3.174 \(\int \frac{\cos ^2(c+d x)}{(a+i a \tan (c+d x))^8} \, dx\)

Optimal. Leaf size=278 \[ -\frac{i}{1024 d \left (a^8-i a^8 \tan (c+d x)\right )}+\frac{9 i}{1024 d \left (a^8+i a^8 \tan (c+d x)\right )}+\frac{i}{128 d \left (a^4+i a^4 \tan (c+d x)\right )^2}+\frac{3 i}{256 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{i}{48 a^2 d (a+i a \tan (c+d x))^6}+\frac{i}{64 a^3 d (a+i a \tan (c+d x))^5}+\frac{7 i}{768 a^5 d (a+i a \tan (c+d x))^3}+\frac{5 x}{512 a^8}+\frac{i a}{36 d (a+i a \tan (c+d x))^9}+\frac{i}{32 d (a+i a \tan (c+d x))^8}+\frac{3 i}{112 a d (a+i a \tan (c+d x))^7} \]

[Out]

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

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Rubi [A]  time = 0.143577, antiderivative size = 278, 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} \[ -\frac{i}{1024 d \left (a^8-i a^8 \tan (c+d x)\right )}+\frac{9 i}{1024 d \left (a^8+i a^8 \tan (c+d x)\right )}+\frac{i}{128 d \left (a^4+i a^4 \tan (c+d x)\right )^2}+\frac{3 i}{256 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{i}{48 a^2 d (a+i a \tan (c+d x))^6}+\frac{i}{64 a^3 d (a+i a \tan (c+d x))^5}+\frac{7 i}{768 a^5 d (a+i a \tan (c+d x))^3}+\frac{5 x}{512 a^8}+\frac{i a}{36 d (a+i a \tan (c+d x))^9}+\frac{i}{32 d (a+i a \tan (c+d x))^8}+\frac{3 i}{112 a d (a+i a \tan (c+d x))^7} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^2/(a + I*a*Tan[c + d*x])^8,x]

[Out]

(5*x)/(512*a^8) + ((I/36)*a)/(d*(a + I*a*Tan[c + d*x])^9) + (I/32)/(d*(a + I*a*Tan[c + d*x])^8) + ((3*I)/112)/
(a*d*(a + I*a*Tan[c + d*x])^7) + (I/48)/(a^2*d*(a + I*a*Tan[c + d*x])^6) + (I/64)/(a^3*d*(a + I*a*Tan[c + d*x]
)^5) + ((7*I)/768)/(a^5*d*(a + I*a*Tan[c + d*x])^3) + ((3*I)/256)/(d*(a^2 + I*a^2*Tan[c + d*x])^4) + (I/128)/(
d*(a^4 + I*a^4*Tan[c + d*x])^2) - (I/1024)/(d*(a^8 - I*a^8*Tan[c + d*x])) + ((9*I)/1024)/(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 ^2(c+d x)}{(a+i a \tan (c+d x))^8} \, dx &=-\frac{\left (i a^3\right ) \operatorname{Subst}\left (\int \frac{1}{(a-x)^2 (a+x)^{10}} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac{\left (i a^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{1024 a^{10} (a-x)^2}+\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{5}{512 a^{10} \left (a^2-x^2\right )}\right ) \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=\frac{i a}{36 d (a+i a \tan (c+d x))^9}+\frac{i}{32 d (a+i a \tan (c+d x))^8}+\frac{3 i}{112 a d (a+i a \tan (c+d x))^7}+\frac{i}{48 a^2 d (a+i a \tan (c+d x))^6}+\frac{i}{64 a^3 d (a+i a \tan (c+d x))^5}+\frac{7 i}{768 a^5 d (a+i a \tan (c+d x))^3}+\frac{3 i}{256 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{i}{128 d \left (a^4+i a^4 \tan (c+d x)\right )^2}-\frac{i}{1024 d \left (a^8-i a^8 \tan (c+d x)\right )}+\frac{9 i}{1024 d \left (a^8+i a^8 \tan (c+d x)\right )}-\frac{(5 i) \operatorname{Subst}\left (\int \frac{1}{a^2-x^2} \, dx,x,i a \tan (c+d x)\right )}{512 a^7 d}\\ &=\frac{5 x}{512 a^8}+\frac{i a}{36 d (a+i a \tan (c+d x))^9}+\frac{i}{32 d (a+i a \tan (c+d x))^8}+\frac{3 i}{112 a d (a+i a \tan (c+d x))^7}+\frac{i}{48 a^2 d (a+i a \tan (c+d x))^6}+\frac{i}{64 a^3 d (a+i a \tan (c+d x))^5}+\frac{7 i}{768 a^5 d (a+i a \tan (c+d x))^3}+\frac{3 i}{256 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{i}{128 d \left (a^4+i a^4 \tan (c+d x)\right )^2}-\frac{i}{1024 d \left (a^8-i a^8 \tan (c+d x)\right )}+\frac{9 i}{1024 d \left (a^8+i a^8 \tan (c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 0.966275, size = 170, normalized size = 0.61 \[ \frac{\sec ^8(c+d x) (-7056 \sin (2 (c+d x))-10080 \sin (4 (c+d x))-9720 \sin (6 (c+d x))+5040 i d x \sin (8 (c+d x))+315 \sin (8 (c+d x))+280 \sin (10 (c+d x))+28224 i \cos (2 (c+d x))+20160 i \cos (4 (c+d x))+12960 i \cos (6 (c+d x))+5040 d x \cos (8 (c+d x))+315 i \cos (8 (c+d x))-224 i \cos (10 (c+d x))+15876 i)}{516096 a^8 d (\tan (c+d x)-i)^8} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^2/(a + I*a*Tan[c + d*x])^8,x]

