3.471 \(\int \frac{\sec ^2(c+d x)}{(a+b \sin (c+d x))^8} \, dx\)

Optimal. Leaf size=529 \[ -\frac{9 a b^2 \left (336 a^4 b^2+280 a^2 b^4+64 a^6+35 b^6\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{8 d \left (a^2-b^2\right )^{17/2}}+\frac{b \left (41484 a^4 b^2+22767 a^2 b^4+9800 a^6+1024 b^6\right ) \sec (c+d x)}{560 d \left (a^2-b^2\right )^7 (a+b \sin (c+d x))}+\frac{11 a b \left (844 a^2 b^2+280 a^4+241 b^4\right ) \sec (c+d x)}{560 d \left (a^2-b^2\right )^6 (a+b \sin (c+d x))^2}+\frac{b \left (1317 a^2 b^2+700 a^4+128 b^4\right ) \sec (c+d x)}{280 d \left (a^2-b^2\right )^5 (a+b \sin (c+d x))^3}+\frac{13 a b \left (28 a^2+27 b^2\right ) \sec (c+d x)}{280 d \left (a^2-b^2\right )^4 (a+b \sin (c+d x))^4}+\frac{b \left (49 a^2+16 b^2\right ) \sec (c+d x)}{70 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))^5}+\frac{5 a b \sec (c+d x)}{14 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^6}+\frac{b \sec (c+d x)}{7 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^7}-\frac{\sec (c+d x) \left (315 a b \left (336 a^4 b^2+280 a^2 b^4+64 a^6+35 b^6\right )-\left (42472 a^6 b^2+125634 a^4 b^4+54511 a^2 b^6+560 a^8+2048 b^8\right ) \sin (c+d x)\right )}{560 d \left (a^2-b^2\right )^8} \]

[Out]

(-9*a*b^2*(64*a^6 + 336*a^4*b^2 + 280*a^2*b^4 + 35*b^6)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(8*(
a^2 - b^2)^(17/2)*d) + (b*Sec[c + d*x])/(7*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^7) + (5*a*b*Sec[c + d*x])/(14*(a
^2 - b^2)^2*d*(a + b*Sin[c + d*x])^6) + (b*(49*a^2 + 16*b^2)*Sec[c + d*x])/(70*(a^2 - b^2)^3*d*(a + b*Sin[c +
d*x])^5) + (13*a*b*(28*a^2 + 27*b^2)*Sec[c + d*x])/(280*(a^2 - b^2)^4*d*(a + b*Sin[c + d*x])^4) + (b*(700*a^4
+ 1317*a^2*b^2 + 128*b^4)*Sec[c + d*x])/(280*(a^2 - b^2)^5*d*(a + b*Sin[c + d*x])^3) + (11*a*b*(280*a^4 + 844*
a^2*b^2 + 241*b^4)*Sec[c + d*x])/(560*(a^2 - b^2)^6*d*(a + b*Sin[c + d*x])^2) + (b*(9800*a^6 + 41484*a^4*b^2 +
22767*a^2*b^4 + 1024*b^6)*Sec[c + d*x])/(560*(a^2 - b^2)^7*d*(a + b*Sin[c + d*x])) - (Sec[c + d*x]*(315*a*b*(
64*a^6 + 336*a^4*b^2 + 280*a^2*b^4 + 35*b^6) - (560*a^8 + 42472*a^6*b^2 + 125634*a^4*b^4 + 54511*a^2*b^6 + 204
8*b^8)*Sin[c + d*x]))/(560*(a^2 - b^2)^8*d)

________________________________________________________________________________________

Rubi [A]  time = 1.76526, antiderivative size = 529, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2694, 2864, 2866, 12, 2660, 618, 204} \[ -\frac{9 a b^2 \left (336 a^4 b^2+280 a^2 b^4+64 a^6+35 b^6\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{8 d \left (a^2-b^2\right )^{17/2}}+\frac{b \left (41484 a^4 b^2+22767 a^2 b^4+9800 a^6+1024 b^6\right ) \sec (c+d x)}{560 d \left (a^2-b^2\right )^7 (a+b \sin (c+d x))}+\frac{11 a b \left (844 a^2 b^2+280 a^4+241 b^4\right ) \sec (c+d x)}{560 d \left (a^2-b^2\right )^6 (a+b \sin (c+d x))^2}+\frac{b \left (1317 a^2 b^2+700 a^4+128 b^4\right ) \sec (c+d x)}{280 d \left (a^2-b^2\right )^5 (a+b \sin (c+d x))^3}+\frac{13 a b \left (28 a^2+27 b^2\right ) \sec (c+d x)}{280 d \left (a^2-b^2\right )^4 (a+b \sin (c+d x))^4}+\frac{b \left (49 a^2+16 b^2\right ) \sec (c+d x)}{70 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))^5}+\frac{5 a b \sec (c+d x)}{14 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^6}+\frac{b \sec (c+d x)}{7 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^7}-\frac{\sec (c+d x) \left (315 a b \left (336 a^4 b^2+280 a^2 b^4+64 a^6+35 b^6\right )-\left (42472 a^6 b^2+125634 a^4 b^4+54511 a^2 b^6+560 a^8+2048 b^8\right ) \sin (c+d x)\right )}{560 d \left (a^2-b^2\right )^8} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^2/(a + b*Sin[c + d*x])^8,x]

[Out]

