3.1.69 \(\int \cos ^{12}(c+d x) (a+i a \tan (c+d x))^5 \, dx\) [69]

Optimal. Leaf size=198 \[ \frac {7 a^5 x}{128}-\frac {i a^{11}}{24 d (a-i a \tan (c+d x))^6}-\frac {i a^{10}}{20 d (a-i a \tan (c+d x))^5}-\frac {3 i a^9}{64 d (a-i a \tan (c+d x))^4}-\frac {i a^8}{24 d (a-i a \tan (c+d x))^3}-\frac {5 i a^7}{128 d (a-i a \tan (c+d x))^2}-\frac {3 i a^6}{64 d (a-i a \tan (c+d x))}+\frac {i a^6}{128 d (a+i a \tan (c+d x))} \]

[Out]

7/128*a^5*x-1/24*I*a^11/d/(a-I*a*tan(d*x+c))^6-1/20*I*a^10/d/(a-I*a*tan(d*x+c))^5-3/64*I*a^9/d/(a-I*a*tan(d*x+
c))^4-1/24*I*a^8/d/(a-I*a*tan(d*x+c))^3-5/128*I*a^7/d/(a-I*a*tan(d*x+c))^2-3/64*I*a^6/d/(a-I*a*tan(d*x+c))+1/1
28*I*a^6/d/(a+I*a*tan(d*x+c))

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Rubi [A]
time = 0.10, antiderivative size = 198, 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^{11}}{24 d (a-i a \tan (c+d x))^6}-\frac {i a^{10}}{20 d (a-i a \tan (c+d x))^5}-\frac {3 i a^9}{64 d (a-i a \tan (c+d x))^4}-\frac {i a^8}{24 d (a-i a \tan (c+d x))^3}-\frac {5 i a^7}{128 d (a-i a \tan (c+d x))^2}-\frac {3 i a^6}{64 d (a-i a \tan (c+d x))}+\frac {i a^6}{128 d (a+i a \tan (c+d x))}+\frac {7 a^5 x}{128} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^12*(a + I*a*Tan[c + d*x])^5,x]

[Out]

(7*a^5*x)/128 - ((I/24)*a^11)/(d*(a - I*a*Tan[c + d*x])^6) - ((I/20)*a^10)/(d*(a - I*a*Tan[c + d*x])^5) - (((3
*I)/64)*a^9)/(d*(a - I*a*Tan[c + d*x])^4) - ((I/24)*a^8)/(d*(a - I*a*Tan[c + d*x])^3) - (((5*I)/128)*a^7)/(d*(
a - I*a*Tan[c + d*x])^2) - (((3*I)/64)*a^6)/(d*(a - I*a*Tan[c + d*x])) + ((I/128)*a^6)/(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 ^{12}(c+d x) (a+i a \tan (c+d x))^5 \, dx &=-\frac {\left (i a^{13}\right ) \text {Subst}\left (\int \frac {1}{(a-x)^7 (a+x)^2} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac {\left (i a^{13}\right ) \text {Subst}\left (\int \left (\frac {1}{4 a^2 (a-x)^7}+\frac {1}{4 a^3 (a-x)^6}+\frac {3}{16 a^4 (a-x)^5}+\frac {1}{8 a^5 (a-x)^4}+\frac {5}{64 a^6 (a-x)^3}+\frac {3}{64 a^7 (a-x)^2}+\frac {1}{128 a^7 (a+x)^2}+\frac {7}{128 a^7 \left (a^2-x^2\right )}\right ) \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac {i a^{11}}{24 d (a-i a \tan (c+d x))^6}-\frac {i a^{10}}{20 d (a-i a \tan (c+d x))^5}-\frac {3 i a^9}{64 d (a-i a \tan (c+d x))^4}-\frac {i a^8}{24 d (a-i a \tan (c+d x))^3}-\frac {5 i a^7}{128 d (a-i a \tan (c+d x))^2}-\frac {3 i a^6}{64 d (a-i a \tan (c+d x))}+\frac {i a^6}{128 d (a+i a \tan (c+d x))}-\frac {\left (7 i a^6\right ) \text {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,i a \tan (c+d x)\right )}{128 d}\\ &=\frac {7 a^5 x}{128}-\frac {i a^{11}}{24 d (a-i a \tan (c+d x))^6}-\frac {i a^{10}}{20 d (a-i a \tan (c+d x))^5}-\frac {3 i a^9}{64 d (a-i a \tan (c+d x))^4}-\frac {i a^8}{24 d (a-i a \tan (c+d x))^3}-\frac {5 i a^7}{128 d (a-i a \tan (c+d x))^2}-\frac {3 i a^6}{64 d (a-i a \tan (c+d x))}+\frac {i a^6}{128 d (a+i a \tan (c+d x))}\\ \end {align*}

