3.1.90 \(\int \cos ^{18}(c+d x) (a+i a \tan (c+d x))^8 \, dx\) [90]

Optimal. Leaf size=279 \[ \frac {5 a^8 x}{512}-\frac {i a^{17}}{36 d (a-i a \tan (c+d x))^9}-\frac {i a^{16}}{32 d (a-i a \tan (c+d x))^8}-\frac {3 i a^{15}}{112 d (a-i a \tan (c+d x))^7}-\frac {i a^{14}}{48 d (a-i a \tan (c+d x))^6}-\frac {i a^{13}}{64 d (a-i a \tan (c+d x))^5}-\frac {3 i a^{12}}{256 d (a-i a \tan (c+d x))^4}-\frac {7 i a^{11}}{768 d (a-i a \tan (c+d x))^3}-\frac {i a^{10}}{128 d (a-i a \tan (c+d x))^2}-\frac {9 i a^9}{1024 d (a-i a \tan (c+d x))}+\frac {i a^9}{1024 d (a+i a \tan (c+d x))} \]

[Out]

5/512*a^8*x-1/36*I*a^17/d/(a-I*a*tan(d*x+c))^9-1/32*I*a^16/d/(a-I*a*tan(d*x+c))^8-3/112*I*a^15/d/(a-I*a*tan(d*
x+c))^7-1/48*I*a^14/d/(a-I*a*tan(d*x+c))^6-1/64*I*a^13/d/(a-I*a*tan(d*x+c))^5-3/256*I*a^12/d/(a-I*a*tan(d*x+c)
)^4-7/768*I*a^11/d/(a-I*a*tan(d*x+c))^3-1/128*I*a^10/d/(a-I*a*tan(d*x+c))^2-9/1024*I*a^9/d/(a-I*a*tan(d*x+c))+
1/1024*I*a^9/d/(a+I*a*tan(d*x+c))

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Rubi [A]
time = 0.13, antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3568, 46, 212} \begin {gather*} -\frac {i a^{17}}{36 d (a-i a \tan (c+d x))^9}-\frac {i a^{16}}{32 d (a-i a \tan (c+d x))^8}-\frac {3 i a^{15}}{112 d (a-i a \tan (c+d x))^7}-\frac {i a^{14}}{48 d (a-i a \tan (c+d x))^6}-\frac {i a^{13}}{64 d (a-i a \tan (c+d x))^5}-\frac {3 i a^{12}}{256 d (a-i a \tan (c+d x))^4}-\frac {7 i a^{11}}{768 d (a-i a \tan (c+d x))^3}-\frac {i a^{10}}{128 d (a-i a \tan (c+d x))^2}-\frac {9 i a^9}{1024 d (a-i a \tan (c+d x))}+\frac {i a^9}{1024 d (a+i a \tan (c+d x))}+\frac {5 a^8 x}{512} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^18*(a + I*a*Tan[c + d*x])^8,x]

[Out]

(5*a^8*x)/512 - ((I/36)*a^17)/(d*(a - I*a*Tan[c + d*x])^9) - ((I/32)*a^16)/(d*(a - I*a*Tan[c + d*x])^8) - (((3
*I)/112)*a^15)/(d*(a - I*a*Tan[c + d*x])^7) - ((I/48)*a^14)/(d*(a - I*a*Tan[c + d*x])^6) - ((I/64)*a^13)/(d*(a
- I*a*Tan[c + d*x])^5) - (((3*I)/256)*a^12)/(d*(a - I*a*Tan[c + d*x])^4) - (((7*I)/768)*a^11)/(d*(a - I*a*Tan
[c + d*x])^3) - ((I/128)*a^10)/(d*(a - I*a*Tan[c + d*x])^2) - (((9*I)/1024)*a^9)/(d*(a - I*a*Tan[c + d*x])) +
((I/1024)*a^9)/(d*(a + I*a*Tan[c + d*x]))

