16}}}{ \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 ) ^{18}}}-{\frac{128}{19\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{19}}}+{\frac{5248}{17\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{17}}}-{\frac{7096}{3\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{15}}}+{\frac{87508}{13\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{13}}}-{\frac{18011}{2\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{11}}}+6215\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{-9}-{\frac{72425}{32\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{7}}}+{\frac{26871}{64\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{5}}}-{\frac{54229}{1536\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{3}}}+{\frac{509}{512\,\tan \left ( 1/2\,dx+c/2 \right ) -512\,i}}+{\frac{{\frac{204605\,i}{192}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{6}}}-{\frac{1}{1536\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +i \right ) ^{3}}}+{\frac{3}{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)^3/(a+I*a*tan(d*x+c))^8,x)

[Out]

2/d/a^8*(-992*I/(tan(1/2*d*x+1/2*c)-I)^16-32525/8*I/(tan(1/2*d*x+1/2*c)-I)^8+7181/1024*I/(tan(1/2*d*x+1/2*c)-I
)^2+32417/4*I/(tan(1/2*d*x+1/2*c)-I)^10-1/1024*I/(tan(1/2*d*x+1/2*c)+I)^2-25468/3*I/(tan(1/2*d*x+1/2*c)-I)^12+
4428*I/(tan(1/2*d*x+1/2*c)-I)^14-2177/16*I/(tan(1/2*d*x+1/2*c)-I)^4+64*I/(tan(1/2*d*x+1/2*c)-I)^18-128/19/(tan
(1/2*d*x+1/2*c)-I)^19+5248/17/(tan(1/2*d*x+1/2*c)-I)^17-7096/3/(tan(1/2*d*x+1/2*c)-I)^15+87508/13/(tan(1/2*d*x
+1/2*c)-I)^13-18011/2/(tan(1/2*d*x+1/2*c)-I)^11+6215/(tan(1/2*d*x+1/2*c)-I)^9-72425/32/(tan(1/2*d*x+1/2*c)-I)^
7+26871/64/(tan(1/2*d*x+1/2*c)-I)^5-54229/1536/(tan(1/2*d*x+1/2*c)-I)^3+509/512/(tan(1/2*d*x+1/2*c)-I)+204605/
192*I/(tan(1/2*d*x+1/2*c)-I)^6-1/1536/(tan(1/2*d*x+1/2*c)+I)^3+3/512/(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)^3/(a+I*a*tan(d*x+c))^8,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 2.54902, size = 543, normalized size = 1.8 \begin{align*} \frac{{\left (-4199 i \, e^{\left (22 i \, d x + 22 i \, c\right )} - 138567 i \, e^{\left (20 i \, d x + 20 i \, c\right )} + 692835 i \, e^{\left (18 i \, d x + 18 i \, c\right )} + 692835 i \, e^{\left (16 i \, d x + 16 i \, c\right )} + 831402 i \, e^{\left (14 i \, d x + 14 i \, c\right )} + 831402 i \, e^{\left (12 i \, d x + 12 i \, c\right )} + 646646 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 377910 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 159885 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 46189 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 8151 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 663 i\right )} e^{\left (-19 i \, d x - 19 i \, c\right )}}{25798656 \, a^{8} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3/(a+I*a*tan(d*x+c))^8,x, algorithm="fricas")

[Out]

1/25798656*(-4199*I*e^(22*I*d*x + 22*I*c) - 138567*I*e^(20*I*d*x + 20*I*c) + 692835*I*e^(18*I*d*x + 18*I*c) +
692835*I*e^(16*I*d*x + 16*I*c) + 831402*I*e^(14*I*d*x + 14*I*c) + 831402*I*e^(12*I*d*x + 12*I*c) + 646646*I*e^
(10*I*d*x + 10*I*c) + 377910*I*e^(8*I*d*x + 8*I*c) + 159885*I*e^(6*I*d*x + 6*I*c) + 46189*I*e^(4*I*d*x + 4*I*c
) + 8151*I*e^(2*I*d*x + 2*I*c) + 663*I)*e^(-19*I*d*x - 19*I*c)/(a^8*d)

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Sympy [A]  time = 4.39548, size = 437, normalized size = 1.45 \begin{align*} \begin{cases} \frac{\left (- 6279106898588469469113471576881812733952 i a^{88} d^{11} e^{103 i c} e^{3 i d x} - 207210527653419492480744562037099820220416 i a^{88} d^{11} e^{101 i c} e^{i d x} + 1036052638267097462403722810185499101102080 i a^{88} d^{11} e^{99 i c} e^{- i d x} + 1036052638267097462403722810185499101102080 i a^{88} d^{11} e^{97 i c} e^{- 3 i d x} + 1243263165920516954884467372222598921322496 i a^{88} d^{11} e^{95 i c} e^{- 5 i d x} + 1243263165920516954884467372222598921322496 i a^{88} d^{11} e^{93 i c} e^{- 7 i d x} + 966982462382624298243474622839799161028608 i a^{88} d^{11} e^{91 i c} e^{- 9 i d x} + 565119620872962252220212441919363146055680 i a^{88} d^{11} e^{89 i c} e^{- 11 i d x} + 239089070369330183631628340812038254100480 i a^{88} d^{11} e^{87 i c} e^{- 13 i d x} + 69070175884473164160248187345699940073472 i a^{88} d^{11} e^{85 i c} e^{- 15 i d x} + 12188854567848205440043797766888224718848 i a^{88} d^{11} e^{83 i c} e^{- 17 i d x} + 991437931356074126702127091086602010624 i a^{88} d^{11} e^{81 i c} e^{- 19 i d x}\right ) e^{- 100 i c}}{38578832784927556418233169368361857437401088 a^{96} d^{12}} & \text{for}\: 38578832784927556418233169368361857437401088 a^{96} d^{12} e^{100 i c} \neq 0 \\\frac{x \left (e^{22 i c} + 11 e^{20 i c} + 55 e^{18 i c} + 165 e^{16 i c} + 330 e^{14 i c} + 462 e^{12 i c} + 462 e^{10 i c} + 330 e^{8 i c} + 165 e^{6 i c} + 55 e^{4 i c} + 11 e^{2 i c} + 1\right ) e^{- 19 i c}}{2048 a^{8}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**3/(a+I*a*tan(d*x+c))**8,x)

