3.150 \(\int \frac{\sec ^{10}(c+d x)}{(a+i a \tan (c+d x))^4} \, dx\)

Optimal. Leaf size=27 \[ \frac{i (a-i a \tan (c+d x))^5}{5 a^9 d} \]

[Out]

((I/5)*(a - I*a*Tan[c + d*x])^5)/(a^9*d)

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Rubi [A]  time = 0.0400154, 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} \[ \frac{i (a-i a \tan (c+d x))^5}{5 a^9 d} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^10/(a + I*a*Tan[c + d*x])^4,x]

[Out]

((I/5)*(a - I*a*Tan[c + d*x])^5)/(a^9*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 ^{10}(c+d x)}{(a+i a \tan (c+d x))^4} \, dx &=-\frac{i \operatorname{Subst}\left (\int (a-x)^4 \, dx,x,i a \tan (c+d x)\right )}{a^9 d}\\ &=\frac{i (a-i a \tan (c+d x))^5}{5 a^9 d}\\ \end{align*}

Mathematica [B]  time = 0.367759, size = 116, normalized size = 4.3 \[ \frac{\sec (c) \sec ^5(c+d x) (-10 \sin (2 c+d x)+5 \sin (2 c+3 d x)-5 \sin (4 c+3 d x)+2 \sin (4 c+5 d x)-10 i \cos (2 c+d x)-5 i \cos (2 c+3 d x)-5 i \cos (4 c+3 d x)+10 \sin (d x)-10 i \cos (d x))}{10 a^4 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^10/(a + I*a*Tan[c + d*x])^4,x]

[Out]

(Sec[c]*Sec[c + d*x]^5*((-10*I)*Cos[d*x] - (10*I)*Cos[2*c + d*x] - (5*I)*Cos[2*c + 3*d*x] - (5*I)*Cos[4*c + 3*
d*x] + 10*Sin[d*x] - 10*Sin[2*c + d*x] + 5*Sin[2*c + 3*d*x] - 5*Sin[4*c + 3*d*x] + 2*Sin[4*c + 5*d*x]))/(10*a^
4*d)

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Maple [B]  time = 0.084, size = 57, normalized size = 2.1 \begin{align*}{\frac{1}{{a}^{4}d} \left ( \tan \left ( dx+c \right ) +{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{5}}{5}}+i \left ( \tan \left ( dx+c \right ) \right ) ^{4}-2\, \left ( \tan \left ( dx+c \right ) \right ) ^{3}-2\,i \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^10/(a+I*a*tan(d*x+c))^4,x)

[Out]

1/d/a^4*(tan(d*x+c)+1/5*tan(d*x+c)^5+I*tan(d*x+c)^4-2*tan(d*x+c)^3-2*I*tan(d*x+c)^2)

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Maxima [B]  time = 0.977301, size = 77, normalized size = 2.85 \begin{align*} \frac{3 \, \tan \left (d x + c\right )^{5} + 15 i \, \tan \left (d x + c\right )^{4} - 30 \, \tan \left (d x + c\right )^{3} - 30 i \, \tan \left (d x + c\right )^{2} + 15 \, \tan \left (d x + c\right )}{15 \, a^{4} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^10/(a+I*a*tan(d*x+c))^4,x, algorithm="maxima")

[Out]

1/15*(3*tan(d*x + c)^5 + 15*I*tan(d*x + c)^4 - 30*tan(d*x + c)^3 - 30*I*tan(d*x + c)^2 + 15*tan(d*x + c))/(a^4
*d)

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Fricas [B]  time = 2.41311, size = 227, normalized size = 8.41 \begin{align*} \frac{32 i}{5 \,{\left (a^{4} d e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, a^{4} d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, a^{4} d e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, a^{4} d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, a^{4} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{4} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^10/(a+I*a*tan(d*x+c))^4,x, algorithm="fricas")

[Out]

32/5*I/(a^4*d*e^(10*I*d*x + 10*I*c) + 5*a^4*d*e^(8*I*d*x + 8*I*c) + 10*a^4*d*e^(6*I*d*x + 6*I*c) + 10*a^4*d*e^
(4*I*d*x + 4*I*c) + 5*a^4*d*e^(2*I*d*x + 2*I*c) + a^4*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)**10/(a+I*a*tan(d*x+c))**4,x)

[Out]

Exception raised: AttributeError

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Giac [B]  time = 1.20862, size = 74, normalized size = 2.74 \begin{align*} \frac{\tan \left (d x + c\right )^{5} + 5 i \, \tan \left (d x + c\right )^{4} - 10 \, \tan \left (d x + c\right )^{3} - 10 i \, \tan \left (d x + c\right )^{2} + 5 \, \tan \left (d x + c\right )}{5 \, a^{4} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^10/(a+I*a*tan(d*x+c))^4,x, algorithm="giac")

[Out]

1/5*(tan(d*x + c)^5 + 5*I*tan(d*x + c)^4 - 10*tan(d*x + c)^3 - 10*I*tan(d*x + c)^2 + 5*tan(d*x + c))/(a^4*d)