\(\int \sec ^2(c+d x) (a+i a \tan (c+d x))^8 \, dx\) [80]

Optimal result

Integrand size = 24, antiderivative size = 27 \[ \int \sec ^2(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {i (a+i a \tan (c+d x))^9}{9 a d} \]

[Out]

-1/9*I*(a+I*a*tan(d*x+c))^9/a/d

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3568, 32} \[ \int \sec ^2(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {i (a+i a \tan (c+d x))^9}{9 a d} \]

[In]

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

[Out]

((-1/9*I)*(a + I*a*Tan[c + d*x])^9)/(a*d)

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]

Rule 3568

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {i \text {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] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(102\) vs. \(2(27)=54\).

Time = 0.67 (sec) , antiderivative size = 102, normalized size of antiderivative = 3.78 \[ \int \sec ^2(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {a^8 \tan (c+d x) \left (9+36 i \tan (c+d x)-84 \tan ^2(c+d x)-126 i \tan ^3(c+d x)+126 \tan ^4(c+d x)+84 i \tan ^5(c+d x)-36 \tan ^6(c+d x)-9 i \tan ^7(c+d x)+\tan ^8(c+d x)\right )}{9 d} \]

[In]

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

[Out]

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (23 ) = 46\).

Time = 111.63 (sec) , antiderivative size = 113, normalized size of antiderivative = 4.19

method	result	size
risch	\(\frac {512 i a^{8} \left (9 \,{\mathrm e}^{16 i \left (d x +c \right )}+36 \,{\mathrm e}^{14 i \left (d x +c \right )}+84 \,{\mathrm e}^{12 i \left (d x +c \right )}+126 \,{\mathrm e}^{10 i \left (d x +c \right )}+126 \,{\mathrm e}^{8 i \left (d x +c \right )}+84 \,{\mathrm e}^{6 i \left (d x +c \right )}+36 \,{\mathrm e}^{4 i \left (d x +c \right )}+9 \,{\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{9 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{9}}\)	\(113\)
derivativedivides	\(\frac {\frac {a^{8} \left (\sin ^{9}\left (d x +c \right )\right )}{9 \cos \left (d x +c \right )^{9}}+\frac {28 i a^{8} \left (\sin ^{6}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )^{6}}-\frac {4 a^{8} \left (\sin ^{7}\left (d x +c \right )\right )}{\cos \left (d x +c \right )^{7}}+\frac {4 i a^{8}}{\cos \left (d x +c \right )^{2}}+\frac {14 a^{8} \left (\sin ^{5}\left (d x +c \right )\right )}{\cos \left (d x +c \right )^{5}}-\frac {14 i a^{8} \left (\sin ^{4}\left (d x +c \right )\right )}{\cos \left (d x +c \right )^{4}}-\frac {28 a^{8} \left (\sin ^{3}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )^{3}}-\frac {i a^{8} \left (\sin ^{8}\left (d x +c \right )\right )}{\cos \left (d x +c \right )^{8}}+a^{8} \tan \left (d x +c \right )}{d}\)	\(180\)
default	\(\frac {\frac {a^{8} \left (\sin ^{9}\left (d x +c \right )\right )}{9 \cos \left (d x +c \right )^{9}}+\frac {28 i a^{8} \left (\sin ^{6}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )^{6}}-\frac {4 a^{8} \left (\sin ^{7}\left (d x +c \right )\right )}{\cos \left (d x +c \right )^{7}}+\frac {4 i a^{8}}{\cos \left (d x +c \right )^{2}}+\frac {14 a^{8} \left (\sin ^{5}\left (d x +c \right )\right )}{\cos \left (d x +c \right )^{5}}-\frac {14 i a^{8} \left (\sin ^{4}\left (d x +c \right )\right )}{\cos \left (d x +c \right )^{4}}-\frac {28 a^{8} \left (\sin ^{3}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )^{3}}-\frac {i a^{8} \left (\sin ^{8}\left (d x +c \right )\right )}{\cos \left (d x +c \right )^{8}}+a^{8} \tan \left (d x +c \right )}{d}\)	\(180\)

[In]

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

[Out]

512/9*I*a^8*(9*exp(16*I*(d*x+c))+36*exp(14*I*(d*x+c))+84*exp(12*I*(d*x+c))+126*exp(10*I*(d*x+c))+126*exp(8*I*(
d*x+c))+84*exp(6*I*(d*x+c))+36*exp(4*I*(d*x+c))+9*exp(2*I*(d*x+c))+1)/d/(exp(2*I*(d*x+c))+1)^9

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 231 vs. \(2 (21) = 42\).