(-9*a*b^2*(64*a^6 + 336*a^4*b^2 + 280*a^2*b^4 + 35*b^6)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(8*(
a^2 - b^2)^(17/2)*d) + (b*Sec[c + d*x])/(7*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^7) + (5*a*b*Sec[c + d*x])/(14*(a
^2 - b^2)^2*d*(a + b*Sin[c + d*x])^6) + (b*(49*a^2 + 16*b^2)*Sec[c + d*x])/(70*(a^2 - b^2)^3*d*(a + b*Sin[c +
d*x])^5) + (13*a*b*(28*a^2 + 27*b^2)*Sec[c + d*x])/(280*(a^2 - b^2)^4*d*(a + b*Sin[c + d*x])^4) + (b*(700*a^4
+ 1317*a^2*b^2 + 128*b^4)*Sec[c + d*x])/(280*(a^2 - b^2)^5*d*(a + b*Sin[c + d*x])^3) + (11*a*b*(280*a^4 + 844*
a^2*b^2 + 241*b^4)*Sec[c + d*x])/(560*(a^2 - b^2)^6*d*(a + b*Sin[c + d*x])^2) + (b*(9800*a^6 + 41484*a^4*b^2 +
22767*a^2*b^4 + 1024*b^6)*Sec[c + d*x])/(560*(a^2 - b^2)^7*d*(a + b*Sin[c + d*x])) - (Sec[c + d*x]*(315*a*b*(
64*a^6 + 336*a^4*b^2 + 280*a^2*b^4 + 35*b^6) - (560*a^8 + 42472*a^6*b^2 + 125634*a^4*b^4 + 54511*a^2*b^6 + 204
8*b^8)*Sin[c + d*x]))/(560*(a^2 - b^2)^8*d)

Rule 2694

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[(b*(g
*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1))/(f*g*(a^2 - b^2)*(m + 1)), x] + Dist[1/((a^2 - b^2)*(m +
1)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1)*(a*(m + 1) - b*(m + p + 2)*Sin[e + f*x]), x], x] /; F
reeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*p]

Rule 2864

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> -Simp[((b*c - a*d)*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1))/(f*g*(a
^2 - b^2)*(m + 1)), x] + Dist[1/((a^2 - b^2)*(m + 1)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1)*Sim
p[(a*c - b*d)*(m + 1) - (b*c - a*d)*(m + p + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x]
&& NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]

Rule 2866

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> Simp[((g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1)*(b*c - a*d - (a*c -
b*d)*Sin[e + f*x]))/(f*g*(a^2 - b^2)*(p + 1)), x] + Dist[1/(g^2*(a^2 - b^2)*(p + 1)), Int[(g*Cos[e + f*x])^(p
+ 2)*(a + b*Sin[e + f*x])^m*Simp[c*(a^2*(p + 2) - b^2*(m + p + 2)) + a*b*d*m + b*(a*c - b*d)*(m + p + 3)*Sin[e
+ f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && LtQ[p, -1] && IntegerQ[2*m]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2660