Rule 46

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 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 ^{18}(c+d x) (a+i a \tan (c+d x))^8 \, dx &=-\frac {\left (i a^{19}\right ) \text {Subst}\left (\int \frac {1}{(a-x)^{10} (a+x)^2} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac {\left (i a^{19}\right ) \text {Subst}\left (\int \left (\frac {1}{4 a^2 (a-x)^{10}}+\frac {1}{4 a^3 (a-x)^9}+\frac {3}{16 a^4 (a-x)^8}+\frac {1}{8 a^5 (a-x)^7}+\frac {5}{64 a^6 (a-x)^6}+\frac {3}{64 a^7 (a-x)^5}+\frac {7}{256 a^8 (a-x)^4}+\frac {1}{64 a^9 (a-x)^3}+\frac {9}{1024 a^{10} (a-x)^2}+\frac {1}{1024 a^{10} (a+x)^2}+\frac {5}{512 a^{10} \left (a^2-x^2\right )}\right ) \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac {i a^{17}}{36 d (a-i a \tan (c+d x))^9}-\frac {i a^{16}}{32 d (a-i a \tan (c+d x))^8}-\frac {3 i a^{15}}{112 d (a-i a \tan (c+d x))^7}-\frac {i a^{14}}{48 d (a-i a \tan (c+d x))^6}-\frac {i a^{13}}{64 d (a-i a \tan (c+d x))^5}-\frac {3 i a^{12}}{256 d (a-i a \tan (c+d x))^4}-\frac {7 i a^{11}}{768 d (a-i a \tan (c+d x))^3}-\frac {i a^{10}}{128 d (a-i a \tan (c+d x))^2}-\frac {9 i a^9}{1024 d (a-i a \tan (c+d x))}+\frac {i a^9}{1024 d (a+i a \tan (c+d x))}-\frac {\left (5 i a^9\right ) \text {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,i a \tan (c+d x)\right )}{512 d}\\ &=\frac {5 a^8 x}{512}-\frac {i a^{17}}{36 d (a-i a \tan (c+d x))^9}-\frac {i a^{16}}{32 d (a-i a \tan (c+d x))^8}-\frac {3 i a^{15}}{112 d (a-i a \tan (c+d x))^7}-\frac {i a^{14}}{48 d (a-i a \tan (c+d x))^6}-\frac {i a^{13}}{64 d (a-i a \tan (c+d x))^5}-\frac {3 i a^{12}}{256 d (a-i a \tan (c+d x))^4}-\frac {7 i a^{11}}{768 d (a-i a \tan (c+d x))^3}-\frac {i a^{10}}{128 d (a-i a \tan (c+d x))^2}-\frac {9 i a^9}{1024 d (a-i a \tan (c+d x))}+\frac {i a^9}{1024 d (a+i a \tan (c+d x))}\\ \end {align*}

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Mathematica [A]
time = 3.98, size = 188, normalized size = 0.67 \begin {gather*} \frac {a^8 (-15876 i-28224 i \cos (2 (c+d x))-20160 i \cos (4 (c+d x))-12960 i \cos (6 (c+d x))-315 i \cos (8 (c+d x))+5040 d x \cos (8 (c+d x))+224 i \cos (10 (c+d x))-7056 \sin (2 (c+d x))-10080 \sin (4 (c+d x))-9720 \sin (6 (c+d x))+315 \sin (8 (c+d x))-5040 i d x \sin (8 (c+d x))+280 \sin (10 (c+d x))) (\cos (8 (c+2 d x))+i \sin (8 (c+2 d x)))}{516096 d (\cos (d x)+i \sin (d x))^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^18*(a + I*a*Tan[c + d*x])^8,x]

[Out]

