3.1.76 \(\int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^5 \, dx\) [76]

Optimal. Leaf size=159 \[ -\frac {5 i a^5 \cos ^7(c+d x)}{231 d}+\frac {5 a^5 \sin (c+d x)}{33 d}-\frac {5 a^5 \sin ^3(c+d x)}{33 d}+\frac {a^5 \sin ^5(c+d x)}{11 d}-\frac {5 a^5 \sin ^7(c+d x)}{231 d}-\frac {2 i a^3 \cos ^9(c+d x) (a+i a \tan (c+d x))^2}{33 d}-\frac {2 i a \cos ^{11}(c+d x) (a+i a \tan (c+d x))^4}{11 d} \]

[Out]

-5/231*I*a^5*cos(d*x+c)^7/d+5/33*a^5*sin(d*x+c)/d-5/33*a^5*sin(d*x+c)^3/d+1/11*a^5*sin(d*x+c)^5/d-5/231*a^5*si
n(d*x+c)^7/d-2/33*I*a^3*cos(d*x+c)^9*(a+I*a*tan(d*x+c))^2/d-2/11*I*a*cos(d*x+c)^11*(a+I*a*tan(d*x+c))^4/d

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Rubi [A]
time = 0.10, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3577, 3567, 2713} \begin {gather*} -\frac {5 a^5 \sin ^7(c+d x)}{231 d}+\frac {a^5 \sin ^5(c+d x)}{11 d}-\frac {5 a^5 \sin ^3(c+d x)}{33 d}+\frac {5 a^5 \sin (c+d x)}{33 d}-\frac {5 i a^5 \cos ^7(c+d x)}{231 d}-\frac {2 i a^3 \cos ^9(c+d x) (a+i a \tan (c+d x))^2}{33 d}-\frac {2 i a \cos ^{11}(c+d x) (a+i a \tan (c+d x))^4}{11 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^11*(a + I*a*Tan[c + d*x])^5,x]

[Out]

(((-5*I)/231)*a^5*Cos[c + d*x]^7)/d + (5*a^5*Sin[c + d*x])/(33*d) - (5*a^5*Sin[c + d*x]^3)/(33*d) + (a^5*Sin[c
+ d*x]^5)/(11*d) - (5*a^5*Sin[c + d*x]^7)/(231*d) - (((2*I)/33)*a^3*Cos[c + d*x]^9*(a + I*a*Tan[c + d*x])^2)/
d - (((2*I)/11)*a*Cos[c + d*x]^11*(a + I*a*Tan[c + d*x])^4)/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 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]

Rubi steps

\begin {align*} \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^5 \, dx &=-\frac {2 i a \cos ^{11}(c+d x) (a+i a \tan (c+d x))^4}{11 d}+\frac {1}{11} \left (3 a^2\right ) \int \cos ^9(c+d x) (a+i a \tan (c+d x))^3 \, dx\\ &=-\frac {2 i a^3 \cos ^9(c+d x) (a+i a \tan (c+d x))^2}{33 d}-\frac {2 i a \cos ^{11}(c+d x) (a+i a \tan (c+d x))^4}{11 d}+\frac {1}{33} \left (5 a^4\right ) \int \cos ^7(c+d x) (a+i a \tan (c+d x)) \, dx\\ &=-\frac {5 i a^5 \cos ^7(c+d x)}{231 d}-\frac {2 i a^3 \cos ^9(c+d x) (a+i a \tan (c+d x))^2}{33 d}-\frac {2 i a \cos ^{11}(c+d x) (a+i a \tan (c+d x))^4}{11 d}+\frac {1}{33} \left (5 a^5\right ) \int \cos ^7(c+d x) \, dx\\ &=-\frac {5 i a^5 \cos ^7(c+d x)}{231 d}-\frac {2 i a^3 \cos ^9(c+d x) (a+i a \tan (c+d x))^2}{33 d}-\frac {2 i a \cos ^{11}(c+d x) (a+i a \tan (c+d x))^4}{11 d}-\frac {\left (5 a^5\right ) \text {Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,-\sin (c+d x)\right )}{33 d}\\ &=-\frac {5 i a^5 \cos ^7(c+d x)}{231 d}+\frac {5 a^5 \sin (c+d x)}{33 d}-\frac {5 a^5 \sin ^3(c+d x)}{33 d}+\frac {a^5 \sin ^5(c+d x)}{11 d}-\frac {5 a^5 \sin ^7(c+d x)}{231 d}-\frac {2 i a^3 \cos ^9(c+d x) (a+i a \tan (c+d x))^2}{33 d}-\frac {2 i a \cos ^{11}(c+d x) (a+i a \tan (c+d x))^4}{11 d}\\ \end {align*}

