3.173 \(\int \frac {1}{(a+i a \tan (c+d x))^8} \, dx\)

Optimal. Leaf size=229 \[ \frac {i}{256 d \left (a^8+i a^8 \tan (c+d x)\right )}+\frac {x}{256 a^8}+\frac {i}{256 d \left (a^4+i a^4 \tan (c+d x)\right )^2}+\frac {i}{80 a^3 d (a+i a \tan (c+d x))^5}+\frac {i}{192 a^2 d \left (a^2+i a^2 \tan (c+d x)\right )^3}+\frac {i}{128 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac {i}{48 a^2 d (a+i a \tan (c+d x))^6}+\frac {i}{28 a d (a+i a \tan (c+d x))^7}+\frac {i}{16 d (a+i a \tan (c+d x))^8} \]

[Out]

1/256*x/a^8+1/16*I/d/(a+I*a*tan(d*x+c))^8+1/28*I/a/d/(a+I*a*tan(d*x+c))^7+1/48*I/a^2/d/(a+I*a*tan(d*x+c))^6+1/
80*I/a^3/d/(a+I*a*tan(d*x+c))^5+1/128*I/d/(a^2+I*a^2*tan(d*x+c))^4+1/192*I/a^2/d/(a^2+I*a^2*tan(d*x+c))^3+1/25
6*I/d/(a^4+I*a^4*tan(d*x+c))^2+1/256*I/d/(a^8+I*a^8*tan(d*x+c))

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Rubi [A]  time = 0.15, antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3479, 8} \[ \frac {i}{256 d \left (a^8+i a^8 \tan (c+d x)\right )}+\frac {i}{256 d \left (a^4+i a^4 \tan (c+d x)\right )^2}+\frac {i}{192 a^2 d \left (a^2+i a^2 \tan (c+d x)\right )^3}+\frac {i}{128 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac {i}{80 a^3 d (a+i a \tan (c+d x))^5}+\frac {i}{48 a^2 d (a+i a \tan (c+d x))^6}+\frac {x}{256 a^8}+\frac {i}{28 a d (a+i a \tan (c+d x))^7}+\frac {i}{16 d (a+i a \tan (c+d x))^8} \]

Antiderivative was successfully verified.

[In]

Int[(a + I*a*Tan[c + d*x])^(-8),x]

[Out]

x/(256*a^8) + (I/16)/(d*(a + I*a*Tan[c + d*x])^8) + (I/28)/(a*d*(a + I*a*Tan[c + d*x])^7) + (I/48)/(a^2*d*(a +
I*a*Tan[c + d*x])^6) + (I/80)/(a^3*d*(a + I*a*Tan[c + d*x])^5) + (I/128)/(d*(a^2 + I*a^2*Tan[c + d*x])^4) + (
I/192)/(a^2*d*(a^2 + I*a^2*Tan[c + d*x])^3) + (I/256)/(d*(a^4 + I*a^4*Tan[c + d*x])^2) + (I/256)/(d*(a^8 + I*a
^8*Tan[c + d*x]))

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3479

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(a*(a + b*Tan[c + d*x])^n)/(2*b*d*n), x] +
Dist[1/(2*a), Int[(a + b*Tan[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0] && LtQ[n
, 0]