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Mathematica [A]
time = 1.52, size = 159, normalized size = 0.80 \begin {gather*} \frac {a^5 (-1750 i \cos (c+d x)-1575 i \cos (3 (c+d x))-84 i \cos (5 (c+d x))+840 d x \cos (5 (c+d x))+50 i \cos (7 (c+d x))-350 \sin (c+d x)-945 \sin (3 (c+d x))+84 \sin (5 (c+d x))-840 i d x \sin (5 (c+d x))+70 \sin (7 (c+d x))) (\cos (5 (c+2 d x))+i \sin (5 (c+2 d x)))}{15360 d (\cos (d x)+i \sin (d x))^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^12*(a + I*a*Tan[c + d*x])^5,x]

[Out]

(a^5*((-1750*I)*Cos[c + d*x] - (1575*I)*Cos[3*(c + d*x)] - (84*I)*Cos[5*(c + d*x)] + 840*d*x*Cos[5*(c + d*x)]
+ (50*I)*Cos[7*(c + d*x)] - 350*Sin[c + d*x] - 945*Sin[3*(c + d*x)] + 84*Sin[5*(c + d*x)] - (840*I)*d*x*Sin[5*
(c + d*x)] + 70*Sin[7*(c + d*x)])*(Cos[5*(c + 2*d*x)] + I*Sin[5*(c + 2*d*x)]))/(15360*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. 360 vs. \(2 (168 ) = 336\).
time = 0.29, size = 361, normalized size = 1.82