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Mathematica [A]
time = 1.01, size = 118, normalized size = 0.74 \begin {gather*} \frac {i a^5 (-462-825 \cos (2 (c+d x))-770 \cos (4 (c+d x))+105 \cos (6 (c+d x))+330 i \sin (2 (c+d x))+616 i \sin (4 (c+d x))-126 i \sin (6 (c+d x))) (\cos (5 (c+2 d x))+i \sin (5 (c+2 d x)))}{7392 d (\cos (d x)+i \sin (d x))^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^11*(a + I*a*Tan[c + d*x])^5,x]

[Out]

((I/7392)*a^5*(-462 - 825*Cos[2*(c + d*x)] - 770*Cos[4*(c + d*x)] + 105*Cos[6*(c + d*x)] + (330*I)*Sin[2*(c +
d*x)] + (616*I)*Sin[4*(c + d*x)] - (126*I)*Sin[6*(c + d*x)])*(Cos[5*(c + 2*d*x)] + I*Sin[5*(c + 2*d*x)]))/(d*(
Cos[d*x] + I*Sin[d*x])^5)

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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. 316 vs. \(2 (140 ) = 280\).
time = 0.30, size = 317, normalized size = 1.99

method	result	size
risch	\(-\frac {i a^{5} {\mathrm e}^{11 i \left (d x +c \right )}}{704 d}-\frac {i a^{5} {\mathrm e}^{9 i \left (d x +c \right )}}{96 d}-\frac {15 i a^{5} {\mathrm e}^{7 i \left (d x +c \right )}}{448 d}-\frac {i a^{5} {\mathrm e}^{5 i \left (d x +c \right )}}{16 d}-\frac {5 i a^{5} {\mathrm e}^{3 i \left (d x +c \right )}}{64 d}-\frac {5 i a^{5} \cos \left (d x +c \right )}{64 d}+\frac {7 a^{5} \sin \left (d x +c \right )}{64 d}\)	\(121\)
derivativedivides	\(\frac {i a^{5} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{11}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{99}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{693}\right )+5 a^{5} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{8}\left (d x +c \right )\right )}{11}-\frac {\sin \left (d x +c \right ) \left (\cos ^{8}\left (d x +c \right )\right )}{33}+\frac {\left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{231}\right )-10 i a^{5} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{9}\left (d x +c \right )\right )}{11}-\frac {2 \left (\cos ^{9}\left (d x +c \right )\right )}{99}\right )-10 a^{5} \left (-\frac {\left (\cos ^{10}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{11}+\frac {\left (\frac {128}{35}+\cos ^{8}\left (d x +c \right )+\frac {8 \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {48 \left (\cos ^{4}\left (d x +c \right )\right )}{35}+\frac {64 \left (\cos ^{2}\left (d x +c \right )\right )}{35}\right ) \sin \left (d x +c \right )}{99}\right )-\frac {5 i a^{5} \left (\cos ^{11}\left (d x +c \right )\right )}{11}+\frac {a^{5} \left (\frac {256}{63}+\cos ^{10}\left (d x +c \right )+\frac {10 \left (\cos ^{8}\left (d x +c \right )\right )}{9}+\frac {80 \left (\cos ^{6}\left (d x +c \right )\right )}{63}+\frac {32 \left (\cos ^{4}\left (d x +c \right )\right )}{21}+\frac {128 \left (\cos ^{2}\left (d x +c \right )\right )}{63}\right ) \sin \left (d x +c \right )}{11}}{d}\)	\(317\)
default	\(\frac {i a^{5} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{11}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{99}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{693}\right )+5 a^{5} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{8}\left (d x +c \right )\right )}{11}-\frac {\sin \left (d x +c \right ) \left (\cos ^{8}\left (d x +c \right )\right )}{33}+\frac {\left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{231}\right )-10 i a^{5} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{9}\left (d x +c \right )\right )}{11}-\frac {2 \left (\cos ^{9}\left (d x +c \right )\right )}{99}\right )-10 a^{5} \left (-\frac {\left (\cos ^{10}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{11}+\frac {\left (\frac {128}{35}+\cos ^{8}\left (d x +c \right )+\frac {8 \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {48 \left (\cos ^{4}\left (d x +c \right )\right )}{35}+\frac {64 \left (\cos ^{2}\left (d x +c \right )\right )}{35}\right ) \sin \left (d x +c \right )}{99}\right )-\frac {5 i a^{5} \left (\cos ^{11}\left (d x +c \right )\right )}{11}+\frac {a^{5} \left (\frac {256}{63}+\cos ^{10}\left (d x +c \right )+\frac {10 \left (\cos ^{8}\left (d x +c \right )\right )}{9}+\frac {80 \left (\cos ^{6}\left (d x +c \right )\right )}{63}+\frac {32 \left (\cos ^{4}\left (d x +c \right )\right )}{21}+\frac {128 \left (\cos ^{2}\left (d x +c \right )\right )}{63}\right ) \sin \left (d x +c \right )}{11}}{d}\)	\(317\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^11*(a+I*a*tan(d*x+c))^5,x,method=_RETURNVERBOSE)

