3.175 \(\int \frac{\cos ^4(c+d x)}{(a+i a \tan (c+d x))^8} \, dx\)

Optimal. Leaf size=333 \[ \frac{i a^2}{80 d (a+i a \tan (c+d x))^{10}}-\frac{11 i}{4096 d \left (a^8-i a^8 \tan (c+d x)\right )}+\frac{55 i}{4096 d \left (a^8+i a^8 \tan (c+d x)\right )}-\frac{i}{4096 d \left (a^4-i a^4 \tan (c+d x)\right )^2}+\frac{45 i}{4096 d \left (a^4+i a^4 \tan (c+d x)\right )^2}+\frac{7 i}{512 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{5 i}{256 a^2 d (a+i a \tan (c+d x))^6}+\frac{21 i}{1280 a^3 d (a+i a \tan (c+d x))^5}+\frac{3 i}{256 a^5 d (a+i a \tan (c+d x))^3}+\frac{33 x}{2048 a^8}+\frac{i a}{48 d (a+i a \tan (c+d x))^9}+\frac{3 i}{128 d (a+i a \tan (c+d x))^8}+\frac{5 i}{224 a d (a+i a \tan (c+d x))^7} \]

[Out]

(33*x)/(2048*a^8) + ((I/80)*a^2)/(d*(a + I*a*Tan[c + d*x])^10) + ((I/48)*a)/(d*(a + I*a*Tan[c + d*x])^9) + ((3
*I)/128)/(d*(a + I*a*Tan[c + d*x])^8) + ((5*I)/224)/(a*d*(a + I*a*Tan[c + d*x])^7) + ((5*I)/256)/(a^2*d*(a + I
*a*Tan[c + d*x])^6) + ((21*I)/1280)/(a^3*d*(a + I*a*Tan[c + d*x])^5) + ((3*I)/256)/(a^5*d*(a + I*a*Tan[c + d*x
])^3) + ((7*I)/512)/(d*(a^2 + I*a^2*Tan[c + d*x])^4) - (I/4096)/(d*(a^4 - I*a^4*Tan[c + d*x])^2) + ((45*I)/409
6)/(d*(a^4 + I*a^4*Tan[c + d*x])^2) - ((11*I)/4096)/(d*(a^8 - I*a^8*Tan[c + d*x])) + ((55*I)/4096)/(d*(a^8 + I
*a^8*Tan[c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 0.180717, antiderivative size = 333, normalized size of antiderivative = 1., 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 = {3487, 44, 206} \[ \frac{i a^2}{80 d (a+i a \tan (c+d x))^{10}}-\frac{11 i}{4096 d \left (a^8-i a^8 \tan (c+d x)\right )}+\frac{55 i}{4096 d \left (a^8+i a^8 \tan (c+d x)\right )}-\frac{i}{4096 d \left (a^4-i a^4 \tan (c+d x)\right )^2}+\frac{45 i}{4096 d \left (a^4+i a^4 \tan (c+d x)\right )^2}+\frac{7 i}{512 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{5 i}{256 a^2 d (a+i a \tan (c+d x))^6}+\frac{21 i}{1280 a^3 d (a+i a \tan (c+d x))^5}+\frac{3 i}{256 a^5 d (a+i a \tan (c+d x))^3}+\frac{33 x}{2048 a^8}+\frac{i a}{48 d (a+i a \tan (c+d x))^9}+\frac{3 i}{128 d (a+i a \tan (c+d x))^8}+\frac{5 i}{224 a d (a+i a \tan (c+d x))^7} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^4/(a + I*a*Tan[c + d*x])^8,x]

[Out]

(33*x)/(2048*a^8) + ((I/80)*a^2)/(d*(a + I*a*Tan[c + d*x])^10) + ((I/48)*a)/(d*(a + I*a*Tan[c + d*x])^9) + ((3
*I)/128)/(d*(a + I*a*Tan[c + d*x])^8) + ((5*I)/224)/(a*d*(a + I*a*Tan[c + d*x])^7) + ((5*I)/256)/(a^2*d*(a + I
*a*Tan[c + d*x])^6) + ((21*I)/1280)/(a^3*d*(a + I*a*Tan[c + d*x])^5) + ((3*I)/256)/(a^5*d*(a + I*a*Tan[c + d*x
])^3) + ((7*I)/512)/(d*(a^2 + I*a^2*Tan[c + d*x])^4) - (I/4096)/(d*(a^4 - I*a^4*Tan[c + d*x])^2) + ((45*I)/409
6)/(d*(a^4 + I*a^4*Tan[c + d*x])^2) - ((11*I)/4096)/(d*(a^8 - I*a^8*Tan[c + d*x])) + ((55*I)/4096)/(d*(a^8 + I
*a^8*Tan[c + d*x]))

