3.1.96 \(\int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^8 \, dx\) [96]

Optimal. Leaf size=136 \[ -\frac {2 i a^3 \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{1155 d}-\frac {2 i a^2 \cos ^7(c+d x) (a+i a \tan (c+d x))^6}{231 d}-\frac {i a \cos ^9(c+d x) (a+i a \tan (c+d x))^7}{33 d}-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^8}{11 d} \]

[Out]

-2/1155*I*a^3*cos(d*x+c)^5*(a+I*a*tan(d*x+c))^5/d-2/231*I*a^2*cos(d*x+c)^7*(a+I*a*tan(d*x+c))^6/d-1/33*I*a*cos
(d*x+c)^9*(a+I*a*tan(d*x+c))^7/d-1/11*I*cos(d*x+c)^11*(a+I*a*tan(d*x+c))^8/d

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Rubi [A]
time = 0.12, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3578, 3569} \begin {gather*} -\frac {2 i a^3 \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{1155 d}-\frac {2 i a^2 \cos ^7(c+d x) (a+i a \tan (c+d x))^6}{231 d}-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^8}{11 d}-\frac {i a \cos ^9(c+d x) (a+i a \tan (c+d x))^7}{33 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^11*(a + I*a*Tan[c + d*x])^8,x]

[Out]

(((-2*I)/1155)*a^3*Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^5)/d - (((2*I)/231)*a^2*Cos[c + d*x]^7*(a + I*a*Tan[c
+ d*x])^6)/d - ((I/33)*a*Cos[c + d*x]^9*(a + I*a*Tan[c + d*x])^7)/d - ((I/11)*Cos[c + d*x]^11*(a + I*a*Tan[c
+ d*x])^8)/d

Rule 3569

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 3578

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(d*S
ec[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 ^{11}(c+d x) (a+i a \tan (c+d x))^8 \, dx &=-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^8}{11 d}+\frac {1}{11} (3 a) \int \cos ^9(c+d x) (a+i a \tan (c+d x))^7 \, dx\\ &=-\frac {i a \cos ^9(c+d x) (a+i a \tan (c+d x))^7}{33 d}-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^8}{11 d}+\frac {1}{33} \left (2 a^2\right ) \int \cos ^7(c+d x) (a+i a \tan (c+d x))^6 \, dx\\ &=-\frac {2 i a^2 \cos ^7(c+d x) (a+i a \tan (c+d x))^6}{231 d}-\frac {i a \cos ^9(c+d x) (a+i a \tan (c+d x))^7}{33 d}-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^8}{11 d}+\frac {1}{231} \left (2 a^3\right ) \int \cos ^5(c+d x) (a+i a \tan (c+d x))^5 \, dx\\ &=-\frac {2 i a^3 \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{1155 d}-\frac {2 i a^2 \cos ^7(c+d x) (a+i a \tan (c+d x))^6}{231 d}-\frac {i a \cos ^9(c+d x) (a+i a \tan (c+d x))^7}{33 d}-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^8}{11 d}\\ \end {align*}

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Mathematica [A]
time = 0.79, size = 73, normalized size = 0.54 \begin {gather*} \frac {a^8 (440 \cos (c+d x)+168 \cos (3 (c+d x))-i (55 \sin (c+d x)+63 \sin (3 (c+d x)))) (-i \cos (8 (c+d x))+\sin (8 (c+d x)))}{4620 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^11*(a + I*a*Tan[c + d*x])^8,x]

[Out]

(a^8*(440*Cos[c + d*x] + 168*Cos[3*(c + d*x)] - I*(55*Sin[c + d*x] + 63*Sin[3*(c + d*x)]))*((-I)*Cos[8*(c + d*
x)] + Sin[8*(c + d*x)]))/(4620*d)

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 566 vs. \(2 (120 ) = 240\).
time = 0.23, size = 567, normalized size = 4.17 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^11*(a+I*a*tan(d*x+c))^8,x,method=_RETURNVERBOSE)

[Out]