[Out]

1/d*(I*a^5*(-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)+5*a^5*(-1/11*si
n(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))-10*I*a^5*(-1/11*sin(d*x+c)^2*cos(d*x+c)^9-2/99*cos(d*x+c)^9)-10*a^5*(-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))-5/11*I*a^5*
cos(d*x+c)^11+1/11*a^5*(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*co
s(d*x+c)^2)*sin(d*x+c))

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Maxima [A]
time = 0.29, size = 246, normalized size = 1.55 \begin {gather*} -\frac {315 i \, a^{5} \cos \left (d x + c\right )^{11} + 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^{5} + 70 i \, {\left (9 \, \cos \left (d x + c\right )^{11} - 11 \, \cos \left (d x + c\right )^{9}\right )} a^{5} + 2 \, {\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^{5} + 3 \, {\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^{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^{5}}{693 \, 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))^5,x, algorithm="maxima")

[Out]

-1/693*(315*I*a^5*cos(d*x + c)^11 + I*(63*cos(d*x + c)^11 - 154*cos(d*x + c)^9 + 99*cos(d*x + c)^7)*a^5 + 70*I
*(9*cos(d*x + c)^11 - 11*cos(d*x + c)^9)*a^5 + 2*(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^5 + 3*(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^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^5)/d

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Fricas [A]
time = 0.41, size = 104, normalized size = 0.65 \begin {gather*} \frac {{\left (-21 i \, a^{5} e^{\left (12 i \, d x + 12 i \, c\right )} - 154 i \, a^{5} e^{\left (10 i \, d x + 10 i \, c\right )} - 495 i \, a^{5} e^{\left (8 i \, d x + 8 i \, c\right )} - 924 i \, a^{5} e^{\left (6 i \, d x + 6 i \, c\right )} - 1155 i \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} - 1386 i \, a^{5} e^{\left (2 i \, d x + 2 i \, c\right )} + 231 i \, a^{5}\right )} e^{\left (-i \, d x - i \, c\right )}}{14784 \, 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))^5,x, algorithm="fricas")

[Out]

1/14784*(-21*I*a^5*e^(12*I*d*x + 12*I*c) - 154*I*a^5*e^(10*I*d*x + 10*I*c) - 495*I*a^5*e^(8*I*d*x + 8*I*c) - 9
24*I*a^5*e^(6*I*d*x + 6*I*c) - 1155*I*a^5*e^(4*I*d*x + 4*I*c) - 1386*I*a^5*e^(2*I*d*x + 2*I*c) + 231*I*a^5)*e^
(-I*d*x - I*c)/d