Rule 3487

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]

Rule 44

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] && L
tQ[m + n + 2, 0])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\cos ^4(c+d x)}{(a+i a \tan (c+d x))^8} \, dx &=-\frac{\left (i a^5\right ) \operatorname{Subst}\left (\int \frac{1}{(a-x)^3 (a+x)^{11}} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac{\left (i a^5\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2048 a^{11} (a-x)^3}+\frac{11}{4096 a^{12} (a-x)^2}+\frac{1}{8 a^3 (a+x)^{11}}+\frac{3}{16 a^4 (a+x)^{10}}+\frac{3}{16 a^5 (a+x)^9}+\frac{5}{32 a^6 (a+x)^8}+\frac{15}{128 a^7 (a+x)^7}+\frac{21}{256 a^8 (a+x)^6}+\frac{7}{128 a^9 (a+x)^5}+\frac{9}{256 a^{10} (a+x)^4}+\frac{45}{2048 a^{11} (a+x)^3}+\frac{55}{4096 a^{12} (a+x)^2}+\frac{33}{2048 a^{12} \left (a^2-x^2\right )}\right ) \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=\frac{i a^2}{80 d (a+i a \tan (c+d x))^{10}}+\frac{i a}{48 d (a+i a \tan (c+d x))^9}+\frac{3 i}{128 d (a+i a \tan (c+d x))^8}+\frac{5 i}{224 a d (a+i a \tan (c+d x))^7}+\frac{5 i}{256 a^2 d (a+i a \tan (c+d x))^6}+\frac{21 i}{1280 a^3 d (a+i a \tan (c+d x))^5}+\frac{3 i}{256 a^5 d (a+i a \tan (c+d x))^3}+\frac{7 i}{512 d \left (a^2+i a^2 \tan (c+d x)\right )^4}-\frac{i}{4096 d \left (a^4-i a^4 \tan (c+d x)\right )^2}+\frac{45 i}{4096 d \left (a^4+i a^4 \tan (c+d x)\right )^2}-\frac{11 i}{4096 d \left (a^8-i a^8 \tan (c+d x)\right )}+\frac{55 i}{4096 d \left (a^8+i a^8 \tan (c+d x)\right )}-\frac{(33 i) \operatorname{Subst}\left (\int \frac{1}{a^2-x^2} \, dx,x,i a \tan (c+d x)\right )}{2048 a^7 d}\\ &=\frac{33 x}{2048 a^8}+\frac{i a^2}{80 d (a+i a \tan (c+d x))^{10}}+\frac{i a}{48 d (a+i a \tan (c+d x))^9}+\frac{3 i}{128 d (a+i a \tan (c+d x))^8}+\frac{5 i}{224 a d (a+i a \tan (c+d x))^7}+\frac{5 i}{256 a^2 d (a+i a \tan (c+d x))^6}+\frac{21 i}{1280 a^3 d (a+i a \tan (c+d x))^5}+\frac{3 i}{256 a^5 d (a+i a \tan (c+d x))^3}+\frac{7 i}{512 d \left (a^2+i a^2 \tan (c+d x)\right )^4}-\frac{i}{4096 d \left (a^4-i a^4 \tan (c+d x)\right )^2}+\frac{45 i}{4096 d \left (a^4+i a^4 \tan (c+d x)\right )^2}-\frac{11 i}{4096 d \left (a^8-i a^8 \tan (c+d x)\right )}+\frac{55 i}{4096 d \left (a^8+i a^8 \tan (c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 1.49975, size = 192, normalized size = 0.58 \[ \frac{\sec ^8(c+d x) (-44352 \sin (2 (c+d x))-69300 \sin (4 (c+d x))-79200 \sin (6 (c+d x))+55440 i d x \sin (8 (c+d x))+3465 \sin (8 (c+d x))+5600 \sin (10 (c+d x))+252 \sin (12 (c+d x))+177408 i \cos (2 (c+d x))+138600 i \cos (4 (c+d x))+105600 i \cos (6 (c+d x))+55440 d x \cos (8 (c+d x))+3465 i \cos (8 (c+d x))-4480 i \cos (10 (c+d x))-168 i \cos (12 (c+d x))+97020 i)}{3440640 a^8 d (\tan (c+d x)-i)^8} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^4/(a + I*a*Tan[c + d*x])^8,x]

