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

Optimal. Leaf size=109 \[ \frac {i (a+i a \tan (c+d x))^{15}}{15 a^7 d}-\frac {3 i (a+i a \tan (c+d x))^{14}}{7 a^6 d}+\frac {12 i (a+i a \tan (c+d x))^{13}}{13 a^5 d}-\frac {2 i (a+i a \tan (c+d x))^{12}}{3 a^4 d} \]

[Out]

-2/3*I*(a+I*a*tan(d*x+c))^12/a^4/d+12/13*I*(a+I*a*tan(d*x+c))^13/a^5/d-3/7*I*(a+I*a*tan(d*x+c))^14/a^6/d+1/15*
I*(a+I*a*tan(d*x+c))^15/a^7/d

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Rubi [A]  time = 0.08, antiderivative size = 109, normalized size of antiderivative = 1.00, 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} \[ \frac {i (a+i a \tan (c+d x))^{15}}{15 a^7 d}-\frac {3 i (a+i a \tan (c+d x))^{14}}{7 a^6 d}+\frac {12 i (a+i a \tan (c+d x))^{13}}{13 a^5 d}-\frac {2 i (a+i a \tan (c+d x))^{12}}{3 a^4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((-2*I)/3)*(a + I*a*Tan[c + d*x])^12)/(a^4*d) + (((12*I)/13)*(a + I*a*Tan[c + d*x])^13)/(a^5*d) - (((3*I)/7)*
(a + I*a*Tan[c + d*x])^14)/(a^6*d) + ((I/15)*(a + I*a*Tan[c + d*x])^15)/(a^7*d)

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])

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]

Rubi steps

\begin {align*} \int \sec ^8(c+d x) (a+i a \tan (c+d x))^8 \, dx &=-\frac {i \operatorname {Subst}\left (\int (a-x)^3 (a+x)^{11} \, dx,x,i a \tan (c+d x)\right )}{a^7 d}\\ &=-\frac {i \operatorname {Subst}\left (\int \left (8 a^3 (a+x)^{11}-12 a^2 (a+x)^{12}+6 a (a+x)^{13}-(a+x)^{14}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^7 d}\\ &=-\frac {2 i (a+i a \tan (c+d x))^{12}}{3 a^4 d}+\frac {12 i (a+i a \tan (c+d x))^{13}}{13 a^5 d}-\frac {3 i (a+i a \tan (c+d x))^{14}}{7 a^6 d}+\frac {i (a+i a \tan (c+d x))^{15}}{15 a^7 d}\\ \end {align*}

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Mathematica [B]  time = 9.50, size = 245, normalized size = 2.25 \[ \frac {a^8 \sec (c) \sec ^{15}(c+d x) (-6435 \sin (2 c+d x)+5005 \sin (2 c+3 d x)-5005 \sin (4 c+3 d x)+3003 \sin (4 c+5 d x)-3003 \sin (6 c+5 d x)+1365 \sin (6 c+7 d x)-1365 \sin (8 c+7 d x)+910 \sin (8 c+9 d x)+210 \sin (10 c+11 d x)+30 \sin (12 c+13 d x)+2 \sin (14 c+15 d x)+6435 i \cos (2 c+d x)+5005 i \cos (2 c+3 d x)+5005 i \cos (4 c+3 d x)+3003 i \cos (4 c+5 d x)+3003 i \cos (6 c+5 d x)+1365 i \cos (6 c+7 d x)+1365 i \cos (8 c+7 d x)+6435 \sin (d x)+6435 i \cos (d x))}{10920 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^8*Sec[c]*Sec[c + d*x]^15*((6435*I)*Cos[d*x] + (6435*I)*Cos[2*c + d*x] + (5005*I)*Cos[2*c + 3*d*x] + (5005*I
)*Cos[4*c + 3*d*x] + (3003*I)*Cos[4*c + 5*d*x] + (3003*I)*Cos[6*c + 5*d*x] + (1365*I)*Cos[6*c + 7*d*x] + (1365
*I)*Cos[8*c + 7*d*x] + 6435*Sin[d*x] - 6435*Sin[2*c + d*x] + 5005*Sin[2*c + 3*d*x] - 5005*Sin[4*c + 3*d*x] + 3
003*Sin[4*c + 5*d*x] - 3003*Sin[6*c + 5*d*x] + 1365*Sin[6*c + 7*d*x] - 1365*Sin[8*c + 7*d*x] + 910*Sin[8*c + 9
*d*x] + 210*Sin[10*c + 11*d*x] + 30*Sin[12*c + 13*d*x] + 2*Sin[14*c + 15*d*x]))/(10920*d)