1/d*(a^8*(-1/11*sin(d*x+c)^7*cos(d*x+c)^4-7/99*cos(d*x+c)^4*sin(d*x+c)^5-5/99*sin(d*x+c)^3*cos(d*x+c)^4-1/33*s
in(d*x+c)*cos(d*x+c)^4+1/99*(cos(d*x+c)^2+2)*sin(d*x+c))-8*I*a^8*(-1/11*sin(d*x+c)^6*cos(d*x+c)^5-2/33*sin(d*x
+c)^4*cos(d*x+c)^5-8/231*sin(d*x+c)^2*cos(d*x+c)^5-16/1155*cos(d*x+c)^5)-28*a^8*(-1/11*sin(d*x+c)^5*cos(d*x+c)
^6-5/99*sin(d*x+c)^3*cos(d*x+c)^6-5/231*sin(d*x+c)*cos(d*x+c)^6+1/231*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(
d*x+c))+56*I*a^8*(-1/11*sin(d*x+c)^4*cos(d*x+c)^7-4/99*sin(d*x+c)^2*cos(d*x+c)^7-8/693*cos(d*x+c)^7)+70*a^8*(-
1/11*sin(d*x+c)^3*cos(d*x+c)^8-1/33*sin(d*x+c)*cos(d*x+c)^8+1/231*(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*(-1/11*sin(d*x+c)^2*cos(d*x+c)^9-2/99*cos(d*x+c)^9)-28*a^8*(-1/11*cos(d*x+c)^10
*sin(d*x+c)+1/99*(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))-8/11
*I*a^8*cos(d*x+c)^11+1/11*a^8*(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+12
8/63*cos(d*x+c)^2)*sin(d*x+c))

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 355 vs. \(2 (112) = 224\).
time = 0.30, size = 355, normalized size = 2.61 \begin {gather*} -\frac {2520 i \, a^{8} \cos \left (d x + c\right )^{11} + 24 i \, {\left (105 \, \cos \left (d x + c\right )^{11} - 385 \, \cos \left (d x + c\right )^{9} + 495 \, \cos \left (d x + c\right )^{7} - 231 \, \cos \left (d x + c\right )^{5}\right )} a^{8} + 280 i \, {\left (63 \, \cos \left (d x + c\right )^{11} - 154 \, \cos \left (d x + c\right )^{9} + 99 \, \cos \left (d x + c\right )^{7}\right )} a^{8} + 1960 i \, {\left (9 \, \cos \left (d x + c\right )^{11} - 11 \, \cos \left (d x + c\right )^{9}\right )} a^{8} + 28 \, {\left (315 \, \sin \left (d x + c\right )^{11} - 1540 \, \sin \left (d x + c\right )^{9} + 2970 \, \sin \left (d x + c\right )^{7} - 2772 \, \sin \left (d x + c\right )^{5} + 1155 \, \sin \left (d x + c\right )^{3}\right )} a^{8} + 210 \, {\left (105 \, \sin \left (d x + c\right )^{11} - 385 \, \sin \left (d x + c\right )^{9} + 495 \, \sin \left (d x + c\right )^{7} - 231 \, \sin \left (d x + c\right )^{5}\right )} a^{8} + 140 \, {\left (63 \, \sin \left (d x + c\right )^{11} - 154 \, \sin \left (d x + c\right )^{9} + 99 \, \sin \left (d x + c\right )^{7}\right )} a^{8} + 5 \, {\left (63 \, \sin \left (d x + c\right )^{11} - 385 \, \sin \left (d x + c\right )^{9} + 990 \, \sin \left (d x + c\right )^{7} - 1386 \, \sin \left (d x + c\right )^{5} + 1155 \, \sin \left (d x + c\right )^{3} - 693 \, \sin \left (d x + c\right )\right )} a^{8} + 35 \, {\left (9 \, \sin \left (d x + c\right )^{11} - 11 \, \sin \left (d x + c\right )^{9}\right )} a^{8}}{3465 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^11*(a+I*a*tan(d*x+c))^8,x, algorithm="maxima")

[Out]

