3.97 \(\int \cos ^{13}(c+d x) (a+i a \tan (c+d x))^8 \, dx\)

Optimal. Leaf size=211 \[ -\frac {20 i a^3 \cos ^7(c+d x) (a+i a \tan (c+d x))^5}{3003 d}-\frac {20 i a^2 \cos ^9(c+d x) (a+i a \tan (c+d x))^6}{1287 d}-\frac {8 i \cos ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^4}{3003 d}-\frac {8 i a^2 \cos ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^3}{9009 d}-\frac {i \cos ^{13}(c+d x) (a+i a \tan (c+d x))^8}{13 d}-\frac {5 i a \cos ^{11}(c+d x) (a+i a \tan (c+d x))^7}{143 d} \]

[Out]

-20/3003*I*a^3*cos(d*x+c)^7*(a+I*a*tan(d*x+c))^5/d-20/1287*I*a^2*cos(d*x+c)^9*(a+I*a*tan(d*x+c))^6/d-5/143*I*a
*cos(d*x+c)^11*(a+I*a*tan(d*x+c))^7/d-1/13*I*cos(d*x+c)^13*(a+I*a*tan(d*x+c))^8/d-8/9009*I*a^2*cos(d*x+c)^3*(a
^2+I*a^2*tan(d*x+c))^3/d-8/3003*I*cos(d*x+c)^5*(a^2+I*a^2*tan(d*x+c))^4/d

________________________________________________________________________________________

Rubi [A]  time = 0.25, antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3497, 3488} \[ -\frac {20 i a^2 \cos ^9(c+d x) (a+i a \tan (c+d x))^6}{1287 d}-\frac {20 i a^3 \cos ^7(c+d x) (a+i a \tan (c+d x))^5}{3003 d}-\frac {8 i \cos ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^4}{3003 d}-\frac {8 i a^2 \cos ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^3}{9009 d}-\frac {i \cos ^{13}(c+d x) (a+i a \tan (c+d x))^8}{13 d}-\frac {5 i a \cos ^{11}(c+d x) (a+i a \tan (c+d x))^7}{143 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^13*(a + I*a*Tan[c + d*x])^8,x]

[Out]

(((-20*I)/3003)*a^3*Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^5)/d - (((20*I)/1287)*a^2*Cos[c + d*x]^9*(a + I*a*Ta
n[c + d*x])^6)/d - (((5*I)/143)*a*Cos[c + d*x]^11*(a + I*a*Tan[c + d*x])^7)/d - ((I/13)*Cos[c + d*x]^13*(a + I
*a*Tan[c + d*x])^8)/d - (((8*I)/9009)*a^2*Cos[c + d*x]^3*(a^2 + I*a^2*Tan[c + d*x])^3)/d - (((8*I)/3003)*Cos[c
+ d*x]^5*(a^2 + I*a^2*Tan[c + d*x])^4)/d

Rule 3488

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)/(a*f*m), x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0] &
& EqQ[Simplify[m + n], 0]

Rule 3497

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)/(a*f*m), x] + Dist[(a*(m + n))/(m*d^2), Int[(d*Sec[e + f*x])^(m + 2)*(
a + b*Tan[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0] && GtQ[n, 0] && LtQ[m, -
1] && IntegersQ[2*m, 2*n]

