3.1.98 \(\int \cos ^{15}(c+d x) (a+i a \tan (c+d x))^8 \, dx\) [98]
Optimal. Leaf size=212 \[ \frac {7 a^8 \sin (c+d x)}{1287 d}-\frac {7 a^8 \sin ^3(c+d x)}{1287 d}+\frac {7 a^8 \sin ^5(c+d x)}{2145 d}-\frac {a^8 \sin ^7(c+d x)}{1287 d}-\frac {2 i a^3 \cos ^{13}(c+d x) (a+i a \tan (c+d x))^5}{195 d}-\frac {2 i a \cos ^{15}(c+d x) (a+i a \tan (c+d x))^7}{15 d}-\frac {2 i a^2 \cos ^{11}(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^3}{715 d}-\frac {2 i \cos ^9(c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{1287 d} \]
[Out]
7/1287*a^8*sin(d*x+c)/d-7/1287*a^8*sin(d*x+c)^3/d+7/2145*a^8*sin(d*x+c)^5/d-1/1287*a^8*sin(d*x+c)^7/d-2/195*I*
a^3*cos(d*x+c)^13*(a+I*a*tan(d*x+c))^5/d-2/15*I*a*cos(d*x+c)^15*(a+I*a*tan(d*x+c))^7/d-2/715*I*a^2*cos(d*x+c)^
11*(a^2+I*a^2*tan(d*x+c))^3/d-2/1287*I*cos(d*x+c)^9*(a^8+I*a^8*tan(d*x+c))/d
________________________________________________________________________________________
Rubi [A]
time = 0.16, antiderivative size = 212, 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 = {3577, 2713}
\begin {gather*} -\frac {a^8 \sin ^7(c+d x)}{1287 d}+\frac {7 a^8 \sin ^5(c+d x)}{2145 d}-\frac {7 a^8 \sin ^3(c+d x)}{1287 d}+\frac {7 a^8 \sin (c+d x)}{1287 d}-\frac {2 i \cos ^9(c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{1287 d}-\frac {2 i a^3 \cos ^{13}(c+d x) (a+i a \tan (c+d x))^5}{195 d}-\frac {2 i a^2 \cos ^{11}(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^3}{715 d}-\frac {2 i a \cos ^{15}(c+d x) (a+i a \tan (c+d x))^7}{15 d} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Cos[c + d*x]^15*(a + I*a*Tan[c + d*x])^8,x]
[Out]
(7*a^8*Sin[c + d*x])/(1287*d) - (7*a^8*Sin[c + d*x]^3)/(1287*d) + (7*a^8*Sin[c + d*x]^5)/(2145*d) - (a^8*Sin[c
+ d*x]^7)/(1287*d) - (((2*I)/195)*a^3*Cos[c + d*x]^13*(a + I*a*Tan[c + d*x])^5)/d - (((2*I)/15)*a*Cos[c + d*x
]^15*(a + I*a*Tan[c + d*x])^7)/d - (((2*I)/715)*a^2*Cos[c + d*x]^11*(a^2 + I*a^2*Tan[c + d*x])^3)/d - (((2*I)/
1287)*Cos[c + d*x]^9*(a^8 + I*a^8*Tan[c + d*x]))/d
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 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]
Rubi steps