method	result	size
risch	\(\frac {7 a^{5} x}{128}-\frac {i a^{5} {\mathrm e}^{12 i \left (d x +c \right )}}{1536 d}-\frac {7 i a^{5} {\mathrm e}^{10 i \left (d x +c \right )}}{1280 d}-\frac {21 i a^{5} {\mathrm e}^{8 i \left (d x +c \right )}}{1024 d}-\frac {35 i a^{5} {\mathrm e}^{6 i \left (d x +c \right )}}{768 d}-\frac {35 i a^{5} {\mathrm e}^{4 i \left (d x +c \right )}}{512 d}-\frac {5 i a^{5} \cos \left (2 d x +2 c \right )}{64 d}+\frac {11 a^{5} \sin \left (2 d x +2 c \right )}{128 d}\)	\(133\)
derivativedivides	\(\frac {i a^{5} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{8}\left (d x +c \right )\right )}{12}-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{8}\left (d x +c \right )\right )}{30}-\frac {\left (\cos ^{8}\left (d x +c \right )\right )}{120}\right )+5 a^{5} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{9}\left (d x +c \right )\right )}{12}-\frac {\left (\cos ^{9}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{40}+\frac {\left (\cos ^{7}\left (d x +c \right )+\frac {7 \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {35 \left (\cos ^{3}\left (d x +c \right )\right )}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{320}+\frac {7 d x}{1024}+\frac {7 c}{1024}\right )-10 i a^{5} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{10}\left (d x +c \right )\right )}{12}-\frac {\left (\cos ^{10}\left (d x +c \right )\right )}{60}\right )-10 a^{5} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{11}\left (d x +c \right )\right )}{12}+\frac {\left (\cos ^{9}\left (d x +c \right )+\frac {9 \left (\cos ^{7}\left (d x +c \right )\right )}{8}+\frac {21 \left (\cos ^{5}\left (d x +c \right )\right )}{16}+\frac {105 \left (\cos ^{3}\left (d x +c \right )\right )}{64}+\frac {315 \cos \left (d x +c \right )}{128}\right ) \sin \left (d x +c \right )}{120}+\frac {21 d x}{1024}+\frac {21 c}{1024}\right )-\frac {5 i a^{5} \left (\cos ^{12}\left (d x +c \right )\right )}{12}+a^{5} \left (\frac {\left (\cos ^{11}\left (d x +c \right )+\frac {11 \left (\cos ^{9}\left (d x +c \right )\right )}{10}+\frac {99 \left (\cos ^{7}\left (d x +c \right )\right )}{80}+\frac {231 \left (\cos ^{5}\left (d x +c \right )\right )}{160}+\frac {231 \left (\cos ^{3}\left (d x +c \right )\right )}{128}+\frac {693 \cos \left (d x +c \right )}{256}\right ) \sin \left (d x +c \right )}{12}+\frac {231 d x}{1024}+\frac {231 c}{1024}\right )}{d}\)	\(361\)
default	\(\frac {i a^{5} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{8}\left (d x +c \right )\right )}{12}-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{8}\left (d x +c \right )\right )}{30}-\frac {\left (\cos ^{8}\left (d x +c \right )\right )}{120}\right )+5 a^{5} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{9}\left (d x +c \right )\right )}{12}-\frac {\left (\cos ^{9}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{40}+\frac {\left (\cos ^{7}\left (d x +c \right )+\frac {7 \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {35 \left (\cos ^{3}\left (d x +c \right )\right )}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{320}+\frac {7 d x}{1024}+\frac {7 c}{1024}\right )-10 i a^{5} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{10}\left (d x +c \right )\right )}{12}-\frac {\left (\cos ^{10}\left (d x +c \right )\right )}{60}\right )-10 a^{5} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{11}\left (d x +c \right )\right )}{12}+\frac {\left (\cos ^{9}\left (d x +c \right )+\frac {9 \left (\cos ^{7}\left (d x +c \right )\right )}{8}+\frac {21 \left (\cos ^{5}\left (d x +c \right )\right )}{16}+\frac {105 \left (\cos ^{3}\left (d x +c \right )\right )}{64}+\frac {315 \cos \left (d x +c \right )}{128}\right ) \sin \left (d x +c \right )}{120}+\frac {21 d x}{1024}+\frac {21 c}{1024}\right )-\frac {5 i a^{5} \left (\cos ^{12}\left (d x +c \right )\right )}{12}+a^{5} \left (\frac {\left (\cos ^{11}\left (d x +c \right )+\frac {11 \left (\cos ^{9}\left (d x +c \right )\right )}{10}+\frac {99 \left (\cos ^{7}\left (d x +c \right )\right )}{80}+\frac {231 \left (\cos ^{5}\left (d x +c \right )\right )}{160}+\frac {231 \left (\cos ^{3}\left (d x +c \right )\right )}{128}+\frac {693 \cos \left (d x +c \right )}{256}\right ) \sin \left (d x +c \right )}{12}+\frac {231 d x}{1024}+\frac {231 c}{1024}\right )}{d}\)	\(361\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^12*(a+I*a*tan(d*x+c))^5,x,method=_RETURNVERBOSE)

[Out]

1/d*(I*a^5*(-1/12*sin(d*x+c)^4*cos(d*x+c)^8-1/30*sin(d*x+c)^2*cos(d*x+c)^8-1/120*cos(d*x+c)^8)+5*a^5*(-1/12*si
n(d*x+c)^3*cos(d*x+c)^9-1/40*cos(d*x+c)^9*sin(d*x+c)+1/320*(cos(d*x+c)^7+7/6*cos(d*x+c)^5+35/24*cos(d*x+c)^3+3
5/16*cos(d*x+c))*sin(d*x+c)+7/1024*d*x+7/1024*c)-10*I*a^5*(-1/12*sin(d*x+c)^2*cos(d*x+c)^10-1/60*cos(d*x+c)^10
)-10*a^5*(-1/12*sin(d*x+c)*cos(d*x+c)^11+1/120*(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+315/128*cos(d*x+c))*sin(d*x+c)+21/1024*d*x+21/1024*c)-5/12*I*a^5*cos(d*x+c)^12+a^5*(1/12*(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
)+231/1024*d*x+231/1024*c))