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[(2*e)/d, Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\sec ^2(c+d x)}{(a+b \sin (c+d x))^8} \, dx &=\frac{b \sec (c+d x)}{7 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^7}-\frac{\int \frac{\sec ^2(c+d x) (-7 a+8 b \sin (c+d x))}{(a+b \sin (c+d x))^7} \, dx}{7 \left (a^2-b^2\right )}\\ &=\frac{b \sec (c+d x)}{7 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^7}+\frac{5 a b \sec (c+d x)}{14 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^6}+\frac{\int \frac{\sec ^2(c+d x) \left (6 \left (7 a^2+8 b^2\right )-105 a b \sin (c+d x)\right )}{(a+b \sin (c+d x))^6} \, dx}{42 \left (a^2-b^2\right )^2}\\ &=\frac{b \sec (c+d x)}{7 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^7}+\frac{5 a b \sec (c+d x)}{14 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^6}+\frac{b \left (49 a^2+16 b^2\right ) \sec (c+d x)}{70 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^5}-\frac{\int \frac{\sec ^2(c+d x) \left (-15 a \left (14 a^2+51 b^2\right )+18 b \left (49 a^2+16 b^2\right ) \sin (c+d x)\right )}{(a+b \sin (c+d x))^5} \, dx}{210 \left (a^2-b^2\right )^3}\\ &=\frac{b \sec (c+d x)}{7 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^7}+\frac{5 a b \sec (c+d x)}{14 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^6}+\frac{b \left (49 a^2+16 b^2\right ) \sec (c+d x)}{70 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^5}+\frac{13 a b \left (28 a^2+27 b^2\right ) \sec (c+d x)}{280 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^4}+\frac{\int \frac{\sec ^2(c+d x) \left (12 \left (70 a^4+549 a^2 b^2+96 b^4\right )-195 a b \left (28 a^2+27 b^2\right ) \sin (c+d x)\right )}{(a+b \sin (c+d x))^4} \, dx}{840 \left (a^2-b^2\right )^4}\\ &=\frac{b \sec (c+d x)}{7 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^7}+\frac{5 a b \sec (c+d x)}{14 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^6}+\frac{b \left (49 a^2+16 b^2\right ) \sec (c+d x)}{70 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^5}+\frac{13 a b \left (28 a^2+27 b^2\right ) \sec (c+d x)}{280 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^4}+\frac{b \left (700 a^4+1317 a^2 b^2+128 b^4\right ) \sec (c+d x)}{280 \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))^3}-\frac{\int \frac{\sec ^2(c+d x) \left (-9 a \left (280 a^4+4016 a^2 b^2+2139 b^4\right )+36 b \left (700 a^4+1317 a^2 b^2+128 b^4\right ) \sin (c+d x)\right )}{(a+b \sin (c+d x))^3} \, dx}{2520 \left (a^2-b^2\right )^5}\\ &=\frac{b \sec (c+d x)}{7 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^7}+\frac{5 a b \sec (c+d x)}{14 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^6}+\frac{b \left (49 a^2+16 b^2\right ) \sec (c+d x)}{70 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^5}+\frac{13 a b \left (28 a^2+27 b^2\right ) \sec (c+d x)}{280 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^4}+\frac{b \left (700 a^4+1317 a^2 b^2+128 b^4\right ) \sec (c+d x)}{280 \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))^3}+\frac{11 a b \left (280 a^4+844 a^2 b^2+241 b^4\right ) \sec (c+d x)}{560 \left (a^2-b^2\right )^6 d (a+b \sin (c+d x))^2}+\frac{\int \frac{\sec ^2(c+d x) \left (18 \left (280 a^6+6816 a^4 b^2+7407 a^2 b^4+512 b^6\right )-297 a b \left (280 a^4+844 a^2 b^2+241 b^4\right ) \sin (c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx}{5040 \left (a^2-b^2\right )^6}\\ &=\frac{b \sec (c+d x)}{7 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^7}+\frac{5 a b \sec (c+d x)}{14 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^6}+\frac{b \left (49 a^2+16 b^2\right ) \sec (c+d x)}{70 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^5}+\frac{13 a b \left (28 a^2+27 b^2\right ) \sec (c+d x)}{280 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^4}+\frac{b \left (700 a^4+1317 a^2 b^2+128 b^4\right ) \sec (c+d x)}{280 \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))^3}+\frac{11 a b \left (280 a^4+844 a^2 b^2+241 b^4\right ) \sec (c+d x)}{560 \left (a^2-b^2\right )^6 d (a+b \sin (c+d x))^2}+\frac{b \left (9800 a^6+41484 a^4 b^2+22767 a^2 b^4+1024 b^6\right ) \sec (c+d x)}{560 \left (a^2-b^2\right )^7 d (a+b \sin (c+d x))}-\frac{\int \frac{\sec ^2(c+d x) \left (-9 a \left (560 a^6+22872 a^4 b^2+42666 a^2 b^4+8977 b^6\right )+18 b \left (9800 a^6+41484 a^4 b^2+22767 a^2 b^4+1024 b^6\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{5040 \left (a^2-b^2\right )^7}\\ &=\frac{b \sec (c+d x)}{7 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^7}+\frac{5 a b \sec (c+d x)}{14 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^6}+\frac{b \left (49 a^2+16 b^2\right ) \sec (c+d x)}{70 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^5}+\frac{13 a b \left (28 a^2+27 b^2\right ) \sec (c+d x)}{280 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^4}+\frac{b \left (700 a^4+1317 a^2 b^2+128 b^4\right ) \sec (c+d x)}{280 \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))^3}+\frac{11 a b \left (280 a^4+844 a^2 b^2+241 b^4\right ) \sec (c+d x)}{560 \left (a^2-b^2\right )^6 d (a+b \sin (c+d x))^2}+\frac{b \left (9800 a^6+41484 a^4 b^2+22767 a^2 b^4+1024 b^6\right ) \sec (c+d x)}{560 \left (a^2-b^2\right )^7 d (a+b \sin (c+d x))}-\frac{\sec (c+d x) \left (315 a b \left (64 a^6+336 a^4 b^2+280 a^2 b^4+35 b^6\right )-\left (560 a^8+42472 a^6 b^2+125634 a^4 b^4+54511 a^2 b^6+2048 b^8\right ) \sin (c+d x)\right )}{560 \left (a^2-b^2\right )^8 d}+\frac{\int -\frac{2835 a b^2 \left (64 a^6+336 a^4 b^2+280 a^2 b^4+35 b^6\right )}{a+b \sin (c+d x)} \, dx}{5040 \left (a^2-b^2\right )^8}\\ &=\frac{b \sec (c+d x)}{7 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^7}+\frac{5 a b \sec (c+d x)}{14 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^6}+\frac{b \left (49 a^2+16 b^2\right ) \sec (c+d x)}{70 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^5}+\frac{13 a b \left (28 a^2+27 b^2\right ) \sec (c+d x)}{280 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^4}+\frac{b \left (700 a^4+1317 a^2 b^2+128 b^4\right ) \sec (c+d x)}{280 \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))^3}+\frac{11 a b \left (280 a^4+844 a^2 b^2+241 b^4\right ) \sec (c+d x)}{560 \left (a^2-b^2\right )^6 d (a+b \sin (c+d x))^2}+\frac{b \left (9800 a^6+41484 a^4 b^2+22767 a^2 b^4+1024 b^6\right ) \sec (c+d x)}{560 \left (a^2-b^2\right )^7 d (a+b \sin (c+d x))}-\frac{\sec (c+d x) \left (315 a b \left (64 a^6+336 a^4 b^2+280 a^2 b^4+35 b^6\right )-\left (560 a^8+42472 a^6 b^2+125634 a^4 b^4+54511 a^2 b^6+2048 b^8\right ) \sin (c+d x)\right )}{560 \left (a^2-b^2\right )^8 d}-\frac{\left (9 a b^2 \left (64 a^6+336 a^4 b^2+280 a^2 b^4+35 b^6\right )\right ) \int \frac{1}{a+b \sin (c+d x)} \, dx}{16 \left (a^2-b^2\right )^8}\\ &=\frac{b \sec (c+d x)}{7 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^7}+\frac{5 a b \sec (c+d x)}{14 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^6}+\frac{b \left (49 a^2+16 b^2\right ) \sec (c+d x)}{70 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^5}+\frac{13 a b \left (28 a^2+27 b^2\right ) \sec (c+d x)}{280 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^4}+\frac{b \left (700 a^4+1317 a^2 b^2+128 b^4\right ) \sec (c+d x)}{280 \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))^3}+\frac{11 a b \left (280 a^4+844 a^2 b^2+241 b^4\right ) \sec (c+d x)}{560 \left (a^2-b^2\right )^6 d (a+b \sin (c+d x))^2}+\frac{b \left (9800 a^6+41484 a^4 b^2+22767 a^2 b^4+1024 b^6\right ) \sec (c+d x)}{560 \left (a^2-b^2\right )^7 d (a+b \sin (c+d x))}-\frac{\sec (c+d x) \left (315 a b \left (64 a^6+336 a^4 b^2+280 a^2 b^4+35 b^6\right )-\left (560 a^8+42472 a^6 b^2+125634 a^4 b^4+54511 a^2 b^6+2048 b^8\right ) \sin (c+d x)\right )}{560 \left (a^2-b^2\right )^8 d}-\frac{\left (9 a b^2 \left (64 a^6+336 a^4 b^2+280 a^2 b^4+35 b^6\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{8 \left (a^2-b^2\right )^8 d}\\ &=\frac{b \sec (c+d x)}{7 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^7}+\frac{5 a b \sec (c+d x)}{14 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^6}+\frac{b \left (49 a^2+16 b^2\right ) \sec (c+d x)}{70 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^5}+\frac{13 a b \left (28 a^2+27 b^2\right ) \sec (c+d x)}{280 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^4}+\frac{b \left (700 a^4+1317 a^2 b^2+128 b^4\right ) \sec (c+d x)}{280 \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))^3}+\frac{11 a b \left (280 a^4+844 a^2 b^2+241 b^4\right ) \sec (c+d x)}{560 \left (a^2-b^2\right )^6 d (a+b \sin (c+d x))^2}+\frac{b \left (9800 a^6+41484 a^4 b^2+22767 a^2 b^4+1024 b^6\right ) \sec (c+d x)}{560 \left (a^2-b^2\right )^7 d (a+b \sin (c+d x))}-\frac{\sec (c+d x) \left (315 a b \left (64 a^6+336 a^4 b^2+280 a^2 b^4+35 b^6\right )-\left (560 a^8+42472 a^6 b^2+125634 a^4 b^4+54511 a^2 b^6+2048 b^8\right ) \sin (c+d x)\right )}{560 \left (a^2-b^2\right )^8 d}+\frac{\left (9 a b^2 \left (64 a^6+336 a^4 b^2+280 a^2 b^4+35 b^6\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac{1}{2} (c+d x)\right )\right )}{4 \left (a^2-b^2\right )^8 d}\\ &=-\frac{9 a b^2 \left (64 a^6+336 a^4 b^2+280 a^2 b^4+35 b^6\right ) \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{8 \left (a^2-b^2\right )^{17/2} d}+\frac{b \sec (c+d x)}{7 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^7}+\frac{5 a b \sec (c+d x)}{14 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^6}+\frac{b \left (49 a^2+16 b^2\right ) \sec (c+d x)}{70 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^5}+\frac{13 a b \left (28 a^2+27 b^2\right ) \sec (c+d x)}{280 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^4}+\frac{b \left (700 a^4+1317 a^2 b^2+128 b^4\right ) \sec (c+d x)}{280 \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))^3}+\frac{11 a b \left (280 a^4+844 a^2 b^2+241 b^4\right ) \sec (c+d x)}{560 \left (a^2-b^2\right )^6 d (a+b \sin (c+d x))^2}+\frac{b \left (9800 a^6+41484 a^4 b^2+22767 a^2 b^4+1024 b^6\right ) \sec (c+d x)}{560 \left (a^2-b^2\right )^7 d (a+b \sin (c+d x))}-\frac{\sec (c+d x) \left (315 a b \left (64 a^6+336 a^4 b^2+280 a^2 b^4+35 b^6\right )-\left (560 a^8+42472 a^6 b^2+125634 a^4 b^4+54511 a^2 b^6+2048 b^8\right ) \sin (c+d x)\right )}{560 \left (a^2-b^2\right )^8 d}\\ \end{align*}

