\(\int \sec ^8(c+d x) (a+i a \tan (c+d x))^8 \, dx\) [77]

Optimal result

Integrand size = 24, antiderivative size = 109 \[ \int \sec ^8(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {2 i (a+i a \tan (c+d x))^{12}}{3 a^4 d}+\frac {12 i (a+i a \tan (c+d x))^{13}}{13 a^5 d}-\frac {3 i (a+i a \tan (c+d x))^{14}}{7 a^6 d}+\frac {i (a+i a \tan (c+d x))^{15}}{15 a^7 d} \]

[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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3568, 45} \[ \int \sec ^8(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {i (a+i a \tan (c+d x))^{15}}{15 a^7 d}-\frac {3 i (a+i a \tan (c+d x))^{14}}{7 a^6 d}+\frac {12 i (a+i a \tan (c+d x))^{13}}{13 a^5 d}-\frac {2 i (a+i a \tan (c+d x))^{12}}{3 a^4 d} \]

[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 45

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 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)^3 (a+x)^{11} \, dx,x,i a \tan (c+d x)\right )}{a^7 d} \\ & = -\frac {i \text {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*}

Mathematica [A] (verified)

Time = 1.93 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.51 \[ \int \sec ^8(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {a^8 (-i+\tan (c+d x))^{12} \left (-144 i-363 \tan (c+d x)+312 i \tan ^2(c+d x)+91 \tan ^3(c+d x)\right )}{1365 d} \]

[In]

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

[Out]

(a^8*(-I + Tan[c + d*x])^12*(-144*I - 363*Tan[c + d*x] + (312*I)*Tan[c + d*x]^2 + 91*Tan[c + d*x]^3))/(1365*d)

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 610 vs. \(2 (93 ) = 186\).

Time = 0.59 (sec) , antiderivative size = 611, normalized size of antiderivative = 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 )+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 ^{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 )+\frac {i a^{8}}{\cos \left (d x +c \right )^{8}}-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}\]

[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)+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*(1/13*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/cos(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)+70*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)+I*a^8/cos(d*x+c)^8-28*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))

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. 345 vs. \(2 (85) = 170\).

Time = 0.24 (sec) , antiderivative size = 345, normalized size of antiderivative = 3.17 \[ \int \sec ^8(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {8192 \, {\left (-1365 i \, a^{8} e^{\left (22 i \, d x + 22 i \, c\right )} - 3003 i \, a^{8} e^{\left (20 i \, d x + 20 i \, c\right )} - 5005 i \, a^{8} e^{\left (18 i \, d x + 18 i \, c\right )} - 6435 i \, a^{8} e^{\left (16 i \, d x + 16 i \, c\right )} - 6435 i \, a^{8} e^{\left (14 i \, d x + 14 i \, c\right )} - 5005 i \, a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} - 3003 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} - 1365 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} - 455 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} - 105 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} - 15 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a^{8}\right )}}{1365 \, {\left (d e^{\left (30 i \, d x + 30 i \, c\right )} + 15 \, d e^{\left (28 i \, d x + 28 i \, c\right )} + 105 \, d e^{\left (26 i \, d x + 26 i \, c\right )} + 455 \, d e^{\left (24 i \, d x + 24 i \, c\right )} + 1365 \, d e^{\left (22 i \, d x + 22 i \, c\right )} + 3003 \, d e^{\left (20 i \, d x + 20 i \, c\right )} + 5005 \, d e^{\left (18 i \, d x + 18 i \, c\right )} + 6435 \, d e^{\left (16 i \, d x + 16 i \, c\right )} + 6435 \, d e^{\left (14 i \, d x + 14 i \, c\right )} + 5005 \, d e^{\left (12 i \, d x + 12 i \, c\right )} + 3003 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 1365 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 455 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 105 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 15 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]

[In]

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

[Out]

-8192/1365*(-1365*I*a^8*e^(22*I*d*x + 22*I*c) - 3003*I*a^8*e^(20*I*d*x + 20*I*c) - 5005*I*a^8*e^(18*I*d*x + 18
*I*c) - 6435*I*a^8*e^(16*I*d*x + 16*I*c) - 6435*I*a^8*e^(14*I*d*x + 14*I*c) - 5005*I*a^8*e^(12*I*d*x + 12*I*c)
- 3003*I*a^8*e^(10*I*d*x + 10*I*c) - 1365*I*a^8*e^(8*I*d*x + 8*I*c) - 455*I*a^8*e^(6*I*d*x + 6*I*c) - 105*I*a
^8*e^(4*I*d*x + 4*I*c) - 15*I*a^8*e^(2*I*d*x + 2*I*c) - I*a^8)/(d*e^(30*I*d*x + 30*I*c) + 15*d*e^(28*I*d*x + 2
8*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) + 1365*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)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 186 vs. \(2 (85) = 170\).