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Maxima [A]
time = 0.52, size = 187, normalized size = 0.94 \begin {gather*} \frac {105 \, {\left (d x + c\right )} a^{5} + \frac {105 \, a^{5} \tan \left (d x + c\right )^{11} + 595 \, a^{5} \tan \left (d x + c\right )^{9} + 1386 \, a^{5} \tan \left (d x + c\right )^{7} + 1686 \, a^{5} \tan \left (d x + c\right )^{5} - 240 i \, a^{5} \tan \left (d x + c\right )^{4} + 45 \, a^{5} \tan \left (d x + c\right )^{3} + 1824 i \, a^{5} \tan \left (d x + c\right )^{2} + 1815 \, a^{5} \tan \left (d x + c\right ) - 496 i \, a^{5}}{\tan \left (d x + c\right )^{12} + 6 \, \tan \left (d x + c\right )^{10} + 15 \, \tan \left (d x + c\right )^{8} + 20 \, \tan \left (d x + c\right )^{6} + 15 \, \tan \left (d x + c\right )^{4} + 6 \, \tan \left (d x + c\right )^{2} + 1}}{1920 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^12*(a+I*a*tan(d*x+c))^5,x, algorithm="maxima")

[Out]

1/1920*(105*(d*x + c)*a^5 + (105*a^5*tan(d*x + c)^11 + 595*a^5*tan(d*x + c)^9 + 1386*a^5*tan(d*x + c)^7 + 1686
*a^5*tan(d*x + c)^5 - 240*I*a^5*tan(d*x + c)^4 + 45*a^5*tan(d*x + c)^3 + 1824*I*a^5*tan(d*x + c)^2 + 1815*a^5*
tan(d*x + c) - 496*I*a^5)/(tan(d*x + c)^12 + 6*tan(d*x + c)^10 + 15*tan(d*x + c)^8 + 20*tan(d*x + c)^6 + 15*ta
n(d*x + c)^4 + 6*tan(d*x + c)^2 + 1))/d

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Fricas [A]
time = 0.39, size = 120, normalized size = 0.61 \begin {gather*} \frac {{\left (840 \, a^{5} d x e^{\left (2 i \, d x + 2 i \, c\right )} - 10 i \, a^{5} e^{\left (14 i \, d x + 14 i \, c\right )} - 84 i \, a^{5} e^{\left (12 i \, d x + 12 i \, c\right )} - 315 i \, a^{5} e^{\left (10 i \, d x + 10 i \, c\right )} - 700 i \, a^{5} e^{\left (8 i \, d x + 8 i \, c\right )} - 1050 i \, a^{5} e^{\left (6 i \, d x + 6 i \, c\right )} - 1260 i \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} + 60 i \, a^{5}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{15360 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^12*(a+I*a*tan(d*x+c))^5,x, algorithm="fricas")

[Out]

1/15360*(840*a^5*d*x*e^(2*I*d*x + 2*I*c) - 10*I*a^5*e^(14*I*d*x + 14*I*c) - 84*I*a^5*e^(12*I*d*x + 12*I*c) - 3
15*I*a^5*e^(10*I*d*x + 10*I*c) - 700*I*a^5*e^(8*I*d*x + 8*I*c) - 1050*I*a^5*e^(6*I*d*x + 6*I*c) - 1260*I*a^5*e
^(4*I*d*x + 4*I*c) + 60*I*a^5)*e^(-2*I*d*x - 2*I*c)/d

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Sympy [A]
time = 0.56, size = 301, normalized size = 1.52 \begin {gather*} \frac {7 a^{5} x}{128} + \begin {cases} \frac {\left (- 33776997205278720 i a^{5} d^{6} e^{14 i c} e^{12 i d x} - 283726776524341248 i a^{5} d^{6} e^{12 i c} e^{10 i d x} - 1063975411966279680 i a^{5} d^{6} e^{10 i c} e^{8 i d x} - 2364389804369510400 i a^{5} d^{6} e^{8 i c} e^{6 i d x} - 3546584706554265600 i a^{5} d^{6} e^{6 i c} e^{4 i d x} - 4255901647865118720 i a^{5} d^{6} e^{4 i c} e^{2 i d x} + 202661983231672320 i a^{5} d^{6} e^{- 2 i d x}\right ) e^{- 2 i c}}{51881467707308113920 d^{7}} & \text {for}\: d^{7} e^{2 i c} \neq 0 \\x \left (- \frac {7 a^{5}}{128} + \frac {\left (a^{5} e^{14 i c} + 7 a^{5} e^{12 i c} + 21 a^{5} e^{10 i c} + 35 a^{5} e^{8 i c} + 35 a^{5} e^{6 i c} + 21 a^{5} e^{4 i c} + 7 a^{5} e^{2 i c} + a^{5}\right ) e^{- 2 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)**12*(a+I*a*tan(d*x+c))**5,x)