Mathematica [A]  time = 4.96575, size = 494, normalized size = 0.93 \[ -\frac{\frac{630 a b^2 \left (336 a^4 b^2+280 a^2 b^4+64 a^6+35 b^6\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{17/2}}+\frac{b^3 \left (86434 a^4 b^2+38831 a^2 b^4+26792 a^6+1488 b^6\right ) \cos (c+d x)}{\left (a^2-b^2\right )^8 (a+b \sin (c+d x))}+\frac{a b^3 \left (23066 a^2 b^2+11112 a^4+5057 b^4\right ) \cos (c+d x)}{\left (a^2-b^2\right )^7 (a+b \sin (c+d x))^2}+\frac{2 b^3 \left (3207 a^2 b^2+2616 a^4+232 b^4\right ) \cos (c+d x)}{\left (a^2-b^2\right )^6 (a+b \sin (c+d x))^3}+\frac{2 a b^3 \left (1216 a^2+739 b^2\right ) \cos (c+d x)}{\left (a^2-b^2\right )^5 (a+b \sin (c+d x))^4}+\frac{8 b^3 \left (129 a^2+26 b^2\right ) \cos (c+d x)}{\left (a^2-b^2\right )^4 (a+b \sin (c+d x))^5}+\frac{360 a b^3 \cos (c+d x)}{\left (a^2-b^2\right )^3 (a+b \sin (c+d x))^6}+\frac{80 b^3 \cos (c+d x)}{\left (a^2-b^2\right )^2 (a+b \sin (c+d x))^7}-\frac{560 \sec (c+d x) \left (\left (28 a^6 b^2+70 a^4 b^4+28 a^2 b^6+a^8+b^8\right ) \sin (c+d x)-8 a b \left (7 a^4 b^2+7 a^2 b^4+a^6+b^6\right )\right )}{\left (a^2-b^2\right )^8}}{560 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^2/(a + b*Sin[c + d*x])^8,x]