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fricas [B]  time = 0.66, size = 345, normalized size = 3.17 \[ \frac {11182080 i \, a^{8} e^{\left (22 i \, d x + 22 i \, c\right )} + 24600576 i \, a^{8} e^{\left (20 i \, d x + 20 i \, c\right )} + 41000960 i \, a^{8} e^{\left (18 i \, d x + 18 i \, c\right )} + 52715520 i \, a^{8} e^{\left (16 i \, d x + 16 i \, c\right )} + 52715520 i \, a^{8} e^{\left (14 i \, d x + 14 i \, c\right )} + 41000960 i \, a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} + 24600576 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} + 11182080 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} + 3727360 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} + 860160 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} + 122880 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} + 8192 i \, a^{8}}{1365 \, {\left (d e^{\left (30 i \, d x + 30 i \, c\right )} + 15 \, d e^{\left (28 i \, d x + 28 i \, c\right )} + 105 \, d e^{\left (26 i \, d x + 26 i \, c\right )} + 455 \, d e^{\left (24 i \, d x + 24 i \, c\right )} + 1365 \, d e^{\left (22 i \, d x + 22 i \, c\right )} + 3003 \, d e^{\left (20 i \, d x + 20 i \, c\right )} + 5005 \, d e^{\left (18 i \, d x + 18 i \, c\right )} + 6435 \, d e^{\left (16 i \, d x + 16 i \, c\right )} + 6435 \, d e^{\left (14 i \, d x + 14 i \, c\right )} + 5005 \, d e^{\left (12 i \, d x + 12 i \, c\right )} + 3003 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 1365 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 455 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 105 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 15 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1365*(11182080*I*a^8*e^(22*I*d*x + 22*I*c) + 24600576*I*a^8*e^(20*I*d*x + 20*I*c) + 41000960*I*a^8*e^(18*I*d
*x + 18*I*c) + 52715520*I*a^8*e^(16*I*d*x + 16*I*c) + 52715520*I*a^8*e^(14*I*d*x + 14*I*c) + 41000960*I*a^8*e^
(12*I*d*x + 12*I*c) + 24600576*I*a^8*e^(10*I*d*x + 10*I*c) + 11182080*I*a^8*e^(8*I*d*x + 8*I*c) + 3727360*I*a^
8*e^(6*I*d*x + 6*I*c) + 860160*I*a^8*e^(4*I*d*x + 4*I*c) + 122880*I*a^8*e^(2*I*d*x + 2*I*c) + 8192*I*a^8)/(d*e
^(30*I*d*x + 30*I*c) + 15*d*e^(28*I*d*x + 28*I*c) + 105*d*e^(26*I*d*x + 26*I*c) + 455*d*e^(24*I*d*x + 24*I*c)
+ 1365*d*e^(22*I*d*x + 22*I*c) + 3003*d*e^(20*I*d*x + 20*I*c) + 5005*d*e^(18*I*d*x + 18*I*c) + 6435*d*e^(16*I*
d*x + 16*I*c) + 6435*d*e^(14*I*d*x + 14*I*c) + 5005*d*e^(12*I*d*x + 12*I*c) + 3003*d*e^(10*I*d*x + 10*I*c) + 1
365*d*e^(8*I*d*x + 8*I*c) + 455*d*e^(6*I*d*x + 6*I*c) + 105*d*e^(4*I*d*x + 4*I*c) + 15*d*e^(2*I*d*x + 2*I*c) +
d)

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giac [B]  time = 5.28, size = 186, normalized size = 1.71 \[ \frac {91 \, a^{8} \tan \left (d x + c\right )^{15} - 780 i \, a^{8} \tan \left (d x + c\right )^{14} - 2625 \, a^{8} \tan \left (d x + c\right )^{13} + 3640 i \, a^{8} \tan \left (d x + c\right )^{12} - 1365 \, a^{8} \tan \left (d x + c\right )^{11} + 12012 i \, a^{8} \tan \left (d x + c\right )^{10} + 15015 \, a^{8} \tan \left (d x + c\right )^{9} + 19305 \, a^{8} \tan \left (d x + c\right )^{7} - 20020 i \, a^{8} \tan \left (d x + c\right )^{6} - 3003 \, a^{8} \tan \left (d x + c\right )^{5} - 10920 i \, a^{8} \tan \left (d x + c\right )^{4} - 11375 \, a^{8} \tan \left (d x + c\right )^{3} + 5460 i \, a^{8} \tan \left (d x + c\right )^{2} + 1365 \, a^{8} \tan \left (d x + c\right )}{1365 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1365*(91*a^8*tan(d*x + c)^15 - 780*I*a^8*tan(d*x + c)^14 - 2625*a^8*tan(d*x + c)^13 + 3640*I*a^8*tan(d*x + c
)^12 - 1365*a^8*tan(d*x + c)^11 + 12012*I*a^8*tan(d*x + c)^10 + 15015*a^8*tan(d*x + c)^9 + 19305*a^8*tan(d*x +
c)^7 - 20020*I*a^8*tan(d*x + c)^6 - 3003*a^8*tan(d*x + c)^5 - 10920*I*a^8*tan(d*x + c)^4 - 11375*a^8*tan(d*x
+ c)^3 + 5460*I*a^8*tan(d*x + c)^2 + 1365*a^8*tan(d*x + c))/d

