3.133 \(\int \frac{\sec ^8(c+d x)}{(a+i a \tan (c+d x))^3} \, dx\)

Optimal. Leaf size=27 \[ \frac{i (a-i a \tan (c+d x))^4}{4 a^7 d} \]

[Out]

((I/4)*(a - I*a*Tan[c + d*x])^4)/(a^7*d)

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Rubi [A]  time = 0.039479, antiderivative size = 27, 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, 32} \[ \frac{i (a-i a \tan (c+d x))^4}{4 a^7 d} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^8/(a + I*a*Tan[c + d*x])^3,x]

[Out]

((I/4)*(a - I*a*Tan[c + d*x])^4)/(a^7*d)

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 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

Mathematica [B]  time = 0.378706, size = 84, normalized size = 3.11 \[ \frac{\sec (c) \sec ^4(c+d x) (2 \sin (c+2 d x)-2 \sin (3 c+2 d x)+\sin (3 c+4 d x)-2 i \cos (c+2 d x)-2 i \cos (3 c+2 d x)-3 \sin (c)-3 i \cos (c))}{4 a^3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^8/(a + I*a*Tan[c + d*x])^3,x]

[Out]

(Sec[c]*Sec[c + d*x]^4*((-3*I)*Cos[c] - (2*I)*Cos[c + 2*d*x] - (2*I)*Cos[3*c + 2*d*x] - 3*Sin[c] + 2*Sin[c + 2
*d*x] - 2*Sin[3*c + 2*d*x] + Sin[3*c + 4*d*x]))/(4*a^3*d)

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Maple [A]  time = 0.073, size = 47, normalized size = 1.7 \begin{align*}{\frac{\tan \left ( dx+c \right ) +{\frac{i}{4}} \left ( \tan \left ( dx+c \right ) \right ) ^{4}- \left ( \tan \left ( dx+c \right ) \right ) ^{3}-{\frac{3\,i}{2}} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{d{a}^{3}}} \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))^3,x)

[Out]

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

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Maxima [B]  time = 0.970975, size = 63, normalized size = 2.33 \begin{align*} -\frac{-i \, \tan \left (d x + c\right )^{4} + 4 \, \tan \left (d x + c\right )^{3} + 6 i \, \tan \left (d x + c\right )^{2} - 4 \, \tan \left (d x + c\right )}{4 \, a^{3} 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))^3,x, algorithm="maxima")

[Out]

-1/4*(-I*tan(d*x + c)^4 + 4*tan(d*x + c)^3 + 6*I*tan(d*x + c)^2 - 4*tan(d*x + c))/(a^3*d)

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Fricas [B]  time = 2.30315, size = 177, normalized size = 6.56 \begin{align*} \frac{4 i}{a^{3} d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, a^{3} d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, a^{3} d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{3} 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))^3,x, algorithm="fricas")

[Out]

4*I/(a^3*d*e^(8*I*d*x + 8*I*c) + 4*a^3*d*e^(6*I*d*x + 6*I*c) + 6*a^3*d*e^(4*I*d*x + 4*I*c) + 4*a^3*d*e^(2*I*d*
x + 2*I*c) + a^3*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))**3,x)

[Out]

Exception raised: AttributeError

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Giac [B]  time = 1.20958, size = 63, normalized size = 2.33 \begin{align*} -\frac{-i \, \tan \left (d x + c\right )^{4} + 4 \, \tan \left (d x + c\right )^{3} + 6 i \, \tan \left (d x + c\right )^{2} - 4 \, \tan \left (d x + c\right )}{4 \, a^{3} 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))^3,x, algorithm="giac")

[Out]

-1/4*(-I*tan(d*x + c)^4 + 4*tan(d*x + c)^3 + 6*I*tan(d*x + c)^2 - 4*tan(d*x + c))/(a^3*d)