Rubi steps

\begin {align*} \int \cos ^{13}(c+d x) (a+i a \tan (c+d x))^8 \, dx &=-\frac {i \cos ^{13}(c+d x) (a+i a \tan (c+d x))^8}{13 d}+\frac {1}{13} (5 a) \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^7 \, dx\\ &=-\frac {5 i a \cos ^{11}(c+d x) (a+i a \tan (c+d x))^7}{143 d}-\frac {i \cos ^{13}(c+d x) (a+i a \tan (c+d x))^8}{13 d}+\frac {1}{143} \left (20 a^2\right ) \int \cos ^9(c+d x) (a+i a \tan (c+d x))^6 \, dx\\ &=-\frac {20 i a^2 \cos ^9(c+d x) (a+i a \tan (c+d x))^6}{1287 d}-\frac {5 i a \cos ^{11}(c+d x) (a+i a \tan (c+d x))^7}{143 d}-\frac {i \cos ^{13}(c+d x) (a+i a \tan (c+d x))^8}{13 d}+\frac {1}{429} \left (20 a^3\right ) \int \cos ^7(c+d x) (a+i a \tan (c+d x))^5 \, dx\\ &=-\frac {20 i a^3 \cos ^7(c+d x) (a+i a \tan (c+d x))^5}{3003 d}-\frac {20 i a^2 \cos ^9(c+d x) (a+i a \tan (c+d x))^6}{1287 d}-\frac {5 i a \cos ^{11}(c+d x) (a+i a \tan (c+d x))^7}{143 d}-\frac {i \cos ^{13}(c+d x) (a+i a \tan (c+d x))^8}{13 d}+\frac {\left (40 a^4\right ) \int \cos ^5(c+d x) (a+i a \tan (c+d x))^4 \, dx}{3003}\\ &=-\frac {20 i a^3 \cos ^7(c+d x) (a+i a \tan (c+d x))^5}{3003 d}-\frac {20 i a^2 \cos ^9(c+d x) (a+i a \tan (c+d x))^6}{1287 d}-\frac {5 i a \cos ^{11}(c+d x) (a+i a \tan (c+d x))^7}{143 d}-\frac {i \cos ^{13}(c+d x) (a+i a \tan (c+d x))^8}{13 d}-\frac {8 i \cos ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^4}{3003 d}+\frac {\left (8 a^5\right ) \int \cos ^3(c+d x) (a+i a \tan (c+d x))^3 \, dx}{3003}\\ &=-\frac {8 i a^5 \cos ^3(c+d x) (a+i a \tan (c+d x))^3}{9009 d}-\frac {20 i a^3 \cos ^7(c+d x) (a+i a \tan (c+d x))^5}{3003 d}-\frac {20 i a^2 \cos ^9(c+d x) (a+i a \tan (c+d x))^6}{1287 d}-\frac {5 i a \cos ^{11}(c+d x) (a+i a \tan (c+d x))^7}{143 d}-\frac {i \cos ^{13}(c+d x) (a+i a \tan (c+d x))^8}{13 d}-\frac {8 i \cos ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^4}{3003 d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 1.65, size = 111, normalized size = 0.53 \[ \frac {a^8 (-1430 i \sin (c+d x)-2457 i \sin (3 (c+d x))-1155 i \sin (5 (c+d x))+11440 \cos (c+d x)+6552 \cos (3 (c+d x))+1848 \cos (5 (c+d x))) (\sin (8 (c+2 d x))-i \cos (8 (c+2 d x)))}{144144 d (\cos (d x)+i \sin (d x))^8} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^13*(a + I*a*Tan[c + d*x])^8,x]

[Out]

(a^8*(11440*Cos[c + d*x] + 6552*Cos[3*(c + d*x)] + 1848*Cos[5*(c + d*x)] - (1430*I)*Sin[c + d*x] - (2457*I)*Si
n[3*(c + d*x)] - (1155*I)*Sin[5*(c + d*x)])*((-I)*Cos[8*(c + 2*d*x)] + Sin[8*(c + 2*d*x)]))/(144144*d*(Cos[d*x
] + I*Sin[d*x])^8)

________________________________________________________________________________________

fricas [A]  time = 0.57, size = 90, normalized size = 0.43 \[ \frac {-693 i \, a^{8} e^{\left (13 i \, d x + 13 i \, c\right )} - 4095 i \, a^{8} e^{\left (11 i \, d x + 11 i \, c\right )} - 10010 i \, a^{8} e^{\left (9 i \, d x + 9 i \, c\right )} - 12870 i \, a^{8} e^{\left (7 i \, d x + 7 i \, c\right )} - 9009 i \, a^{8} e^{\left (5 i \, d x + 5 i \, c\right )} - 3003 i \, a^{8} e^{\left (3 i \, d x + 3 i \, c\right )}}{288288 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^13*(a+I*a*tan(d*x+c))^8,x, algorithm="fricas")

[Out]

1/288288*(-693*I*a^8*e^(13*I*d*x + 13*I*c) - 4095*I*a^8*e^(11*I*d*x + 11*I*c) - 10010*I*a^8*e^(9*I*d*x + 9*I*c
) - 12870*I*a^8*e^(7*I*d*x + 7*I*c) - 9009*I*a^8*e^(5*I*d*x + 5*I*c) - 3003*I*a^8*e^(3*I*d*x + 3*I*c))/d

