\(\int \cos (c+d x) (a+i a \tan (c+d x))^8 \, dx\) [91]

Optimal result

Integrand size = 22, antiderivative size = 235 \[ \int \cos (c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {3003 a^8 \text {arctanh}(\sin (c+d x))}{16 d}-\frac {3003 i a^8 \sec (c+d x)}{16 d}-\frac {13 i a^3 \sec (c+d x) (a+i a \tan (c+d x))^5}{6 d}-\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^7}{d}-\frac {429 i a^2 \sec (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^3}{40 d}-\frac {143 i \sec (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^4}{30 d}-\frac {1001 i \sec (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )^2}{40 d}-\frac {1001 i \sec (c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{16 d} \]

[Out]

-3003/16*a^8*arctanh(sin(d*x+c))/d-3003/16*I*a^8*sec(d*x+c)/d-13/6*I*a^3*sec(d*x+c)*(a+I*a*tan(d*x+c))^5/d-2*I
*a*cos(d*x+c)*(a+I*a*tan(d*x+c))^7/d-429/40*I*a^2*sec(d*x+c)*(a^2+I*a^2*tan(d*x+c))^3/d-143/30*I*sec(d*x+c)*(a
^2+I*a^2*tan(d*x+c))^4/d-1001/40*I*sec(d*x+c)*(a^4+I*a^4*tan(d*x+c))^2/d-1001/16*I*sec(d*x+c)*(a^8+I*a^8*tan(d
*x+c))/d

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3577, 3579, 3567, 3855} \[ \int \cos (c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {3003 a^8 \text {arctanh}(\sin (c+d x))}{16 d}-\frac {3003 i a^8 \sec (c+d x)}{16 d}-\frac {1001 i \sec (c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{16 d}-\frac {1001 i \sec (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )^2}{40 d}-\frac {13 i a^3 \sec (c+d x) (a+i a \tan (c+d x))^5}{6 d}-\frac {429 i a^2 \sec (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^3}{40 d}-\frac {143 i \sec (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^4}{30 d}-\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^7}{d} \]

[In]

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

[Out]

(-3003*a^8*ArcTanh[Sin[c + d*x]])/(16*d) - (((3003*I)/16)*a^8*Sec[c + d*x])/d - (((13*I)/6)*a^3*Sec[c + d*x]*(
a + I*a*Tan[c + d*x])^5)/d - ((2*I)*a*Cos[c + d*x]*(a + I*a*Tan[c + d*x])^7)/d - (((429*I)/40)*a^2*Sec[c + d*x
]*(a^2 + I*a^2*Tan[c + d*x])^3)/d - (((143*I)/30)*Sec[c + d*x]*(a^2 + I*a^2*Tan[c + d*x])^4)/d - (((1001*I)/40
)*Sec[c + d*x]*(a^4 + I*a^4*Tan[c + d*x])^2)/d - (((1001*I)/16)*Sec[c + d*x]*(a^8 + I*a^8*Tan[c + d*x]))/d

Rule 3567

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 3577

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[2*b*(d
*Sec[e + f*x])^m*((a + b*Tan[e + f*x])^(n - 1)/(f*m)), x] - Dist[b^2*((m + 2*n - 2)/(d^2*m)), Int[(d*Sec[e + f
*x])^(m + 2)*(a + b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0] && GtQ[n,
1] && ((IGtQ[n/2, 0] && ILtQ[m - 1/2, 0]) || (EqQ[n, 2] && LtQ[m, 0]) || (LeQ[m, -1] && GtQ[m + n, 0]) || (ILt
Q[m, 0] && LtQ[m/2 + n - 1, 0] && IntegerQ[n]) || (EqQ[n, 3/2] && EqQ[m, -2^(-1)])) && IntegerQ[2*m]