(a^8*(-15876*I - (28224*I)*Cos[2*(c + d*x)] - (20160*I)*Cos[4*(c + d*x)] - (12960*I)*Cos[6*(c + d*x)] - (315*I
)*Cos[8*(c + d*x)] + 5040*d*x*Cos[8*(c + d*x)] + (224*I)*Cos[10*(c + d*x)] - 7056*Sin[2*(c + d*x)] - 10080*Sin
[4*(c + d*x)] - 9720*Sin[6*(c + d*x)] + 315*Sin[8*(c + d*x)] - (5040*I)*d*x*Sin[8*(c + d*x)] + 280*Sin[10*(c +
d*x)])*(Cos[8*(c + 2*d*x)] + I*Sin[8*(c + 2*d*x)]))/(516096*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. 788 vs. \(2 (237 ) = 474\).
time = 0.32, size = 789, normalized size = 2.83

method	result	size
risch	\(\frac {5 a^{8} x}{512}-\frac {i a^{8} {\mathrm e}^{18 i \left (d x +c \right )}}{18432 d}-\frac {5 i a^{8} {\mathrm e}^{16 i \left (d x +c \right )}}{8192 d}-\frac {45 i a^{8} {\mathrm e}^{14 i \left (d x +c \right )}}{14336 d}-\frac {5 i a^{8} {\mathrm e}^{12 i \left (d x +c \right )}}{512 d}-\frac {21 i a^{8} {\mathrm e}^{10 i \left (d x +c \right )}}{1024 d}-\frac {63 i a^{8} {\mathrm e}^{8 i \left (d x +c \right )}}{2048 d}-\frac {35 i a^{8} {\mathrm e}^{6 i \left (d x +c \right )}}{1024 d}-\frac {15 i a^{8} {\mathrm e}^{4 i \left (d x +c \right )}}{512 d}-\frac {11 i a^{8} \cos \left (2 d x +2 c \right )}{512 d}+\frac {23 a^{8} \sin \left (2 d x +2 c \right )}{1024 d}\)	\(187\)
derivativedivides	\(\text {Expression too large to display}\)	\(789\)
default	\(\text {Expression too large to display}\)	\(789\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^18*(a+I*a*tan(d*x+c))^8,x,method=_RETURNVERBOSE)

[Out]

1/d*(a^8*(-1/18*sin(d*x+c)^7*cos(d*x+c)^11-7/288*sin(d*x+c)^5*cos(d*x+c)^11-5/576*sin(d*x+c)^3*cos(d*x+c)^11-5
/2304*sin(d*x+c)*cos(d*x+c)^11+1/4608*(cos(d*x+c)^9+9/8*cos(d*x+c)^7+21/16*cos(d*x+c)^5+105/64*cos(d*x+c)^3+31
5/128*cos(d*x+c))*sin(d*x+c)+35/65536*d*x+35/65536*c)-8*I*a^8*(-1/18*sin(d*x+c)^6*cos(d*x+c)^12-1/48*sin(d*x+c
)^4*cos(d*x+c)^12-1/168*sin(d*x+c)^2*cos(d*x+c)^12-1/1008*cos(d*x+c)^12)-28*a^8*(-1/18*sin(d*x+c)^5*cos(d*x+c)
^13-5/288*sin(d*x+c)^3*cos(d*x+c)^13-5/1344*cos(d*x+c)^13*sin(d*x+c)+5/16128*(cos(d*x+c)^11+11/10*cos(d*x+c)^9
+99/80*cos(d*x+c)^7+231/160*cos(d*x+c)^5+231/128*cos(d*x+c)^3+693/256*cos(d*x+c))*sin(d*x+c)+55/65536*d*x+55/6
5536*c)+56*I*a^8*(-1/18*sin(d*x+c)^4*cos(d*x+c)^14-1/72*sin(d*x+c)^2*cos(d*x+c)^14-1/504*cos(d*x+c)^14)+70*a^8
*(-1/18*sin(d*x+c)^3*cos(d*x+c)^15-1/96*cos(d*x+c)^15*sin(d*x+c)+1/1344*(cos(d*x+c)^13+13/12*cos(d*x+c)^11+143
/120*cos(d*x+c)^9+429/320*cos(d*x+c)^7+1001/640*cos(d*x+c)^5+1001/512*cos(d*x+c)^3+3003/1024*cos(d*x+c))*sin(d
*x+c)+143/65536*d*x+143/65536*c)-56*I*a^8*(-1/18*sin(d*x+c)^2*cos(d*x+c)^16-1/144*cos(d*x+c)^16)-28*a^8*(-1/18
*cos(d*x+c)^17*sin(d*x+c)+1/288*(cos(d*x+c)^15+15/14*cos(d*x+c)^13+65/56*cos(d*x+c)^11+143/112*cos(d*x+c)^9+12
87/896*cos(d*x+c)^7+429/256*cos(d*x+c)^5+2145/1024*cos(d*x+c)^3+6435/2048*cos(d*x+c))*sin(d*x+c)+715/65536*d*x
+715/65536*c)-4/9*I*a^8*cos(d*x+c)^18+a^8*(1/18*(cos(d*x+c)^17+17/16*cos(d*x+c)^15+255/224*cos(d*x+c)^13+1105/
896*cos(d*x+c)^11+2431/1792*cos(d*x+c)^9+21879/14336*cos(d*x+c)^7+7293/4096*cos(d*x+c)^5+36465/16384*cos(d*x+c
)^3+109395/32768*cos(d*x+c))*sin(d*x+c)+12155/65536*d*x+12155/65536*c))