\begin {align*} \int \cos ^{15}(c+d x) (a+i a \tan (c+d x))^8 \, dx &=-\frac {2 i a \cos ^{15}(c+d x) (a+i a \tan (c+d x))^7}{15 d}+\frac {1}{15} a^2 \int \cos ^{13}(c+d x) (a+i a \tan (c+d x))^6 \, dx\\ &=-\frac {2 i a^3 \cos ^{13}(c+d x) (a+i a \tan (c+d x))^5}{195 d}-\frac {2 i a \cos ^{15}(c+d x) (a+i a \tan (c+d x))^7}{15 d}+\frac {1}{65} a^4 \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^4 \, dx\\ &=-\frac {2 i a^5 \cos ^{11}(c+d x) (a+i a \tan (c+d x))^3}{715 d}-\frac {2 i a^3 \cos ^{13}(c+d x) (a+i a \tan (c+d x))^5}{195 d}-\frac {2 i a \cos ^{15}(c+d x) (a+i a \tan (c+d x))^7}{15 d}+\frac {1}{143} a^6 \int \cos ^9(c+d x) (a+i a \tan (c+d x))^2 \, dx\\ &=-\frac {2 i a^5 \cos ^{11}(c+d x) (a+i a \tan (c+d x))^3}{715 d}-\frac {2 i a^3 \cos ^{13}(c+d x) (a+i a \tan (c+d x))^5}{195 d}-\frac {2 i a \cos ^{15}(c+d x) (a+i a \tan (c+d x))^7}{15 d}-\frac {2 i \cos ^9(c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{1287 d}+\frac {\left (7 a^8\right ) \int \cos ^7(c+d x) \, dx}{1287}\\ &=-\frac {2 i a^5 \cos ^{11}(c+d x) (a+i a \tan (c+d x))^3}{715 d}-\frac {2 i a^3 \cos ^{13}(c+d x) (a+i a \tan (c+d x))^5}{195 d}-\frac {2 i a \cos ^{15}(c+d x) (a+i a \tan (c+d x))^7}{15 d}-\frac {2 i \cos ^9(c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{1287 d}-\frac {\left (7 a^8\right ) \text {Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,-\sin (c+d x)\right )}{1287 d}\\ &=\frac {7 a^8 \sin (c+d x)}{1287 d}-\frac {7 a^8 \sin ^3(c+d x)}{1287 d}+\frac {7 a^8 \sin ^5(c+d x)}{2145 d}-\frac {a^8 \sin ^7(c+d x)}{1287 d}-\frac {2 i a^5 \cos ^{11}(c+d x) (a+i a \tan (c+d x))^3}{715 d}-\frac {2 i a^3 \cos ^{13}(c+d x) (a+i a \tan (c+d x))^5}{195 d}-\frac {2 i a \cos ^{15}(c+d x) (a+i a \tan (c+d x))^7}{15 d}-\frac {2 i \cos ^9(c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{1287 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 1.81, size = 133, normalized size = 0.63 \begin {gather*} \frac {a^8 (28600 \cos (c+d x)+19656 \cos (3 (c+d x))+9240 \cos (5 (c+d x))+3432 \cos (7 (c+d x))-3575 i \sin (c+d x)-7371 i \sin (3 (c+d x))-5775 i \sin (5 (c+d x))-3003 i \sin (7 (c+d x))) (-i \cos (8 (c+2 d x))+\sin (8 (c+2 d x)))}{411840 d (\cos (d x)+i \sin (d x))^8} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Cos[c + d*x]^15*(a + I*a*Tan[c + d*x])^8,x]
[Out]
(a^8*(28600*Cos[c + d*x] + 19656*Cos[3*(c + d*x)] + 9240*Cos[5*(c + d*x)] + 3432*Cos[7*(c + d*x)] - (3575*I)*S
in[c + d*x] - (7371*I)*Sin[3*(c + d*x)] - (5775*I)*Sin[5*(c + d*x)] - (3003*I)*Sin[7*(c + d*x)])*((-I)*Cos[8*(
c + 2*d*x)] + Sin[8*(c + 2*d*x)]))/(411840*d*(Cos[d*x] + I*Sin[d*x])^8)
________________________________________________________________________________________
Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 666 vs. \(2 (188 ) = 376\).