[Out]

(Sec[c + d*x]^8*(97020*I + (177408*I)*Cos[2*(c + d*x)] + (138600*I)*Cos[4*(c + d*x)] + (105600*I)*Cos[6*(c + d
*x)] + (3465*I)*Cos[8*(c + d*x)] + 55440*d*x*Cos[8*(c + d*x)] - (4480*I)*Cos[10*(c + d*x)] - (168*I)*Cos[12*(c
+ d*x)] - 44352*Sin[2*(c + d*x)] - 69300*Sin[4*(c + d*x)] - 79200*Sin[6*(c + d*x)] + 3465*Sin[8*(c + d*x)] +
(55440*I)*d*x*Sin[8*(c + d*x)] + 5600*Sin[10*(c + d*x)] + 252*Sin[12*(c + d*x)]))/(3440640*a^8*d*(-I + Tan[c +
d*x])^8)

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Maple [A]  time = 0.122, size = 274, normalized size = 0.8 \begin{align*}{\frac{-{\frac{33\,i}{4096}}\ln \left ( \tan \left ( dx+c \right ) -i \right ) }{d{a}^{8}}}+{\frac{{\frac{7\,i}{512}}}{d{a}^{8} \left ( \tan \left ( dx+c \right ) -i \right ) ^{4}}}+{\frac{{\frac{3\,i}{128}}}{d{a}^{8} \left ( \tan \left ( dx+c \right ) -i \right ) ^{8}}}-{\frac{{\frac{i}{80}}}{d{a}^{8} \left ( \tan \left ( dx+c \right ) -i \right ) ^{10}}}-{\frac{{\frac{5\,i}{256}}}{d{a}^{8} \left ( \tan \left ( dx+c \right ) -i \right ) ^{6}}}-{\frac{{\frac{45\,i}{4096}}}{d{a}^{8} \left ( \tan \left ( dx+c \right ) -i \right ) ^{2}}}+{\frac{1}{48\,d{a}^{8} \left ( \tan \left ( dx+c \right ) -i \right ) ^{9}}}-{\frac{5}{224\,d{a}^{8} \left ( \tan \left ( dx+c \right ) -i \right ) ^{7}}}+{\frac{21}{1280\,d{a}^{8} \left ( \tan \left ( dx+c \right ) -i \right ) ^{5}}}-{\frac{3}{256\,d{a}^{8} \left ( \tan \left ( dx+c \right ) -i \right ) ^{3}}}+{\frac{55}{4096\,d{a}^{8} \left ( \tan \left ( dx+c \right ) -i \right ) }}+{\frac{{\frac{i}{4096}}}{d{a}^{8} \left ( \tan \left ( dx+c \right ) +i \right ) ^{2}}}+{\frac{{\frac{33\,i}{4096}}\ln \left ( \tan \left ( dx+c \right ) +i \right ) }{d{a}^{8}}}+{\frac{11}{4096\,d{a}^{8} \left ( \tan \left ( dx+c \right ) +i \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4/(a+I*a*tan(d*x+c))^8,x)

[Out]

-33/4096*I/d/a^8*ln(tan(d*x+c)-I)+7/512*I/d/a^8/(tan(d*x+c)-I)^4+3/128*I/d/a^8/(tan(d*x+c)-I)^8-1/80*I/d/a^8/(
tan(d*x+c)-I)^10-5/256*I/d/a^8/(tan(d*x+c)-I)^6-45/4096*I/d/a^8/(tan(d*x+c)-I)^2+1/48/d/a^8/(tan(d*x+c)-I)^9-5
/224/d/a^8/(tan(d*x+c)-I)^7+21/1280/d/a^8/(tan(d*x+c)-I)^5-3/256/d/a^8/(tan(d*x+c)-I)^3+55/4096/d/a^8/(tan(d*x
+c)-I)+1/4096*I/d/a^8/(tan(d*x+c)+I)^2+33/4096*I/d/a^8*ln(tan(d*x+c)+I)+11/4096/d/a^8/(tan(d*x+c)+I)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4/(a+I*a*tan(d*x+c))^8,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 2.55239, size = 571, normalized size = 1.71 \begin{align*} \frac{{\left (55440 \, d x e^{\left (20 i \, d x + 20 i \, c\right )} - 210 i \, e^{\left (24 i \, d x + 24 i \, c\right )} - 5040 i \, e^{\left (22 i \, d x + 22 i \, c\right )} + 92400 i \, e^{\left (18 i \, d x + 18 i \, c\right )} + 103950 i \, e^{\left (16 i \, d x + 16 i \, c\right )} + 110880 i \, e^{\left (14 i \, d x + 14 i \, c\right )} + 97020 i \, e^{\left (12 i \, d x + 12 i \, c\right )} + 66528 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 34650 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 13200 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 3465 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 560 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 42 i\right )} e^{\left (-20 i \, d x - 20 i \, c\right )}}{3440640 \, a^{8} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4/(a+I*a*tan(d*x+c))^8,x, algorithm="fricas")

