3.80 \(\int \sec ^2(c+d x) (a+i a \tan (c+d x))^8 \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0377482, 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} \[ -\frac{i (a+i a \tan (c+d x))^9}{9 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-I/9)*(a + I*a*Tan[c + d*x])^9)/(a*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 \sec ^2(c+d x) (a+i a \tan (c+d x))^8 \, dx &=-\frac{i \operatorname{Subst}\left (\int (a+x)^8 \, dx,x,i a \tan (c+d x)\right )}{a d}\\ &=-\frac{i (a+i a \tan (c+d x))^9}{9 a d}\\ \end{align*}

Mathematica [B]  time = 2.86396, size = 212, normalized size = 7.85 \[ \frac{a^8 \sec (c) \sec ^9(c+d x) (-126 \sin (2 c+d x)+84 \sin (2 c+3 d x)-84 \sin (4 c+3 d x)+36 \sin (4 c+5 d x)-36 \sin (6 c+5 d x)+9 \sin (6 c+7 d x)-9 \sin (8 c+7 d x)+2 \sin (8 c+9 d x)+126 i \cos (2 c+d x)+84 i \cos (2 c+3 d x)+84 i \cos (4 c+3 d x)+36 i \cos (4 c+5 d x)+36 i \cos (6 c+5 d x)+9 i \cos (6 c+7 d x)+9 i \cos (8 c+7 d x)+126 \sin (d x)+126 i \cos (d x))}{18 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^8*Sec[c]*Sec[c + d*x]^9*((126*I)*Cos[d*x] + (126*I)*Cos[2*c + d*x] + (84*I)*Cos[2*c + 3*d*x] + (84*I)*Cos[4
*c + 3*d*x] + (36*I)*Cos[4*c + 5*d*x] + (36*I)*Cos[6*c + 5*d*x] + (9*I)*Cos[6*c + 7*d*x] + (9*I)*Cos[8*c + 7*d
*x] + 126*Sin[d*x] - 126*Sin[2*c + d*x] + 84*Sin[2*c + 3*d*x] - 84*Sin[4*c + 3*d*x] + 36*Sin[4*c + 5*d*x] - 36
*Sin[6*c + 5*d*x] + 9*Sin[6*c + 7*d*x] - 9*Sin[8*c + 7*d*x] + 2*Sin[8*c + 9*d*x]))/(18*d)

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Maple [B]  time = 0.089, size = 180, normalized size = 6.7 \begin{align*}{\frac{1}{d} \left ({\frac{{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{9}}{9\, \left ( \cos \left ( dx+c \right ) \right ) ^{9}}}-{\frac{14\,i{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}-4\,{\frac{{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{7}}}+{\frac{{\frac{28\,i}{3}}{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}}+14\,{\frac{{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}-{\frac{i{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}}-{\frac{28\,{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+{\frac{4\,i{a}^{8}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{a}^{8}\tan \left ( dx+c \right ) \right ) } \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/d*(1/9*a^8*sin(d*x+c)^9/cos(d*x+c)^9-14*I*a^8*sin(d*x+c)^4/cos(d*x+c)^4-4*a^8*sin(d*x+c)^7/cos(d*x+c)^7+28/3
*I*a^8*sin(d*x+c)^6/cos(d*x+c)^6+14*a^8*sin(d*x+c)^5/cos(d*x+c)^5-I*a^8*sin(d*x+c)^8/cos(d*x+c)^8-28/3*a^8*sin
(d*x+c)^3/cos(d*x+c)^3+4*I*a^8/cos(d*x+c)^2+a^8*tan(d*x+c))

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Maxima [A]  time = 1.16394, size = 28, normalized size = 1.04 \begin{align*} -\frac{i \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{9}}{9 \, 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]

-1/9*I*(I*a*tan(d*x + c) + a)^9/(a*d)