time = 0.30, size = 667, normalized size = 3.15
| | |
| method |
result |
size |
| | |
| risch |
\(-\frac {i a^{8} {\mathrm e}^{15 i \left (d x +c \right )}}{1920 d}-\frac {7 i a^{8} {\mathrm e}^{13 i \left (d x +c \right )}}{1664 d}-\frac {21 i a^{8} {\mathrm e}^{11 i \left (d x +c \right )}}{1408 d}-\frac {35 i a^{8} {\mathrm e}^{9 i \left (d x +c \right )}}{1152 d}-\frac {5 i a^{8} {\mathrm e}^{7 i \left (d x +c \right )}}{128 d}-\frac {21 i a^{8} {\mathrm e}^{5 i \left (d x +c \right )}}{640 d}-\frac {7 i a^{8} {\mathrm e}^{3 i \left (d x +c \right )}}{384 d}-\frac {i a^{8} {\mathrm e}^{i \left (d x +c \right )}}{128 d}\) |
\(146\) |
| derivativedivides |
\(\text {Expression too large to display}\) |
\(667\) |
| default |
\(\text {Expression too large to display}\) |
\(667\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(d*x+c)^15*(a+I*a*tan(d*x+c))^8,x,method=_RETURNVERBOSE)
[Out]
1/d*(a^8*(-1/15*sin(d*x+c)^7*cos(d*x+c)^8-7/195*sin(d*x+c)^5*cos(d*x+c)^8-7/429*sin(d*x+c)^3*cos(d*x+c)^8-7/12
87*sin(d*x+c)*cos(d*x+c)^8+1/1287*(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))-8*I*a^8*(-
1/15*sin(d*x+c)^6*cos(d*x+c)^9-2/65*sin(d*x+c)^4*cos(d*x+c)^9-8/715*sin(d*x+c)^2*cos(d*x+c)^9-16/6435*cos(d*x+
c)^9)-28*a^8*(-1/15*sin(d*x+c)^5*cos(d*x+c)^10-1/39*sin(d*x+c)^3*cos(d*x+c)^10-1/143*cos(d*x+c)^10*sin(d*x+c)+
1/1287*(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
5*sin(d*x+c)^4*cos(d*x+c)^11-4/195*sin(d*x+c)^2*cos(d*x+c)^11-8/2145*cos(d*x+c)^11)+70*a^8*(-1/15*sin(d*x+c)^3
*cos(d*x+c)^12-1/65*sin(d*x+c)*cos(d*x+c)^12+1/715*(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))-56*I*a^8*(-1/15*sin(d*x+c)^2*cos(d*x+c)^13-2/195*cos(d*x+c
)^13)-28*a^8*(-1/15*sin(d*x+c)*cos(d*x+c)^14+1/195*(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))-8/15*I*a^8*cos(d*x+c)^15+1/15*a
^8*(2048/429+cos(d*x+c)^14+14/13*cos(d*x+c)^12+168/143*cos(d*x+c)^10+560/429*cos(d*x+c)^8+640/429*cos(d*x+c)^6
+256/143*cos(d*x+c)^4+1024/429*cos(d*x+c)^2)*sin(d*x+c))
________________________________________________________________________________________
Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 453 vs. \(2 (180) = 360\).
time = 0.30, size = 453, normalized size = 2.14 \begin {gather*} -\frac {3432 i \, a^{8} \cos \left (d x + c\right )^{15} + 8 i \, {\left (429 \, \cos \left (d x + c\right )^{15} - 1485 \, \cos \left (d x + c\right )^{13} + 1755 \, \cos \left (d x + c\right )^{11} - 715 \, \cos \left (d x + c\right )^{9}\right )} a^{8} + 168 i \, {\left (143 \, \cos \left (d x + c\right )^{15} - 330 \, \cos \left (d x + c\right )^{13} + 195 \, \cos \left (d x + c\right )^{11}\right )} a^{8} + 1848 i \, {\left (13 \, \cos \left (d x + c\right )^{15} - 15 \, \cos \left (d x + c\right )^{13}\right )} a^{8} + 4 \, {\left (3003 \, \sin \left (d x + c\right )^{15} - 13860 \, \sin \left (d x + c\right )^{13} + 24570 \, \sin \left (d x + c\right )^{11} - 20020 \, \sin \left (d x + c\right )^{9} + 6435 \, \sin \left (d x + c\right )^{7}\right )} a^{8} + 10 \, {\left (3003 \, \sin \left (d x + c\right )^{15} - 17325 \, \sin \left (d x + c\right )^{13} + 40950 \, \sin \left (d x + c\right )^{11} - 50050 \, \sin \left (d x + c\right )^{9} + 32175 \, \sin \left (d x + c\right )^{7} - 9009 \, \sin \left (d x + c\right )^{5}\right )} a^{8} + 4 \, {\left (3003 \, \sin \left (d x + c\right )^{15} - 20790 \, \sin \left (d x + c\right )^{13} + 61425 \, \sin \left (d x + c\right )^{11} - 100100 \, \sin \left (d x + c\right )^{9} + 96525 \, \sin \left (d x + c\right )^{7} - 54054 \, \sin \left (d x + c\right )^{5} + 15015 \, \sin \left (d x + c\right )^{3}\right )} a^{8} + {\left (429 \, \sin \left (d x + c\right )^{15} - 1485 \, \sin \left (d x + c\right )^{13} + 1755 \, \sin \left (d x + c\right )^{11} - 715 \, \sin \left (d x + c\right )^{9}\right )} a^{8} + {\left (429 \, \sin \left (d x + c\right )^{15} - 3465 \, \sin \left (d x + c\right )^{13} + 12285 \, \sin \left (d x + c\right )^{11} - 25025 \, \sin \left (d x + c\right )^{9} + 32175 \, \sin \left (d x + c\right )^{7} - 27027 \, \sin \left (d x + c\right )^{5} + 15015 \, \sin \left (d x + c\right )^{3} - 6435 \, \sin \left (d x + c\right )\right )} a^{8}}{6435 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^15*(a+I*a*tan(d*x+c))^8,x, algorithm="maxima")
[Out]
-1/6435*(3432*I*a^8*cos(d*x + c)^15 + 8*I*(429*cos(d*x + c)^15 - 1485*cos(d*x + c)^13 + 1755*cos(d*x + c)^11 -
715*cos(d*x + c)^9)*a^8 + 168*I*(143*cos(d*x + c)^15 - 330*cos(d*x + c)^13 + 195*cos(d*x + c)^11)*a^8 + 1848*
I*(13*cos(d*x + c)^15 - 15*cos(d*x + c)^13)*a^8 + 4*(3003*sin(d*x + c)^15 - 13860*sin(d*x + c)^13 + 24570*sin(
d*x + c)^11 - 20020*sin(d*x + c)^9 + 6435*sin(d*x + c)^7)*a^8 + 10*(3003*sin(d*x + c)^15 - 17325*sin(d*x + c)^
13 + 40950*sin(d*x + c)^11 - 50050*sin(d*x + c)^9 + 32175*sin(d*x + c)^7 - 9009*sin(d*x + c)^5)*a^8 + 4*(3003*
sin(d*x + c)^15 - 20790*sin(d*x + c)^13 + 61425*sin(d*x + c)^11 - 100100*sin(d*x + c)^9 + 96525*sin(d*x + c)^7
- 54054*sin(d*x + c)^5 + 15015*sin(d*x + c)^3)*a^8 + (429*sin(d*x + c)^15 - 1485*sin(d*x + c)^13 + 1755*sin(d
*x + c)^11 - 715*sin(d*x + c)^9)*a^8 + (429*sin(d*x + c)^15 - 3465*sin(d*x + c)^13 + 12285*sin(d*x + c)^11 - 2
5025*sin(d*x + c)^9 + 32175*sin(d*x + c)^7 - 27027*sin(d*x + c)^5 + 15015*sin(d*x + c)^3 - 6435*sin(d*x + c))*
a^8)/d
________________________________________________________________________________________
Fricas [A]
time = 0.43, size = 118, normalized size = 0.56 \begin {gather*} \frac {-429 i \, a^{8} e^{\left (15 i \, d x + 15 i \, c\right )} - 3465 i \, a^{8} e^{\left (13 i \, d x + 13 i \, c\right )} - 12285 i \, a^{8} e^{\left (11 i \, d x + 11 i \, c\right )} - 25025 i \, a^{8} e^{\left (9 i \, d x + 9 i \, c\right )} - 32175 i \, a^{8} e^{\left (7 i \, d x + 7 i \, c\right )} - 27027 i \, a^{8} e^{\left (5 i \, d x + 5 i \, c\right )} - 15015 i \, a^{8} e^{\left (3 i \, d x + 3 i \, c\right )} - 6435 i \, a^{8} e^{\left (i \, d x + i \, c\right )}}{823680 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^15*(a+I*a*tan(d*x+c))^8,x, algorithm="fricas")
[Out]
1/823680*(-429*I*a^8*e^(15*I*d*x + 15*I*c) - 3465*I*a^8*e^(13*I*d*x + 13*I*c) - 12285*I*a^8*e^(11*I*d*x + 11*I
*c) - 25025*I*a^8*e^(9*I*d*x + 9*I*c) - 32175*I*a^8*e^(7*I*d*x + 7*I*c) - 27027*I*a^8*e^(5*I*d*x + 5*I*c) - 15
015*I*a^8*e^(3*I*d*x + 3*I*c) - 6435*I*a^8*e^(I*d*x + I*c))/d
________________________________________________________________________________________
Sympy [A]