Rule 3579

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

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^7}{d}-\left (13 a^2\right ) \int \sec (c+d x) (a+i a \tan (c+d x))^6 \, dx \\ & = -\frac {13 i a^3 \sec (c+d x) (a+i a \tan (c+d x))^5}{6 d}-\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^7}{d}-\frac {1}{6} \left (143 a^3\right ) \int \sec (c+d x) (a+i a \tan (c+d x))^5 \, dx \\ & = -\frac {13 i a^3 \sec (c+d x) (a+i a \tan (c+d x))^5}{6 d}-\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^7}{d}-\frac {143 i \sec (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^4}{30 d}-\frac {1}{10} \left (429 a^4\right ) \int \sec (c+d x) (a+i a \tan (c+d x))^4 \, dx \\ & = -\frac {429 i a^5 \sec (c+d x) (a+i a \tan (c+d x))^3}{40 d}-\frac {13 i a^3 \sec (c+d x) (a+i a \tan (c+d x))^5}{6 d}-\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^7}{d}-\frac {143 i \sec (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^4}{30 d}-\frac {1}{40} \left (3003 a^5\right ) \int \sec (c+d x) (a+i a \tan (c+d x))^3 \, dx \\ & = -\frac {429 i a^5 \sec (c+d x) (a+i a \tan (c+d x))^3}{40 d}-\frac {13 i a^3 \sec (c+d x) (a+i a \tan (c+d x))^5}{6 d}-\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^7}{d}-\frac {143 i \sec (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^4}{30 d}-\frac {1001 i \sec (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )^2}{40 d}-\frac {1}{8} \left (1001 a^6\right ) \int \sec (c+d x) (a+i a \tan (c+d x))^2 \, dx \\ & = -\frac {429 i a^5 \sec (c+d x) (a+i a \tan (c+d x))^3}{40 d}-\frac {13 i a^3 \sec (c+d x) (a+i a \tan (c+d x))^5}{6 d}-\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^7}{d}-\frac {143 i \sec (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^4}{30 d}-\frac {1001 i \sec (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )^2}{40 d}-\frac {1001 i \sec (c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{16 d}-\frac {1}{16} \left (3003 a^7\right ) \int \sec (c+d x) (a+i a \tan (c+d x)) \, dx \\ & = -\frac {3003 i a^8 \sec (c+d x)}{16 d}-\frac {429 i a^5 \sec (c+d x) (a+i a \tan (c+d x))^3}{40 d}-\frac {13 i a^3 \sec (c+d x) (a+i a \tan (c+d x))^5}{6 d}-\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^7}{d}-\frac {143 i \sec (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^4}{30 d}-\frac {1001 i \sec (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )^2}{40 d}-\frac {1001 i \sec (c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{16 d}-\frac {1}{16} \left (3003 a^8\right ) \int \sec (c+d x) \, dx \\ & = -\frac {3003 a^8 \text {arctanh}(\sin (c+d x))}{16 d}-\frac {3003 i a^8 \sec (c+d x)}{16 d}-\frac {429 i a^5 \sec (c+d x) (a+i a \tan (c+d x))^3}{40 d}-\frac {13 i a^3 \sec (c+d x) (a+i a \tan (c+d x))^5}{6 d}-\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^7}{d}-\frac {143 i \sec (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^4}{30 d}-\frac {1001 i \sec (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )^2}{40 d}-\frac {1001 i \sec (c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{16 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.76 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.87 \[ \int \cos (c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {a^8 \cos ^2(c+d x) (\cos (8 c)-i \sin (8 c)) \left (-658944 i \cos (c+d x)+720720 \cos ^6(c+d x) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+5 (-73216 i \cos (3 (c+d x))-19968 i \cos (5 (c+d x))-1536 i \cos (7 (c+d x))+12870 \sin (c+d x)+22165 \sin (3 (c+d x))+10959 \sin (5 (c+d x))+1536 \sin (7 (c+d x)))\right ) (-i+\tan (c+d x))^8}{3840 d (\cos (d x)+i \sin (d x))^8} \]

[In]

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

[Out]

(a^8*Cos[c + d*x]^2*(Cos[8*c] - I*Sin[8*c])*((-658944*I)*Cos[c + d*x] + 720720*Cos[c + d*x]^6*(Log[Cos[(c + d*
x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 5*((-73216*I)*Cos[3*(c + d*x)] - (1996
8*I)*Cos[5*(c + d*x)] - (1536*I)*Cos[7*(c + d*x)] + 12870*Sin[c + d*x] + 22165*Sin[3*(c + d*x)] + 10959*Sin[5*
(c + d*x)] + 1536*Sin[7*(c + d*x)]))*(-I + Tan[c + d*x])^8)/(3840*d*(Cos[d*x] + I*Sin[d*x])^8)