________________________________________________________________________________________

giac [B]  time = 18.20, size = 2891, normalized size = 13.70 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^13*(a+I*a*tan(d*x+c))^8,x, algorithm="giac")

[Out]

1/755729694720*(9725263833285*a^8*e^(28*I*d*x + 14*I*c)*log(I*e^(I*d*x + I*c) + 1) + 136153693665990*a^8*e^(26
*I*d*x + 12*I*c)*log(I*e^(I*d*x + I*c) + 1) + 884999008828935*a^8*e^(24*I*d*x + 10*I*c)*log(I*e^(I*d*x + I*c)
+ 1) + 3539996035315740*a^8*e^(22*I*d*x + 8*I*c)*log(I*e^(I*d*x + I*c) + 1) + 9734989097118285*a^8*e^(20*I*d*x
+ 6*I*c)*log(I*e^(I*d*x + I*c) + 1) + 19469978194236570*a^8*e^(18*I*d*x + 4*I*c)*log(I*e^(I*d*x + I*c) + 1) +
29204967291354855*a^8*e^(16*I*d*x + 2*I*c)*log(I*e^(I*d*x + I*c) + 1) + 29204967291354855*a^8*e^(12*I*d*x - 2
*I*c)*log(I*e^(I*d*x + I*c) + 1) + 19469978194236570*a^8*e^(10*I*d*x - 4*I*c)*log(I*e^(I*d*x + I*c) + 1) + 973
4989097118285*a^8*e^(8*I*d*x - 6*I*c)*log(I*e^(I*d*x + I*c) + 1) + 3539996035315740*a^8*e^(6*I*d*x - 8*I*c)*lo
g(I*e^(I*d*x + I*c) + 1) + 884999008828935*a^8*e^(4*I*d*x - 10*I*c)*log(I*e^(I*d*x + I*c) + 1) + 1361536936659
90*a^8*e^(2*I*d*x - 12*I*c)*log(I*e^(I*d*x + I*c) + 1) + 33377105475834120*a^8*e^(14*I*d*x)*log(I*e^(I*d*x + I
*c) + 1) + 9725263833285*a^8*e^(-14*I*c)*log(I*e^(I*d*x + I*c) + 1) + 9720402036345*a^8*e^(28*I*d*x + 14*I*c)*
log(I*e^(I*d*x + I*c) - 1) + 136085628508830*a^8*e^(26*I*d*x + 12*I*c)*log(I*e^(I*d*x + I*c) - 1) + 8845565853
07395*a^8*e^(24*I*d*x + 10*I*c)*log(I*e^(I*d*x + I*c) - 1) + 3538226341229580*a^8*e^(22*I*d*x + 8*I*c)*log(I*e
^(I*d*x + I*c) - 1) + 9730122438381345*a^8*e^(20*I*d*x + 6*I*c)*log(I*e^(I*d*x + I*c) - 1) + 19460244876762690
*a^8*e^(18*I*d*x + 4*I*c)*log(I*e^(I*d*x + I*c) - 1) + 29190367315144035*a^8*e^(16*I*d*x + 2*I*c)*log(I*e^(I*d
*x + I*c) - 1) + 29190367315144035*a^8*e^(12*I*d*x - 2*I*c)*log(I*e^(I*d*x + I*c) - 1) + 19460244876762690*a^8
*e^(10*I*d*x - 4*I*c)*log(I*e^(I*d*x + I*c) - 1) + 9730122438381345*a^8*e^(8*I*d*x - 6*I*c)*log(I*e^(I*d*x + I
*c) - 1) + 3538226341229580*a^8*e^(6*I*d*x - 8*I*c)*log(I*e^(I*d*x + I*c) - 1) + 884556585307395*a^8*e^(4*I*d*
x - 10*I*c)*log(I*e^(I*d*x + I*c) - 1) + 136085628508830*a^8*e^(2*I*d*x - 12*I*c)*log(I*e^(I*d*x + I*c) - 1) +
33360419788736040*a^8*e^(14*I*d*x)*log(I*e^(I*d*x + I*c) - 1) + 9720402036345*a^8*e^(-14*I*c)*log(I*e^(I*d*x
+ I*c) - 1) - 9725263833285*a^8*e^(28*I*d*x + 14*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 136153693665990*a^8*e^(26*
I*d*x + 12*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 884999008828935*a^8*e^(24*I*d*x + 10*I*c)*log(-I*e^(I*d*x + I*c)
+ 1) - 3539996035315740*a^8*e^(22*I*d*x + 8*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 9734989097118285*a^8*e^(20*I*d
*x + 6*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 19469978194236570*a^8*e^(18*I*d*x + 4*I*c)*log(-I*e^(I*d*x + I*c) +
1) - 29204967291354855*a^8*e^(16*I*d*x + 2*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 29204967291354855*a^8*e^(12*I*d*
x - 2*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 19469978194236570*a^8*e^(10*I*d*x - 4*I*c)*log(-I*e^(I*d*x + I*c) + 1
) - 9734989097118285*a^8*e^(8*I*d*x - 6*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 3539996035315740*a^8*e^(6*I*d*x - 8
*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 884999008828935*a^8*e^(4*I*d*x - 10*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 136
153693665990*a^8*e^(2*I*d*x - 12*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 33377105475834120*a^8*e^(14*I*d*x)*log(-I*
e^(I*d*x + I*c) + 1) - 9725263833285*a^8*e^(-14*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 9720402036345*a^8*e^(28*I*d
*x + 14*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 136085628508830*a^8*e^(26*I*d*x + 12*I*c)*log(-I*e^(I*d*x + I*c) -
1) - 884556585307395*a^8*e^(24*I*d*x + 10*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 3538226341229580*a^8*e^(22*I*d*x
+ 8*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 9730122438381345*a^8*e^(20*I*d*x + 6*I*c)*log(-I*e^(I*d*x + I*c) - 1) -