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Maxima [A]
time = 0.53, size = 269, normalized size = 0.96 \begin {gather*} \frac {315 \, {\left (d x + c\right )} a^{8} + \frac {315 \, a^{8} \tan \left (d x + c\right )^{17} + 2730 \, a^{8} \tan \left (d x + c\right )^{15} + 10458 \, a^{8} \tan \left (d x + c\right )^{13} + 23202 \, a^{8} \tan \left (d x + c\right )^{11} + 32768 \, a^{8} \tan \left (d x + c\right )^{9} + 27486 \, a^{8} \tan \left (d x + c\right )^{7} + 21504 i \, a^{8} \tan \left (d x + c\right )^{6} + 86310 \, a^{8} \tan \left (d x + c\right )^{5} - 119808 i \, a^{8} \tan \left (d x + c\right )^{4} - 121002 \, a^{8} \tan \left (d x + c\right )^{3} + 82944 i \, a^{8} \tan \left (d x + c\right )^{2} + 31941 \, a^{8} \tan \left (d x + c\right ) - 5120 i \, a^{8}}{\tan \left (d x + c\right )^{18} + 9 \, \tan \left (d x + c\right )^{16} + 36 \, \tan \left (d x + c\right )^{14} + 84 \, \tan \left (d x + c\right )^{12} + 126 \, \tan \left (d x + c\right )^{10} + 126 \, \tan \left (d x + c\right )^{8} + 84 \, \tan \left (d x + c\right )^{6} + 36 \, \tan \left (d x + c\right )^{4} + 9 \, \tan \left (d x + c\right )^{2} + 1}}{32256 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^18*(a+I*a*tan(d*x+c))^8,x, algorithm="maxima")

[Out]

1/32256*(315*(d*x + c)*a^8 + (315*a^8*tan(d*x + c)^17 + 2730*a^8*tan(d*x + c)^15 + 10458*a^8*tan(d*x + c)^13 +
23202*a^8*tan(d*x + c)^11 + 32768*a^8*tan(d*x + c)^9 + 27486*a^8*tan(d*x + c)^7 + 21504*I*a^8*tan(d*x + c)^6
+ 86310*a^8*tan(d*x + c)^5 - 119808*I*a^8*tan(d*x + c)^4 - 121002*a^8*tan(d*x + c)^3 + 82944*I*a^8*tan(d*x + c
)^2 + 31941*a^8*tan(d*x + c) - 5120*I*a^8)/(tan(d*x + c)^18 + 9*tan(d*x + c)^16 + 36*tan(d*x + c)^14 + 84*tan(
d*x + c)^12 + 126*tan(d*x + c)^10 + 126*tan(d*x + c)^8 + 84*tan(d*x + c)^6 + 36*tan(d*x + c)^4 + 9*tan(d*x + c
)^2 + 1))/d