time = 0.94, size = 313, normalized size = 1.48 \begin {gather*} \begin {cases} \frac {- 10867748850798428160 i a^{8} d^{7} e^{15 i c} e^{15 i d x} - 87777971487218073600 i a^{8} d^{7} e^{13 i c} e^{13 i d x} - 311212808000136806400 i a^{8} d^{7} e^{11 i c} e^{11 i d x} - 633952016296574976000 i a^{8} d^{7} e^{9 i c} e^{9 i d x} - 815081163809882112000 i a^{8} d^{7} e^{7 i c} e^{7 i d x} - 684668177600300974080 i a^{8} d^{7} e^{5 i c} e^{5 i d x} - 380371209777944985600 i a^{8} d^{7} e^{3 i c} e^{3 i d x} - 163016232761976422400 i a^{8} d^{7} e^{i c} e^{i d x}}{20866077793532982067200 d^{8}} & \text {for}\: d^{8} \neq 0 \\x \left (\frac {a^{8} e^{15 i c}}{128} + \frac {7 a^{8} e^{13 i c}}{128} + \frac {21 a^{8} e^{11 i c}}{128} + \frac {35 a^{8} e^{9 i c}}{128} + \frac {35 a^{8} e^{7 i c}}{128} + \frac {21 a^{8} e^{5 i c}}{128} + \frac {7 a^{8} e^{3 i c}}{128} + \frac {a^{8} e^{i c}}{128}\right ) & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)**15*(a+I*a*tan(d*x+c))**8,x)
[Out]
Piecewise(((-10867748850798428160*I*a**8*d**7*exp(15*I*c)*exp(15*I*d*x) - 87777971487218073600*I*a**8*d**7*exp
(13*I*c)*exp(13*I*d*x) - 311212808000136806400*I*a**8*d**7*exp(11*I*c)*exp(11*I*d*x) - 633952016296574976000*I
*a**8*d**7*exp(9*I*c)*exp(9*I*d*x) - 815081163809882112000*I*a**8*d**7*exp(7*I*c)*exp(7*I*d*x) - 6846681776003
00974080*I*a**8*d**7*exp(5*I*c)*exp(5*I*d*x) - 380371209777944985600*I*a**8*d**7*exp(3*I*c)*exp(3*I*d*x) - 163
016232761976422400*I*a**8*d**7*exp(I*c)*exp(I*d*x))/(20866077793532982067200*d**8), Ne(d**8, 0)), (x*(a**8*exp
(15*I*c)/128 + 7*a**8*exp(13*I*c)/128 + 21*a**8*exp(11*I*c)/128 + 35*a**8*exp(9*I*c)/128 + 35*a**8*exp(7*I*c)/
128 + 21*a**8*exp(5*I*c)/128 + 7*a**8*exp(3*I*c)/128 + a**8*exp(I*c)/128), True))
________________________________________________________________________________________
Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 2919 vs. \(2 (180) = 360\).
time = 1.81, size = 2919, normalized size = 13.77 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^15*(a+I*a*tan(d*x+c))^8,x, algorithm="giac")
[Out]
1/863691079680*(5682101344920*a^8*e^(28*I*d*x + 14*I*c)*log(I*e^(I*d*x + I*c) + 1) + 79549418828880*a^8*e^(26*
I*d*x + 12*I*c)*log(I*e^(I*d*x + I*c) + 1) + 517071222387720*a^8*e^(24*I*d*x + 10*I*c)*log(I*e^(I*d*x + I*c) +
1) + 2068284889550880*a^8*e^(22*I*d*x + 8*I*c)*log(I*e^(I*d*x + I*c) + 1) + 5687783446264920*a^8*e^(20*I*d*x
+ 6*I*c)*log(I*e^(I*d*x + I*c) + 1) + 11375566892529840*a^8*e^(18*I*d*x + 4*I*c)*log(I*e^(I*d*x + I*c) + 1) +
17063350338794760*a^8*e^(16*I*d*x + 2*I*c)*log(I*e^(I*d*x + I*c) + 1) + 17063350338794760*a^8*e^(12*I*d*x - 2*
I*c)*log(I*e^(I*d*x + I*c) + 1) + 11375566892529840*a^8*e^(10*I*d*x - 4*I*c)*log(I*e^(I*d*x + I*c) + 1) + 5687
783446264920*a^8*e^(8*I*d*x - 6*I*c)*log(I*e^(I*d*x + I*c) + 1) + 2068284889550880*a^8*e^(6*I*d*x - 8*I*c)*log
(I*e^(I*d*x + I*c) + 1) + 517071222387720*a^8*e^(4*I*d*x - 10*I*c)*log(I*e^(I*d*x + I*c) + 1) + 79549418828880
*a^8*e^(2*I*d*x - 12*I*c)*log(I*e^(I*d*x + I*c) + 1) + 19500971815765440*a^8*e^(14*I*d*x)*log(I*e^(I*d*x + I*c
) + 1) + 5682101344920*a^8*e^(-14*I*c)*log(I*e^(I*d*x + I*c) + 1) + 5674116082635*a^8*e^(28*I*d*x + 14*I*c)*lo
g(I*e^(I*d*x + I*c) - 1) + 79437625156890*a^8*e^(26*I*d*x + 12*I*c)*log(I*e^(I*d*x + I*c) - 1) + 5163445635197