19460244876762690*a^8*e^(18*I*d*x + 4*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 29190367315144035*a^8*e^(16*I*d*x +
2*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 29190367315144035*a^8*e^(12*I*d*x - 2*I*c)*log(-I*e^(I*d*x + I*c) - 1) -
19460244876762690*a^8*e^(10*I*d*x - 4*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 9730122438381345*a^8*e^(8*I*d*x - 6*I
*c)*log(-I*e^(I*d*x + I*c) - 1) - 3538226341229580*a^8*e^(6*I*d*x - 8*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 88455
6585307395*a^8*e^(4*I*d*x - 10*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 136085628508830*a^8*e^(2*I*d*x - 12*I*c)*log
(-I*e^(I*d*x + I*c) - 1) - 33360419788736040*a^8*e^(14*I*d*x)*log(-I*e^(I*d*x + I*c) - 1) - 9720402036345*a^8*
e^(-14*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 4861796940*a^8*e^(28*I*d*x + 14*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 6
8065157160*a^8*e^(26*I*d*x + 12*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 442423521540*a^8*e^(24*I*d*x + 10*I*c)*log(
I*e^(I*d*x) + e^(-I*c)) - 1769694086160*a^8*e^(22*I*d*x + 8*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 4866658736940*a
^8*e^(20*I*d*x + 6*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 9733317473880*a^8*e^(18*I*d*x + 4*I*c)*log(I*e^(I*d*x) +
e^(-I*c)) - 14599976210820*a^8*e^(16*I*d*x + 2*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 14599976210820*a^8*e^(12*I*
d*x - 2*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 9733317473880*a^8*e^(10*I*d*x - 4*I*c)*log(I*e^(I*d*x) + e^(-I*c))
- 4866658736940*a^8*e^(8*I*d*x - 6*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 1769694086160*a^8*e^(6*I*d*x - 8*I*c)*lo
g(I*e^(I*d*x) + e^(-I*c)) - 442423521540*a^8*e^(4*I*d*x - 10*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 68065157160*a^
8*e^(2*I*d*x - 12*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 16685687098080*a^8*e^(14*I*d*x)*log(I*e^(I*d*x) + e^(-I*c
)) - 4861796940*a^8*e^(-14*I*c)*log(I*e^(I*d*x) + e^(-I*c)) + 4861796940*a^8*e^(28*I*d*x + 14*I*c)*log(-I*e^(I
*d*x) + e^(-I*c)) + 68065157160*a^8*e^(26*I*d*x + 12*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 442423521540*a^8*e^(2
4*I*d*x + 10*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 1769694086160*a^8*e^(22*I*d*x + 8*I*c)*log(-I*e^(I*d*x) + e^(
-I*c)) + 4866658736940*a^8*e^(20*I*d*x + 6*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 9733317473880*a^8*e^(18*I*d*x +
4*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 14599976210820*a^8*e^(16*I*d*x + 2*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) +
14599976210820*a^8*e^(12*I*d*x - 2*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 9733317473880*a^8*e^(10*I*d*x - 4*I*c)*
log(-I*e^(I*d*x) + e^(-I*c)) + 4866658736940*a^8*e^(8*I*d*x - 6*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 1769694086
160*a^8*e^(6*I*d*x - 8*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 442423521540*a^8*e^(4*I*d*x - 10*I*c)*log(-I*e^(I*d
*x) + e^(-I*c)) + 68065157160*a^8*e^(2*I*d*x - 12*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 16685687098080*a^8*e^(14
*I*d*x)*log(-I*e^(I*d*x) + e^(-I*c)) + 4861796940*a^8*e^(-14*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) - 1816657920*I*
a^8*e^(41*I*d*x + 27*I*c) - 36168007680*I*a^8*e^(39*I*d*x + 25*I*c) - 341843640320*I*a^8*e^(37*I*d*x + 23*I*c)
- 2039236526080*I*a^8*e^(35*I*d*x + 21*I*c) - 8609784135680*I*a^8*e^(33*I*d*x + 19*I*c) - 27342720204800*I*a^
8*e^(31*I*d*x + 17*I*c) - 67753266380800*I*a^8*e^(29*I*d*x + 15*I*c) - 134089539584000*I*a^8*e^(27*I*d*x + 13*
I*c) - 215146797465600*I*a^8*e^(25*I*d*x + 11*I*c) - 282406740295680*I*a^8*e^(23*I*d*x + 9*I*c) - 304579309731
840*I*a^8*e^(21*I*d*x + 7*I*c) - 269947696578560*I*a^8*e^(19*I*d*x + 5*I*c) - 195820823511040*I*a^8*e^(17*I*d*
x + 3*I*c) - 115246062632960*I*a^8*e^(15*I*d*x + I*c) - 54220889784320*I*a^8*e^(13*I*d*x - I*c) - 199247370649
60*I*a^8*e^(11*I*d*x - 3*I*c) - 5513153085440*I*a^8*e^(9*I*d*x - 5*I*c) - 1080738447360*I*a^8*e^(7*I*d*x - 7*I
*c) - 133827133440*I*a^8*e^(5*I*d*x - 9*I*c) - 7872184320*I*a^8*e^(3*I*d*x - 11*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))