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Fricas [A]
time = 0.47, size = 162, normalized size = 0.58 \begin {gather*} \frac {{\left (5040 \, a^{8} d x e^{\left (2 i \, d x + 2 i \, c\right )} - 28 i \, a^{8} e^{\left (20 i \, d x + 20 i \, c\right )} - 315 i \, a^{8} e^{\left (18 i \, d x + 18 i \, c\right )} - 1620 i \, a^{8} e^{\left (16 i \, d x + 16 i \, c\right )} - 5040 i \, a^{8} e^{\left (14 i \, d x + 14 i \, c\right )} - 10584 i \, a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} - 15876 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} - 17640 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} - 15120 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} - 11340 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} + 252 i \, a^{8}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{516096 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^18*(a+I*a*tan(d*x+c))^8,x, algorithm="fricas")

[Out]

1/516096*(5040*a^8*d*x*e^(2*I*d*x + 2*I*c) - 28*I*a^8*e^(20*I*d*x + 20*I*c) - 315*I*a^8*e^(18*I*d*x + 18*I*c)
- 1620*I*a^8*e^(16*I*d*x + 16*I*c) - 5040*I*a^8*e^(14*I*d*x + 14*I*c) - 10584*I*a^8*e^(12*I*d*x + 12*I*c) - 15
876*I*a^8*e^(10*I*d*x + 10*I*c) - 17640*I*a^8*e^(8*I*d*x + 8*I*c) - 15120*I*a^8*e^(6*I*d*x + 6*I*c) - 11340*I*
a^8*e^(4*I*d*x + 4*I*c) + 252*I*a^8)*e^(-2*I*d*x - 2*I*c)/d

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Sympy [A]
time = 1.05, size = 413, normalized size = 1.48 \begin {gather*} \frac {5 a^{8} x}{512} + \begin {cases} \frac {\left (- 277298568799925181577403826176 i a^{8} d^{9} e^{20 i c} e^{18 i d x} - 3119608898999158292745793044480 i a^{8} d^{9} e^{18 i c} e^{16 i d x} - 16043702909138528362692649943040 i a^{8} d^{9} e^{16 i c} e^{14 i d x} - 49913742383986532683932688711680 i a^{8} d^{9} e^{14 i c} e^{12 i d x} - 104818859006371718636258646294528 i a^{8} d^{9} e^{12 i c} e^{10 i d x} - 157228288509557577954387969441792 i a^{8} d^{9} e^{10 i c} e^{8 i d x} - 174698098343952864393764410490880 i a^{8} d^{9} e^{8 i c} e^{6 i d x} - 149741227151959598051798066135040 i a^{8} d^{9} e^{6 i c} e^{4 i d x} - 112305920363969698538848549601280 i a^{8} d^{9} e^{4 i c} e^{2 i d x} + 2495687119199326634196634435584 i a^{8} d^{9} e^{- 2 i d x}\right ) e^{- 2 i c}}{5111167220120220946834707324076032 d^{10}} & \text {for}\: d^{10} e^{2 i c} \neq 0 \\x \left (- \frac {5 a^{8}}{512} + \frac {\left (a^{8} e^{20 i c} + 10 a^{8} e^{18 i c} + 45 a^{8} e^{16 i c} + 120 a^{8} e^{14 i c} + 210 a^{8} e^{12 i c} + 252 a^{8} e^{10 i c} + 210 a^{8} e^{8 i c} + 120 a^{8} e^{6 i c} + 45 a^{8} e^{4 i c} + 10 a^{8} e^{2 i c} + a^{8}\right ) e^{- 2 i c}}{1024}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**18*(a+I*a*tan(d*x+c))**8,x)

[Out]