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maple [B]  time = 0.55, size = 611, normalized size = 5.61 \[ \frac {a^{8} \left (\frac {\sin ^{9}\left (d x +c \right )}{15 \cos \left (d x +c \right )^{15}}+\frac {2 \left (\sin ^{9}\left (d x +c \right )\right )}{65 \cos \left (d x +c \right )^{13}}+\frac {8 \left (\sin ^{9}\left (d x +c \right )\right )}{715 \cos \left (d x +c \right )^{11}}+\frac {16 \left (\sin ^{9}\left (d x +c \right )\right )}{6435 \cos \left (d x +c \right )^{9}}\right )+\frac {i a^{8}}{\cos \left (d x +c \right )^{8}}-28 a^{8} \left (\frac {\sin ^{7}\left (d x +c \right )}{13 \cos \left (d x +c \right )^{13}}+\frac {6 \left (\sin ^{7}\left (d x +c \right )\right )}{143 \cos \left (d x +c \right )^{11}}+\frac {8 \left (\sin ^{7}\left (d x +c \right )\right )}{429 \cos \left (d x +c \right )^{9}}+\frac {16 \left (\sin ^{7}\left (d x +c \right )\right )}{3003 \cos \left (d x +c \right )^{7}}\right )-8 i a^{8} \left (\frac {\sin ^{8}\left (d x +c \right )}{14 \cos \left (d x +c \right )^{14}}+\frac {\sin ^{8}\left (d x +c \right )}{28 \cos \left (d x +c \right )^{12}}+\frac {\sin ^{8}\left (d x +c \right )}{70 \cos \left (d x +c \right )^{10}}+\frac {\sin ^{8}\left (d x +c \right )}{280 \cos \left (d x +c \right )^{8}}\right )+70 a^{8} \left (\frac {\sin ^{5}\left (d x +c \right )}{11 \cos \left (d x +c \right )^{11}}+\frac {2 \left (\sin ^{5}\left (d x +c \right )\right )}{33 \cos \left (d x +c \right )^{9}}+\frac {8 \left (\sin ^{5}\left (d x +c \right )\right )}{231 \cos \left (d x +c \right )^{7}}+\frac {16 \left (\sin ^{5}\left (d x +c \right )\right )}{1155 \cos \left (d x +c \right )^{5}}\right )+56 i a^{8} \left (\frac {\sin ^{6}\left (d x +c \right )}{12 \cos \left (d x +c \right )^{12}}+\frac {\sin ^{6}\left (d x +c \right )}{20 \cos \left (d x +c \right )^{10}}+\frac {\sin ^{6}\left (d x +c \right )}{40 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{6}\left (d x +c \right )}{120 \cos \left (d x +c \right )^{6}}\right )-28 a^{8} \left (\frac {\sin ^{3}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{21 \cos \left (d x +c \right )^{7}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{5}}+\frac {16 \left (\sin ^{3}\left (d x +c \right )\right )}{315 \cos \left (d x +c \right )^{3}}\right )-56 i a^{8} \left (\frac {\sin ^{4}\left (d x +c \right )}{10 \cos \left (d x +c \right )^{10}}+\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{40 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{4}\left (d x +c \right )}{20 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{4}\left (d x +c \right )}{40 \cos \left (d x +c \right )^{4}}\right )-a^{8} \left (-\frac {16}{35}-\frac {\left (\sec ^{6}\left (d x +c \right )\right )}{7}-\frac {6 \left (\sec ^{4}\left (d x +c \right )\right )}{35}-\frac {8 \left (\sec ^{2}\left (d x +c \right )\right )}{35}\right ) \tan \left (d x +c \right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(a^8*(1/15*sin(d*x+c)^9/cos(d*x+c)^15+2/65*sin(d*x+c)^9/cos(d*x+c)^13+8/715*sin(d*x+c)^9/cos(d*x+c)^11+16/
6435*sin(d*x+c)^9/cos(d*x+c)^9)+I*a^8/cos(d*x+c)^8-28*a^8*(1/13*sin(d*x+c)^7/cos(d*x+c)^13+6/143*sin(d*x+c)^7/
cos(d*x+c)^11+8/429*sin(d*x+c)^7/cos(d*x+c)^9+16/3003*sin(d*x+c)^7/cos(d*x+c)^7)-8*I*a^8*(1/14*sin(d*x+c)^8/co
s(d*x+c)^14+1/28*sin(d*x+c)^8/cos(d*x+c)^12+1/70*sin(d*x+c)^8/cos(d*x+c)^10+1/280*sin(d*x+c)^8/cos(d*x+c)^8)+7
0*a^8*(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)+56*I*a^8*(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+1/120*sin(d*x+c)^6/cos(d*x+c)^6)-28*a^8*(1/9*sin(d*x+c)^3/cos(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)-56*I*a^8*(1/10*sin(d*x+c)^4/co
s(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)-a^8*
(-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 [B]  time = 0.32, size = 186, normalized size = 1.71 \[ \frac {3003 \, a^{8} \tan \left (d x + c\right )^{15} - 25740 i \, a^{8} \tan \left (d x + c\right )^{14} - 86625 \, a^{8} \tan \left (d x + c\right )^{13} + 120120 i \, a^{8} \tan \left (d x + c\right )^{12} - 45045 \, a^{8} \tan \left (d x + c\right )^{11} + 396396 i \, a^{8} \tan \left (d x + c\right )^{10} + 495495 \, a^{8} \tan \left (d x + c\right )^{9} + 637065 \, a^{8} \tan \left (d x + c\right )^{7} - 660660 i \, a^{8} \tan \left (d x + c\right )^{6} - 99099 \, a^{8} \tan \left (d x + c\right )^{5} - 360360 i \, a^{8} \tan \left (d x + c\right )^{4} - 375375 \, a^{8} \tan \left (d x + c\right )^{3} + 180180 i \, a^{8} \tan \left (d x + c\right )^{2} + 45045 \, a^{8} \tan \left (d x + c\right )}{45045 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/45045*(3003*a^8*tan(d*x + c)^15 - 25740*I*a^8*tan(d*x + c)^14 - 86625*a^8*tan(d*x + c)^13 + 120120*I*a^8*tan
(d*x + c)^12 - 45045*a^8*tan(d*x + c)^11 + 396396*I*a^8*tan(d*x + c)^10 + 495495*a^8*tan(d*x + c)^9 + 637065*a
^8*tan(d*x + c)^7 - 660660*I*a^8*tan(d*x + c)^6 - 99099*a^8*tan(d*x + c)^5 - 360360*I*a^8*tan(d*x + c)^4 - 375
375*a^8*tan(d*x + c)^3 + 180180*I*a^8*tan(d*x + c)^2 + 45045*a^8*tan(d*x + c))/d

