3.172 \(\int \frac{\sec ^2(c+d x)}{(a+i a \tan (c+d x))^8} \, dx\)

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

[Out]

(I/7)/(a*d*(a + I*a*Tan[c + d*x])^7)

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Rubi [A]  time = 0.0388354, 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} \[ \frac{i}{7 a d (a+i a \tan (c+d x))^7} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^2/(a + I*a*Tan[c + d*x])^8,x]

[Out]

(I/7)/(a*d*(a + I*a*Tan[c + d*x])^7)

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 ^2(c+d x)}{(a+i a \tan (c+d x))^8} \, dx &=-\frac{i \operatorname{Subst}\left (\int \frac{1}{(a+x)^8} \, dx,x,i a \tan (c+d x)\right )}{a d}\\ &=\frac{i}{7 a d (a+i a \tan (c+d x))^7}\\ \end{align*}

Mathematica [B]  time = 0.228745, size = 100, normalized size = 3.7 \[ \frac{i \sec ^8(c+d x) (14 i \sin (2 (c+d x))+14 i \sin (4 (c+d x))+6 i \sin (6 (c+d x))+56 \cos (2 (c+d x))+28 \cos (4 (c+d x))+8 \cos (6 (c+d x))+35)}{896 a^8 d (\tan (c+d x)-i)^8} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^2/(a + I*a*Tan[c + d*x])^8,x]

[Out]

((I/896)*Sec[c + d*x]^8*(35 + 56*Cos[2*(c + d*x)] + 28*Cos[4*(c + d*x)] + 8*Cos[6*(c + d*x)] + (14*I)*Sin[2*(c
+ d*x)] + (14*I)*Sin[4*(c + d*x)] + (6*I)*Sin[6*(c + d*x)]))/(a^8*d*(-I + Tan[c + d*x])^8)

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Maple [A]  time = 0.054, size = 24, normalized size = 0.9 \begin{align*}{\frac{{\frac{i}{7}}}{ad \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{7}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^2/(a+I*a*tan(d*x+c))^8,x)

[Out]

1/7*I/a/d/(a+I*a*tan(d*x+c))^7

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Maxima [A]  time = 1.19162, size = 28, normalized size = 1.04 \begin{align*} \frac{i}{7 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{7} a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^2/(a+I*a*tan(d*x+c))^8,x, algorithm="maxima")

[Out]

1/7*I/((I*a*tan(d*x + c) + a)^7*a*d)

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Fricas [B]  time = 2.35175, size = 278, normalized size = 10.3 \begin{align*} \frac{{\left (7 i \, e^{\left (12 i \, d x + 12 i \, c\right )} + 21 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 35 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 35 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 21 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 7 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )} e^{\left (-14 i \, d x - 14 i \, c\right )}}{896 \, a^{8} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^2/(a+I*a*tan(d*x+c))^8,x, algorithm="fricas")

[Out]

1/896*(7*I*e^(12*I*d*x + 12*I*c) + 21*I*e^(10*I*d*x + 10*I*c) + 35*I*e^(8*I*d*x + 8*I*c) + 35*I*e^(6*I*d*x + 6
*I*c) + 21*I*e^(4*I*d*x + 4*I*c) + 7*I*e^(2*I*d*x + 2*I*c) + I)*e^(-14*I*d*x - 14*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)**2/(a+I*a*tan(d*x+c))**8,x)

[Out]

Exception raised: AttributeError

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Giac [B]  time = 1.24341, size = 255, normalized size = 9.44 \begin{align*} -\frac{2 \,{\left (7 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} - 42 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{12} - 182 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 490 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} + 1001 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 1484 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 1716 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 1484 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 1001 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 490 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 182 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 42 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 7 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{7 \, a^{8} d{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - i\right )}^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^2/(a+I*a*tan(d*x+c))^8,x, algorithm="giac")

[Out]

-2/7*(7*tan(1/2*d*x + 1/2*c)^13 - 42*I*tan(1/2*d*x + 1/2*c)^12 - 182*tan(1/2*d*x + 1/2*c)^11 + 490*I*tan(1/2*d
*x + 1/2*c)^10 + 1001*tan(1/2*d*x + 1/2*c)^9 - 1484*I*tan(1/2*d*x + 1/2*c)^8 - 1716*tan(1/2*d*x + 1/2*c)^7 + 1
484*I*tan(1/2*d*x + 1/2*c)^6 + 1001*tan(1/2*d*x + 1/2*c)^5 - 490*I*tan(1/2*d*x + 1/2*c)^4 - 182*tan(1/2*d*x +
1/2*c)^3 + 42*I*tan(1/2*d*x + 1/2*c)^2 + 7*tan(1/2*d*x + 1/2*c))/(a^8*d*(tan(1/2*d*x + 1/2*c) - I)^14)