5*a**8*x/512 + Piecewise(((-277298568799925181577403826176*I*a**8*d**9*exp(20*I*c)*exp(18*I*d*x) - 31196088989
99158292745793044480*I*a**8*d**9*exp(18*I*c)*exp(16*I*d*x) - 16043702909138528362692649943040*I*a**8*d**9*exp(
16*I*c)*exp(14*I*d*x) - 49913742383986532683932688711680*I*a**8*d**9*exp(14*I*c)*exp(12*I*d*x) - 1048188590063
71718636258646294528*I*a**8*d**9*exp(12*I*c)*exp(10*I*d*x) - 157228288509557577954387969441792*I*a**8*d**9*exp
(10*I*c)*exp(8*I*d*x) - 174698098343952864393764410490880*I*a**8*d**9*exp(8*I*c)*exp(6*I*d*x) - 14974122715195
9598051798066135040*I*a**8*d**9*exp(6*I*c)*exp(4*I*d*x) - 112305920363969698538848549601280*I*a**8*d**9*exp(4*
I*c)*exp(2*I*d*x) + 2495687119199326634196634435584*I*a**8*d**9*exp(-2*I*d*x))*exp(-2*I*c)/(511116722012022094
6834707324076032*d**10), Ne(d**10*exp(2*I*c), 0)), (x*(-5*a**8/512 + (a**8*exp(20*I*c) + 10*a**8*exp(18*I*c) +
45*a**8*exp(16*I*c) + 120*a**8*exp(14*I*c) + 210*a**8*exp(12*I*c) + 252*a**8*exp(10*I*c) + 210*a**8*exp(8*I*c
) + 120*a**8*exp(6*I*c) + 45*a**8*exp(4*I*c) + 10*a**8*exp(2*I*c) + a**8)*exp(-2*I*c)/1024), 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. 1514 vs. \(2 (217) = 434\).
time = 1.52, size = 1514, normalized size = 5.43 \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)^18*(a+I*a*tan(d*x+c))^8,x, algorithm="giac")

[Out]