[Out]

(Sec[c + d*x]^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*S
in[10*(c + d*x)]))/(516096*a^8*d*(-I + Tan[c + d*x])^8)

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Maple [A]  time = 0.119, size = 234, normalized size = 0.8 \begin{align*}{\frac{-{\frac{5\,i}{1024}}\ln \left ( \tan \left ( dx+c \right ) -i \right ) }{d{a}^{8}}}+{\frac{{\frac{3\,i}{256}}}{d{a}^{8} \left ( \tan \left ( dx+c \right ) -i \right ) ^{4}}}+{\frac{{\frac{i}{32}}}{d{a}^{8} \left ( \tan \left ( dx+c \right ) -i \right ) ^{8}}}-{\frac{{\frac{i}{48}}}{d{a}^{8} \left ( \tan \left ( dx+c \right ) -i \right ) ^{6}}}-{\frac{{\frac{i}{128}}}{d{a}^{8} \left ( \tan \left ( dx+c \right ) -i \right ) ^{2}}}+{\frac{1}{36\,d{a}^{8} \left ( \tan \left ( dx+c \right ) -i \right ) ^{9}}}-{\frac{3}{112\,d{a}^{8} \left ( \tan \left ( dx+c \right ) -i \right ) ^{7}}}+{\frac{1}{64\,d{a}^{8} \left ( \tan \left ( dx+c \right ) -i \right ) ^{5}}}-{\frac{7}{768\,d{a}^{8} \left ( \tan \left ( dx+c \right ) -i \right ) ^{3}}}+{\frac{9}{1024\,d{a}^{8} \left ( \tan \left ( dx+c \right ) -i \right ) }}+{\frac{{\frac{5\,i}{1024}}\ln \left ( \tan \left ( dx+c \right ) +i \right ) }{d{a}^{8}}}+{\frac{1}{1024\,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)^2/(a+I*a*tan(d*x+c))^8,x)

[Out]