Maple [A] (verified)

Time = 90.46 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.64

method	result	size
risch	\(-\frac {128 i a^{8} {\mathrm e}^{i \left (d x +c \right )}}{d}-\frac {i a^{8} \left (62475 \,{\mathrm e}^{11 i \left (d x +c \right )}+246505 \,{\mathrm e}^{9 i \left (d x +c \right )}+416094 \,{\mathrm e}^{7 i \left (d x +c \right )}+364194 \,{\mathrm e}^{5 i \left (d x +c \right )}+163095 \,{\mathrm e}^{3 i \left (d x +c \right )}+29685 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{120 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{6}}-\frac {3003 a^{8} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{16 d}+\frac {3003 a^{8} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{16 d}\)	\(151\)
derivativedivides	\(\frac {a^{8} \left (\frac {\sin ^{9}\left (d x +c \right )}{6 \cos \left (d x +c \right )^{6}}-\frac {\sin ^{9}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{4}}+\frac {5 \left (\sin ^{9}\left (d x +c \right )\right )}{16 \cos \left (d x +c \right )^{2}}+\frac {5 \left (\sin ^{7}\left (d x +c \right )\right )}{16}+\frac {7 \left (\sin ^{5}\left (d x +c \right )\right )}{16}+\frac {35 \left (\sin ^{3}\left (d x +c \right )\right )}{48}+\frac {35 \sin \left (d x +c \right )}{16}-\frac {35 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )-8 i a^{8} \left (\frac {\sin ^{8}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}-\frac {\sin ^{8}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{3}}+\frac {\sin ^{8}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\frac {16}{5}+\sin ^{6}\left (d x +c \right )+\frac {6 \left (\sin ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (d x +c \right )\right )}{5}\right ) \cos \left (d x +c \right )\right )-28 a^{8} \left (\frac {\sin ^{7}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {3 \left (\sin ^{7}\left (d x +c \right )\right )}{8 \cos \left (d x +c \right )^{2}}-\frac {3 \left (\sin ^{5}\left (d x +c \right )\right )}{8}-\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{8}-\frac {15 \sin \left (d x +c \right )}{8}+\frac {15 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+56 i a^{8} \left (\frac {\sin ^{6}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )+70 a^{8} \left (\frac {\sin ^{5}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{2}+\frac {3 \sin \left (d x +c \right )}{2}-\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-56 i a^{8} \left (\frac {\sin ^{4}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )\right )-28 a^{8} \left (-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )-8 i a^{8} \cos \left (d x +c \right )+a^{8} \sin \left (d x +c \right )}{d}\)	\(522\)
default	\(\frac {a^{8} \left (\frac {\sin ^{9}\left (d x +c \right )}{6 \cos \left (d x +c \right )^{6}}-\frac {\sin ^{9}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{4}}+\frac {5 \left (\sin ^{9}\left (d x +c \right )\right )}{16 \cos \left (d x +c \right )^{2}}+\frac {5 \left (\sin ^{7}\left (d x +c \right )\right )}{16}+\frac {7 \left (\sin ^{5}\left (d x +c \right )\right )}{16}+\frac {35 \left (\sin ^{3}\left (d x +c \right )\right )}{48}+\frac {35 \sin \left (d x +c \right )}{16}-\frac {35 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )-8 i a^{8} \left (\frac {\sin ^{8}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}-\frac {\sin ^{8}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{3}}+\frac {\sin ^{8}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\frac {16}{5}+\sin ^{6}\left (d x +c \right )+\frac {6 \left (\sin ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (d x +c \right )\right )}{5}\right ) \cos \left (d x +c \right )\right )-28 a^{8} \left (\frac {\sin ^{7}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {3 \left (\sin ^{7}\left (d x +c \right )\right )}{8 \cos \left (d x +c \right )^{2}}-\frac {3 \left (\sin ^{5}\left (d x +c \right )\right )}{8}-\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{8}-\frac {15 \sin \left (d x +c \right )}{8}+\frac {15 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+56 i a^{8} \left (\frac {\sin ^{6}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )+70 a^{8} \left (\frac {\sin ^{5}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{2}+\frac {3 \sin \left (d x +c \right )}{2}-\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-56 i a^{8} \left (\frac {\sin ^{4}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )\right )-28 a^{8} \left (-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )-8 i a^{8} \cos \left (d x +c \right )+a^{8} \sin \left (d x +c \right )}{d}\)	\(522\)