[Out]

-((630*a*b^2*(64*a^6 + 336*a^4*b^2 + 280*a^2*b^4 + 35*b^6)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(
a^2 - b^2)^(17/2) + (80*b^3*Cos[c + d*x])/((a^2 - b^2)^2*(a + b*Sin[c + d*x])^7) + (360*a*b^3*Cos[c + d*x])/((
a^2 - b^2)^3*(a + b*Sin[c + d*x])^6) + (8*b^3*(129*a^2 + 26*b^2)*Cos[c + d*x])/((a^2 - b^2)^4*(a + b*Sin[c + d
*x])^5) + (2*a*b^3*(1216*a^2 + 739*b^2)*Cos[c + d*x])/((a^2 - b^2)^5*(a + b*Sin[c + d*x])^4) + (2*b^3*(2616*a^
4 + 3207*a^2*b^2 + 232*b^4)*Cos[c + d*x])/((a^2 - b^2)^6*(a + b*Sin[c + d*x])^3) + (a*b^3*(11112*a^4 + 23066*a
^2*b^2 + 5057*b^4)*Cos[c + d*x])/((a^2 - b^2)^7*(a + b*Sin[c + d*x])^2) + (b^3*(26792*a^6 + 86434*a^4*b^2 + 38
831*a^2*b^4 + 1488*b^6)*Cos[c + d*x])/((a^2 - b^2)^8*(a + b*Sin[c + d*x])) - (560*Sec[c + d*x]*(-8*a*b*(a^6 +
7*a^4*b^2 + 7*a^2*b^4 + b^6) + (a^8 + 28*a^6*b^2 + 70*a^4*b^4 + 28*a^2*b^6 + b^8)*Sin[c + d*x]))/(a^2 - b^2)^8
)/(560*d)

________________________________________________________________________________________

Maple [B]  time = 0.216, size = 7675, normalized size = 14.5 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^2/(a+b*sin(d*x+c))^8,x)

[Out]

result too large to display

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^2/(a+b*sin(d*x+c))^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 10.0989, size = 9913, normalized size = 18.74 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^2/(a+b*sin(d*x+c))^8,x, algorithm="fricas")

[Out]

