3.2.83 \(\int \frac {\cos (c+d x)}{(a+i a \tan (c+d x))^8} \, dx\) [183]

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

[Out]

192/12155*sin(d*x+c)/a^8/d-64/12155*sin(d*x+c)^3/a^8/d+1/17*I*cos(d*x+c)/d/(a+I*a*tan(d*x+c))^8+3/85*I*cos(d*x
+c)/a/d/(a+I*a*tan(d*x+c))^7+24/1105*I*cos(d*x+c)/a^2/d/(a+I*a*tan(d*x+c))^6+168/12155*I*cos(d*x+c)/a^3/d/(a+I
*a*tan(d*x+c))^5+112/12155*I*cos(d*x+c)/d/(a^2+I*a^2*tan(d*x+c))^4+16/2431*I*cos(d*x+c)/a^2/d/(a^2+I*a^2*tan(d
*x+c))^3+128/12155*I*cos(d*x+c)^3/d/(a^8+I*a^8*tan(d*x+c))

________________________________________________________________________________________

Rubi [A]
time = 0.25, antiderivative size = 271, normalized size of antiderivative = 1.00, 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 = {3583, 3581, 2713} \begin {gather*} -\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 {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^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 {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} \end {gather*}

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 2713

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]

Rule 3581

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 3583

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]

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 \text {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 = 1.20, size = 139, normalized size = 0.51 \begin {gather*} -\frac {i \sec ^8(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))-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)))}{3111680 a^8 d (-i+\tan (c+d x))^8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-1/3111680*I)*Sec[c + d*x]^8*(-194480*Cos[c + d*x] - 148512*Cos[3*(c + d*x)] - 89760*Cos[5*(c + d*x)] - 5834
4*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)*S
in[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)

________________________________________________________________________________________

Maple [A]
time = 0.31, size = 306, normalized size = 1.13

method	result	size
risch	\(\frac {3 i {\mathrm e}^{-3 i \left (d x +c \right )}}{128 a^{8} d}+\frac {21 i {\mathrm e}^{-5 i \left (d x +c \right )}}{640 a^{8} d}+\frac {9 i {\mathrm e}^{-7 i \left (d x +c \right )}}{256 a^{8} d}+\frac {7 i {\mathrm e}^{-9 i \left (d x +c \right )}}{256 a^{8} d}+\frac {21 i {\mathrm e}^{-11 i \left (d x +c \right )}}{1408 a^{8} d}+\frac {9 i {\mathrm e}^{-13 i \left (d x +c \right )}}{1664 a^{8} d}+\frac {3 i {\mathrm e}^{-15 i \left (d x +c \right )}}{2560 a^{8} d}+\frac {i {\mathrm e}^{-17 i \left (d x +c \right )}}{8704 a^{8} d}+\frac {i \cos \left (d x +c \right )}{64 a^{8} d}+\frac {5 \sin \left (d x +c \right )}{256 a^{8} d}\)	\(175\)
derivativedivides	\(\frac {\frac {2}{512 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+512 i}-\frac {128 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{16}}-\frac {7937 i}{32 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {10241 i}{2 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}-\frac {5384 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{12}}+\frac {38218 i}{5 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}+\frac {1793 i}{128 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {1568 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{14}}+\frac {13313 i}{8 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {256}{17 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{17}}-\frac {2752}{5 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{15}}+\frac {42800}{13 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{13}}-\frac {77908}{11 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}+\frac {6847}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}-\frac {12799}{4 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {57083}{80 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {4351}{64 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {511}{256 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}{d \,a^{8}}\)	\(306\)
default	\(\frac {\frac {2}{512 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+512 i}-\frac {128 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{16}}-\frac {7937 i}{32 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {10241 i}{2 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}-\frac {5384 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{12}}+\frac {38218 i}{5 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}+\frac {1793 i}{128 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {1568 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{14}}+\frac {13313 i}{8 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {256}{17 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{17}}-\frac {2752}{5 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{15}}+\frac {42800}{13 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{13}}-\frac {77908}{11 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}+\frac {6847}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}-\frac {12799}{4 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {57083}{80 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {4351}{64 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {511}{256 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}{d \,a^{8}}\)	\(306\)

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,method=_RETURNVERBOSE)

[Out]