Time = 0.28 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.71 \[ \int \sec ^8(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {91 \, a^{8} \tan \left (d x + c\right )^{15} - 780 i \, a^{8} \tan \left (d x + c\right )^{14} - 2625 \, a^{8} \tan \left (d x + c\right )^{13} + 3640 i \, a^{8} \tan \left (d x + c\right )^{12} - 1365 \, a^{8} \tan \left (d x + c\right )^{11} + 12012 i \, a^{8} \tan \left (d x + c\right )^{10} + 15015 \, a^{8} \tan \left (d x + c\right )^{9} + 19305 \, a^{8} \tan \left (d x + c\right )^{7} - 20020 i \, a^{8} \tan \left (d x + c\right )^{6} - 3003 \, a^{8} \tan \left (d x + c\right )^{5} - 10920 i \, a^{8} \tan \left (d x + c\right )^{4} - 11375 \, a^{8} \tan \left (d x + c\right )^{3} + 5460 i \, a^{8} \tan \left (d x + c\right )^{2} + 1365 \, a^{8} \tan \left (d x + c\right )}{1365 \, d} \]

[In]

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

[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

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. 186 vs. \(2 (85) = 170\).

Time = 1.20 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.71 \[ \int \sec ^8(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {91 \, a^{8} \tan \left (d x + c\right )^{15} - 780 i \, a^{8} \tan \left (d x + c\right )^{14} - 2625 \, a^{8} \tan \left (d x + c\right )^{13} + 3640 i \, a^{8} \tan \left (d x + c\right )^{12} - 1365 \, a^{8} \tan \left (d x + c\right )^{11} + 12012 i \, a^{8} \tan \left (d x + c\right )^{10} + 15015 \, a^{8} \tan \left (d x + c\right )^{9} + 19305 \, a^{8} \tan \left (d x + c\right )^{7} - 20020 i \, a^{8} \tan \left (d x + c\right )^{6} - 3003 \, a^{8} \tan \left (d x + c\right )^{5} - 10920 i \, a^{8} \tan \left (d x + c\right )^{4} - 11375 \, a^{8} \tan \left (d x + c\right )^{3} + 5460 i \, a^{8} \tan \left (d x + c\right )^{2} + 1365 \, a^{8} \tan \left (d x + c\right )}{1365 \, d} \]

[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

Mupad [B] (verification not implemented)

Time = 5.60 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.40 \[ \int \sec ^8(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {a^8\,\left (\frac {\sin \left (9\,c+9\,d\,x\right )}{12}+\frac {\sin \left (11\,c+11\,d\,x\right )}{52}+\frac {\sin \left (13\,c+13\,d\,x\right )}{364}+\frac {\sin \left (15\,c+15\,d\,x\right )}{5460}+\frac {\cos \left (c+d\,x\right )\,297{}\mathrm {i}}{7168}+\frac {\cos \left (3\,c+3\,d\,x\right )\,33{}\mathrm {i}}{1024}+\frac {\cos \left (5\,c+5\,d\,x\right )\,99{}\mathrm {i}}{5120}+\frac {\cos \left (7\,c+7\,d\,x\right )\,9{}\mathrm {i}}{1024}-\frac {\cos \left (9\,c+9\,d\,x\right )\,247{}\mathrm {i}}{3072}-\frac {\cos \left (11\,c+11\,d\,x\right )\,19{}\mathrm {i}}{1024}-\frac {\cos \left (13\,c+13\,d\,x\right )\,19{}\mathrm {i}}{7168}-\frac {\cos \left (15\,c+15\,d\,x\right )\,19{}\mathrm {i}}{107520}\right )}{d\,{\cos \left (c+d\,x\right )}^{15}} \]

[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)