[1/1120*(1120*a^16*b - 8960*a^14*b^3 + 31360*a^12*b^5 - 62720*a^10*b^7 + 78400*a^8*b^9 - 62720*a^6*b^11 + 3136
0*a^4*b^13 - 8960*a^2*b^15 + 1120*b^17 - 2*(560*a^10*b^7 + 41912*a^8*b^9 + 83162*a^6*b^11 - 71123*a^4*b^13 - 5
2463*a^2*b^15 - 2048*b^17)*cos(d*x + c)^8 + 28*(840*a^12*b^5 + 53648*a^10*b^7 + 95441*a^8*b^9 - 77704*a^6*b^11
- 60644*a^4*b^13 - 11069*a^2*b^15 - 512*b^17)*cos(d*x + c)^6 - 70*(560*a^14*b^3 + 27440*a^12*b^5 + 71064*a^10
*b^7 + 29927*a^8*b^9 - 81421*a^6*b^11 - 43131*a^4*b^13 - 4183*a^2*b^15 - 256*b^17)*cos(d*x + c)^4 + 140*(56*a^
16*b + 1400*a^14*b^3 + 13832*a^12*b^5 + 24080*a^10*b^7 - 4591*a^8*b^9 - 23443*a^6*b^11 - 10717*a^4*b^13 - 553*
a^2*b^15 - 64*b^17)*cos(d*x + c)^2 - 315*(7*(64*a^8*b^8 + 336*a^6*b^10 + 280*a^4*b^12 + 35*a^2*b^14)*cos(d*x +
c)^7 - 7*(320*a^10*b^6 + 1872*a^8*b^8 + 2408*a^6*b^10 + 1015*a^4*b^12 + 105*a^2*b^14)*cos(d*x + c)^5 + 7*(192
*a^12*b^4 + 1648*a^10*b^6 + 4392*a^8*b^8 + 3913*a^6*b^10 + 1190*a^4*b^12 + 105*a^2*b^14)*cos(d*x + c)^3 - (64*
a^14*b^2 + 1680*a^12*b^4 + 9576*a^10*b^6 + 18123*a^8*b^8 + 12887*a^6*b^10 + 3185*a^4*b^12 + 245*a^2*b^14)*cos(
d*x + c) + ((64*a^7*b^9 + 336*a^5*b^11 + 280*a^3*b^13 + 35*a*b^15)*cos(d*x + c)^7 - 3*(448*a^9*b^7 + 2416*a^7*
b^9 + 2296*a^5*b^11 + 525*a^3*b^13 + 35*a*b^15)*cos(d*x + c)^5 + (2240*a^11*b^5 + 14448*a^9*b^7 + 24104*a^7*b^
9 + 13993*a^5*b^11 + 2310*a^3*b^13 + 105*a*b^15)*cos(d*x + c)^3 - (448*a^13*b^3 + 4592*a^11*b^5 + 15064*a^9*b^
7 + 17165*a^7*b^9 + 7441*a^5*b^11 + 1015*a^3*b^13 + 35*a*b^15)*cos(d*x + c))*sin(d*x + c))*sqrt(-a^2 + b^2)*lo
g(-((2*a^2 - b^2)*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2 - 2*(a*cos(d*x + c)*sin(d*x + c) + b*cos(d*x
+ c))*sqrt(-a^2 + b^2))/(b^2*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2)) - 14*(80*a^17 - 640*a^15*b^2 +
2240*a^13*b^4 - 4480*a^11*b^6 + 5600*a^9*b^8 - 4480*a^7*b^10 + 2240*a^5*b^12 - 640*a^3*b^14 + 80*a*b^16 - (56
0*a^11*b^6 + 39032*a^9*b^8 + 70922*a^7*b^10 - 68603*a^5*b^12 - 41438*a^3*b^14 - 473*a*b^16)*cos(d*x + c)^6 + 1
0*(280*a^13*b^4 + 15960*a^11*b^6 + 29463*a^9*b^8 - 13541*a^7*b^10 - 23679*a^5*b^12 - 8391*a^3*b^14 - 92*a*b^16
)*cos(d*x + c)^4 - 15*(112*a^15*b^2 + 4256*a^13*b^4 + 13272*a^11*b^6 + 11977*a^9*b^8 - 15634*a^7*b^10 - 11088*
a^5*b^12 - 2870*a^3*b^14 - 25*a*b^16)*cos(d*x + c)^2)*sin(d*x + c))/(7*(a^19*b^6 - 9*a^17*b^8 + 36*a^15*b^10 -
84*a^13*b^12 + 126*a^11*b^14 - 126*a^9*b^16 + 84*a^7*b^18 - 36*a^5*b^20 + 9*a^3*b^22 - a*b^24)*d*cos(d*x + c)
^7 - 7*(5*a^21*b^4 - 42*a^19*b^6 + 153*a^17*b^8 - 312*a^15*b^10 + 378*a^13*b^12 - 252*a^11*b^14 + 42*a^9*b^16
+ 72*a^7*b^18 - 63*a^5*b^20 + 22*a^3*b^22 - 3*a*b^24)*d*cos(d*x + c)^5 + 7*(3*a^23*b^2 - 17*a^21*b^4 + 21*a^19
*b^6 + 81*a^17*b^8 - 354*a^15*b^10 + 630*a^13*b^12 - 630*a^11*b^14 + 354*a^9*b^16 - 81*a^7*b^18 - 21*a^5*b^20
+ 17*a^3*b^22 - 3*a*b^24)*d*cos(d*x + c)^3 - (a^25 + 12*a^23*b^2 - 118*a^21*b^4 + 364*a^19*b^6 - 441*a^17*b^8
- 168*a^15*b^10 + 1260*a^13*b^12 - 1800*a^11*b^14 + 1311*a^9*b^16 - 484*a^7*b^18 + 42*a^5*b^20 + 28*a^3*b^22 -
7*a*b^24)*d*cos(d*x + c) + ((a^18*b^7 - 9*a^16*b^9 + 36*a^14*b^11 - 84*a^12*b^13 + 126*a^10*b^15 - 126*a^8*b^
17 + 84*a^6*b^19 - 36*a^4*b^21 + 9*a^2*b^23 - b^25)*d*cos(d*x + c)^7 - 3*(7*a^20*b^5 - 62*a^18*b^7 + 243*a^16*
b^9 - 552*a^14*b^11 + 798*a^12*b^13 - 756*a^10*b^15 + 462*a^8*b^17 - 168*a^6*b^19 + 27*a^4*b^21 + 2*a^2*b^23 -
b^25)*d*cos(d*x + c)^5 + (35*a^22*b^3 - 273*a^20*b^5 + 885*a^18*b^7 - 1455*a^16*b^9 + 990*a^14*b^11 + 630*a^1
2*b^13 - 1974*a^10*b^15 + 1890*a^8*b^17 - 945*a^6*b^19 + 235*a^4*b^21 - 15*a^2*b^23 - 3*b^25)*d*cos(d*x + c)^3
- (7*a^24*b - 28*a^22*b^3 - 42*a^20*b^5 + 484*a^18*b^7 - 1311*a^16*b^9 + 1800*a^14*b^11 - 1260*a^12*b^13 + 16