[Out]

1/3440640*(55440*d*x*e^(20*I*d*x + 20*I*c) - 210*I*e^(24*I*d*x + 24*I*c) - 5040*I*e^(22*I*d*x + 22*I*c) + 9240
0*I*e^(18*I*d*x + 18*I*c) + 103950*I*e^(16*I*d*x + 16*I*c) + 110880*I*e^(14*I*d*x + 14*I*c) + 97020*I*e^(12*I*
d*x + 12*I*c) + 66528*I*e^(10*I*d*x + 10*I*c) + 34650*I*e^(8*I*d*x + 8*I*c) + 13200*I*e^(6*I*d*x + 6*I*c) + 34
65*I*e^(4*I*d*x + 4*I*c) + 560*I*e^(2*I*d*x + 2*I*c) + 42*I)*e^(-20*I*d*x - 20*I*c)/(a^8*d)

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Sympy [A]  time = 2.0115, size = 464, normalized size = 1.39 \begin{align*} \begin{cases} \frac{\left (- 11433487528543532372369386809707411904921600 i a^{88} d^{11} e^{114 i c} e^{4 i d x} - 274403700685044776936865283432977885718118400 i a^{88} d^{11} e^{112 i c} e^{2 i d x} + 5030734512559154243842530196271261238165504000 i a^{88} d^{11} e^{108 i c} e^{- 2 i d x} + 5659576326629048524322846470805168892936192000 i a^{88} d^{11} e^{106 i c} e^{- 4 i d x} + 6036881415070985092611036235525513485798604800 i a^{88} d^{11} e^{104 i c} e^{- 6 i d x} + 5282271238187111956034656706084824300073779200 i a^{88} d^{11} e^{102 i c} e^{- 8 i d x} + 3622128849042591055566621741315308091479162880 i a^{88} d^{11} e^{100 i c} e^{- 10 i d x} + 1886525442209682841440948823601722964312064000 i a^{88} d^{11} e^{98 i c} e^{- 12 i d x} + 718676358937022034834647170895894462595072000 i a^{88} d^{11} e^{96 i c} e^{- 14 i d x} + 188652544220968284144094882360172296431206400 i a^{88} d^{11} e^{94 i c} e^{- 16 i d x} + 30489300076116086326318364825886431746457600 i a^{88} d^{11} e^{92 i c} e^{- 18 i d x} + 2286697505708706474473877361941482380984320 i a^{88} d^{11} e^{90 i c} e^{- 20 i d x}\right ) e^{- 110 i c}}{187326259667657234388900033490246236650235494400 a^{96} d^{12}} & \text{for}\: 187326259667657234388900033490246236650235494400 a^{96} d^{12} e^{110 i c} \neq 0 \\x \left (\frac{\left (e^{24 i c} + 12 e^{22 i c} + 66 e^{20 i c} + 220 e^{18 i c} + 495 e^{16 i c} + 792 e^{14 i c} + 924 e^{12 i c} + 792 e^{10 i c} + 495 e^{8 i c} + 220 e^{6 i c} + 66 e^{4 i c} + 12 e^{2 i c} + 1\right ) e^{- 20 i c}}{4096 a^{8}} - \frac{33}{2048 a^{8}}\right ) & \text{otherwise} \end{cases} + \frac{33 x}{2048 a^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**4/(a+I*a*tan(d*x+c))**8,x)

[Out]