2/d/a^8*(1/512/(tan(1/2*d*x+1/2*c)+I)-64*I/(-I+tan(1/2*d*x+1/2*c))^16-7937/64*I/(-I+tan(1/2*d*x+1/2*c))^4-1024
1/4*I/(-I+tan(1/2*d*x+1/2*c))^8-2692*I/(-I+tan(1/2*d*x+1/2*c))^12+19109/5*I/(-I+tan(1/2*d*x+1/2*c))^10+1793/25
6*I/(-I+tan(1/2*d*x+1/2*c))^2+784*I/(-I+tan(1/2*d*x+1/2*c))^14+13313/16*I/(-I+tan(1/2*d*x+1/2*c))^6+128/17/(-I
+tan(1/2*d*x+1/2*c))^17-1376/5/(-I+tan(1/2*d*x+1/2*c))^15+21400/13/(-I+tan(1/2*d*x+1/2*c))^13-38954/11/(-I+tan
(1/2*d*x+1/2*c))^11+6847/2/(-I+tan(1/2*d*x+1/2*c))^9-12799/8/(-I+tan(1/2*d*x+1/2*c))^7+57083/160/(-I+tan(1/2*d
*x+1/2*c))^5-4351/128/(-I+tan(1/2*d*x+1/2*c))^3+511/512/(-I+tan(1/2*d*x+1/2*c)))

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

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 >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

________________________________________________________________________________________

Fricas [A]
time = 0.37, size = 118, normalized size = 0.44 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

Sympy [A]
time = 0.60, size = 367, normalized size = 1.35 \begin {gather*} \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}\: 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 {gather*}

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(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*exp(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))

________________________________________________________________________________________

Giac [A]
time = 1.25, size = 249, normalized size = 0.92 \begin {gather*} \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 {gather*}

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

________________________________________________________________________________________

Mupad [B]
time = 6.66, size = 262, normalized size = 0.97 \begin {gather*} \frac {\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {152329\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128}-\frac {41121\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{32}+\frac {41121\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{32}-\frac {96165\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{64}+\frac {96165\,\sin \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{64}-\frac {55095\,\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{32}+\frac {55095\,\sin \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )}{32}-\frac {491811\,\sin \left (\frac {15\,c}{2}+\frac {15\,d\,x}{2}\right )}{256}+\frac {6435\,\sin \left (\frac {17\,c}{2}+\frac {17\,d\,x}{2}\right )}{256}+\frac {\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )\,12155{}\mathrm {i}}{16}-\frac {\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )\,12155{}\mathrm {i}}{16}+\frac {\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )\,21437{}\mathrm {i}}{16}-\frac {\cos \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )\,21437{}\mathrm {i}}{16}+\frac {\cos \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )\,27047{}\mathrm {i}}{16}-\frac {\cos \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )\,27047{}\mathrm {i}}{16}+\frac {\cos \left (\frac {15\,c}{2}+\frac {15\,d\,x}{2}\right )\,61387{}\mathrm {i}}{32}-\frac {\cos \left (\frac {17\,c}{2}+\frac {17\,d\,x}{2}\right )\,715{}\mathrm {i}}{32}\right )\,2{}\mathrm {i}}{12155\,a^8\,d\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}\right )}^{17}\,\left (\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(cos(c/2 + (d*x)/2)*((cos((3*c)/2 + (3*d*x)/2)*12155i)/16 - (cos((5*c)/2 + (5*d*x)/2)*12155i)/16 + (cos((7*c)/
2 + (7*d*x)/2)*21437i)/16 - (cos((9*c)/2 + (9*d*x)/2)*21437i)/16 + (cos((11*c)/2 + (11*d*x)/2)*27047i)/16 - (c
os((13*c)/2 + (13*d*x)/2)*27047i)/16 + (cos((15*c)/2 + (15*d*x)/2)*61387i)/32 - (cos((17*c)/2 + (17*d*x)/2)*71
5i)/32 + (152329*sin(c/2 + (d*x)/2))/128 - (41121*sin((3*c)/2 + (3*d*x)/2))/32 + (41121*sin((5*c)/2 + (5*d*x)/
2))/32 - (96165*sin((7*c)/2 + (7*d*x)/2))/64 + (96165*sin((9*c)/2 + (9*d*x)/2))/64 - (55095*sin((11*c)/2 + (11
*d*x)/2))/32 + (55095*sin((13*c)/2 + (13*d*x)/2))/32 - (491811*sin((15*c)/2 + (15*d*x)/2))/256 + (6435*sin((17
*c)/2 + (17*d*x)/2))/256)*2i)/(12155*a^8*d*(cos(c/2 + (d*x)/2) + sin(c/2 + (d*x)/2)*1i)^17*(cos(c/2 + (d*x)/2)
*1i + sin(c/2 + (d*x)/2)))

________________________________________________________________________________________