[In]

int(cos(d*x+c)*(a+I*a*tan(d*x+c))^8,x,method=_RETURNVERBOSE)

[Out]

-128*I/d*a^8*exp(I*(d*x+c))-1/120*I*a^8/d/(exp(2*I*(d*x+c))+1)^6*(62475*exp(11*I*(d*x+c))+246505*exp(9*I*(d*x+
c))+416094*exp(7*I*(d*x+c))+364194*exp(5*I*(d*x+c))+163095*exp(3*I*(d*x+c))+29685*exp(I*(d*x+c)))-3003/16/d*a^
8*ln(exp(I*(d*x+c))+I)+3003/16/d*a^8*ln(exp(I*(d*x+c))-I)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.61 \[ \int \cos (c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {-30720 i \, a^{8} e^{\left (13 i \, d x + 13 i \, c\right )} - 309270 i \, a^{8} e^{\left (11 i \, d x + 11 i \, c\right )} - 953810 i \, a^{8} e^{\left (9 i \, d x + 9 i \, c\right )} - 1446588 i \, a^{8} e^{\left (7 i \, d x + 7 i \, c\right )} - 1189188 i \, a^{8} e^{\left (5 i \, d x + 5 i \, c\right )} - 510510 i \, a^{8} e^{\left (3 i \, d x + 3 i \, c\right )} - 90090 i \, a^{8} e^{\left (i \, d x + i \, c\right )} - 45045 \, {\left (a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} + 6 \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} + 20 \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} + 6 \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{8}\right )} \log \left (e^{\left (i \, d x + i \, c\right )} + i\right ) + 45045 \, {\left (a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} + 6 \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} + 20 \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} + 6 \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{8}\right )} \log \left (e^{\left (i \, d x + i \, c\right )} - i\right )}{240 \, {\left (d e^{\left (12 i \, d x + 12 i \, c\right )} + 6 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 20 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 6 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]

[In]

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

[Out]

1/240*(-30720*I*a^8*e^(13*I*d*x + 13*I*c) - 309270*I*a^8*e^(11*I*d*x + 11*I*c) - 953810*I*a^8*e^(9*I*d*x + 9*I
*c) - 1446588*I*a^8*e^(7*I*d*x + 7*I*c) - 1189188*I*a^8*e^(5*I*d*x + 5*I*c) - 510510*I*a^8*e^(3*I*d*x + 3*I*c)
- 90090*I*a^8*e^(I*d*x + I*c) - 45045*(a^8*e^(12*I*d*x + 12*I*c) + 6*a^8*e^(10*I*d*x + 10*I*c) + 15*a^8*e^(8*
I*d*x + 8*I*c) + 20*a^8*e^(6*I*d*x + 6*I*c) + 15*a^8*e^(4*I*d*x + 4*I*c) + 6*a^8*e^(2*I*d*x + 2*I*c) + a^8)*lo
g(e^(I*d*x + I*c) + I) + 45045*(a^8*e^(12*I*d*x + 12*I*c) + 6*a^8*e^(10*I*d*x + 10*I*c) + 15*a^8*e^(8*I*d*x +
8*I*c) + 20*a^8*e^(6*I*d*x + 6*I*c) + 15*a^8*e^(4*I*d*x + 4*I*c) + 6*a^8*e^(2*I*d*x + 2*I*c) + a^8)*log(e^(I*d
*x + I*c) - I))/(d*e^(12*I*d*x + 12*I*c) + 6*d*e^(10*I*d*x + 10*I*c) + 15*d*e^(8*I*d*x + 8*I*c) + 20*d*e^(6*I*
d*x + 6*I*c) + 15*d*e^(4*I*d*x + 4*I*c) + 6*d*e^(2*I*d*x + 2*I*c) + d)

Sympy [A] (verification not implemented)