[Out]

7*a**5*x/128 + Piecewise(((-33776997205278720*I*a**5*d**6*exp(14*I*c)*exp(12*I*d*x) - 283726776524341248*I*a**
5*d**6*exp(12*I*c)*exp(10*I*d*x) - 1063975411966279680*I*a**5*d**6*exp(10*I*c)*exp(8*I*d*x) - 2364389804369510
400*I*a**5*d**6*exp(8*I*c)*exp(6*I*d*x) - 3546584706554265600*I*a**5*d**6*exp(6*I*c)*exp(4*I*d*x) - 4255901647
865118720*I*a**5*d**6*exp(4*I*c)*exp(2*I*d*x) + 202661983231672320*I*a**5*d**6*exp(-2*I*d*x))*exp(-2*I*c)/(518
81467707308113920*d**7), Ne(d**7*exp(2*I*c), 0)), (x*(-7*a**5/128 + (a**5*exp(14*I*c) + 7*a**5*exp(12*I*c) + 2
1*a**5*exp(10*I*c) + 35*a**5*exp(8*I*c) + 35*a**5*exp(6*I*c) + 21*a**5*exp(4*I*c) + 7*a**5*exp(2*I*c) + a**5)*
exp(-2*I*c)/128), 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. 914 vs. \(2 (154) = 308\).
time = 0.96, size = 914, normalized size = 4.62 \begin {gather*} \frac {6720 \, a^{5} d x e^{\left (18 i \, d x + 10 i \, c\right )} + 53760 \, a^{5} d x e^{\left (16 i \, d x + 8 i \, c\right )} + 188160 \, a^{5} d x e^{\left (14 i \, d x + 6 i \, c\right )} + 376320 \, a^{5} d x e^{\left (12 i \, d x + 4 i \, c\right )} + 470400 \, a^{5} d x e^{\left (10 i \, d x + 2 i \, c\right )} + 188160 \, a^{5} d x e^{\left (6 i \, d x - 2 i \, c\right )} + 53760 \, a^{5} d x e^{\left (4 i \, d x - 4 i \, c\right )} + 6720 \, a^{5} d x e^{\left (2 i \, d x - 6 i \, c\right )} + 376320 \, a^{5} d x e^{\left (8 i \, d x\right )} - 2355 i \, a^{5} e^{\left (18 i \, d x + 10 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 18840 i \, a^{5} e^{\left (16 i \, d x + 8 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 65940 i \, a^{5} e^{\left (14 i \, d x + 6 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 131880 i \, a^{5} e^{\left (12 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 164850 i \, a^{5} e^{\left (10 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 65940 i \, a^{5} e^{\left (6 i \, d x - 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 18840 i \, a^{5} e^{\left (4 i \, d x - 4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 2355 i \, a^{5} e^{\left (2 i \, d x - 6 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 131880 i \, a^{5} e^{\left (8 i \, d x\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 2355 i \, a^{5} e^{\left (18 i \, d x + 10 i \, c\right )} \log \left (e^{\left (2 i \, d x\right )} + e^{\left (-2 i \, c\right )}\right ) + 18840 i \, a^{5} e^{\left (16 i \, d x + 8 i \, c\right )} \log \left (e^{\left (2 i \, d x\right )} + e^{\left (-2 