Rubi steps

\begin {align*} \int \frac {1}{(a+i a \tan (c+d x))^8} \, dx &=\frac {i}{16 d (a+i a \tan (c+d x))^8}+\frac {\int \frac {1}{(a+i a \tan (c+d x))^7} \, dx}{2 a}\\ &=\frac {i}{16 d (a+i a \tan (c+d x))^8}+\frac {i}{28 a d (a+i a \tan (c+d x))^7}+\frac {\int \frac {1}{(a+i a \tan (c+d x))^6} \, dx}{4 a^2}\\ &=\frac {i}{16 d (a+i a \tan (c+d x))^8}+\frac {i}{28 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 {\int \frac {1}{(a+i a \tan (c+d x))^5} \, dx}{8 a^3}\\ &=\frac {i}{16 d (a+i a \tan (c+d x))^8}+\frac {i}{28 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}{80 a^3 d (a+i a \tan (c+d x))^5}+\frac {\int \frac {1}{(a+i a \tan (c+d x))^4} \, dx}{16 a^4}\\ &=\frac {i}{16 d (a+i a \tan (c+d x))^8}+\frac {i}{28 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}{80 a^3 d (a+i a \tan (c+d x))^5}+\frac {i}{128 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac {\int \frac {1}{(a+i a \tan (c+d x))^3} \, dx}{32 a^5}\\ &=\frac {i}{16 d (a+i a \tan (c+d x))^8}+\frac {i}{28 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}{80 a^3 d (a+i a \tan (c+d x))^5}+\frac {i}{192 a^5 d (a+i a \tan (c+d x))^3}+\frac {i}{128 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac {\int \frac {1}{(a+i a \tan (c+d x))^2} \, dx}{64 a^6}\\ &=\frac {i}{16 d (a+i a \tan (c+d x))^8}+\frac {i}{28 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}{80 a^3 d (a+i a \tan (c+d x))^5}+\frac {i}{192 a^5 d (a+i a \tan (c+d x))^3}+\frac {i}{128 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac {i}{256 d \left (a^4+i a^4 \tan (c+d x)\right )^2}+\frac {\int \frac {1}{a+i a \tan (c+d x)} \, dx}{128 a^7}\\ &=\frac {i}{16 d (a+i a \tan (c+d x))^8}+\frac {i}{28 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}{80 a^3 d (a+i a \tan (c+d x))^5}+\frac {i}{192 a^5 d (a+i a \tan (c+d x))^3}+\frac {i}{128 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac {i}{256 d \left (a^4+i a^4 \tan (c+d x)\right )^2}+\frac {i}{256 d \left (a^8+i a^8 \tan (c+d x)\right )}+\frac {\int 1 \, dx}{256 a^8}\\ &=\frac {x}{256 a^8}+\frac {i}{16 d (a+i a \tan (c+d x))^8}+\frac {i}{28 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}{80 a^3 d (a+i a \tan (c+d x))^5}+\frac {i}{192 a^5 d (a+i a \tan (c+d x))^3}+\frac {i}{128 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac {i}{256 d \left (a^4+i a^4 \tan (c+d x)\right )^2}+\frac {i}{256 d \left (a^8+i a^8 \tan (c+d x)\right )}\\ \end {align*}

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Mathematica [A]  time = 0.72, size = 148, normalized size = 0.65 \[ \frac {\sec ^8(c+d x) (-6272 \sin (2 (c+d x))-7840 \sin (4 (c+d x))-5760 \sin (6 (c+d x))+1680 i d x \sin (8 (c+d x))+105 \sin (8 (c+d x))+25088 i \cos (2 (c+d x))+15680 i \cos (4 (c+d x))+7680 i \cos (6 (c+d x))+1680 d x \cos (8 (c+d x))+105 i \cos (8 (c+d x))+14700 i)}{430080 a^8 d (\tan (c+d x)-i)^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sec[c + d*x]^8*(14700*I + (25088*I)*Cos[2*(c + d*x)] + (15680*I)*Cos[4*(c + d*x)] + (7680*I)*Cos[6*(c + d*x)]
+ (105*I)*Cos[8*(c + d*x)] + 1680*d*x*Cos[8*(c + d*x)] - 6272*Sin[2*(c + d*x)] - 7840*Sin[4*(c + d*x)] - 5760
*Sin[6*(c + d*x)] + 105*Sin[8*(c + d*x)] + (1680*I)*d*x*Sin[8*(c + d*x)]))/(430080*a^8*d*(-I + Tan[c + d*x])^8
)

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fricas [A]  time = 0.69, size = 109, normalized size = 0.48 \[ \frac {{\left (1680 \, d x e^{\left (16 i \, d x + 16 i \, c\right )} + 6720 i \, e^{\left (14 i \, d x + 14 i \, c\right )} + 11760 i \, e^{\left (12 i \, d x + 12 i \, c\right )} + 15680 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 14700 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 9408 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 3920 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 960 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 105 i\right )} e^{\left (-16 i \, d x - 16 i \, c\right )}}{430080 \, a^{8} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+I*a*tan(d*x+c))^8,x, algorithm="fricas")

[Out]

1/430080*(1680*d*x*e^(16*I*d*x + 16*I*c) + 6720*I*e^(14*I*d*x + 14*I*c) + 11760*I*e^(12*I*d*x + 12*I*c) + 1568
0*I*e^(10*I*d*x + 10*I*c) + 14700*I*e^(8*I*d*x + 8*I*c) + 9408*I*e^(6*I*d*x + 6*I*c) + 3920*I*e^(4*I*d*x + 4*I
*c) + 960*I*e^(2*I*d*x + 2*I*c) + 105*I)*e^(-16*I*d*x - 16*I*c)/(a^8*d)