Time = 0.43 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.36 \[ \int \cos (c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {3003 a^{8} \left (\frac {\log {\left (e^{i d x} - i e^{- i c} \right )}}{16} - \frac {\log {\left (e^{i d x} + i e^{- i c} \right )}}{16}\right )}{d} + \frac {- 62475 i a^{8} e^{11 i c} e^{11 i d x} - 246505 i a^{8} e^{9 i c} e^{9 i d x} - 416094 i a^{8} e^{7 i c} e^{7 i d x} - 364194 i a^{8} e^{5 i c} e^{5 i d x} - 163095 i a^{8} e^{3 i c} e^{3 i d x} - 29685 i a^{8} e^{i c} e^{i d x}}{120 d e^{12 i c} e^{12 i d x} + 720 d e^{10 i c} e^{10 i d x} + 1800 d e^{8 i c} e^{8 i d x} + 2400 d e^{6 i c} e^{6 i d x} + 1800 d e^{4 i c} e^{4 i d x} + 720 d e^{2 i c} e^{2 i d x} + 120 d} + \begin {cases} - \frac {128 i a^{8} e^{i c} e^{i d x}}{d} & \text {for}\: d \neq 0 \\128 a^{8} x e^{i c} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

3003*a**8*(log(exp(I*d*x) - I*exp(-I*c))/16 - log(exp(I*d*x) + I*exp(-I*c))/16)/d + (-62475*I*a**8*exp(11*I*c)
*exp(11*I*d*x) - 246505*I*a**8*exp(9*I*c)*exp(9*I*d*x) - 416094*I*a**8*exp(7*I*c)*exp(7*I*d*x) - 364194*I*a**8
*exp(5*I*c)*exp(5*I*d*x) - 163095*I*a**8*exp(3*I*c)*exp(3*I*d*x) - 29685*I*a**8*exp(I*c)*exp(I*d*x))/(120*d*ex
p(12*I*c)*exp(12*I*d*x) + 720*d*exp(10*I*c)*exp(10*I*d*x) + 1800*d*exp(8*I*c)*exp(8*I*d*x) + 2400*d*exp(6*I*c)
*exp(6*I*d*x) + 1800*d*exp(4*I*c)*exp(4*I*d*x) + 720*d*exp(2*I*c)*exp(2*I*d*x) + 120*d) + Piecewise((-128*I*a*
*8*exp(I*c)*exp(I*d*x)/d, Ne(d, 0)), (128*a**8*x*exp(I*c), True))

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 396 vs. \(2 (195) = 390\).

Time = 0.27 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.69 \[ \int \cos (c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {5 \, a^{8} {\left (\frac {2 \, {\left (87 \, \sin \left (d x + c\right )^{5} - 136 \, \sin \left (d x + c\right )^{3} + 57 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} + 105 \, \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, \log \left (\sin \left (d x + c\right ) - 1\right ) - 96 \, \sin \left (d x + c\right )\right )} + 840 \, a^{8} {\left (\frac {2 \, {\left (9 \, \sin \left (d x + c\right )^{3} - 7 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} + 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\sin \left (d x + c\right ) - 1\right ) - 16 \, \sin \left (d x + c\right )\right )} + 8400 \, a^{8} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} + 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\sin \left (d x + c\right ) - 1\right ) - 4 \, \sin \left (d x + c\right )\right )} + 26880 i \, a^{8} {\left (\frac {1}{\cos \left (d x + c\right )} + \cos \left (d x + c\right )\right )} + 8960 i \, a^{8} {\left (\frac {6 \, \cos \left (d x + c\right )^{2} - 1}{\cos \left (d x + c\right )^{3}} + 3 \, \cos \left (d x + c\right )\right )} + 768 i \, a^{8} {\left (\frac {15 \, \cos \left (d x + c\right )^{4} - 5 \, \cos \left (d x + c\right )^{2} + 1}{\cos \left (d x + c\right )^{5}} + 5 \, \cos \left (d x + c\right )\right )} + 6720 \, a^{8} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right ) - 2 \, \sin \left (d x + c\right )\right )} + 3840 i \, a^{8} \cos \left (d x + c\right ) - 480 \, a^{8} \sin \left (d x + c\right )}{480 \, d} \]

[In]

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

[Out]