-5/1024*I/d/a^8*ln(tan(d*x+c)-I)+3/256*I/d/a^8/(tan(d*x+c)-I)^4+1/32*I/d/a^8/(tan(d*x+c)-I)^8-1/48*I/d/a^8/(ta
n(d*x+c)-I)^6-1/128*I/d/a^8/(tan(d*x+c)-I)^2+1/36/d/a^8/(tan(d*x+c)-I)^9-3/112/d/a^8/(tan(d*x+c)-I)^7+1/64/d/a
^8/(tan(d*x+c)-I)^5-7/768/d/a^8/(tan(d*x+c)-I)^3+9/1024/d/a^8/(tan(d*x+c)-I)+5/1024*I/d/a^8*ln(tan(d*x+c)+I)+1
/1024/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)^2/(a+I*a*tan(d*x+c))^8,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 2.44525, size = 479, normalized size = 1.72 \begin{align*} \frac{{\left (5040 \, d x e^{\left (18 i \, d x + 18 i \, c\right )} - 252 i \, e^{\left (20 i \, d x + 20 i \, c\right )} + 11340 i \, e^{\left (16 i \, d x + 16 i \, c\right )} + 15120 i \, e^{\left (14 i \, d x + 14 i \, c\right )} + 17640 i \, e^{\left (12 i \, d x + 12 i \, c\right )} + 15876 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 10584 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 5040 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 1620 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 315 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 28 i\right )} e^{\left (-18 i \, d x - 18 i \, c\right )}}{516096 \, a^{8} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2/(a+I*a*tan(d*x+c))^8,x, algorithm="fricas")

[Out]

1/516096*(5040*d*x*e^(18*I*d*x + 18*I*c) - 252*I*e^(20*I*d*x + 20*I*c) + 11340*I*e^(16*I*d*x + 16*I*c) + 15120
*I*e^(14*I*d*x + 14*I*c) + 17640*I*e^(12*I*d*x + 12*I*c) + 15876*I*e^(10*I*d*x + 10*I*c) + 10584*I*e^(8*I*d*x
+ 8*I*c) + 5040*I*e^(6*I*d*x + 6*I*c) + 1620*I*e^(4*I*d*x + 4*I*c) + 315*I*e^(2*I*d*x + 2*I*c) + 28*I)*e^(-18*
I*d*x - 18*I*c)/(a^8*d)

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Sympy [A]  time = 2.735, size = 396, normalized size = 1.42 \begin{align*} \begin{cases} \frac{\left (- 2495687119199326634196634435584 i a^{72} d^{9} e^{92 i c} e^{2 i d x} + 112305920363969698538848549601280 i a^{72} d^{9} e^{88 i c} e^{- 2 i d x} + 149741227151959598051798066135040 i a^{72} d^{9} e^{86 i c} e^{- 4 i d x} + 174698098343952864393764410490880 i a^{72} d^{9} e^{84 i c} e^{- 6 i d x} + 157228288509557577954387969441792 i a^{72} d^{9} e^{82 i c} e^{- 8 i d x} + 104818859006371718636258646294528 i a^{72} d^{9} e^{80 i c} e^{- 10 i d x} + 49913742383986532683932688711680 i a^{72} d^{9} e^{78 i c} e^{- 12 i d x} + 16043702909138528362692649943040 i a^{72} d^{9} e^{76 i c} e^{- 14 i d x} + 3119608898999158292745793044480 i a^{72} d^{9} e^{74 i c} e^{- 16 i d x} + 277298568799925181577403826176 i a^{72} d^{9} e^{72 i c} e^{- 18 i d x}\right ) e^{- 90 i c}}{5111167220120220946834707324076032 a^{80} d^{10}} & \text{for}\: 5111167220120220946834707324076032 a^{80} d^{10} e^{90 i c} \neq 0 \\x \left (\frac{\left (e^{20 i c} + 10 e^{18 i c} + 45 e^{16 i c} + 120 e^{14 i c} + 210 e^{12 i c} + 252 e^{10 i c} + 210 e^{8 i c} + 120 e^{6 i c} + 45 e^{4 i c} + 10 e^{2 i c} + 1\right ) e^{- 18 i c}}{1024 a^{8}} - \frac{5}{512 a^{8}}\right ) & \text{otherwise} \end{cases} + \frac{5 x}{512 a^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**2/(a+I*a*tan(d*x+c))**8,x)