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Fricas [B]  time = 1.29112, size = 741, normalized size = 27.44 \begin{align*} \frac{4608 i \, a^{8} e^{\left (16 i \, d x + 16 i \, c\right )} + 18432 i \, a^{8} e^{\left (14 i \, d x + 14 i \, c\right )} + 43008 i \, a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} + 64512 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} + 64512 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} + 43008 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} + 18432 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} + 4608 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} + 512 i \, a^{8}}{9 \,{\left (d e^{\left (18 i \, d x + 18 i \, c\right )} + 9 \, d e^{\left (16 i \, d x + 16 i \, c\right )} + 36 \, d e^{\left (14 i \, d x + 14 i \, c\right )} + 84 \, d e^{\left (12 i \, d x + 12 i \, c\right )} + 126 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 126 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 84 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 36 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 9 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \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]

1/9*(4608*I*a^8*e^(16*I*d*x + 16*I*c) + 18432*I*a^8*e^(14*I*d*x + 14*I*c) + 43008*I*a^8*e^(12*I*d*x + 12*I*c)
+ 64512*I*a^8*e^(10*I*d*x + 10*I*c) + 64512*I*a^8*e^(8*I*d*x + 8*I*c) + 43008*I*a^8*e^(6*I*d*x + 6*I*c) + 1843
2*I*a^8*e^(4*I*d*x + 4*I*c) + 4608*I*a^8*e^(2*I*d*x + 2*I*c) + 512*I*a^8)/(d*e^(18*I*d*x + 18*I*c) + 9*d*e^(16
*I*d*x + 16*I*c) + 36*d*e^(14*I*d*x + 14*I*c) + 84*d*e^(12*I*d*x + 12*I*c) + 126*d*e^(10*I*d*x + 10*I*c) + 126
*d*e^(8*I*d*x + 8*I*c) + 84*d*e^(6*I*d*x + 6*I*c) + 36*d*e^(4*I*d*x + 4*I*c) + 9*d*e^(2*I*d*x + 2*I*c) + d)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} a^{8} \left (\int - 28 \tan ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 70 \tan ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int - 28 \tan ^{6}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \tan ^{8}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 8 i \tan{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int - 56 i \tan ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 56 i \tan ^{5}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int - 8 i \tan ^{7}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \sec ^{2}{\left (c + d x \right )}\, dx\right ) \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]

a**8*(Integral(-28*tan(c + d*x)**2*sec(c + d*x)**2, x) + Integral(70*tan(c + d*x)**4*sec(c + d*x)**2, x) + Int
egral(-28*tan(c + d*x)**6*sec(c + d*x)**2, x) + Integral(tan(c + d*x)**8*sec(c + d*x)**2, x) + Integral(8*I*ta
n(c + d*x)*sec(c + d*x)**2, x) + Integral(-56*I*tan(c + d*x)**3*sec(c + d*x)**2, x) + Integral(56*I*tan(c + d*
x)**5*sec(c + d*x)**2, x) + Integral(-8*I*tan(c + d*x)**7*sec(c + d*x)**2, x) + Integral(sec(c + d*x)**2, x))

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Giac [B]  time = 1.79316, size = 162, normalized size = 6. \begin{align*} \frac{a^{8} \tan \left (d x + c\right )^{9} - 9 i \, a^{8} \tan \left (d x + c\right )^{8} - 36 \, a^{8} \tan \left (d x + c\right )^{7} + 84 i \, a^{8} \tan \left (d x + c\right )^{6} + 126 \, a^{8} \tan \left (d x + c\right )^{5} - 126 i \, a^{8} \tan \left (d x + c\right )^{4} - 84 \, a^{8} \tan \left (d x + c\right )^{3} + 36 i \, a^{8} \tan \left (d x + c\right )^{2} + 9 \, a^{8} \tan \left (d x + c\right )}{9 \, 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="giac")

[Out]

1/9*(a^8*tan(d*x + c)^9 - 9*I*a^8*tan(d*x + c)^8 - 36*a^8*tan(d*x + c)^7 + 84*I*a^8*tan(d*x + c)^6 + 126*a^8*t
an(d*x + c)^5 - 126*I*a^8*tan(d*x + c)^4 - 84*a^8*tan(d*x + c)^3 + 36*I*a^8*tan(d*x + c)^2 + 9*a^8*tan(d*x + c
))/d