-1/480*(5*a^8*(2*(87*sin(d*x + c)^5 - 136*sin(d*x + c)^3 + 57*sin(d*x + c))/(sin(d*x + c)^6 - 3*sin(d*x + c)^4
+ 3*sin(d*x + c)^2 - 1) + 105*log(sin(d*x + c) + 1) - 105*log(sin(d*x + c) - 1) - 96*sin(d*x + c)) + 840*a^8*
(2*(9*sin(d*x + c)^3 - 7*sin(d*x + c))/(sin(d*x + c)^4 - 2*sin(d*x + c)^2 + 1) + 15*log(sin(d*x + c) + 1) - 15
*log(sin(d*x + c) - 1) - 16*sin(d*x + c)) + 8400*a^8*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) + 3*log(sin(d*x + c)
+ 1) - 3*log(sin(d*x + c) - 1) - 4*sin(d*x + c)) + 26880*I*a^8*(1/cos(d*x + c) + cos(d*x + c)) + 8960*I*a^8*(
(6*cos(d*x + c)^2 - 1)/cos(d*x + c)^3 + 3*cos(d*x + c)) + 768*I*a^8*((15*cos(d*x + c)^4 - 5*cos(d*x + c)^2 + 1
)/cos(d*x + c)^5 + 5*cos(d*x + c)) + 6720*a^8*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1) - 2*sin(d*x + c))
+ 3840*I*a^8*cos(d*x + c) - 480*a^8*sin(d*x + c))/d

Giac [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 924 vs. \(2 (195) = 390\).

Time = 1.44 (sec) , antiderivative size = 924, normalized size of antiderivative = 3.93 \[ \int \cos (c+d x) (a+i a \tan (c+d x))^8 \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/61440*(11512215*a^8*e^(12*I*d*x + 12*I*c)*log(I*e^(I*d*x + I*c) + 1) + 69073290*a^8*e^(10*I*d*x + 10*I*c)*lo
g(I*e^(I*d*x + I*c) + 1) + 172683225*a^8*e^(8*I*d*x + 8*I*c)*log(I*e^(I*d*x + I*c) + 1) + 230244300*a^8*e^(6*I
*d*x + 6*I*c)*log(I*e^(I*d*x + I*c) + 1) + 172683225*a^8*e^(4*I*d*x + 4*I*c)*log(I*e^(I*d*x + I*c) + 1) + 6907
3290*a^8*e^(2*I*d*x + 2*I*c)*log(I*e^(I*d*x + I*c) + 1) - 19305*a^8*e^(12*I*d*x + 12*I*c)*log(I*e^(I*d*x + I*c
) - 1) - 115830*a^8*e^(10*I*d*x + 10*I*c)*log(I*e^(I*d*x + I*c) - 1) - 289575*a^8*e^(8*I*d*x + 8*I*c)*log(I*e^
(I*d*x + I*c) - 1) - 386100*a^8*e^(6*I*d*x + 6*I*c)*log(I*e^(I*d*x + I*c) - 1) - 289575*a^8*e^(4*I*d*x + 4*I*c
)*log(I*e^(I*d*x + I*c) - 1) - 115830*a^8*e^(2*I*d*x + 2*I*c)*log(I*e^(I*d*x + I*c) - 1) - 11512215*a^8*e^(12*
I*d*x + 12*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 69073290*a^8*e^(10*I*d*x + 10*I*c)*log(-I*e^(I*d*x + I*c) + 1) -
172683225*a^8*e^(8*I*d*x + 8*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 230244300*a^8*e^(6*I*d*x + 6*I*c)*log(-I*e^(I
*d*x + I*c) + 1) - 172683225*a^8*e^(4*I*d*x + 4*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 69073290*a^8*e^(2*I*d*x + 2
*I*c)*log(-I*e^(I*d*x + I*c) + 1) + 19305*a^8*e^(12*I*d*x + 12*I*c)*log(-I*e^(I*d*x + I*c) - 1) + 115830*a^8*e
^(10*I*d*x + 10*I*c)*log(-I*e^(I*d*x + I*c) - 1) + 289575*a^8*e^(8*I*d*x + 8*I*c)*log(-I*e^(I*d*x + I*c) - 1)
+ 386100*a^8*e^(6*I*d*x + 6*I*c)*log(-I*e^(I*d*x + I*c) - 1) + 289575*a^8*e^(4*I*d*x + 4*I*c)*log(-I*e^(I*d*x
+ I*c) - 1) + 115830*a^8*e^(2*I*d*x + 2*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 7864320*I*a^8*e^(13*I*d*x + 13*I*c)
- 79173120*I*a^8*e^(11*I*d*x + 11*I*c) - 244175360*I*a^8*e^(9*I*d*x + 9*I*c) - 370326528*I*a^8*e^(7*I*d*x + 7
*I*c) - 304432128*I*a^8*e^(5*I*d*x + 5*I*c) - 130690560*I*a^8*e^(3*I*d*x + 3*I*c) - 23063040*I*a^8*e^(I*d*x +
I*c) + 11512215*a^8*log(I*e^(I*d*x + I*c) + 1) - 19305*a^8*log(I*e^(I*d*x + I*c) - 1) - 11512215*a^8*log(-I*e^
(I*d*x + I*c) + 1) + 19305*a^8*log(-I*e^(I*d*x + I*c) - 1))/(d*e^(12*I*d*x + 12*I*c) + 6*d*e^(10*I*d*x + 10*I*
c) + 15*d*e^(8*I*d*x + 8*I*c) + 20*d*e^(6*I*d*x + 6*I*c) + 15*d*e^(4*I*d*x + 4*I*c) + 6*d*e^(2*I*d*x + 2*I*c)
+ d)