[Out]

Piecewise(((-2495687119199326634196634435584*I*a**72*d**9*exp(92*I*c)*exp(2*I*d*x) + 1123059203639696985388485
49601280*I*a**72*d**9*exp(88*I*c)*exp(-2*I*d*x) + 149741227151959598051798066135040*I*a**72*d**9*exp(86*I*c)*e
xp(-4*I*d*x) + 174698098343952864393764410490880*I*a**72*d**9*exp(84*I*c)*exp(-6*I*d*x) + 15722828850955757795
4387969441792*I*a**72*d**9*exp(82*I*c)*exp(-8*I*d*x) + 104818859006371718636258646294528*I*a**72*d**9*exp(80*I
*c)*exp(-10*I*d*x) + 49913742383986532683932688711680*I*a**72*d**9*exp(78*I*c)*exp(-12*I*d*x) + 16043702909138
528362692649943040*I*a**72*d**9*exp(76*I*c)*exp(-14*I*d*x) + 3119608898999158292745793044480*I*a**72*d**9*exp(
74*I*c)*exp(-16*I*d*x) + 277298568799925181577403826176*I*a**72*d**9*exp(72*I*c)*exp(-18*I*d*x))*exp(-90*I*c)/
(5111167220120220946834707324076032*a**80*d**10), Ne(5111167220120220946834707324076032*a**80*d**10*exp(90*I*c
), 0)), (x*((exp(20*I*c) + 10*exp(18*I*c) + 45*exp(16*I*c) + 120*exp(14*I*c) + 210*exp(12*I*c) + 252*exp(10*I*
c) + 210*exp(8*I*c) + 120*exp(6*I*c) + 45*exp(4*I*c) + 10*exp(2*I*c) + 1)*exp(-18*I*c)/(1024*a**8) - 5/(512*a*
*8)), True)) + 5*x/(512*a**8)

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Giac [A]  time = 1.1867, size = 220, normalized size = 0.79 \begin{align*} -\frac{-\frac{2520 i \, \log \left (\tan \left (d x + c\right ) + i\right )}{a^{8}} + \frac{2520 i \, \log \left (\tan \left (d x + c\right ) - i\right )}{a^{8}} + \frac{504 \,{\left (5 i \, \tan \left (d x + c\right ) - 6\right )}}{a^{8}{\left (\tan \left (d x + c\right ) + i\right )}} + \frac{-7129 i \, \tan \left (d x + c\right )^{9} - 68697 \, \tan \left (d x + c\right )^{8} + 296964 i \, \tan \left (d x + c\right )^{7} + 758772 \, \tan \left (d x + c\right )^{6} - 1271214 i \, \tan \left (d x + c\right )^{5} - 1465758 \, \tan \left (d x + c\right )^{4} + 1191540 i \, \tan \left (d x + c\right )^{3} + 693828 \, \tan \left (d x + c\right )^{2} - 295425 i \, \tan \left (d x + c\right ) - 89553}{a^{8}{\left (\tan \left (d x + c\right ) - i\right )}^{9}}}{516096 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2/(a+I*a*tan(d*x+c))^8,x, algorithm="giac")

[Out]

-1/516096*(-2520*I*log(tan(d*x + c) + I)/a^8 + 2520*I*log(tan(d*x + c) - I)/a^8 + 504*(5*I*tan(d*x + c) - 6)/(
a^8*(tan(d*x + c) + I)) + (-7129*I*tan(d*x + c)^9 - 68697*tan(d*x + c)^8 + 296964*I*tan(d*x + c)^7 + 758772*ta
n(d*x + c)^6 - 1271214*I*tan(d*x + c)^5 - 1465758*tan(d*x + c)^4 + 1191540*I*tan(d*x + c)^3 + 693828*tan(d*x +
c)^2 - 295425*I*tan(d*x + c) - 89553)/(a^8*(tan(d*x + c) - I)^9))/d