3.1.84 \(\int \cos ^6(c+d x) (a+i a \tan (c+d x))^8 \, dx\) [84]

Optimal. Leaf size=114 \[ -8 a^8 x+\frac {8 i a^8 \log (\cos (c+d x))}{d}+\frac {a^8 \tan (c+d x)}{d}-\frac {16 i a^{11}}{3 d (a-i a \tan (c+d x))^3}+\frac {16 i a^{10}}{d (a-i a \tan (c+d x))^2}-\frac {24 i a^9}{d (a-i a \tan (c+d x))} \]

[Out]

-8*a^8*x+8*I*a^8*ln(cos(d*x+c))/d+a^8*tan(d*x+c)/d-16/3*I*a^11/d/(a-I*a*tan(d*x+c))^3+16*I*a^10/d/(a-I*a*tan(d
*x+c))^2-24*I*a^9/d/(a-I*a*tan(d*x+c))

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Rubi [A]
time = 0.06, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3568, 45} \begin {gather*} -\frac {16 i a^{11}}{3 d (a-i a \tan (c+d x))^3}+\frac {16 i a^{10}}{d (a-i a \tan (c+d x))^2}-\frac {24 i a^9}{d (a-i a \tan (c+d x))}+\frac {a^8 \tan (c+d x)}{d}+\frac {8 i a^8 \log (\cos (c+d x))}{d}-8 a^8 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^6*(a + I*a*Tan[c + d*x])^8,x]

[Out]

-8*a^8*x + ((8*I)*a^8*Log[Cos[c + d*x]])/d + (a^8*Tan[c + d*x])/d - (((16*I)/3)*a^11)/(d*(a - I*a*Tan[c + d*x]
)^3) + ((16*I)*a^10)/(d*(a - I*a*Tan[c + d*x])^2) - ((24*I)*a^9)/(d*(a - I*a*Tan[c + d*x]))

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 3568

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[1/(a^(m - 2)*b
*f), Subst[Int[(a - x)^(m/2 - 1)*(a + x)^(n + m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x
] && EqQ[a^2 + b^2, 0] && IntegerQ[m/2]

Rubi steps

\begin {align*} \int \cos ^6(c+d x) (a+i a \tan (c+d x))^8 \, dx &=-\frac {\left (i a^7\right ) \text {Subst}\left (\int \frac {(a+x)^4}{(a-x)^4} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac {\left (i a^7\right ) \text {Subst}\left (\int \left (1+\frac {16 a^4}{(a-x)^4}-\frac {32 a^3}{(a-x)^3}+\frac {24 a^2}{(a-x)^2}-\frac {8 a}{a-x}\right ) \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-8 a^8 x+\frac {8 i a^8 \log (\cos (c+d x))}{d}+\frac {a^8 \tan (c+d x)}{d}-\frac {16 i a^{11}}{3 d (a-i a \tan (c+d x))^3}+\frac {16 i a^{10}}{d (a-i a \tan (c+d x))^2}-\frac {24 i a^9}{d (a-i a \tan (c+d x))}\\ \end {align*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(414\) vs. \(2(114)=228\).
time = 1.75, size = 414, normalized size = 3.63 \begin {gather*} -\frac {a^8 \sec (c) \sec (c+d x) \left (12 i \cos (c)+10 i \cos (3 c+2 d x)+12 d x \cos (3 c+2 d x)-2 i \cos (3 c+4 d x)+12 d x \cos (3 c+4 d x)+i \cos (5 c+4 d x)+12 d x \cos (5 c+4 d x)+\cos (c+2 d x) \left (7 i+12 d x-6 i \log \left (\cos ^2(c+d x)\right )\right )-6 i \cos (3 c+2 d x) \log \left (\cos ^2(c+d x)\right )-6 i \cos (3 c+4 d x) \log \left (\cos ^2(c+d x)\right )-6 i \cos (5 c+4 d x) \log \left (\cos ^2(c+d x)\right )+11 \sin (c+2 d x)-12 i d x \sin (c+2 d x)-6 \log \left (\cos ^2(c+d x)\right ) \sin (c+2 d x)+14 \sin (3 c+2 d x)-12 i d x \sin (3 c+2 d x)-6 \log \left (\cos ^2(c+d x)\right ) \sin (3 c+2 d x)-4 \sin (3 c+4 d x)-12 i d x \sin (3 c+4 d x)-6 \log \left (\cos ^2(c+d x)\right ) \sin (3 c+4 d x)-\sin (5 c+4 d x)-12 i d x \sin (5 c+4 d x)-6 \log \left (\cos ^2(c+d x)\right ) \sin (5 c+4 d x)\right ) (\cos (3 c+11 d x)+i \sin (3 c+11 d x))}{6 d (\cos (d x)+i \sin (d x))^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^6*(a + I*a*Tan[c + d*x])^8,x]