________________________________________________________________________________________

maple [B]  time = 0.73, size = 617, normalized size = 2.92 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^13*(a+I*a*tan(d*x+c))^8,x)

[Out]

1/d*(a^8*(-1/13*sin(d*x+c)^7*cos(d*x+c)^6-7/143*sin(d*x+c)^5*cos(d*x+c)^6-35/1287*sin(d*x+c)^3*cos(d*x+c)^6-5/
429*sin(d*x+c)*cos(d*x+c)^6+1/429*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c))-8*I*a^8*(-1/13*sin(d*x+c)^6*
cos(d*x+c)^7-6/143*sin(d*x+c)^4*cos(d*x+c)^7-8/429*sin(d*x+c)^2*cos(d*x+c)^7-16/3003*cos(d*x+c)^7)-28*a^8*(-1/
13*sin(d*x+c)^5*cos(d*x+c)^8-5/143*sin(d*x+c)^3*cos(d*x+c)^8-5/429*sin(d*x+c)*cos(d*x+c)^8+5/3003*(16/5+cos(d*
x+c)^6+6/5*cos(d*x+c)^4+8/5*cos(d*x+c)^2)*sin(d*x+c))+56*I*a^8*(-1/13*sin(d*x+c)^4*cos(d*x+c)^9-4/143*sin(d*x+
c)^2*cos(d*x+c)^9-8/1287*cos(d*x+c)^9)+70*a^8*(-1/13*sin(d*x+c)^3*cos(d*x+c)^10-3/143*sin(d*x+c)*cos(d*x+c)^10
+1/429*(128/35+cos(d*x+c)^8+8/7*cos(d*x+c)^6+48/35*cos(d*x+c)^4+64/35*cos(d*x+c)^2)*sin(d*x+c))-56*I*a^8*(-1/1
3*sin(d*x+c)^2*cos(d*x+c)^11-2/143*cos(d*x+c)^11)-28*a^8*(-1/13*cos(d*x+c)^12*sin(d*x+c)+1/143*(256/63+cos(d*x
+c)^10+10/9*cos(d*x+c)^8+80/63*cos(d*x+c)^6+32/21*cos(d*x+c)^4+128/63*cos(d*x+c)^2)*sin(d*x+c))-8/13*I*a^8*cos
(d*x+c)^13+1/13*a^8*(1024/231+cos(d*x+c)^12+12/11*cos(d*x+c)^10+40/33*cos(d*x+c)^8+320/231*cos(d*x+c)^6+128/77
*cos(d*x+c)^4+512/231*cos(d*x+c)^2)*sin(d*x+c))