i \, c\right )}\right ) + 65940 i \, a^{5} e^{\left (14 i \, d x + 6 i \, c\right )} \log \left (e^{\left (2 i \, d x\right )} + e^{\left (-2 i \, c\right )}\right ) + 131880 i \, a^{5} e^{\left (12 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x\right )} + e^{\left (-2 i \, c\right )}\right ) + 164850 i \, a^{5} e^{\left (10 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x\right )} + e^{\left (-2 i \, c\right )}\right ) + 65940 i \, a^{5} e^{\left (6 i \, d x - 2 i \, c\right )} \log \left (e^{\left (2 i \, d x\right )} + e^{\left (-2 i \, c\right )}\right ) + 18840 i \, a^{5} e^{\left (4 i \, d x - 4 i \, c\right )} \log \left (e^{\left (2 i \, d x\right )} + e^{\left (-2 i \, c\right )}\right ) + 2355 i \, a^{5} e^{\left (2 i \, d x - 6 i \, c\right )} \log \left (e^{\left (2 i \, d x\right )} + e^{\left (-2 i \, c\right )}\right ) + 131880 i \, a^{5} e^{\left (8 i \, d x\right )} \log \left (e^{\left (2 i \, d x\right )} + e^{\left (-2 i \, c\right )}\right ) - 80 i \, a^{5} e^{\left (30 i \, d x + 22 i \, c\right )} - 1312 i \, a^{5} e^{\left (28 i \, d x + 20 i \, c\right )} - 10136 i \, a^{5} e^{\left (26 i \, d x + 18 i \, c\right )} - 49056 i \, a^{5} e^{\left (24 i \, d x + 16 i \, c\right )} - 166992 i \, a^{5} e^{\left (22 i \, d x + 14 i \, c\right )} - 426720 i \, a^{5} e^{\left (20 i \, d x + 12 i \, c\right )} - 845712 i \, a^{5} e^{\left (18 i \, d x + 10 i \, c\right )} - 1304736 i \, a^{5} e^{\left (16 i \, d x + 8 i \, c\right )} - 1538256 i \, a^{5} e^{\left (14 i \, d x + 6 i \, c\right )} - 1340192 i \, a^{5} e^{\left (12 i \, d x + 4 i \, c\right )} - 820120 i \, a^{5} e^{\left (10 i \, d x + 2 i \, c\right )} - 62160 i \, a^{5} e^{\left (6 i \, d x - 2 i \, c\right )} + 3360 i \, a^{5} e^{\left (4 i \, d x - 4 i \, c\right )} + 3840 i \, a^{5} e^{\left (2 i \, d x - 6 i \, c\right )} - 321440 i \, a^{5} e^{\left (8 i \, d x\right )} + 480 i \, a^{5} e^{\left (-8 i \, c\right )}}{122880 \, {\left (d e^{\left (18 i \, d x + 10 i \, c\right )} + 8 \, d e^{\left (16 i \, d x + 8 i \, c\right )} + 28 \, d e^{\left (14 i \, d x + 6 i \, c\right )} + 56 \, d e^{\left (12 i \, d x + 4 i \, c\right )} + 70 \, d e^{\left (10 i \, d x + 2 i \, c\right )} + 28 \, d e^{\left (6 i \, d x - 2 i \, c\right )} + 8 \, d e^{\left (4 i \, d x - 4 i \, c\right )} + d e^{\left (2 i \, d x - 6 i \, c\right )} + 56 \, d e^{\left (8 i \, d x\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^12*(a+I*a*tan(d*x+c))^5,x, algorithm="giac")