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Sympy [A]
time = 0.57, size = 265, normalized size = 1.67 \begin {gather*} \begin {cases} \frac {\left (- 90194313216 i a^{5} d^{6} e^{12 i c} e^{11 i d x} - 661424963584 i a^{5} d^{6} e^{10 i c} e^{9 i d x} - 2126008811520 i a^{5} d^{6} e^{8 i c} e^{7 i d x} - 3968549781504 i a^{5} d^{6} e^{6 i c} e^{5 i d x} - 4960687226880 i a^{5} d^{6} e^{4 i c} e^{3 i d x} - 5952824672256 i a^{5} d^{6} e^{2 i c} e^{i d x} + 992137445376 i a^{5} d^{6} e^{- i d x}\right ) e^{- i c}}{63496796504064 d^{7}} & \text {for}\: d^{7} e^{i c} \neq 0 \\\frac {x \left (a^{5} e^{12 i c} + 6 a^{5} e^{10 i c} + 15 a^{5} e^{8 i c} + 20 a^{5} e^{6 i c} + 15 a^{5} e^{4 i c} + 6 a^{5} e^{2 i c} + a^{5}\right ) e^{- i c}}{64} & \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))**5,x)

[Out]

Piecewise(((-90194313216*I*a**5*d**6*exp(12*I*c)*exp(11*I*d*x) - 661424963584*I*a**5*d**6*exp(10*I*c)*exp(9*I*
d*x) - 2126008811520*I*a**5*d**6*exp(8*I*c)*exp(7*I*d*x) - 3968549781504*I*a**5*d**6*exp(6*I*c)*exp(5*I*d*x) -
4960687226880*I*a**5*d**6*exp(4*I*c)*exp(3*I*d*x) - 5952824672256*I*a**5*d**6*exp(2*I*c)*exp(I*d*x) + 9921374
45376*I*a**5*d**6*exp(-I*d*x))*exp(-I*c)/(63496796504064*d**7), Ne(d**7*exp(I*c), 0)), (x*(a**5*exp(12*I*c) +
6*a**5*exp(10*I*c) + 15*a**5*exp(8*I*c) + 20*a**5*exp(6*I*c) + 15*a**5*exp(4*I*c) + 6*a**5*exp(2*I*c) + a**5)*
exp(-I*c)/64, 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. 1807 vs. \(2 (135) = 270\).
time = 1.12, size = 1807, normalized size = 11.36 \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))^5,x, algorithm="giac")

[Out]