[Out]

-1/6*(a^8*Sec[c]*Sec[c + d*x]*((12*I)*Cos[c] + (10*I)*Cos[3*c + 2*d*x] + 12*d*x*Cos[3*c + 2*d*x] - (2*I)*Cos[3
*c + 4*d*x] + 12*d*x*Cos[3*c + 4*d*x] + I*Cos[5*c + 4*d*x] + 12*d*x*Cos[5*c + 4*d*x] + Cos[c + 2*d*x]*(7*I + 1
2*d*x - (6*I)*Log[Cos[c + d*x]^2]) - (6*I)*Cos[3*c + 2*d*x]*Log[Cos[c + d*x]^2] - (6*I)*Cos[3*c + 4*d*x]*Log[C
os[c + d*x]^2] - (6*I)*Cos[5*c + 4*d*x]*Log[Cos[c + d*x]^2] + 11*Sin[c + 2*d*x] - (12*I)*d*x*Sin[c + 2*d*x] -
6*Log[Cos[c + d*x]^2]*Sin[c + 2*d*x] + 14*Sin[3*c + 2*d*x] - (12*I)*d*x*Sin[3*c + 2*d*x] - 6*Log[Cos[c + d*x]^
2]*Sin[3*c + 2*d*x] - 4*Sin[3*c + 4*d*x] - (12*I)*d*x*Sin[3*c + 4*d*x] - 6*Log[Cos[c + d*x]^2]*Sin[3*c + 4*d*x
] - Sin[5*c + 4*d*x] - (12*I)*d*x*Sin[5*c + 4*d*x] - 6*Log[Cos[c + d*x]^2]*Sin[5*c + 4*d*x])*(Cos[3*c + 11*d*x
] + I*Sin[3*c + 11*d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^8)

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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. 397 vs. \(2 (105 ) = 210\).
time = 0.20, size = 398, normalized size = 3.49

method	result	size
risch	\(-\frac {2 i a^{8} {\mathrm e}^{6 i \left (d x +c \right )}}{3 d}+\frac {2 i a^{8} {\mathrm e}^{4 i \left (d x +c \right )}}{d}-\frac {6 i a^{8} {\mathrm e}^{2 i \left (d x +c \right )}}{d}+\frac {16 a^{8} c}{d}+\frac {2 i a^{8}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {8 i a^{8} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\)	\(108\)
derivativedivides	\(\frac {a^{8} \left (\frac {\sin ^{9}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\sin ^{7}\left (d x +c \right )+\frac {7 \left (\sin ^{5}\left (d x +c \right )\right )}{6}+\frac {35 \left (\sin ^{3}\left (d x +c \right )\right )}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )-\frac {35 d x}{16}-\frac {35 c}{16}\right )+\frac {28 i a^{8} \left (\sin ^{6}\left (d x +c \right )\right )}{3}-28 a^{8} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )-56 i a^{8} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{6}-\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{12}\right )+70 a^{8} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{6}-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )-\frac {4 i a^{8} \left (\cos ^{6}\left (d x +c \right )\right )}{3}-28 a^{8} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )-8 i a^{8} \left (-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{6}-\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )+a^{8} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\)	\(398\)
default	\(\frac {a^{8} \left (\frac {\sin ^{9}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\sin ^{7}\left (d x +c \right )+\frac {7 \left (\sin ^{5}\left (d x +c \right )\right )}{6}+\frac {35 \left (\sin ^{3}\left (d x +c \right )\right )}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )-\frac {35 d x}{16}-\frac {35 c}{16}\right )+\frac {28 i a^{8} \left (\sin ^{6}\left (d x +c \right )\right )}{3}-28 a^{8} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )-56 i a^{8} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{6}-\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{12}\right )+70 a^{8} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{6}-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )-\frac {4 i a^{8} \left (\cos ^{6}\left (d x +c \right )\right )}{3}-28 a^{8} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )-8 i a^{8} \left (-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{6}-\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )+a^{8} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\)	\(398\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*(a+I*a*tan(d*x+c))^8,x,method=_RETURNVERBOSE)

[Out]

