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

Optimal. Leaf size=43 \[ \frac{i (a-i a \tan (c+d x))^4}{8 d \left (a^3+i a^3 \tan (c+d x)\right )^4} \]

[Out]

((I/8)*(a - I*a*Tan[c + d*x])^4)/(d*(a^3 + I*a^3*Tan[c + d*x])^4)

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Rubi [A]  time = 0.0431671, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {3487, 37} \[ \frac{i (a-i a \tan (c+d x))^4}{8 d \left (a^3+i a^3 \tan (c+d x)\right )^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((I/8)*(a - I*a*Tan[c + d*x])^4)/(d*(a^3 + I*a^3*Tan[c + d*x])^4)

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 37

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

Rubi steps

\begin{align*} \int \frac{\sec ^8(c+d x)}{(a+i a \tan (c+d x))^8} \, dx &=-\frac{i \operatorname{Subst}\left (\int \frac{(a-x)^3}{(a+x)^5} \, dx,x,i a \tan (c+d x)\right )}{a^7 d}\\ &=\frac{i (a-i a \tan (c+d x))^4}{8 d \left (a^3+i a^3 \tan (c+d x)\right )^4}\\ \end{align*}

Mathematica [A]  time = 0.0594251, size = 32, normalized size = 0.74 \[ \frac{i \sec ^8(c+d x)}{8 d (a+i a \tan (c+d x))^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((I/8)*Sec[c + d*x]^8)/(d*(a + I*a*Tan[c + d*x])^8)

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Maple [A]  time = 0.115, size = 63, normalized size = 1.5 \begin{align*}{\frac{1}{d{a}^{8}} \left ( - \left ( \tan \left ( dx+c \right ) -i \right ) ^{-1}-{\frac{3\,i}{ \left ( \tan \left ( dx+c \right ) -i \right ) ^{2}}}+4\, \left ( \tan \left ( dx+c \right ) -i \right ) ^{-3}+{\frac{2\,i}{ \left ( \tan \left ( dx+c \right ) -i \right ) ^{4}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d/a^8*(-1/(tan(d*x+c)-I)-3*I/(tan(d*x+c)-I)^2+4/(tan(d*x+c)-I)^3+2*I/(tan(d*x+c)-I)^4)

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Maxima [B]  time = 1.14773, size = 217, normalized size = 5.05 \begin{align*} -\frac{35 \, \tan \left (d x + c\right )^{6} - 105 i \, \tan \left (d x + c\right )^{5} - 140 \, \tan \left (d x + c\right )^{4} + 140 i \, \tan \left (d x + c\right )^{3} + 105 \, \tan \left (d x + c\right )^{2} - 35 i \, \tan \left (d x + c\right )}{{\left (35 \, a^{8} \tan \left (d x + c\right )^{7} - 245 i \, a^{8} \tan \left (d x + c\right )^{6} - 735 \, a^{8} \tan \left (d x + c\right )^{5} + 1225 i \, a^{8} \tan \left (d x + c\right )^{4} + 1225 \, a^{8} \tan \left (d x + c\right )^{3} - 735 i \, a^{8} \tan \left (d x + c\right )^{2} - 245 \, a^{8} \tan \left (d x + c\right ) + 35 i \, a^{8}\right )} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(35*tan(d*x + c)^6 - 105*I*tan(d*x + c)^5 - 140*tan(d*x + c)^4 + 140*I*tan(d*x + c)^3 + 105*tan(d*x + c)^2 -
35*I*tan(d*x + c))/((35*a^8*tan(d*x + c)^7 - 245*I*a^8*tan(d*x + c)^6 - 735*a^8*tan(d*x + c)^5 + 1225*I*a^8*ta
n(d*x + c)^4 + 1225*a^8*tan(d*x + c)^3 - 735*I*a^8*tan(d*x + c)^2 - 245*a^8*tan(d*x + c) + 35*I*a^8)*d)

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Fricas [A]  time = 2.52265, size = 49, normalized size = 1.14 \begin{align*} \frac{i \, e^{\left (-8 i \, d x - 8 i \, c\right )}}{8 \, a^{8} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*I*e^(-8*I*d*x - 8*I*c)/(a^8*d)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: AttributeError

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Giac [B]  time = 1.19459, size = 95, normalized size = 2.21 \begin{align*} -\frac{2 \,{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 7 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 7 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{a^{8} d{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - i\right )}^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(tan(1/2*d*x + 1/2*c)^7 - 7*tan(1/2*d*x + 1/2*c)^5 + 7*tan(1/2*d*x + 1/2*c)^3 - tan(1/2*d*x + 1/2*c))/(a^8*
d*(tan(1/2*d*x + 1/2*c) - I)^8)