8*a^10*b^15 + 441*a^8*b^17 - 364*a^6*b^19 + 118*a^4*b^21 - 12*a^2*b^23 - b^25)*d*cos(d*x + c))*sin(d*x + c)),
1/560*(560*a^16*b - 4480*a^14*b^3 + 15680*a^12*b^5 - 31360*a^10*b^7 + 39200*a^8*b^9 - 31360*a^6*b^11 + 15680*a
^4*b^13 - 4480*a^2*b^15 + 560*b^17 - (560*a^10*b^7 + 41912*a^8*b^9 + 83162*a^6*b^11 - 71123*a^4*b^13 - 52463*a
^2*b^15 - 2048*b^17)*cos(d*x + c)^8 + 14*(840*a^12*b^5 + 53648*a^10*b^7 + 95441*a^8*b^9 - 77704*a^6*b^11 - 606
44*a^4*b^13 - 11069*a^2*b^15 - 512*b^17)*cos(d*x + c)^6 - 35*(560*a^14*b^3 + 27440*a^12*b^5 + 71064*a^10*b^7 +
29927*a^8*b^9 - 81421*a^6*b^11 - 43131*a^4*b^13 - 4183*a^2*b^15 - 256*b^17)*cos(d*x + c)^4 + 70*(56*a^16*b +
1400*a^14*b^3 + 13832*a^12*b^5 + 24080*a^10*b^7 - 4591*a^8*b^9 - 23443*a^6*b^11 - 10717*a^4*b^13 - 553*a^2*b^1
5 - 64*b^17)*cos(d*x + c)^2 + 315*(7*(64*a^8*b^8 + 336*a^6*b^10 + 280*a^4*b^12 + 35*a^2*b^14)*cos(d*x + c)^7 -
7*(320*a^10*b^6 + 1872*a^8*b^8 + 2408*a^6*b^10 + 1015*a^4*b^12 + 105*a^2*b^14)*cos(d*x + c)^5 + 7*(192*a^12*b
^4 + 1648*a^10*b^6 + 4392*a^8*b^8 + 3913*a^6*b^10 + 1190*a^4*b^12 + 105*a^2*b^14)*cos(d*x + c)^3 - (64*a^14*b^
2 + 1680*a^12*b^4 + 9576*a^10*b^6 + 18123*a^8*b^8 + 12887*a^6*b^10 + 3185*a^4*b^12 + 245*a^2*b^14)*cos(d*x + c
) + ((64*a^7*b^9 + 336*a^5*b^11 + 280*a^3*b^13 + 35*a*b^15)*cos(d*x + c)^7 - 3*(448*a^9*b^7 + 2416*a^7*b^9 + 2
296*a^5*b^11 + 525*a^3*b^13 + 35*a*b^15)*cos(d*x + c)^5 + (2240*a^11*b^5 + 14448*a^9*b^7 + 24104*a^7*b^9 + 139
93*a^5*b^11 + 2310*a^3*b^13 + 105*a*b^15)*cos(d*x + c)^3 - (448*a^13*b^3 + 4592*a^11*b^5 + 15064*a^9*b^7 + 171
65*a^7*b^9 + 7441*a^5*b^11 + 1015*a^3*b^13 + 35*a*b^15)*cos(d*x + c))*sin(d*x + c))*sqrt(a^2 - b^2)*arctan(-(a
*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c))) - 7*(80*a^17 - 640*a^15*b^2 + 2240*a^13*b^4 - 4480*a^11*b^6
+ 5600*a^9*b^8 - 4480*a^7*b^10 + 2240*a^5*b^12 - 640*a^3*b^14 + 80*a*b^16 - (560*a^11*b^6 + 39032*a^9*b^8 + 7
0922*a^7*b^10 - 68603*a^5*b^12 - 41438*a^3*b^14 - 473*a*b^16)*cos(d*x + c)^6 + 10*(280*a^13*b^4 + 15960*a^11*b
^6 + 29463*a^9*b^8 - 13541*a^7*b^10 - 23679*a^5*b^12 - 8391*a^3*b^14 - 92*a*b^16)*cos(d*x + c)^4 - 15*(112*a^1
5*b^2 + 4256*a^13*b^4 + 13272*a^11*b^6 + 11977*a^9*b^8 - 15634*a^7*b^10 - 11088*a^5*b^12 - 2870*a^3*b^14 - 25*
a*b^16)*cos(d*x + c)^2)*sin(d*x + c))/(7*(a^19*b^6 - 9*a^17*b^8 + 36*a^15*b^10 - 84*a^13*b^12 + 126*a^11*b^14
- 126*a^9*b^16 + 84*a^7*b^18 - 36*a^5*b^20 + 9*a^3*b^22 - a*b^24)*d*cos(d*x + c)^7 - 7*(5*a^21*b^4 - 42*a^19*b
^6 + 153*a^17*b^8 - 312*a^15*b^10 + 378*a^13*b^12 - 252*a^11*b^14 + 42*a^9*b^16 + 72*a^7*b^18 - 63*a^5*b^20 +
22*a^3*b^22 - 3*a*b^24)*d*cos(d*x + c)^5 + 7*(3*a^23*b^2 - 17*a^21*b^4 + 21*a^19*b^6 + 81*a^17*b^8 - 354*a^15*
b^10 + 630*a^13*b^12 - 630*a^11*b^14 + 354*a^9*b^16 - 81*a^7*b^18 - 21*a^5*b^20 + 17*a^3*b^22 - 3*a*b^24)*d*co
s(d*x + c)^3 - (a^25 + 12*a^23*b^2 - 118*a^21*b^4 + 364*a^19*b^6 - 441*a^17*b^8 - 168*a^15*b^10 + 1260*a^13*b^
12 - 1800*a^11*b^14 + 1311*a^9*b^16 - 484*a^7*b^18 + 42*a^5*b^20 + 28*a^3*b^22 - 7*a*b^24)*d*cos(d*x + c) + ((
a^18*b^7 - 9*a^16*b^9 + 36*a^14*b^11 - 84*a^12*b^13 + 126*a^10*b^15 - 126*a^8*b^17 + 84*a^6*b^19 - 36*a^4*b^21
+ 9*a^2*b^23 - b^25)*d*cos(d*x + c)^7 - 3*(7*a^20*b^5 - 62*a^18*b^7 + 243*a^16*b^9 - 552*a^14*b^11 + 798*a^12
*b^13 - 756*a^10*b^15 + 462*a^8*b^17 - 168*a^6*b^19 + 27*a^4*b^21 + 2*a^2*b^23 - b^25)*d*cos(d*x + c)^5 + (35*
a^22*b^3 - 273*a^20*b^5 + 885*a^18*b^7 - 1455*a^16*b^9 + 990*a^14*b^11 + 630*a^12*b^13 - 1974*a^10*b^15 + 1890
*a^8*b^17 - 945*a^6*b^19 + 235*a^4*b^21 - 15*a^2*b^23 - 3*b^25)*d*cos(d*x + c)^3 - (7*a^24*b - 28*a^22*b^3 - 4
2*a^20*b^5 + 484*a^18*b^7 - 1311*a^16*b^9 + 1800*a^14*b^11 - 1260*a^12*b^13 + 168*a^10*b^15 + 441*a^8*b^17 - 3
64*a^6*b^19 + 118*a^4*b^21 - 12*a^2*b^23 - b^25)*d*cos(d*x + c))*sin(d*x + c))]