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giac [A]  time = 1.60, size = 132, normalized size = 0.58 \[ -\frac {-\frac {840 i \, \log \left (-i \, \tan \left (d x + c\right ) + 1\right )}{a^{8}} + \frac {840 i \, \log \left (-i \, \tan \left (d x + c\right ) - 1\right )}{a^{8}} + \frac {-2283 i \, \tan \left (d x + c\right )^{8} - 19944 \, \tan \left (d x + c\right )^{7} + 77364 i \, \tan \left (d x + c\right )^{6} + 175448 \, \tan \left (d x + c\right )^{5} - 258370 i \, \tan \left (d x + c\right )^{4} - 261464 \, \tan \left (d x + c\right )^{3} + 192052 i \, \tan \left (d x + c\right )^{2} + 114152 \, \tan \left (d x + c\right ) - 67819 i}{a^{8} {\left (\tan \left (d x + c\right ) - i\right )}^{8}}}{430080 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+I*a*tan(d*x+c))^8,x, algorithm="giac")

[Out]

-1/430080*(-840*I*log(-I*tan(d*x + c) + 1)/a^8 + 840*I*log(-I*tan(d*x + c) - 1)/a^8 + (-2283*I*tan(d*x + c)^8
- 19944*tan(d*x + c)^7 + 77364*I*tan(d*x + c)^6 + 175448*tan(d*x + c)^5 - 258370*I*tan(d*x + c)^4 - 261464*tan
(d*x + c)^3 + 192052*I*tan(d*x + c)^2 + 114152*tan(d*x + c) - 67819*I)/(a^8*(tan(d*x + c) - I)^8))/d

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maple [A]  time = 0.11, size = 196, normalized size = 0.86 \[ \frac {i \ln \left (\tan \left (d x +c \right )+i\right )}{512 d \,a^{8}}+\frac {i}{16 d \,a^{8} \left (\tan \left (d x +c \right )-i\right )^{8}}+\frac {i}{128 d \,a^{8} \left (\tan \left (d x +c \right )-i\right )^{4}}-\frac {i \ln \left (\tan \left (d x +c \right )-i\right )}{512 a^{8} d}-\frac {i}{48 d \,a^{8} \left (\tan \left (d x +c \right )-i\right )^{6}}-\frac {i}{256 a^{8} d \left (\tan \left (d x +c \right )-i\right )^{2}}-\frac {1}{28 d \,a^{8} \left (\tan \left (d x +c \right )-i\right )^{7}}+\frac {1}{80 d \,a^{8} \left (\tan \left (d x +c \right )-i\right )^{5}}-\frac {1}{192 a^{8} d \left (\tan \left (d x +c \right )-i\right )^{3}}+\frac {1}{256 a^{8} d \left (\tan \left (d x +c \right )-i\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+I*a*tan(d*x+c))^8,x)

[Out]

1/512*I/d/a^8*ln(tan(d*x+c)+I)+1/16*I/d/a^8/(tan(d*x+c)-I)^8+1/128*I/d/a^8/(tan(d*x+c)-I)^4-1/512*I/d/a^8*ln(t
an(d*x+c)-I)-1/48*I/d/a^8/(tan(d*x+c)-I)^6-1/256*I/d/a^8/(tan(d*x+c)-I)^2-1/28/d/a^8/(tan(d*x+c)-I)^7+1/80/d/a
^8/(tan(d*x+c)-I)^5-1/192/a^8/d/(tan(d*x+c)-I)^3+1/256/a^8/d/(tan(d*x+c)-I)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+I*a*tan(d*x+c))^8,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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mupad [B]  time = 4.96, size = 198, normalized size = 0.86 \[ \frac {x}{256\,a^8}-\frac {\frac {\mathrm {tan}\left (c+d\,x\right )\,5993{}\mathrm {i}}{26880\,a^8}+\frac {16}{105\,a^8}-\frac {143\,{\mathrm {tan}\left (c+d\,x\right )}^2}{480\,a^8}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,1193{}\mathrm {i}}{3840\,a^8}+\frac {11\,{\mathrm {tan}\left (c+d\,x\right )}^4}{48\,a^8}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^5\,85{}\mathrm {i}}{768\,a^8}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^6}{32\,a^8}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^7\,1{}\mathrm {i}}{256\,a^8}}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^8\,1{}\mathrm {i}+8\,{\mathrm {tan}\left (c+d\,x\right )}^7-{\mathrm {tan}\left (c+d\,x\right )}^6\,28{}\mathrm {i}-56\,{\mathrm {tan}\left (c+d\,x\right )}^5+{\mathrm {tan}\left (c+d\,x\right )}^4\,70{}\mathrm {i}+56\,{\mathrm {tan}\left (c+d\,x\right )}^3-{\mathrm {tan}\left (c+d\,x\right )}^2\,28{}\mathrm {i}-8\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + a*tan(c + d*x)*1i)^8,x)