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mupad [B]  time = 4.83, size = 153, normalized size = 1.40 \[ \frac {a^8\,\left (\frac {\sin \left (9\,c+9\,d\,x\right )}{12}+\frac {\sin \left (11\,c+11\,d\,x\right )}{52}+\frac {\sin \left (13\,c+13\,d\,x\right )}{364}+\frac {\sin \left (15\,c+15\,d\,x\right )}{5460}+\frac {\cos \left (c+d\,x\right )\,297{}\mathrm {i}}{7168}+\frac {\cos \left (3\,c+3\,d\,x\right )\,33{}\mathrm {i}}{1024}+\frac {\cos \left (5\,c+5\,d\,x\right )\,99{}\mathrm {i}}{5120}+\frac {\cos \left (7\,c+7\,d\,x\right )\,9{}\mathrm {i}}{1024}-\frac {\cos \left (9\,c+9\,d\,x\right )\,247{}\mathrm {i}}{3072}-\frac {\cos \left (11\,c+11\,d\,x\right )\,19{}\mathrm {i}}{1024}-\frac {\cos \left (13\,c+13\,d\,x\right )\,19{}\mathrm {i}}{7168}-\frac {\cos \left (15\,c+15\,d\,x\right )\,19{}\mathrm {i}}{107520}\right )}{d\,{\cos \left (c+d\,x\right )}^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a^8*((cos(c + d*x)*297i)/7168 + (cos(3*c + 3*d*x)*33i)/1024 + (cos(5*c + 5*d*x)*99i)/5120 + (cos(7*c + 7*d*x)
*9i)/1024 - (cos(9*c + 9*d*x)*247i)/3072 - (cos(11*c + 11*d*x)*19i)/1024 - (cos(13*c + 13*d*x)*19i)/7168 - (co
s(15*c + 15*d*x)*19i)/107520 + sin(9*c + 9*d*x)/12 + sin(11*c + 11*d*x)/52 + sin(13*c + 13*d*x)/364 + sin(15*c
+ 15*d*x)/5460))/(d*cos(c + d*x)^15)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**8*(a+I*a*tan(d*x+c))**8,x)

[Out]

Timed out

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