-1/121110528*(168111405*a^5*e^(17*I*d*x + 9*I*c)*log(I*e^(I*d*x + I*c) + 1) + 1344891240*a^5*e^(15*I*d*x + 7*I
*c)*log(I*e^(I*d*x + I*c) + 1) + 4707119340*a^5*e^(13*I*d*x + 5*I*c)*log(I*e^(I*d*x + I*c) + 1) + 9414238680*a
^5*e^(11*I*d*x + 3*I*c)*log(I*e^(I*d*x + I*c) + 1) + 11767798350*a^5*e^(9*I*d*x + I*c)*log(I*e^(I*d*x + I*c) +
1) + 9414238680*a^5*e^(7*I*d*x - I*c)*log(I*e^(I*d*x + I*c) + 1) + 4707119340*a^5*e^(5*I*d*x - 3*I*c)*log(I*e
^(I*d*x + I*c) + 1) + 1344891240*a^5*e^(3*I*d*x - 5*I*c)*log(I*e^(I*d*x + I*c) + 1) + 168111405*a^5*e^(I*d*x -
7*I*c)*log(I*e^(I*d*x + I*c) + 1) + 170251620*a^5*e^(17*I*d*x + 9*I*c)*log(I*e^(I*d*x + I*c) - 1) + 136201296
0*a^5*e^(15*I*d*x + 7*I*c)*log(I*e^(I*d*x + I*c) - 1) + 4767045360*a^5*e^(13*I*d*x + 5*I*c)*log(I*e^(I*d*x + I
*c) - 1) + 9534090720*a^5*e^(11*I*d*x + 3*I*c)*log(I*e^(I*d*x + I*c) - 1) + 11917613400*a^5*e^(9*I*d*x + I*c)*
log(I*e^(I*d*x + I*c) - 1) + 9534090720*a^5*e^(7*I*d*x - I*c)*log(I*e^(I*d*x + I*c) - 1) + 4767045360*a^5*e^(5
*I*d*x - 3*I*c)*log(I*e^(I*d*x + I*c) - 1) + 1362012960*a^5*e^(3*I*d*x - 5*I*c)*log(I*e^(I*d*x + I*c) - 1) + 1
70251620*a^5*e^(I*d*x - 7*I*c)*log(I*e^(I*d*x + I*c) - 1) - 168111405*a^5*e^(17*I*d*x + 9*I*c)*log(-I*e^(I*d*x
+ I*c) + 1) - 1344891240*a^5*e^(15*I*d*x + 7*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 4707119340*a^5*e^(13*I*d*x +
5*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 9414238680*a^5*e^(11*I*d*x + 3*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 1176779
8350*a^5*e^(9*I*d*x + I*c)*log(-I*e^(I*d*x + I*c) + 1) - 9414238680*a^5*e^(7*I*d*x - I*c)*log(-I*e^(I*d*x + I*
c) + 1) - 4707119340*a^5*e^(5*I*d*x - 3*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 1344891240*a^5*e^(3*I*d*x - 5*I*c)*
log(-I*e^(I*d*x + I*c) + 1) - 168111405*a^5*e^(I*d*x - 7*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 170251620*a^5*e^(1
7*I*d*x + 9*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 1362012960*a^5*e^(15*I*d*x + 7*I*c)*log(-I*e^(I*d*x + I*c) - 1)
- 4767045360*a^5*e^(13*I*d*x + 5*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 9534090720*a^5*e^(11*I*d*x + 3*I*c)*log(-
I*e^(I*d*x + I*c) - 1) - 11917613400*a^5*e^(9*I*d*x + I*c)*log(-I*e^(I*d*x + I*c) - 1) - 9534090720*a^5*e^(7*I
*d*x - I*c)*log(-I*e^(I*d*x + I*c) - 1) - 4767045360*a^5*e^(5*I*d*x - 3*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 136
2012960*a^5*e^(3*I*d*x - 5*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 170251620*a^5*e^(I*d*x - 7*I*c)*log(-I*e^(I*d*x
+ I*c) - 1) + 2140215*a^5*e^(17*I*d*x + 9*I*c)*log(I*e^(I*d*x) + e^(-I*c)) + 17121720*a^5*e^(15*I*d*x + 7*I*c)
*log(I*e^(I*d*x) + e^(-I*c)) + 59926020*a^5*e^(13*I*d*x + 5*I*c)*log(I*e^(I*d*x) + e^(-I*c)) + 119852040*a^5*e
^(11*I*d*x + 3*I*c)*log(I*e^(I*d*x) + e^(-I*c)) + 149815050*a^5*e^(9*I*d*x + I*c)*log(I*e^(I*d*x) + e^(-I*c))
+ 119852040*a^5*e^(7*I*d*x - I*c)*log(I*e^(I*d*x) + e^(-I*c)) + 59926020*a^5*e^(5*I*d*x - 3*I*c)*log(I*e^(I*d*
x) + e^(-I*c)) + 17121720*a^5*e^(3*I*d*x - 5*I*c)*log(I*e^(I*d*x) + e^(-I*c)) + 2140215*a^5*e^(I*d*x - 7*I*c)*
log(I*e^(I*d*x) + e^(-I*c)) - 2140215*a^5*e^(17*I*d*x + 9*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) - 17121720*a^5*e^(
15*I*d*x + 7*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) - 59926020*a^5*e^(13*I*d*x + 5*I*c)*log(-I*e^(I*d*x) + e^(-I*c)
) - 119852040*a^5*e^(11*I*d*x + 3*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) - 149815050*a^5*e^(9*I*d*x + I*c)*log(-I*e
^(I*d*x) + e^(-I*c)) - 119852040*a^5*e^(7*I*d*x - I*c)*log(-I*e^(I*d*x) + e^(-I*c)) - 59926020*a^5*e^(5*I*d*x
- 3*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) - 17121720*a^5*e^(3*I*d*x - 5*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) - 214021
5*a^5*e^(I*d*x - 7*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 172032*I*a^5*e^(28*I*d*x + 20*I*c) + 2637824*I*a^5*e^(2
6*I*d*x + 18*I*c) + 18964480*I*a^5*e^(24*I*d*x + 16*I*c) + 84967424*I*a^5*e^(22*I*d*x + 14*I*c) + 266248192*I*
a^5*e^(20*I*d*x + 12*I*c) + 624017408*I*a^5*e^(18*I*d*x + 10*I*c) + 1137074176*I*a^5*e^(16*I*d*x + 8*I*c) + 16
26275840*I*a^5*e^(14*I*d*x + 6*I*c) + 1792860160*I*a^5*e^(12*I*d*x + 4*I*c) + 1464320000*I*a^5*e^(10*I*d*x + 2
*I*c) + 295206912*I*a^5*e^(6*I*d*x - 2*I*c) + 47308800*I*a^5*e^(4*I*d*x - 4*I*c) - 3784704*I*a^5*e^(2*I*d*x -
6*I*c) + 832905216*I*a^5*e^(8*I*d*x) - 1892352*I*a^5*e^(-8*I*c))/(d*e^(17*I*d*x + 9*I*c) + 8*d*e^(15*I*d*x + 7
*I*c) + 28*d*e^(13*I*d*x + 5*I*c) + 56*d*e^(11*I*d*x + 3*I*c) + 70*d*e^(9*I*d*x + I*c) + 56*d*e^(7*I*d*x - I*c
) + 28*d*e^(5*I*d*x - 3*I*c) + 8*d*e^(3*I*d*x - 5*I*c) + d*e^(I*d*x - 7*I*c))