1/16515072*(161280*a^8*d*x*e^(30*I*d*x + 16*I*c) + 2257920*a^8*d*x*e^(28*I*d*x + 14*I*c) + 14676480*a^8*d*x*e^
(26*I*d*x + 12*I*c) + 58705920*a^8*d*x*e^(24*I*d*x + 10*I*c) + 161441280*a^8*d*x*e^(22*I*d*x + 8*I*c) + 322882
560*a^8*d*x*e^(20*I*d*x + 6*I*c) + 484323840*a^8*d*x*e^(18*I*d*x + 4*I*c) + 553512960*a^8*d*x*e^(16*I*d*x + 2*
I*c) + 322882560*a^8*d*x*e^(12*I*d*x - 2*I*c) + 161441280*a^8*d*x*e^(10*I*d*x - 4*I*c) + 58705920*a^8*d*x*e^(8
*I*d*x - 6*I*c) + 14676480*a^8*d*x*e^(6*I*d*x - 8*I*c) + 2257920*a^8*d*x*e^(4*I*d*x - 10*I*c) + 161280*a^8*d*x
*e^(2*I*d*x - 12*I*c) + 484323840*a^8*d*x*e^(14*I*d*x) - 75789*I*a^8*e^(30*I*d*x + 16*I*c)*log(e^(2*I*d*x + 2*
I*c) + 1) - 1061046*I*a^8*e^(28*I*d*x + 14*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 6896799*I*a^8*e^(26*I*d*x + 12*
I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 27587196*I*a^8*e^(24*I*d*x + 10*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 758647
89*I*a^8*e^(22*I*d*x + 8*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 151729578*I*a^8*e^(20*I*d*x + 6*I*c)*log(e^(2*I*d
*x + 2*I*c) + 1) - 227594367*I*a^8*e^(18*I*d*x + 4*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 260107848*I*a^8*e^(16*I
*d*x + 2*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 151729578*I*a^8*e^(12*I*d*x - 2*I*c)*log(e^(2*I*d*x + 2*I*c) + 1)
- 75864789*I*a^8*e^(10*I*d*x - 4*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 27587196*I*a^8*e^(8*I*d*x - 6*I*c)*log(e
^(2*I*d*x + 2*I*c) + 1) - 6896799*I*a^8*e^(6*I*d*x - 8*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 1061046*I*a^8*e^(4*
I*d*x - 10*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 75789*I*a^8*e^(2*I*d*x - 12*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) -
227594367*I*a^8*e^(14*I*d*x)*log(e^(2*I*d*x + 2*I*c) + 1) + 75789*I*a^8*e^(30*I*d*x + 16*I*c)*log(e^(2*I*d*x)
+ e^(-2*I*c)) + 1061046*I*a^8*e^(28*I*d*x + 14*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 6896799*I*a^8*e^(26*I*d*x
+ 12*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 27587196*I*a^8*e^(24*I*d*x + 10*I*c)*log(e^(2*I*d*x) + e^(-2*I*c))
+ 75864789*I*a^8*e^(22*I*d*x + 8*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 151729578*I*a^8*e^(20*I*d*x + 6*I*c)*log
(e^(2*I*d*x) + e^(-2*I*c)) + 227594367*I*a^8*e^(18*I*d*x + 4*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 260107848*I*
a^8*e^(16*I*d*x + 2*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 151729578*I*a^8*e^(12*I*d*x - 2*I*c)*log(e^(2*I*d*x)
+ e^(-2*I*c)) + 75864789*I*a^8*e^(10*I*d*x - 4*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 27587196*I*a^8*e^(8*I*d*x
- 6*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 6896799*I*a^8*e^(6*I*d*x - 8*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 106
1046*I*a^8*e^(4*I*d*x - 10*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 75789*I*a^8*e^(2*I*d*x - 12*I*c)*log(e^(2*I*d*
x) + e^(-2*I*c)) + 227594367*I*a^8*e^(14*I*d*x)*log(e^(2*I*d*x) + e^(-2*I*c)) - 896*I*a^8*e^(48*I*d*x + 34*I*c
) - 22624*I*a^8*e^(46*I*d*x + 32*I*c) - 274496*I*a^8*e^(44*I*d*x + 30*I*c) - 2130464*I*a^8*e^(42*I*d*x + 28*I*
c) - 11880064*I*a^8*e^(40*I*d*x + 26*I*c) - 50679776*I*a^8*e^(38*I*d*x + 24*I*c) - 171966144*I*a^8*e^(36*I*d*x
+ 22*I*c) - 476470176*I*a^8*e^(34*I*d*x + 20*I*c) - 1098297984*I*a^8*e^(32*I*d*x + 18*I*c) - 2135476640*I*a^8
*e^(30*I*d*x + 16*I*c) - 3538601920*I*a^8*e^(28*I*d*x + 14*I*c) - 5032909280*I*a^8*e^(26*I*d*x + 12*I*c) - 616
5461120*I*a^8*e^(24*I*d*x + 10*I*c) - 6498731680*I*a^8*e^(22*I*d*x + 8*I*c) - 5857001024*I*a^8*e^(20*I*d*x + 6
*I*c) - 4459555296*I*a^8*e^(18*I*d*x + 4*I*c) - 2817258624*I*a^8*e^(16*I*d*x + 2*I*c) - 573963264*I*a^8*e^(12*
I*d*x - 2*I*c) - 168384384*I*a^8*e^(10*I*d*x - 4*I*c) - 32288256*I*a^8*e^(8*I*d*x - 6*I*c) - 2628864*I*a^8*e^(
6*I*d*x - 8*I*c) + 370944*I*a^8*e^(4*I*d*x - 10*I*c) + 112896*I*a^8*e^(2*I*d*x - 12*I*c) - 1439738496*I*a^8*e^
(14*I*d*x) + 8064*I*a^8*e^(-14*I*c))/(d*e^(30*I*d*x + 16*I*c) + 14*d*e^(28*I*d*x + 14*I*c) + 91*d*e^(26*I*d*x
+ 12*I*c) + 364*d*e^(24*I*d*x + 10*I*c) + 1001*d*e^(22*I*d*x + 8*I*c) + 2002*d*e^(20*I*d*x + 6*I*c) + 3003*d*e
^(18*I*d*x + 4*I*c) + 3432*d*e^(16*I*d*x + 2*I*c) + 2002*d*e^(12*I*d*x - 2*I*c) + 1001*d*e^(10*I*d*x - 4*I*c)
+ 364*d*e^(8*I*d*x - 6*I*c) + 91*d*e^(6*I*d*x - 8*I*c) + 14*d*e^(4*I*d*x - 10*I*c) + d*e^(2*I*d*x - 12*I*c) +
3003*d*e^(14*I*d*x))