85*a^8*e^(24*I*d*x + 10*I*c)*log(I*e^(I*d*x + I*c) - 1) + 2065378254079140*a^8*e^(22*I*d*x + 8*I*c)*log(I*e^(I
*d*x + I*c) - 1) + 5679790198717635*a^8*e^(20*I*d*x + 6*I*c)*log(I*e^(I*d*x + I*c) - 1) + 11359580397435270*a^
8*e^(18*I*d*x + 4*I*c)*log(I*e^(I*d*x + I*c) - 1) + 17039370596152905*a^8*e^(16*I*d*x + 2*I*c)*log(I*e^(I*d*x
+ I*c) - 1) + 17039370596152905*a^8*e^(12*I*d*x - 2*I*c)*log(I*e^(I*d*x + I*c) - 1) + 11359580397435270*a^8*e^
(10*I*d*x - 4*I*c)*log(I*e^(I*d*x + I*c) - 1) + 5679790198717635*a^8*e^(8*I*d*x - 6*I*c)*log(I*e^(I*d*x + I*c)
- 1) + 2065378254079140*a^8*e^(6*I*d*x - 8*I*c)*log(I*e^(I*d*x + I*c) - 1) + 516344563519785*a^8*e^(4*I*d*x -
10*I*c)*log(I*e^(I*d*x + I*c) - 1) + 79437625156890*a^8*e^(2*I*d*x - 12*I*c)*log(I*e^(I*d*x + I*c) - 1) + 194
73566395603320*a^8*e^(14*I*d*x)*log(I*e^(I*d*x + I*c) - 1) + 5674116082635*a^8*e^(-14*I*c)*log(I*e^(I*d*x + I*
c) - 1) - 5682101344920*a^8*e^(28*I*d*x + 14*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 79549418828880*a^8*e^(26*I*d*x
+ 12*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 517071222387720*a^8*e^(24*I*d*x + 10*I*c)*log(-I*e^(I*d*x + I*c) + 1)
- 2068284889550880*a^8*e^(22*I*d*x + 8*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 5687783446264920*a^8*e^(20*I*d*x +
6*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 11375566892529840*a^8*e^(18*I*d*x + 4*I*c)*log(-I*e^(I*d*x + I*c) + 1) -
17063350338794760*a^8*e^(16*I*d*x + 2*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 17063350338794760*a^8*e^(12*I*d*x - 2
*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 11375566892529840*a^8*e^(10*I*d*x - 4*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 5
687783446264920*a^8*e^(8*I*d*x - 6*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 2068284889550880*a^8*e^(6*I*d*x - 8*I*c)
*log(-I*e^(I*d*x + I*c) + 1) - 517071222387720*a^8*e^(4*I*d*x - 10*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 79549418
828880*a^8*e^(2*I*d*x - 12*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 19500971815765440*a^8*e^(14*I*d*x)*log(-I*e^(I*d
*x + I*c) + 1) - 5682101344920*a^8*e^(-14*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 5674116082635*a^8*e^(28*I*d*x + 1
4*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 79437625156890*a^8*e^(26*I*d*x + 12*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 51
6344563519785*a^8*e^(24*I*d*x + 10*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 2065378254079140*a^8*e^(22*I*d*x + 8*I*c
)*log(-I*e^(I*d*x + I*c) - 1) - 5679790198717635*a^8*e^(20*I*d*x + 6*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 113595
80397435270*a^8*e^(18*I*d*x + 4*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 17039370596152905*a^8*e^(16*I*d*x + 2*I*c)*
log(-I*e^(I*d*x + I*c) - 1) - 17039370596152905*a^8*e^(12*I*d*x - 2*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 1135958
0397435270*a^8*e^(10*I*d*x - 4*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 5679790198717635*a^8*e^(8*I*d*x - 6*I*c)*log
(-I*e^(I*d*x + I*c) - 1) - 2065378254079140*a^8*e^(6*I*d*x - 8*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 516344563519
785*a^8*e^(4*I*d*x - 10*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 79437625156890*a^8*e^(2*I*d*x - 12*I*c)*log(-I*e^(I