Piecewise(((-11433487528543532372369386809707411904921600*I*a**88*d**11*exp(114*I*c)*exp(4*I*d*x) - 2744037006
85044776936865283432977885718118400*I*a**88*d**11*exp(112*I*c)*exp(2*I*d*x) + 50307345125591542438425301962712
61238165504000*I*a**88*d**11*exp(108*I*c)*exp(-2*I*d*x) + 5659576326629048524322846470805168892936192000*I*a**
88*d**11*exp(106*I*c)*exp(-4*I*d*x) + 6036881415070985092611036235525513485798604800*I*a**88*d**11*exp(104*I*c
)*exp(-6*I*d*x) + 5282271238187111956034656706084824300073779200*I*a**88*d**11*exp(102*I*c)*exp(-8*I*d*x) + 36
22128849042591055566621741315308091479162880*I*a**88*d**11*exp(100*I*c)*exp(-10*I*d*x) + 188652544220968284144
0948823601722964312064000*I*a**88*d**11*exp(98*I*c)*exp(-12*I*d*x) + 71867635893702203483464717089589446259507
2000*I*a**88*d**11*exp(96*I*c)*exp(-14*I*d*x) + 188652544220968284144094882360172296431206400*I*a**88*d**11*ex
p(94*I*c)*exp(-16*I*d*x) + 30489300076116086326318364825886431746457600*I*a**88*d**11*exp(92*I*c)*exp(-18*I*d*
x) + 2286697505708706474473877361941482380984320*I*a**88*d**11*exp(90*I*c)*exp(-20*I*d*x))*exp(-110*I*c)/(1873
26259667657234388900033490246236650235494400*a**96*d**12), Ne(187326259667657234388900033490246236650235494400
*a**96*d**12*exp(110*I*c), 0)), (x*((exp(24*I*c) + 12*exp(22*I*c) + 66*exp(20*I*c) + 220*exp(18*I*c) + 495*exp
(16*I*c) + 792*exp(14*I*c) + 924*exp(12*I*c) + 792*exp(10*I*c) + 495*exp(8*I*c) + 220*exp(6*I*c) + 66*exp(4*I*
c) + 12*exp(2*I*c) + 1)*exp(-20*I*c)/(4096*a**8) - 33/(2048*a**8)), True)) + 33*x/(2048*a**8)

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Giac [A]  time = 1.17036, size = 254, normalized size = 0.76 \begin{align*} -\frac{-\frac{27720 i \, \log \left (-i \, \tan \left (d x + c\right ) + 1\right )}{a^{8}} + \frac{27720 i \, \log \left (-i \, \tan \left (d x + c\right ) - 1\right )}{a^{8}} + \frac{420 \,{\left (99 i \, \tan \left (d x + c\right )^{2} - 220 \, \tan \left (d x + c\right ) - 123 i\right )}}{a^{8}{\left (\tan \left (d x + c\right ) + i\right )}^{2}} - \frac{81191 i \, \tan \left (d x + c\right )^{10} + 858110 \, \tan \left (d x + c\right )^{9} - 4107195 i \, \tan \left (d x + c\right )^{8} - 11748840 \, \tan \left (d x + c\right )^{7} + 22318590 i \, \tan \left (d x + c\right )^{6} + 29583540 \, \tan \left (d x + c\right )^{5} - 27983550 i \, \tan \left (d x + c\right )^{4} - 19002600 \, \tan \left (d x + c\right )^{3} + 9206235 i \, \tan \left (d x + c\right )^{2} + 3108990 \, \tan \left (d x + c\right ) - 648327 i}{a^{8}{\left (\tan \left (d x + c\right ) - i\right )}^{10}}}{3440640 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4/(a+I*a*tan(d*x+c))^8,x, algorithm="giac")

[Out]

-1/3440640*(-27720*I*log(-I*tan(d*x + c) + 1)/a^8 + 27720*I*log(-I*tan(d*x + c) - 1)/a^8 + 420*(99*I*tan(d*x +
c)^2 - 220*tan(d*x + c) - 123*I)/(a^8*(tan(d*x + c) + I)^2) - (81191*I*tan(d*x + c)^10 + 858110*tan(d*x + c)^
9 - 4107195*I*tan(d*x + c)^8 - 11748840*tan(d*x + c)^7 + 22318590*I*tan(d*x + c)^6 + 29583540*tan(d*x + c)^5 -
27983550*I*tan(d*x + c)^4 - 19002600*tan(d*x + c)^3 + 9206235*I*tan(d*x + c)^2 + 3108990*tan(d*x + c) - 64832
7*I)/(a^8*(tan(d*x + c) - I)^10))/d