________________________________________________________________________________________

maxima [B]  time = 0.42, size = 405, normalized size = 1.92 \[ -\frac {5544 i \, a^{8} \cos \left (d x + c\right )^{13} + 24 i \, {\left (231 \, \cos \left (d x + c\right )^{13} - 819 \, \cos \left (d x + c\right )^{11} + 1001 \, \cos \left (d x + c\right )^{9} - 429 \, \cos \left (d x + c\right )^{7}\right )} a^{8} + 392 i \, {\left (99 \, \cos \left (d x + c\right )^{13} - 234 \, \cos \left (d x + c\right )^{11} + 143 \, \cos \left (d x + c\right )^{9}\right )} a^{8} + 3528 i \, {\left (11 \, \cos \left (d x + c\right )^{13} - 13 \, \cos \left (d x + c\right )^{11}\right )} a^{8} - 42 \, {\left (1155 \, \sin \left (d x + c\right )^{13} - 5460 \, \sin \left (d x + c\right )^{11} + 10010 \, \sin \left (d x + c\right )^{9} - 8580 \, \sin \left (d x + c\right )^{7} + 3003 \, \sin \left (d x + c\right )^{5}\right )} a^{8} - 28 \, {\left (693 \, \sin \left (d x + c\right )^{13} - 4095 \, \sin \left (d x + c\right )^{11} + 10010 \, \sin \left (d x + c\right )^{9} - 12870 \, \sin \left (d x + c\right )^{7} + 9009 \, \sin \left (d x + c\right )^{5} - 3003 \, \sin \left (d x + c\right )^{3}\right )} a^{8} - 84 \, {\left (231 \, \sin \left (d x + c\right )^{13} - 819 \, \sin \left (d x + c\right )^{11} + 1001 \, \sin \left (d x + c\right )^{9} - 429 \, \sin \left (d x + c\right )^{7}\right )} a^{8} - 3 \, {\left (231 \, \sin \left (d x + c\right )^{13} - 1638 \, \sin \left (d x + c\right )^{11} + 5005 \, \sin \left (d x + c\right )^{9} - 8580 \, \sin \left (d x + c\right )^{7} + 9009 \, \sin \left (d x + c\right )^{5} - 6006 \, \sin \left (d x + c\right )^{3} + 3003 \, \sin \left (d x + c\right )\right )} a^{8} - 7 \, {\left (99 \, \sin \left (d x + c\right )^{13} - 234 \, \sin \left (d x + c\right )^{11} + 143 \, \sin \left (d x + c\right )^{9}\right )} a^{8}}{9009 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^13*(a+I*a*tan(d*x+c))^8,x, algorithm="maxima")

[Out]

-1/9009*(5544*I*a^8*cos(d*x + c)^13 + 24*I*(231*cos(d*x + c)^13 - 819*cos(d*x + c)^11 + 1001*cos(d*x + c)^9 -
429*cos(d*x + c)^7)*a^8 + 392*I*(99*cos(d*x + c)^13 - 234*cos(d*x + c)^11 + 143*cos(d*x + c)^9)*a^8 + 3528*I*(
11*cos(d*x + c)^13 - 13*cos(d*x + c)^11)*a^8 - 42*(1155*sin(d*x + c)^13 - 5460*sin(d*x + c)^11 + 10010*sin(d*x
+ c)^9 - 8580*sin(d*x + c)^7 + 3003*sin(d*x + c)^5)*a^8 - 28*(693*sin(d*x + c)^13 - 4095*sin(d*x + c)^11 + 10
010*sin(d*x + c)^9 - 12870*sin(d*x + c)^7 + 9009*sin(d*x + c)^5 - 3003*sin(d*x + c)^3)*a^8 - 84*(231*sin(d*x +
c)^13 - 819*sin(d*x + c)^11 + 1001*sin(d*x + c)^9 - 429*sin(d*x + c)^7)*a^8 - 3*(231*sin(d*x + c)^13 - 1638*s
in(d*x + c)^11 + 5005*sin(d*x + c)^9 - 8580*sin(d*x + c)^7 + 9009*sin(d*x + c)^5 - 6006*sin(d*x + c)^3 + 3003*
sin(d*x + c))*a^8 - 7*(99*sin(d*x + c)^13 - 234*sin(d*x + c)^11 + 143*sin(d*x + c)^9)*a^8)/d