[Out]

Piecewise(((-6279106898588469469113471576881812733952*I*a**88*d**11*exp(103*I*c)*exp(3*I*d*x) - 20721052765341
9492480744562037099820220416*I*a**88*d**11*exp(101*I*c)*exp(I*d*x) + 10360526382670974624037228101854991011020
80*I*a**88*d**11*exp(99*I*c)*exp(-I*d*x) + 1036052638267097462403722810185499101102080*I*a**88*d**11*exp(97*I*
c)*exp(-3*I*d*x) + 1243263165920516954884467372222598921322496*I*a**88*d**11*exp(95*I*c)*exp(-5*I*d*x) + 12432
63165920516954884467372222598921322496*I*a**88*d**11*exp(93*I*c)*exp(-7*I*d*x) + 96698246238262429824347462283
9799161028608*I*a**88*d**11*exp(91*I*c)*exp(-9*I*d*x) + 565119620872962252220212441919363146055680*I*a**88*d**
11*exp(89*I*c)*exp(-11*I*d*x) + 239089070369330183631628340812038254100480*I*a**88*d**11*exp(87*I*c)*exp(-13*I
*d*x) + 69070175884473164160248187345699940073472*I*a**88*d**11*exp(85*I*c)*exp(-15*I*d*x) + 12188854567848205
440043797766888224718848*I*a**88*d**11*exp(83*I*c)*exp(-17*I*d*x) + 991437931356074126702127091086602010624*I*
a**88*d**11*exp(81*I*c)*exp(-19*I*d*x))*exp(-100*I*c)/(38578832784927556418233169368361857437401088*a**96*d**1
2), Ne(38578832784927556418233169368361857437401088*a**96*d**12*exp(100*I*c), 0)), (x*(exp(22*I*c) + 11*exp(20
*I*c) + 55*exp(18*I*c) + 165*exp(16*I*c) + 330*exp(14*I*c) + 462*exp(12*I*c) + 462*exp(10*I*c) + 330*exp(8*I*c
) + 165*exp(6*I*c) + 55*exp(4*I*c) + 11*exp(2*I*c) + 1)*exp(-19*I*c)/(2048*a**8), True))

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Giac [A]  time = 1.19953, size = 406, normalized size = 1.35 \begin{align*} \frac{\frac{4199 \,{\left (18 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 33 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 17\right )}}{a^{8}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + i\right )}^{3}} + \frac{12823746 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{18} - 140368371 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{17} - 879644311 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{16} + 3693272440 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{15} + 11467502592 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{14} - 27403194676 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} - 51919375300 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{12} + 79183835016 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 98304418212 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} - 99750226290 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 82860874122 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 56110430792 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 30766700912 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 13462452660 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 4616712644 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 1197851960 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 226248618 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 27911475 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 2143959}{a^{8}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - i\right )}^{19}}}{6449664 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3/(a+I*a*tan(d*x+c))^8,x, algorithm="giac")

[Out]

1/6449664*(4199*(18*tan(1/2*d*x + 1/2*c)^2 + 33*I*tan(1/2*d*x + 1/2*c) - 17)/(a^8*(tan(1/2*d*x + 1/2*c) + I)^3
) + (12823746*tan(1/2*d*x + 1/2*c)^18 - 140368371*I*tan(1/2*d*x + 1/2*c)^17 - 879644311*tan(1/2*d*x + 1/2*c)^1
6 + 3693272440*I*tan(1/2*d*x + 1/2*c)^15 + 11467502592*tan(1/2*d*x + 1/2*c)^14 - 27403194676*I*tan(1/2*d*x + 1
/2*c)^13 - 51919375300*tan(1/2*d*x + 1/2*c)^12 + 79183835016*I*tan(1/2*d*x + 1/2*c)^11 + 98304418212*tan(1/2*d
*x + 1/2*c)^10 - 99750226290*I*tan(1/2*d*x + 1/2*c)^9 - 82860874122*tan(1/2*d*x + 1/2*c)^8 + 56110430792*I*tan
(1/2*d*x + 1/2*c)^7 + 30766700912*tan(1/2*d*x + 1/2*c)^6 - 13462452660*I*tan(1/2*d*x + 1/2*c)^5 - 4616712644*t
an(1/2*d*x + 1/2*c)^4 + 1197851960*I*tan(1/2*d*x + 1/2*c)^3 + 226248618*tan(1/2*d*x + 1/2*c)^2 - 27911475*I*ta
n(1/2*d*x + 1/2*c) - 2143959)/(a^8*(tan(1/2*d*x + 1/2*c) - I)^19))/d