3.59 \(\int \sec ^8(c+d x) (a+i a \tan (c+d x))^5 \, dx\)

Optimal. Leaf size=109 \[ \frac{i (a+i a \tan (c+d x))^{12}}{12 a^7 d}-\frac{6 i (a+i a \tan (c+d x))^{11}}{11 a^6 d}+\frac{6 i (a+i a \tan (c+d x))^{10}}{5 a^5 d}-\frac{8 i (a+i a \tan (c+d x))^9}{9 a^4 d} \]

[Out]

(((-8*I)/9)*(a + I*a*Tan[c + d*x])^9)/(a^4*d) + (((6*I)/5)*(a + I*a*Tan[c + d*x])^10)/(a^5*d) - (((6*I)/11)*(a
+ I*a*Tan[c + d*x])^11)/(a^6*d) + ((I/12)*(a + I*a*Tan[c + d*x])^12)/(a^7*d)

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Rubi [A]  time = 0.073068, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {3487, 43} \[ \frac{i (a+i a \tan (c+d x))^{12}}{12 a^7 d}-\frac{6 i (a+i a \tan (c+d x))^{11}}{11 a^6 d}+\frac{6 i (a+i a \tan (c+d x))^{10}}{5 a^5 d}-\frac{8 i (a+i a \tan (c+d x))^9}{9 a^4 d} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^8*(a + I*a*Tan[c + d*x])^5,x]

[Out]

(((-8*I)/9)*(a + I*a*Tan[c + d*x])^9)/(a^4*d) + (((6*I)/5)*(a + I*a*Tan[c + d*x])^10)/(a^5*d) - (((6*I)/11)*(a
+ I*a*Tan[c + d*x])^11)/(a^6*d) + ((I/12)*(a + I*a*Tan[c + d*x])^12)/(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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \sec ^8(c+d x) (a+i a \tan (c+d x))^5 \, dx &=-\frac{i \operatorname{Subst}\left (\int (a-x)^3 (a+x)^8 \, dx,x,i a \tan (c+d x)\right )}{a^7 d}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (8 a^3 (a+x)^8-12 a^2 (a+x)^9+6 a (a+x)^{10}-(a+x)^{11}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^7 d}\\ &=-\frac{8 i (a+i a \tan (c+d x))^9}{9 a^4 d}+\frac{6 i (a+i a \tan (c+d x))^{10}}{5 a^5 d}-\frac{6 i (a+i a \tan (c+d x))^{11}}{11 a^6 d}+\frac{i (a+i a \tan (c+d x))^{12}}{12 a^7 d}\\ \end{align*}

Mathematica [A]  time = 3.58874, size = 167, normalized size = 1.53 \[ \frac{a^5 \sec (c) \sec ^{12}(c+d x) (792 \sin (c+2 d x)-792 \sin (3 c+2 d x)+495 \sin (3 c+4 d x)-495 \sin (5 c+4 d x)+440 \sin (5 c+6 d x)+132 \sin (7 c+8 d x)+24 \sin (9 c+10 d x)+2 \sin (11 c+12 d x)+792 i \cos (c+2 d x)+792 i \cos (3 c+2 d x)+495 i \cos (3 c+4 d x)+495 i \cos (5 c+4 d x)-924 \sin (c)+924 i \cos (c))}{3960 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^8*(a + I*a*Tan[c + d*x])^5,x]

[Out]

(a^5*Sec[c]*Sec[c + d*x]^12*((924*I)*Cos[c] + (792*I)*Cos[c + 2*d*x] + (792*I)*Cos[3*c + 2*d*x] + (495*I)*Cos[
3*c + 4*d*x] + (495*I)*Cos[5*c + 4*d*x] - 924*Sin[c] + 792*Sin[c + 2*d*x] - 792*Sin[3*c + 2*d*x] + 495*Sin[3*c
+ 4*d*x] - 495*Sin[5*c + 4*d*x] + 440*Sin[5*c + 6*d*x] + 132*Sin[7*c + 8*d*x] + 24*Sin[9*c + 10*d*x] + 2*Sin[
11*c + 12*d*x]))/(3960*d)