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Mupad [B]
time = 4.60, size = 139, normalized size = 0.87 \begin {gather*} \frac {a^5\,\left (\frac {5\,\sin \left (3\,c+3\,d\,x\right )}{64}-\frac {\cos \left (5\,c+5\,d\,x\right )\,1{}\mathrm {i}}{16}-\frac {\cos \left (7\,c+7\,d\,x\right )\,15{}\mathrm {i}}{448}-\frac {\cos \left (9\,c+9\,d\,x\right )\,1{}\mathrm {i}}{96}-\frac {\cos \left (11\,c+11\,d\,x\right )\,1{}\mathrm {i}}{704}-\frac {\cos \left (3\,c+3\,d\,x\right )\,5{}\mathrm {i}}{64}+\frac {\sin \left (5\,c+5\,d\,x\right )}{16}+\frac {15\,\sin \left (7\,c+7\,d\,x\right )}{448}+\frac {\sin \left (9\,c+9\,d\,x\right )}{96}+\frac {\sin \left (11\,c+11\,d\,x\right )}{704}+\frac {\sqrt {24}\,\cos \left (c+d\,x-\mathrm {atanh}\left (\frac {7}{5}\right )\,1{}\mathrm {i}\right )}{64}\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)^5,x)

[Out]

(a^5*((5*sin(3*c + 3*d*x))/64 - (cos(5*c + 5*d*x)*1i)/16 - (cos(7*c + 7*d*x)*15i)/448 - (cos(9*c + 9*d*x)*1i)/
96 - (cos(11*c + 11*d*x)*1i)/704 - (cos(3*c + 3*d*x)*5i)/64 + sin(5*c + 5*d*x)/16 + (15*sin(7*c + 7*d*x))/448
+ sin(9*c + 9*d*x)/96 + sin(11*c + 11*d*x)/704 + (24^(1/2)*cos(c - atanh(7/5)*1i + d*x))/64))/d

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