1/d*(a^8*(sin(d*x+c)^9/cos(d*x+c)+(sin(d*x+c)^7+7/6*sin(d*x+c)^5+35/24*sin(d*x+c)^3+35/16*sin(d*x+c))*cos(d*x+
c)-35/16*d*x-35/16*c)+28/3*I*a^8*sin(d*x+c)^6-28*a^8*(-1/6*(sin(d*x+c)^5+5/4*sin(d*x+c)^3+15/8*sin(d*x+c))*cos
(d*x+c)+5/16*d*x+5/16*c)-56*I*a^8*(-1/6*sin(d*x+c)^2*cos(d*x+c)^4-1/12*cos(d*x+c)^4)+70*a^8*(-1/6*sin(d*x+c)^3
*cos(d*x+c)^3-1/8*sin(d*x+c)*cos(d*x+c)^3+1/16*sin(d*x+c)*cos(d*x+c)+1/16*d*x+1/16*c)-4/3*I*a^8*cos(d*x+c)^6-2
8*a^8*(-1/6*sin(d*x+c)*cos(d*x+c)^5+1/24*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+1/16*d*x+1/16*c)-8*I*a^8*(-1
/6*sin(d*x+c)^6-1/4*sin(d*x+c)^4-1/2*sin(d*x+c)^2-ln(cos(d*x+c)))+a^8*(1/6*(cos(d*x+c)^5+5/4*cos(d*x+c)^3+15/8
*cos(d*x+c))*sin(d*x+c)+5/16*d*x+5/16*c))

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Maxima [A]
time = 0.50, size = 146, normalized size = 1.28 \begin {gather*} -\frac {24 \, {\left (d x + c\right )} a^{8} + 12 i \, a^{8} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 3 \, a^{8} \tan \left (d x + c\right ) - \frac {8 \, {\left (9 \, a^{8} \tan \left (d x + c\right )^{5} - 15 i \, a^{8} \tan \left (d x + c\right )^{4} + 4 \, a^{8} \tan \left (d x + c\right )^{3} - 12 i \, a^{8} \tan \left (d x + c\right )^{2} + 3 \, a^{8} \tan \left (d x + c\right ) - 5 i \, a^{8}\right )}}{\tan \left (d x + c\right )^{6} + 3 \, \tan \left (d x + c\right )^{4} + 3 \, \tan \left (d x + c\right )^{2} + 1}}{3 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*(a+I*a*tan(d*x+c))^8,x, algorithm="maxima")

[Out]

-1/3*(24*(d*x + c)*a^8 + 12*I*a^8*log(tan(d*x + c)^2 + 1) - 3*a^8*tan(d*x + c) - 8*(9*a^8*tan(d*x + c)^5 - 15*
I*a^8*tan(d*x + c)^4 + 4*a^8*tan(d*x + c)^3 - 12*I*a^8*tan(d*x + c)^2 + 3*a^8*tan(d*x + c) - 5*I*a^8)/(tan(d*x
+ c)^6 + 3*tan(d*x + c)^4 + 3*tan(d*x + c)^2 + 1))/d

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Fricas [A]
time = 0.38, size = 113, normalized size = 0.99 \begin {gather*} -\frac {2 \, {\left (i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} - 2 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} + 6 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} + 9 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} - 3 i \, a^{8} + 12 \, {\left (-i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a^{8}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )\right )}}{3 \, {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*(a+I*a*tan(d*x+c))^8,x, algorithm="fricas")

[Out]

-2/3*(I*a^8*e^(8*I*d*x + 8*I*c) - 2*I*a^8*e^(6*I*d*x + 6*I*c) + 6*I*a^8*e^(4*I*d*x + 4*I*c) + 9*I*a^8*e^(2*I*d
*x + 2*I*c) - 3*I*a^8 + 12*(-I*a^8*e^(2*I*d*x + 2*I*c) - I*a^8)*log(e^(2*I*d*x + 2*I*c) + 1))/(d*e^(2*I*d*x +
2*I*c) + d)

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Sympy [A]
time = 0.51, size = 172, normalized size = 1.51 \begin {gather*} \frac {2 i a^{8}}{d e^{2 i c} e^{2 i d x} + d} + \frac {8 i a^{8} \log {\left (e^{2 i d x} + e^{- 2 i c} \right )}}{d} + \begin {cases} \frac {- 2 i a^{8} d^{2} e^{6 i c} e^{6 i d x} + 6 i a^{8} d^{2} e^{4 i c} e^{4 i d x} - 18 i a^{8} d^{2} e^{2 i c} e^{2 i d x}}{3 d^{3}} & \text {for}\: d^{3} \neq 0 \\x \left (4 a^{8} e^{6 i c} - 8 a^{8} e^{4 i c} + 12 a^{8} e^{2 i c}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**6*(a+I*a*tan(d*x+c))**8,x)