*d*x + I*c) - 1) - 19473566395603320*a^8*e^(14*I*d*x)*log(-I*e^(I*d*x + I*c) - 1) - 5674116082635*a^8*e^(-14*I
*c)*log(-I*e^(I*d*x + I*c) - 1) - 7985262285*a^8*e^(28*I*d*x + 14*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 111793671
990*a^8*e^(26*I*d*x + 12*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 726658867935*a^8*e^(24*I*d*x + 10*I*c)*log(I*e^(I*
d*x) + e^(-I*c)) - 2906635471740*a^8*e^(22*I*d*x + 8*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 7993247547285*a^8*e^(2
0*I*d*x + 6*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 15986495094570*a^8*e^(18*I*d*x + 4*I*c)*log(I*e^(I*d*x) + e^(-I
*c)) - 23979742641855*a^8*e^(16*I*d*x + 2*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 23979742641855*a^8*e^(12*I*d*x -
2*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 15986495094570*a^8*e^(10*I*d*x - 4*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 799
3247547285*a^8*e^(8*I*d*x - 6*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 2906635471740*a^8*e^(6*I*d*x - 8*I*c)*log(I*e
^(I*d*x) + e^(-I*c)) - 726658867935*a^8*e^(4*I*...
________________________________________________________________________________________
Mupad [B]
time = 5.26, size = 222, normalized size = 1.05 \begin {gather*} \frac {2\,a^8\,\left (2\,{\sin \left (\frac {c}{4}+\frac {d\,x}{4}\right )}^2-1\right )\,\left (-\frac {44779\,{\sin \left (c+d\,x\right )}^2}{32}+\frac {\sin \left (c+d\,x\right )\,32175{}\mathrm {i}}{128}-\frac {26075\,{\sin \left (2\,c+2\,d\,x\right )}^2}{16}-\frac {\sin \left (2\,c+2\,d\,x\right )\,3575{}\mathrm {i}}{8}+\frac {114583\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{64}-\frac {57925\,{\sin \left (3\,c+3\,d\,x\right )}^2}{32}+\frac {\sin \left (3\,c+3\,d\,x\right )\,84227{}\mathrm {i}}{128}+\frac {116585\,{\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}^2}{64}+\frac {119315\,{\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}^2}{64}+\frac {122285\,{\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}^2}{64}-\sin \left (4\,c+4\,d\,x\right )\,754{}\mathrm {i}+\frac {\sin \left (5\,c+5\,d\,x\right )\,111527{}\mathrm {i}}{128}-\frac {\sin \left (6\,c+6\,d\,x\right )\,7187{}\mathrm {i}}{8}+\frac {\sin \left (7\,c+7\,d\,x\right )\,121427{}\mathrm {i}}{128}-952\right )}{6435\,d\,\left (-{\sin \left (\frac {15\,c}{4}+\frac {15\,d\,x}{4}\right )}^2\,2{}\mathrm {i}+\sin \left (\frac {15\,c}{2}+\frac {15\,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)^15*(a + a*tan(c + d*x)*1i)^8,x)
[Out]
(2*a^8*(2*sin(c/4 + (d*x)/4)^2 - 1)*((sin(c + d*x)*32175i)/128 - (sin(2*c + 2*d*x)*3575i)/8 + (sin(3*c + 3*d*x
)*84227i)/128 - sin(4*c + 4*d*x)*754i + (sin(5*c + 5*d*x)*111527i)/128 - (sin(6*c + 6*d*x)*7187i)/8 + (sin(7*c
+ 7*d*x)*121427i)/128 - (26075*sin(2*c + 2*d*x)^2)/16 + (114583*sin(c/2 + (d*x)/2)^2)/64 - (57925*sin(3*c + 3
*d*x)^2)/32 + (116585*sin((3*c)/2 + (3*d*x)/2)^2)/64 + (119315*sin((5*c)/2 + (5*d*x)/2)^2)/64 + (122285*sin((7
*c)/2 + (7*d*x)/2)^2)/64 - (44779*sin(c + d*x)^2)/32 - 952))/(6435*d*(sin((15*c)/2 + (15*d*x)/2) - sin((15*c)/
4 + (15*d*x)/4)^2*2i + 1i))
________________________________________________________________________________________