-1/3465*(2520*I*a^8*cos(d*x + c)^11 + 24*I*(105*cos(d*x + c)^11 - 385*cos(d*x + c)^9 + 495*cos(d*x + c)^7 - 23
1*cos(d*x + c)^5)*a^8 + 280*I*(63*cos(d*x + c)^11 - 154*cos(d*x + c)^9 + 99*cos(d*x + c)^7)*a^8 + 1960*I*(9*co
s(d*x + c)^11 - 11*cos(d*x + c)^9)*a^8 + 28*(315*sin(d*x + c)^11 - 1540*sin(d*x + c)^9 + 2970*sin(d*x + c)^7 -
2772*sin(d*x + c)^5 + 1155*sin(d*x + c)^3)*a^8 + 210*(105*sin(d*x + c)^11 - 385*sin(d*x + c)^9 + 495*sin(d*x
+ c)^7 - 231*sin(d*x + c)^5)*a^8 + 140*(63*sin(d*x + c)^11 - 154*sin(d*x + c)^9 + 99*sin(d*x + c)^7)*a^8 + 5*(
63*sin(d*x + c)^11 - 385*sin(d*x + c)^9 + 990*sin(d*x + c)^7 - 1386*sin(d*x + c)^5 + 1155*sin(d*x + c)^3 - 693
*sin(d*x + c))*a^8 + 35*(9*sin(d*x + c)^11 - 11*sin(d*x + c)^9)*a^8)/d

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Fricas [A]
time = 0.39, size = 62, normalized size = 0.46 \begin {gather*} \frac {-105 i \, a^{8} e^{\left (11 i \, d x + 11 i \, c\right )} - 385 i \, a^{8} e^{\left (9 i \, d x + 9 i \, c\right )} - 495 i \, a^{8} e^{\left (7 i \, d x + 7 i \, c\right )} - 231 i \, a^{8} e^{\left (5 i \, d x + 5 i \, c\right )}}{9240 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^11*(a+I*a*tan(d*x+c))^8,x, algorithm="fricas")

[Out]

1/9240*(-105*I*a^8*e^(11*I*d*x + 11*I*c) - 385*I*a^8*e^(9*I*d*x + 9*I*c) - 495*I*a^8*e^(7*I*d*x + 7*I*c) - 231
*I*a^8*e^(5*I*d*x + 5*I*c))/d

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Sympy [A]
time = 0.66, size = 162, normalized size = 1.19 \begin {gather*} \begin {cases} \frac {- 53760 i a^{8} d^{3} e^{11 i c} e^{11 i d x} - 197120 i a^{8} d^{3} e^{9 i c} e^{9 i d x} - 253440 i a^{8} d^{3} e^{7 i c} e^{7 i d x} - 118272 i a^{8} d^{3} e^{5 i c} e^{5 i d x}}{4730880 d^{4}} & \text {for}\: d^{4} \neq 0 \\x \left (\frac {a^{8} e^{11 i c}}{8} + \frac {3 a^{8} e^{9 i c}}{8} + \frac {3 a^{8} e^{7 i c}}{8} + \frac {a^{8} e^{5 i c}}{8}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**11*(a+I*a*tan(d*x+c))**8,x)

[Out]

Piecewise(((-53760*I*a**8*d**3*exp(11*I*c)*exp(11*I*d*x) - 197120*I*a**8*d**3*exp(9*I*c)*exp(9*I*d*x) - 253440
*I*a**8*d**3*exp(7*I*c)*exp(7*I*d*x) - 118272*I*a**8*d**3*exp(5*I*c)*exp(5*I*d*x))/(4730880*d**4), Ne(d**4, 0)
), (x*(a**8*exp(11*I*c)/8 + 3*a**8*exp(9*I*c)/8 + 3*a**8*exp(7*I*c)/8 + a**8*exp(5*I*c)/8), True))

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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2863 vs. \(2 (112) = 224\).
time = 1.80, size = 2863, normalized size = 21.05 \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)^11*(a+I*a*tan(d*x+c))^8,x, algorithm="giac")

[Out]