[Out]

2*I*a**8/(d*exp(2*I*c)*exp(2*I*d*x) + d) + 8*I*a**8*log(exp(2*I*d*x) + exp(-2*I*c))/d + Piecewise(((-2*I*a**8*
d**2*exp(6*I*c)*exp(6*I*d*x) + 6*I*a**8*d**2*exp(4*I*c)*exp(4*I*d*x) - 18*I*a**8*d**2*exp(2*I*c)*exp(2*I*d*x))
/(3*d**3), Ne(d**3, 0)), (x*(4*a**8*exp(6*I*c) - 8*a**8*exp(4*I*c) + 12*a**8*exp(2*I*c)), 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. 799 vs. \(2 (98) = 196\).
time = 1.18, size = 799, normalized size = 7.01 \begin {gather*} -\frac {2 \, {\left (-12 i \, a^{8} e^{\left (28 i \, d x + 14 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 168 i \, a^{8} e^{\left (26 i \, d x + 12 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 1092 i \, a^{8} e^{\left (24 i \, d x + 10 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 4368 i \, a^{8} e^{\left (22 i \, d x + 8 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 12012 i \, a^{8} e^{\left (20 i \, d x + 6 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 24024 i \, a^{8} e^{\left (18 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 36036 i \, a^{8} e^{\left (16 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 36036 i \, a^{8} e^{\left (12 i \, d x - 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 24024 i \, a^{8} e^{\left (10 i \, d x - 4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 12012 i \, a^{8} e^{\left (8 i \, d x - 6 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 4368 i \, a^{8} e^{\left (6 i \, d x - 8 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 1092 i \, a^{8} e^{\left (4 i \, d x - 10 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 168 i \, a^{8} e^{\left (2 i \, d x - 12 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 41184 i \, a^{8} e^{\left (14 i \, d x\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 12 i \, a^{8} e^{\left (-14 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + i \, a^{8} e^{\left (34 i \, d x + 20 i \, c\right )} + 11 i \, a^{8} e^{\left (32 i \, d x + 18 i \, c\right )} + 58 i \, a^{8} e^{\left (30 i \, d x + 16 i \, c\right )} + 217 i \, a^{8} e^{\left (28 i \, d x + 14 i \, c\right )} + 725 i \, a^{8} e^{\left (26 i \, d x + 12 i \, c\right )} + 2236 i \, a^{8} e^{\left (24 i \, d x + 10 i \, c\right )} + 5772 i \, a^{8} e^{\left (22 i \, d x + 8 i \, c\right )} + 11583 i \, a^{8} e^{\left (20 i \, d x + 6 i \, c\right )} + 17589 i \, a^{8} e^{\left (18 i \, d x + 4 i \, c\right )} + 20020 i \, a^{8} e^{\left (16 i \, d x + 2 i \, c\right )} + 10231 i \, a^{8} e^{\left (12 i \, d x - 2 i \, c\right )} + 4147 i \, a^{8} e^{\left (10 i \, d x - 4 i \, c\right )} + 872 i \, a^{8} e^{\left (8 i \, d x - 6 i \, c\right )} - 80 i \, a^{8} e^{\left (6 i \, d x - 8 i \, c\right )} - 111 i \, a^{8} e^{\left (4 i \, d x - 10 i \, c\right )} - 30 i \, a^{8} e^{\left (2 i \, d x - 12 i \, c\right )} + 16874 i \, a^{8} e^{\left (14 i \, d x\right )} - 3 i \, a^{8} e^{\left (-14 i \, c\right )}\right )}}{3 \, {\left (d e^{\left (28 i \, d x + 14 i \, c\right )} + 14 \, d e^{\left (26 i \, d x + 12 i \, c\right )} + 91 \, d e^{\left (24 i \, d x + 10 i \, c\right )} + 364 \, d e^{\left (22 i \, d x + 8 i \, c\right )} + 1001 \, d e^{\left (20 i \, d x + 6 i \, c\right )} + 2002 \, d e^{\left (18 i \, d x + 4 i \, c\right )} + 3003 \, d e^{\left (16 i \, d x + 2 i \, c\right )} + 3003 \, d e^{\left (12 i \, d x - 2 i \, c\right )} + 2002 \, d e^{\left (10 i \, d x - 4 i \, c\right )} + 1001 \, d e^{\left (8 i \, d x - 6 i \, c\right )} + 364 \, d e^{\left (6 i \, d x - 8 i \, c\right )} + 91 \, d e^{\left (4 i \, d x - 10 i \, c\right )} + 14 \, d e^{\left (2 i \, d x - 12 i \, c\right )} + 3432 \, d e^{\left (14 i \, d x\right )} + d e^{\left (-14 i \, c\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*(a+I*a*tan(d*x+c))^8,x, algorithm="giac")

