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

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

[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]))

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Rubi [A]  time = 0.153116, antiderivative size = 229, normalized size of antiderivative = 1., 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 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]

Rule 8

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

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*}

Mathematica [A]  time = 0.483862, 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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Maple [A]  time = 0.029, size = 196, normalized size = 0.9 \begin{align*}{\frac{{\frac{i}{16}}}{d{a}^{8} \left ( \tan \left ( dx+c \right ) -i \right ) ^{8}}}+{\frac{{\frac{i}{128}}}{d{a}^{8} \left ( \tan \left ( dx+c \right ) -i \right ) ^{4}}}-{\frac{{\frac{i}{512}}\ln \left ( \tan \left ( dx+c \right ) -i \right ) }{d{a}^{8}}}-{\frac{{\frac{i}{48}}}{d{a}^{8} \left ( \tan \left ( dx+c \right ) -i \right ) ^{6}}}-{\frac{{\frac{i}{256}}}{d{a}^{8} \left ( \tan \left ( dx+c \right ) -i \right ) ^{2}}}-{\frac{1}{28\,d{a}^{8} \left ( \tan \left ( dx+c \right ) -i \right ) ^{7}}}+{\frac{1}{80\,d{a}^{8} \left ( \tan \left ( dx+c \right ) -i \right ) ^{5}}}-{\frac{1}{192\,d{a}^{8} \left ( \tan \left ( dx+c \right ) -i \right ) ^{3}}}+{\frac{1}{256\,d{a}^{8} \left ( \tan \left ( dx+c \right ) -i \right ) }}+{\frac{{\frac{i}{512}}\ln \left ( \tan \left ( dx+c \right ) +i \right ) }{d{a}^{8}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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(tan(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/d/a^8
/(tan(d*x+c)-I)^3+1/256/d/a^8/(tan(d*x+c)-I)+1/512*I/d/a^8*ln(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(1/(a+I*a*tan(d*x+c))^8,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 2.40363, size = 396, normalized size = 1.73 \begin{align*} \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} \end{align*}

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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Sympy [A]  time = 2.11228, size = 326, normalized size = 1.42 \begin{align*} \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}} \end{align*}

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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Giac [A]  time = 1.10688, size = 178, normalized size = 0.78 \begin{align*} -\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} \end{align*}

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