[Out]

x/(256*a^8) - ((tan(c + d*x)*5993i)/(26880*a^8) + 16/(105*a^8) - (143*tan(c + d*x)^2)/(480*a^8) - (tan(c + d*x
)^3*1193i)/(3840*a^8) + (11*tan(c + d*x)^4)/(48*a^8) + (tan(c + d*x)^5*85i)/(768*a^8) - tan(c + d*x)^6/(32*a^8
) - (tan(c + d*x)^7*1i)/(256*a^8))/(d*(56*tan(c + d*x)^3 - tan(c + d*x)^2*28i - 8*tan(c + d*x) + tan(c + d*x)^
4*70i - 56*tan(c + d*x)^5 - tan(c + d*x)^6*28i + 8*tan(c + d*x)^7 + tan(c + d*x)^8*1i + 1i))

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sympy [A]  time = 0.75, size = 326, normalized size = 1.42 \[ \begin {cases} \frac {\left (22698142121947299840 i a^{56} d^{7} e^{70 i c} e^{- 2 i d x} + 39721748713407774720 i a^{56} d^{7} e^{68 i c} e^{- 4 i d x} + 52962331617877032960 i a^{56} d^{7} e^{66 i c} e^{- 6 i d x} + 49652185891759718400 i a^{56} d^{7} e^{64 i c} e^{- 8 i d x} + 31777398970726219776 i a^{56} d^{7} e^{62 i c} e^{- 10 i d x} + 13240582904469258240 i a^{56} d^{7} e^{60 i c} e^{- 12 i d x} + 3242591731706757120 i a^{56} d^{7} e^{58 i c} e^{- 14 i d x} + 354658470655426560 i a^{56} d^{7} e^{56 i c} e^{- 16 i d x}\right ) e^{- 72 i c}}{1452681095804627189760 a^{64} d^{8}} & \text {for}\: 1452681095804627189760 a^{64} d^{8} e^{72 i c} \neq 0 \\x \left (\frac {\left (e^{16 i c} + 8 e^{14 i c} + 28 e^{12 i c} + 56 e^{10 i c} + 70 e^{8 i c} + 56 e^{6 i c} + 28 e^{4 i c} + 8 e^{2 i c} + 1\right ) e^{- 16 i c}}{256 a^{8}} - \frac {1}{256 a^{8}}\right ) & \text {otherwise} \end {cases} + \frac {x}{256 a^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+I*a*tan(d*x+c))**8,x)

[Out]

Piecewise(((22698142121947299840*I*a**56*d**7*exp(70*I*c)*exp(-2*I*d*x) + 39721748713407774720*I*a**56*d**7*ex
p(68*I*c)*exp(-4*I*d*x) + 52962331617877032960*I*a**56*d**7*exp(66*I*c)*exp(-6*I*d*x) + 49652185891759718400*I
*a**56*d**7*exp(64*I*c)*exp(-8*I*d*x) + 31777398970726219776*I*a**56*d**7*exp(62*I*c)*exp(-10*I*d*x) + 1324058
2904469258240*I*a**56*d**7*exp(60*I*c)*exp(-12*I*d*x) + 3242591731706757120*I*a**56*d**7*exp(58*I*c)*exp(-14*I
*d*x) + 354658470655426560*I*a**56*d**7*exp(56*I*c)*exp(-16*I*d*x))*exp(-72*I*c)/(1452681095804627189760*a**64
*d**8), Ne(1452681095804627189760*a**64*d**8*exp(72*I*c), 0)), (x*((exp(16*I*c) + 8*exp(14*I*c) + 28*exp(12*I*
c) + 56*exp(10*I*c) + 70*exp(8*I*c) + 56*exp(6*I*c) + 28*exp(4*I*c) + 8*exp(2*I*c) + 1)*exp(-16*I*c)/(256*a**8
) - 1/(256*a**8)), True)) + x/(256*a**8)

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