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Maple [B]  time = 0.083, size = 377, normalized size = 3.5 \begin{align*} \text{result too large to display} \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))^5,x)

[Out]

1/d*(I*a^5*(1/12*sin(d*x+c)^6/cos(d*x+c)^12+1/20*sin(d*x+c)^6/cos(d*x+c)^10+1/40*sin(d*x+c)^6/cos(d*x+c)^8+1/1
20*sin(d*x+c)^6/cos(d*x+c)^6)+5*a^5*(1/11*sin(d*x+c)^5/cos(d*x+c)^11+2/33*sin(d*x+c)^5/cos(d*x+c)^9+8/231*sin(
d*x+c)^5/cos(d*x+c)^7+16/1155*sin(d*x+c)^5/cos(d*x+c)^5)-10*I*a^5*(1/10*sin(d*x+c)^4/cos(d*x+c)^10+3/40*sin(d*
x+c)^4/cos(d*x+c)^8+1/20*sin(d*x+c)^4/cos(d*x+c)^6+1/40*sin(d*x+c)^4/cos(d*x+c)^4)-10*a^5*(1/9*sin(d*x+c)^3/co
s(d*x+c)^9+2/21*sin(d*x+c)^3/cos(d*x+c)^7+8/105*sin(d*x+c)^3/cos(d*x+c)^5+16/315*sin(d*x+c)^3/cos(d*x+c)^3)+5/
8*I*a^5/cos(d*x+c)^8-a^5*(-16/35-1/7*sec(d*x+c)^6-6/35*sec(d*x+c)^4-8/35*sec(d*x+c)^2)*tan(d*x+c))

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Maxima [A]  time = 1.12479, size = 216, normalized size = 1.98 \begin{align*} \frac{2310 i \, a^{5} \tan \left (d x + c\right )^{12} + 12600 \, a^{5} \tan \left (d x + c\right )^{11} - 19404 i \, a^{5} \tan \left (d x + c\right )^{10} + 15400 \, a^{5} \tan \left (d x + c\right )^{9} - 76230 i \, a^{5} \tan \left (d x + c\right )^{8} - 55440 \, a^{5} \tan \left (d x + c\right )^{7} - 64680 i \, a^{5} \tan \left (d x + c\right )^{6} - 121968 \, a^{5} \tan \left (d x + c\right )^{5} + 34650 i \, a^{5} \tan \left (d x + c\right )^{4} - 64680 \, a^{5} \tan \left (d x + c\right )^{3} + 69300 i \, a^{5} \tan \left (d x + c\right )^{2} + 27720 \, a^{5} \tan \left (d x + c\right )}{27720 \, 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))^5,x, algorithm="maxima")

[Out]

1/27720*(2310*I*a^5*tan(d*x + c)^12 + 12600*a^5*tan(d*x + c)^11 - 19404*I*a^5*tan(d*x + c)^10 + 15400*a^5*tan(
d*x + c)^9 - 76230*I*a^5*tan(d*x + c)^8 - 55440*a^5*tan(d*x + c)^7 - 64680*I*a^5*tan(d*x + c)^6 - 121968*a^5*t
an(d*x + c)^5 + 34650*I*a^5*tan(d*x + c)^4 - 64680*a^5*tan(d*x + c)^3 + 69300*I*a^5*tan(d*x + c)^2 + 27720*a^5
*tan(d*x + c))/d