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Mupad [B]
time = 5.23, size = 231, normalized size = 0.83 \begin {gather*} \frac {5\,a^8\,x}{512}+\frac {\frac {5\,a^8\,{\mathrm {tan}\left (c+d\,x\right )}^9}{512}+\frac {a^8\,{\mathrm {tan}\left (c+d\,x\right )}^8\,5{}\mathrm {i}}{64}-\frac {205\,a^8\,{\mathrm {tan}\left (c+d\,x\right )}^7}{768}-\frac {a^8\,{\mathrm {tan}\left (c+d\,x\right )}^6\,95{}\mathrm {i}}{192}+\frac {a^8\,{\mathrm {tan}\left (c+d\,x\right )}^5}{2}+\frac {a^8\,{\mathrm {tan}\left (c+d\,x\right )}^4\,11{}\mathrm {i}}{64}+\frac {393\,a^8\,{\mathrm {tan}\left (c+d\,x\right )}^3}{1792}+\frac {a^8\,{\mathrm {tan}\left (c+d\,x\right )}^2\,163{}\mathrm {i}}{448}-\frac {9019\,a^8\,\mathrm {tan}\left (c+d\,x\right )}{32256}-\frac {a^8\,10{}\mathrm {i}}{63}}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^{10}+{\mathrm {tan}\left (c+d\,x\right )}^9\,8{}\mathrm {i}-27\,{\mathrm {tan}\left (c+d\,x\right )}^8-{\mathrm {tan}\left (c+d\,x\right )}^7\,48{}\mathrm {i}+42\,{\mathrm {tan}\left (c+d\,x\right )}^6+42\,{\mathrm {tan}\left (c+d\,x\right )}^4+{\mathrm {tan}\left (c+d\,x\right )}^3\,48{}\mathrm {i}-27\,{\mathrm {tan}\left (c+d\,x\right )}^2-\mathrm {tan}\left (c+d\,x\right )\,8{}\mathrm {i}+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^18*(a + a*tan(c + d*x)*1i)^8,x)

[Out]

(5*a^8*x)/512 + ((a^8*tan(c + d*x)^2*163i)/448 - (a^8*10i)/63 - (9019*a^8*tan(c + d*x))/32256 + (393*a^8*tan(c
+ d*x)^3)/1792 + (a^8*tan(c + d*x)^4*11i)/64 + (a^8*tan(c + d*x)^5)/2 - (a^8*tan(c + d*x)^6*95i)/192 - (205*a
^8*tan(c + d*x)^7)/768 + (a^8*tan(c + d*x)^8*5i)/64 + (5*a^8*tan(c + d*x)^9)/512)/(d*(tan(c + d*x)^3*48i - 27*
tan(c + d*x)^2 - tan(c + d*x)*8i + 42*tan(c + d*x)^4 + 42*tan(c + d*x)^6 - tan(c + d*x)^7*48i - 27*tan(c + d*x
)^8 + tan(c + d*x)^9*8i + tan(c + d*x)^10 + 1))

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