1/4844421120*(82027951005*a^8*e^(28*I*d*x + 14*I*c)*log(I*e^(I*d*x + I*c) + 1) + 1148391314070*a^8*e^(26*I*d*x
+ 12*I*c)*log(I*e^(I*d*x + I*c) + 1) + 7464543541455*a^8*e^(24*I*d*x + 10*I*c)*log(I*e^(I*d*x + I*c) + 1) + 2
9858174165820*a^8*e^(22*I*d*x + 8*I*c)*log(I*e^(I*d*x + I*c) + 1) + 82109978956005*a^8*e^(20*I*d*x + 6*I*c)*lo
g(I*e^(I*d*x + I*c) + 1) + 164219957912010*a^8*e^(18*I*d*x + 4*I*c)*log(I*e^(I*d*x + I*c) + 1) + 2463299368680
15*a^8*e^(16*I*d*x + 2*I*c)*log(I*e^(I*d*x + I*c) + 1) + 246329936868015*a^8*e^(12*I*d*x - 2*I*c)*log(I*e^(I*d
*x + I*c) + 1) + 164219957912010*a^8*e^(10*I*d*x - 4*I*c)*log(I*e^(I*d*x + I*c) + 1) + 82109978956005*a^8*e^(8
*I*d*x - 6*I*c)*log(I*e^(I*d*x + I*c) + 1) + 29858174165820*a^8*e^(6*I*d*x - 8*I*c)*log(I*e^(I*d*x + I*c) + 1)
+ 7464543541455*a^8*e^(4*I*d*x - 10*I*c)*log(I*e^(I*d*x + I*c) + 1) + 1148391314070*a^8*e^(2*I*d*x - 12*I*c)*
log(I*e^(I*d*x + I*c) + 1) + 281519927849160*a^8*e^(14*I*d*x)*log(I*e^(I*d*x + I*c) + 1) + 82027951005*a^8*e^(
-14*I*c)*log(I*e^(I*d*x + I*c) + 1) + 82004266575*a^8*e^(28*I*d*x + 14*I*c)*log(I*e^(I*d*x + I*c) - 1) + 11480
59732050*a^8*e^(26*I*d*x + 12*I*c)*log(I*e^(I*d*x + I*c) - 1) + 7462388258325*a^8*e^(24*I*d*x + 10*I*c)*log(I*
e^(I*d*x + I*c) - 1) + 29849553033300*a^8*e^(22*I*d*x + 8*I*c)*log(I*e^(I*d*x + I*c) - 1) + 82086270841575*a^8
*e^(20*I*d*x + 6*I*c)*log(I*e^(I*d*x + I*c) - 1) + 164172541683150*a^8*e^(18*I*d*x + 4*I*c)*log(I*e^(I*d*x + I
*c) - 1) + 246258812524725*a^8*e^(16*I*d*x + 2*I*c)*log(I*e^(I*d*x + I*c) - 1) + 246258812524725*a^8*e^(12*I*d
*x - 2*I*c)*log(I*e^(I*d*x + I*c) - 1) + 164172541683150*a^8*e^(10*I*d*x - 4*I*c)*log(I*e^(I*d*x + I*c) - 1) +
82086270841575*a^8*e^(8*I*d*x - 6*I*c)*log(I*e^(I*d*x + I*c) - 1) + 29849553033300*a^8*e^(6*I*d*x - 8*I*c)*lo
g(I*e^(I*d*x + I*c) - 1) + 7462388258325*a^8*e^(4*I*d*x - 10*I*c)*log(I*e^(I*d*x + I*c) - 1) + 1148059732050*a
^8*e^(2*I*d*x - 12*I*c)*log(I*e^(I*d*x + I*c) - 1) + 281438642885400*a^8*e^(14*I*d*x)*log(I*e^(I*d*x + I*c) -
1) + 82004266575*a^8*e^(-14*I*c)*log(I*e^(I*d*x + I*c) - 1) - 82027951005*a^8*e^(28*I*d*x + 14*I*c)*log(-I*e^(
I*d*x + I*c) + 1) - 1148391314070*a^8*e^(26*I*d*x + 12*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 7464543541455*a^8*e^
(24*I*d*x + 10*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 29858174165820*a^8*e^(22*I*d*x + 8*I*c)*log(-I*e^(I*d*x + I*
c) + 1) - 82109978956005*a^8*e^(20*I*d*x + 6*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 164219957912010*a^8*e^(18*I*d*
x + 4*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 246329936868015*a^8*e^(16*I*d*x + 2*I*c)*log(-I*e^(I*d*x + I*c) + 1)
- 246329936868015*a^8*e^(12*I*d*x - 2*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 164219957912010*a^8*e^(10*I*d*x - 4*I
*c)*log(-I*e^(I*d*x + I*c) + 1) - 82109978956005*a^8*e^(8*I*d*x - 6*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 2985817
4165820*a^8*e^(6*I*d*x - 8*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 7464543541455*a^8*e^(4*I*d*x - 10*I*c)*log(-I*e^
(I*d*x + I*c) + 1) - 1148391314070*a^8*e^(2*I*d*x - 12*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 281519927849160*a^8*