Mupad [B] (verification not implemented)

Time = 9.23 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.70 \[ \int \cos (c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {\frac {3019\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{8}+\frac {a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,2891{}\mathrm {i}}{8}-\frac {52795\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{24}-\frac {a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,45115{}\mathrm {i}}{24}+\frac {22415\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{4}+\frac {a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,43757{}\mathrm {i}}{12}-\frac {97811\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{12}-\frac {a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,12977{}\mathrm {i}}{4}+\frac {167237\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{24}+\frac {a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,160729{}\mathrm {i}}{120}-\frac {127113\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{40}-\frac {a^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,25499{}\mathrm {i}}{120}+\frac {8848\,a^8}{15}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,1{}\mathrm {i}-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,6{}\mathrm {i}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,15{}\mathrm {i}-20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,20{}\mathrm {i}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,15{}\mathrm {i}-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,6{}\mathrm {i}+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1{}\mathrm {i}\right )}-\frac {3003\,a^8\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{8\,d} \]

[In]

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

[Out]

((a^8*tan(c/2 + (d*x)/2)^3*160729i)/120 - (127113*a^8*tan(c/2 + (d*x)/2)^2)/40 + (167237*a^8*tan(c/2 + (d*x)/2
)^4)/24 - (a^8*tan(c/2 + (d*x)/2)^5*12977i)/4 - (97811*a^8*tan(c/2 + (d*x)/2)^6)/12 + (a^8*tan(c/2 + (d*x)/2)^
7*43757i)/12 + (22415*a^8*tan(c/2 + (d*x)/2)^8)/4 - (a^8*tan(c/2 + (d*x)/2)^9*45115i)/24 - (52795*a^8*tan(c/2
+ (d*x)/2)^10)/24 + (a^8*tan(c/2 + (d*x)/2)^11*2891i)/8 + (3019*a^8*tan(c/2 + (d*x)/2)^12)/8 + (8848*a^8)/15 -
(a^8*tan(c/2 + (d*x)/2)*25499i)/120)/(d*(tan(c/2 + (d*x)/2) - tan(c/2 + (d*x)/2)^2*6i - 6*tan(c/2 + (d*x)/2)^
3 + tan(c/2 + (d*x)/2)^4*15i + 15*tan(c/2 + (d*x)/2)^5 - tan(c/2 + (d*x)/2)^6*20i - 20*tan(c/2 + (d*x)/2)^7 +
tan(c/2 + (d*x)/2)^8*15i + 15*tan(c/2 + (d*x)/2)^9 - tan(c/2 + (d*x)/2)^10*6i - 6*tan(c/2 + (d*x)/2)^11 + tan(
c/2 + (d*x)/2)^12*1i + tan(c/2 + (d*x)/2)^13 + 1i)) - (3003*a^8*atanh(tan(c/2 + (d*x)/2)))/(8*d)