[Out]

-2/3*(-12*I*a^8*e^(28*I*d*x + 14*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 168*I*a^8*e^(26*I*d*x + 12*I*c)*log(e^(2*
I*d*x + 2*I*c) + 1) - 1092*I*a^8*e^(24*I*d*x + 10*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 4368*I*a^8*e^(22*I*d*x +
8*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 12012*I*a^8*e^(20*I*d*x + 6*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 24024*I
*a^8*e^(18*I*d*x + 4*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 36036*I*a^8*e^(16*I*d*x + 2*I*c)*log(e^(2*I*d*x + 2*I
*c) + 1) - 36036*I*a^8*e^(12*I*d*x - 2*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 24024*I*a^8*e^(10*I*d*x - 4*I*c)*lo
g(e^(2*I*d*x + 2*I*c) + 1) - 12012*I*a^8*e^(8*I*d*x - 6*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 4368*I*a^8*e^(6*I*
d*x - 8*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 1092*I*a^8*e^(4*I*d*x - 10*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 168
*I*a^8*e^(2*I*d*x - 12*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 41184*I*a^8*e^(14*I*d*x)*log(e^(2*I*d*x + 2*I*c) +
1) - 12*I*a^8*e^(-14*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) + I*a^8*e^(34*I*d*x + 20*I*c) + 11*I*a^8*e^(32*I*d*x +
18*I*c) + 58*I*a^8*e^(30*I*d*x + 16*I*c) + 217*I*a^8*e^(28*I*d*x + 14*I*c) + 725*I*a^8*e^(26*I*d*x + 12*I*c) +
2236*I*a^8*e^(24*I*d*x + 10*I*c) + 5772*I*a^8*e^(22*I*d*x + 8*I*c) + 11583*I*a^8*e^(20*I*d*x + 6*I*c) + 17589
*I*a^8*e^(18*I*d*x + 4*I*c) + 20020*I*a^8*e^(16*I*d*x + 2*I*c) + 10231*I*a^8*e^(12*I*d*x - 2*I*c) + 4147*I*a^8
*e^(10*I*d*x - 4*I*c) + 872*I*a^8*e^(8*I*d*x - 6*I*c) - 80*I*a^8*e^(6*I*d*x - 8*I*c) - 111*I*a^8*e^(4*I*d*x -
10*I*c) - 30*I*a^8*e^(2*I*d*x - 12*I*c) + 16874*I*a^8*e^(14*I*d*x) - 3*I*a^8*e^(-14*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) + 1
4*d*e^(2*I*d*x - 12*I*c) + 3432*d*e^(14*I*d*x) + d*e^(-14*I*c))

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Mupad [B]
time = 3.41, size = 103, normalized size = 0.90 \begin {gather*} \frac {a^8\,\mathrm {tan}\left (c+d\,x\right )}{d}-\frac {24\,a^8\,{\mathrm {tan}\left (c+d\,x\right )}^2+a^8\,\mathrm {tan}\left (c+d\,x\right )\,32{}\mathrm {i}-\frac {40\,a^8}{3}}{d\,\left (-{\mathrm {tan}\left (c+d\,x\right )}^3-{\mathrm {tan}\left (c+d\,x\right )}^2\,3{}\mathrm {i}+3\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}-\frac {a^8\,\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,8{}\mathrm {i}}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^6*(a + a*tan(c + d*x)*1i)^8,x)

[Out]

(a^8*tan(c + d*x))/d - (a^8*log(tan(c + d*x) + 1i)*8i)/d - (a^8*tan(c + d*x)*32i - (40*a^8)/3 + 24*a^8*tan(c +
d*x)^2)/(d*(3*tan(c + d*x) - tan(c + d*x)^2*3i - tan(c + d*x)^3 + 1i))

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