[Out]

1/122880*(6720*a^5*d*x*e^(18*I*d*x + 10*I*c) + 53760*a^5*d*x*e^(16*I*d*x + 8*I*c) + 188160*a^5*d*x*e^(14*I*d*x
+ 6*I*c) + 376320*a^5*d*x*e^(12*I*d*x + 4*I*c) + 470400*a^5*d*x*e^(10*I*d*x + 2*I*c) + 188160*a^5*d*x*e^(6*I*
d*x - 2*I*c) + 53760*a^5*d*x*e^(4*I*d*x - 4*I*c) + 6720*a^5*d*x*e^(2*I*d*x - 6*I*c) + 376320*a^5*d*x*e^(8*I*d*
x) - 2355*I*a^5*e^(18*I*d*x + 10*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 18840*I*a^5*e^(16*I*d*x + 8*I*c)*log(e^(2
*I*d*x + 2*I*c) + 1) - 65940*I*a^5*e^(14*I*d*x + 6*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 131880*I*a^5*e^(12*I*d*
x + 4*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 164850*I*a^5*e^(10*I*d*x + 2*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 659
40*I*a^5*e^(6*I*d*x - 2*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 18840*I*a^5*e^(4*I*d*x - 4*I*c)*log(e^(2*I*d*x + 2
*I*c) + 1) - 2355*I*a^5*e^(2*I*d*x - 6*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 131880*I*a^5*e^(8*I*d*x)*log(e^(2*I
*d*x + 2*I*c) + 1) + 2355*I*a^5*e^(18*I*d*x + 10*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 18840*I*a^5*e^(16*I*d*x
+ 8*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 65940*I*a^5*e^(14*I*d*x + 6*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 1318
80*I*a^5*e^(12*I*d*x + 4*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 164850*I*a^5*e^(10*I*d*x + 2*I*c)*log(e^(2*I*d*x
) + e^(-2*I*c)) + 65940*I*a^5*e^(6*I*d*x - 2*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 18840*I*a^5*e^(4*I*d*x - 4*I
*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 2355*I*a^5*e^(2*I*d*x - 6*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 131880*I*a^
5*e^(8*I*d*x)*log(e^(2*I*d*x) + e^(-2*I*c)) - 80*I*a^5*e^(30*I*d*x + 22*I*c) - 1312*I*a^5*e^(28*I*d*x + 20*I*c
) - 10136*I*a^5*e^(26*I*d*x + 18*I*c) - 49056*I*a^5*e^(24*I*d*x + 16*I*c) - 166992*I*a^5*e^(22*I*d*x + 14*I*c)
- 426720*I*a^5*e^(20*I*d*x + 12*I*c) - 845712*I*a^5*e^(18*I*d*x + 10*I*c) - 1304736*I*a^5*e^(16*I*d*x + 8*I*c
) - 1538256*I*a^5*e^(14*I*d*x + 6*I*c) - 1340192*I*a^5*e^(12*I*d*x + 4*I*c) - 820120*I*a^5*e^(10*I*d*x + 2*I*c
) - 62160*I*a^5*e^(6*I*d*x - 2*I*c) + 3360*I*a^5*e^(4*I*d*x - 4*I*c) + 3840*I*a^5*e^(2*I*d*x - 6*I*c) - 321440
*I*a^5*e^(8*I*d*x) + 480*I*a^5*e^(-8*I*c))/(d*e^(18*I*d*x + 10*I*c) + 8*d*e^(16*I*d*x + 8*I*c) + 28*d*e^(14*I*
d*x + 6*I*c) + 56*d*e^(12*I*d*x + 4*I*c) + 70*d*e^(10*I*d*x + 2*I*c) + 28*d*e^(6*I*d*x - 2*I*c) + 8*d*e^(4*I*d
*x - 4*I*c) + d*e^(2*I*d*x - 6*I*c) + 56*d*e^(8*I*d*x))

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Mupad [B]
time = 4.90, size = 171, normalized size = 0.86 \begin {gather*} \frac {7\,a^5\,x}{128}-\frac {-\frac {7\,a^5\,{\mathrm {tan}\left (c+d\,x\right )}^6}{128}-\frac {a^5\,{\mathrm {tan}\left (c+d\,x\right )}^5\,35{}\mathrm {i}}{128}+\frac {49\,a^5\,{\mathrm {tan}\left (c+d\,x\right )}^4}{96}+\frac {a^5\,{\mathrm {tan}\left (c+d\,x\right )}^3\,35{}\mathrm {i}}{96}+\frac {63\,a^5\,{\mathrm {tan}\left (c+d\,x\right )}^2}{640}+\frac {a^5\,\mathrm {tan}\left (c+d\,x\right )\,133{}\mathrm {i}}{384}-\frac {31\,a^5}{120}}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^7+{\mathrm {tan}\left (c+d\,x\right )}^6\,5{}\mathrm {i}-9\,{\mathrm {tan}\left (c+d\,x\right )}^5-{\mathrm {tan}\left (c+d\,x\right )}^4\,5{}\mathrm {i}-5\,{\mathrm {tan}\left (c+d\,x\right )}^3-{\mathrm {tan}\left (c+d\,x\right )}^2\,9{}\mathrm {i}+5\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^12*(a + a*tan(c + d*x)*1i)^5,x)

[Out]

(7*a^5*x)/128 - ((a^5*tan(c + d*x)*133i)/384 - (31*a^5)/120 + (63*a^5*tan(c + d*x)^2)/640 + (a^5*tan(c + d*x)^
3*35i)/96 + (49*a^5*tan(c + d*x)^4)/96 - (a^5*tan(c + d*x)^5*35i)/128 - (7*a^5*tan(c + d*x)^6)/128)/(d*(5*tan(
c + d*x) - tan(c + d*x)^2*9i - 5*tan(c + d*x)^3 - tan(c + d*x)^4*5i - 9*tan(c + d*x)^5 + tan(c + d*x)^6*5i + t
an(c + d*x)^7 + 1i))

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