________________________________________________________________________________________

mupad [B]  time = 4.08, size = 93, normalized size = 0.44 \[ -\frac {a^8\,\left (\frac {{\mathrm {e}}^{c\,3{}\mathrm {i}+d\,x\,3{}\mathrm {i}}\,1{}\mathrm {i}}{96}+\frac {{\mathrm {e}}^{c\,5{}\mathrm {i}+d\,x\,5{}\mathrm {i}}\,1{}\mathrm {i}}{32}+\frac {{\mathrm {e}}^{c\,7{}\mathrm {i}+d\,x\,7{}\mathrm {i}}\,5{}\mathrm {i}}{112}+\frac {{\mathrm {e}}^{c\,9{}\mathrm {i}+d\,x\,9{}\mathrm {i}}\,5{}\mathrm {i}}{144}+\frac {{\mathrm {e}}^{c\,11{}\mathrm {i}+d\,x\,11{}\mathrm {i}}\,5{}\mathrm {i}}{352}+\frac {{\mathrm {e}}^{c\,13{}\mathrm {i}+d\,x\,13{}\mathrm {i}}\,1{}\mathrm {i}}{416}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^13*(a + a*tan(c + d*x)*1i)^8,x)

[Out]

-(a^8*((exp(c*3i + d*x*3i)*1i)/96 + (exp(c*5i + d*x*5i)*1i)/32 + (exp(c*7i + d*x*7i)*5i)/112 + (exp(c*9i + d*x
*9i)*5i)/144 + (exp(c*11i + d*x*11i)*5i)/352 + (exp(c*13i + d*x*13i)*1i)/416))/d

________________________________________________________________________________________

sympy [A]  time = 1.40, size = 241, normalized size = 1.14 \[ \begin {cases} \frac {- 17439916032 i a^{8} d^{5} e^{13 i c} e^{13 i d x} - 103054049280 i a^{8} d^{5} e^{11 i c} e^{11 i d x} - 251909898240 i a^{8} d^{5} e^{9 i c} e^{9 i d x} - 323884154880 i a^{8} d^{5} e^{7 i c} e^{7 i d x} - 226718908416 i a^{8} d^{5} e^{5 i c} e^{5 i d x} - 75572969472 i a^{8} d^{5} e^{3 i c} e^{3 i d x}}{7255005069312 d^{6}} & \text {for}\: 7255005069312 d^{6} \neq 0 \\x \left (\frac {a^{8} e^{13 i c}}{32} + \frac {5 a^{8} e^{11 i c}}{32} + \frac {5 a^{8} e^{9 i c}}{16} + \frac {5 a^{8} e^{7 i c}}{16} + \frac {5 a^{8} e^{5 i c}}{32} + \frac {a^{8} e^{3 i c}}{32}\right ) & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**13*(a+I*a*tan(d*x+c))**8,x)

[Out]

Piecewise(((-17439916032*I*a**8*d**5*exp(13*I*c)*exp(13*I*d*x) - 103054049280*I*a**8*d**5*exp(11*I*c)*exp(11*I
*d*x) - 251909898240*I*a**8*d**5*exp(9*I*c)*exp(9*I*d*x) - 323884154880*I*a**8*d**5*exp(7*I*c)*exp(7*I*d*x) -
226718908416*I*a**8*d**5*exp(5*I*c)*exp(5*I*d*x) - 75572969472*I*a**8*d**5*exp(3*I*c)*exp(3*I*d*x))/(725500506
9312*d**6), Ne(7255005069312*d**6, 0)), (x*(a**8*exp(13*I*c)/32 + 5*a**8*exp(11*I*c)/32 + 5*a**8*exp(9*I*c)/16
+ 5*a**8*exp(7*I*c)/16 + 5*a**8*exp(5*I*c)/32 + a**8*exp(3*I*c)/32), True))

________________________________________________________________________________________