e^(14*I*d*x)*log(-I*e^(I*d*x + I*c) + 1) - 82027951005*a^8*e^(-14*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 820042665
75*a^8*e^(28*I*d*x + 14*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 1148059732050*a^8*e^(26*I*d*x + 12*I*c)*log(-I*e^(I
*d*x + I*c) - 1) - 7462388258325*a^8*e^(24*I*d*x + 10*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 29849553033300*a^8*e^
(22*I*d*x + 8*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 82086270841575*a^8*e^(20*I*d*x + 6*I*c)*log(-I*e^(I*d*x + I*c
) - 1) - 164172541683150*a^8*e^(18*I*d*x + 4*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 246258812524725*a^8*e^(16*I*d*
x + 2*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 246258812524725*a^8*e^(12*I*d*x - 2*I*c)*log(-I*e^(I*d*x + I*c) - 1)
- 164172541683150*a^8*e^(10*I*d*x - 4*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 82086270841575*a^8*e^(8*I*d*x - 6*I*c
)*log(-I*e^(I*d*x + I*c) - 1) - 29849553033300*a^8*e^(6*I*d*x - 8*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 746238825
8325*a^8*e^(4*I*d*x - 10*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 1148059732050*a^8*e^(2*I*d*x - 12*I*c)*log(-I*e^(I
*d*x + I*c) - 1) - 281438642885400*a^8*e^(14*I*d*x)*log(-I*e^(I*d*x + I*c) - 1) - 82004266575*a^8*e^(-14*I*c)*
log(-I*e^(I*d*x + I*c) - 1) - 23684430*a^8*e^(28*I*d*x + 14*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 331582020*a^8*e
^(26*I*d*x + 12*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 2155283130*a^8*e^(24*I*d*x + 10*I*c)*log(I*e^(I*d*x) + e^(-
I*c)) - 8621132520*a^8*e^(22*I*d*x + 8*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 23708114430*a^8*e^(20*I*d*x + 6*I*c)
*log(I*e^(I*d*x) + e^(-I*c)) - 47416228860*a^8*e^(18*I*d*x + 4*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 71124343290*
a^8*e^(16*I*d*x + 2*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 71124343290*a^8*e^(12*I*d*x - 2*I*c)*log(I*e^(I*d*x) +
e^(-I*c)) - 47416228860*a^8*e^(10*I*d*x - 4*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 23708114430*a^8*e^(8*I*d*x - 6*
I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 8621132520*a^8*e^(6*I*d*x - 8*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 2155283130
*a^8*e^(4*I*d*x - 10*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 331582020*a^8*e^(2*I*d*x - 12*I*c)*log(I*e^(I*d*x) + e
^(-I*c)) - 81284963760*a^8*e^(14*I*d*x)*log(I*e...

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Mupad [B]
time = 3.80, size = 65, normalized size = 0.48 \begin {gather*} -\frac {a^8\,\left (\frac {{\mathrm {e}}^{c\,5{}\mathrm {i}+d\,x\,5{}\mathrm {i}}\,1{}\mathrm {i}}{40}+\frac {{\mathrm {e}}^{c\,7{}\mathrm {i}+d\,x\,7{}\mathrm {i}}\,3{}\mathrm {i}}{56}+\frac {{\mathrm {e}}^{c\,9{}\mathrm {i}+d\,x\,9{}\mathrm {i}}\,1{}\mathrm {i}}{24}+\frac {{\mathrm {e}}^{c\,11{}\mathrm {i}+d\,x\,11{}\mathrm {i}}\,1{}\mathrm {i}}{88}\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^11*(a + a*tan(c + d*x)*1i)^8,x)

[Out]

-(a^8*((exp(c*5i + d*x*5i)*1i)/40 + (exp(c*7i + d*x*7i)*3i)/56 + (exp(c*9i + d*x*9i)*1i)/24 + (exp(c*11i + d*x
*11i)*1i)/88))/d

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