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Fricas [B]  time = 1.19618, size = 883, normalized size = 8.1 \begin{align*} \frac{506880 i \, a^{5} e^{\left (16 i \, d x + 16 i \, c\right )} + 811008 i \, a^{5} e^{\left (14 i \, d x + 14 i \, c\right )} + 946176 i \, a^{5} e^{\left (12 i \, d x + 12 i \, c\right )} + 811008 i \, a^{5} e^{\left (10 i \, d x + 10 i \, c\right )} + 506880 i \, a^{5} e^{\left (8 i \, d x + 8 i \, c\right )} + 225280 i \, a^{5} e^{\left (6 i \, d x + 6 i \, c\right )} + 67584 i \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} + 12288 i \, a^{5} e^{\left (2 i \, d x + 2 i \, c\right )} + 1024 i \, a^{5}}{495 \,{\left (d e^{\left (24 i \, d x + 24 i \, c\right )} + 12 \, d e^{\left (22 i \, d x + 22 i \, c\right )} + 66 \, d e^{\left (20 i \, d x + 20 i \, c\right )} + 220 \, d e^{\left (18 i \, d x + 18 i \, c\right )} + 495 \, d e^{\left (16 i \, d x + 16 i \, c\right )} + 792 \, d e^{\left (14 i \, d x + 14 i \, c\right )} + 924 \, d e^{\left (12 i \, d x + 12 i \, c\right )} + 792 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 495 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 220 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 66 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 12 \, 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)^8*(a+I*a*tan(d*x+c))^5,x, algorithm="fricas")

[Out]

1/495*(506880*I*a^5*e^(16*I*d*x + 16*I*c) + 811008*I*a^5*e^(14*I*d*x + 14*I*c) + 946176*I*a^5*e^(12*I*d*x + 12
*I*c) + 811008*I*a^5*e^(10*I*d*x + 10*I*c) + 506880*I*a^5*e^(8*I*d*x + 8*I*c) + 225280*I*a^5*e^(6*I*d*x + 6*I*
c) + 67584*I*a^5*e^(4*I*d*x + 4*I*c) + 12288*I*a^5*e^(2*I*d*x + 2*I*c) + 1024*I*a^5)/(d*e^(24*I*d*x + 24*I*c)
+ 12*d*e^(22*I*d*x + 22*I*c) + 66*d*e^(20*I*d*x + 20*I*c) + 220*d*e^(18*I*d*x + 18*I*c) + 495*d*e^(16*I*d*x +
16*I*c) + 792*d*e^(14*I*d*x + 14*I*c) + 924*d*e^(12*I*d*x + 12*I*c) + 792*d*e^(10*I*d*x + 10*I*c) + 495*d*e^(8
*I*d*x + 8*I*c) + 220*d*e^(6*I*d*x + 6*I*c) + 66*d*e^(4*I*d*x + 4*I*c) + 12*d*e^(2*I*d*x + 2*I*c) + d)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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))**5,x)

[Out]

Timed out

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Giac [A]  time = 1.43765, size = 216, normalized size = 1.98 \begin{align*} -\frac{-165 i \, a^{5} \tan \left (d x + c\right )^{12} - 900 \, a^{5} \tan \left (d x + c\right )^{11} + 1386 i \, a^{5} \tan \left (d x + c\right )^{10} - 1100 \, a^{5} \tan \left (d x + c\right )^{9} + 5445 i \, a^{5} \tan \left (d x + c\right )^{8} + 3960 \, a^{5} \tan \left (d x + c\right )^{7} + 4620 i \, a^{5} \tan \left (d x + c\right )^{6} + 8712 \, a^{5} \tan \left (d x + c\right )^{5} - 2475 i \, a^{5} \tan \left (d x + c\right )^{4} + 4620 \, a^{5} \tan \left (d x + c\right )^{3} - 4950 i \, a^{5} \tan \left (d x + c\right )^{2} - 1980 \, a^{5} \tan \left (d x + c\right )}{1980 \, 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))^5,x, algorithm="giac")

[Out]

-1/1980*(-165*I*a^5*tan(d*x + c)^12 - 900*a^5*tan(d*x + c)^11 + 1386*I*a^5*tan(d*x + c)^10 - 1100*a^5*tan(d*x
+ c)^9 + 5445*I*a^5*tan(d*x + c)^8 + 3960*a^5*tan(d*x + c)^7 + 4620*I*a^5*tan(d*x + c)^6 + 8712*a^5*tan(d*x +
c)^5 - 2475*I*a^5*tan(d*x + c)^4 + 4620*a^5*tan(d*x + c)^3 - 4950*I*a^5*tan(d*x + c)^2 - 1980*a^5*tan(d*x + c)
)/d