Time = 0.24 (sec) , antiderivative size = 231, normalized size of antiderivative = 8.56 \[ \int \sec ^2(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {512 \, {\left (-9 i \, a^{8} e^{\left (16 i \, d x + 16 i \, c\right )} - 36 i \, a^{8} e^{\left (14 i \, d x + 14 i \, c\right )} - 84 i \, a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} - 126 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} - 126 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} - 84 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} - 36 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} - 9 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a^{8}\right )}}{9 \, {\left (d e^{\left (18 i \, d x + 18 i \, c\right )} + 9 \, d e^{\left (16 i \, d x + 16 i \, c\right )} + 36 \, d e^{\left (14 i \, d x + 14 i \, c\right )} + 84 \, d e^{\left (12 i \, d x + 12 i \, c\right )} + 126 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 126 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 84 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 36 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 9 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]

[In]

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

[Out]

-512/9*(-9*I*a^8*e^(16*I*d*x + 16*I*c) - 36*I*a^8*e^(14*I*d*x + 14*I*c) - 84*I*a^8*e^(12*I*d*x + 12*I*c) - 126
*I*a^8*e^(10*I*d*x + 10*I*c) - 126*I*a^8*e^(8*I*d*x + 8*I*c) - 84*I*a^8*e^(6*I*d*x + 6*I*c) - 36*I*a^8*e^(4*I*
d*x + 4*I*c) - 9*I*a^8*e^(2*I*d*x + 2*I*c) - 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)

Sympy [F]

\[ \int \sec ^2(c+d x) (a+i a \tan (c+d x))^8 \, dx=a^{8} \left (\int \left (- 28 \tan ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\right )\, dx + \int 70 \tan ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \left (- 28 \tan ^{6}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\right )\, dx + \int \tan ^{8}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 8 i \tan {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \left (- 56 i \tan ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\right )\, dx + \int 56 i \tan ^{5}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \left (- 8 i \tan ^{7}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\right )\, dx + \int \sec ^{2}{\left (c + d x \right )}\, dx\right ) \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int \sec ^2(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {i \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{9}}{9 \, a d} \]

[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)

Giac [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (21) = 42\).

Time = 1.06 (sec) , antiderivative size = 120, normalized size of antiderivative = 4.44 \[ \int \sec ^2(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {a^{8} \tan \left (d x + c\right )^{9} - 9 i \, a^{8} \tan \left (d x + c\right )^{8} - 36 \, a^{8} \tan \left (d x + c\right )^{7} + 84 i \, a^{8} \tan \left (d x + c\right )^{6} + 126 \, a^{8} \tan \left (d x + c\right )^{5} - 126 i \, a^{8} \tan \left (d x + c\right )^{4} - 84 \, a^{8} \tan \left (d x + c\right )^{3} + 36 i \, a^{8} \tan \left (d x + c\right )^{2} + 9 \, a^{8} \tan \left (d x + c\right )}{9 \, d} \]

[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

Mupad [B] (verification not implemented)

Time = 4.38 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.07 \[ \int \sec ^2(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {a^8\,\left (\sin \left (9\,c+9\,d\,x\right )+\frac {\cos \left (c+d\,x\right )\,63{}\mathrm {i}}{128}+\frac {\cos \left (3\,c+3\,d\,x\right )\,21{}\mathrm {i}}{64}+\frac {\cos \left (5\,c+5\,d\,x\right )\,9{}\mathrm {i}}{64}+\frac {\cos \left (7\,c+7\,d\,x\right )\,9{}\mathrm {i}}{256}-\frac {\cos \left (9\,c+9\,d\,x\right )\,255{}\mathrm {i}}{256}\right )}{9\,d\,{\cos \left (c+d\,x\right )}^9} \]

[In]

int((a + a*tan(c + d*x)*1i)^8/cos(c + d*x)^2,x)

[Out]

(a^8*((cos(c + d*x)*63i)/128 + (cos(3*c + 3*d*x)*21i)/64 + (cos(5*c + 5*d*x)*9i)/64 + (cos(7*c + 7*d*x)*9i)/25
6 - (cos(9*c + 9*d*x)*255i)/256 + sin(9*c + 9*d*x)))/(9*d*cos(c + d*x)^9)