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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)**2/(a+b*sin(d*x+c))**8,x)

[Out]

Timed out

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Giac [B]  time = 1.85505, size = 3524, normalized size = 6.66 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^2/(a+b*sin(d*x+c))^8,x, algorithm="giac")

[Out]

-1/280*(315*(64*a^7*b^2 + 336*a^5*b^4 + 280*a^3*b^6 + 35*a*b^8)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(a) + arc
tan((a*tan(1/2*d*x + 1/2*c) + b)/sqrt(a^2 - b^2)))/((a^16 - 8*a^14*b^2 + 28*a^12*b^4 - 56*a^10*b^6 + 70*a^8*b^
8 - 56*a^6*b^10 + 28*a^4*b^12 - 8*a^2*b^14 + b^16)*sqrt(a^2 - b^2)) + 560*(a^8*tan(1/2*d*x + 1/2*c) + 28*a^6*b
^2*tan(1/2*d*x + 1/2*c) + 70*a^4*b^4*tan(1/2*d*x + 1/2*c) + 28*a^2*b^6*tan(1/2*d*x + 1/2*c) + b^8*tan(1/2*d*x
+ 1/2*c) - 8*a^7*b - 56*a^5*b^3 - 56*a^3*b^5 - 8*a*b^7)/((a^16 - 8*a^14*b^2 + 28*a^12*b^4 - 56*a^10*b^6 + 70*a
^8*b^8 - 56*a^6*b^10 + 28*a^4*b^12 - 8*a^2*b^14 + b^16)*(tan(1/2*d*x + 1/2*c)^2 - 1)) + (82320*a^18*b^4*tan(1/
2*d*x + 1/2*c)^13 + 41160*a^16*b^6*tan(1/2*d*x + 1/2*c)^13 + 49665*a^14*b^8*tan(1/2*d*x + 1/2*c)^13 - 31360*a^
12*b^10*tan(1/2*d*x + 1/2*c)^13 + 15680*a^10*b^12*tan(1/2*d*x + 1/2*c)^13 - 4480*a^8*b^14*tan(1/2*d*x + 1/2*c)
^13 + 560*a^6*b^16*tan(1/2*d*x + 1/2*c)^13 + 47040*a^19*b^3*tan(1/2*d*x + 1/2*c)^12 + 952560*a^17*b^5*tan(1/2*
d*x + 1/2*c)^12 + 743400*a^15*b^7*tan(1/2*d*x + 1/2*c)^12 + 370685*a^13*b^9*tan(1/2*d*x + 1/2*c)^12 - 188160*a
^11*b^11*tan(1/2*d*x + 1/2*c)^12 + 94080*a^9*b^13*tan(1/2*d*x + 1/2*c)^12 - 26880*a^7*b^15*tan(1/2*d*x + 1/2*c
)^12 + 3360*a^5*b^17*tan(1/2*d*x + 1/2*c)^12 + 987840*a^18*b^4*tan(1/2*d*x + 1/2*c)^11 + 5221440*a^16*b^6*tan(
1/2*d*x + 1/2*c)^11 + 4792620*a^14*b^8*tan(1/2*d*x + 1/2*c)^11 + 1272530*a^12*b^10*tan(1/2*d*x + 1/2*c)^11 - 5
01760*a^10*b^12*tan(1/2*d*x + 1/2*c)^11 + 277760*a^8*b^14*tan(1/2*d*x + 1/2*c)^11 - 85120*a^6*b^16*tan(1/2*d*x
+ 1/2*c)^11 + 11200*a^4*b^18*tan(1/2*d*x + 1/2*c)^11 + 282240*a^19*b^3*tan(1/2*d*x + 1/2*c)^10 + 7056000*a^17
*b^5*tan(1/2*d*x + 1/2*c)^10 + 18695040*a^15*b^7*tan(1/2*d*x + 1/2*c)^10 + 15575140*a^13*b^9*tan(1/2*d*x + 1/2
*c)^10 + 2689610*a^11*b^11*tan(1/2*d*x + 1/2*c)^10 - 721280*a^9*b^13*tan(1/2*d*x + 1/2*c)^10 + 474880*a^7*b^15
*tan(1/2*d*x + 1/2*c)^10 - 160160*a^5*b^17*tan(1/2*d*x + 1/2*c)^10 + 22400*a^3*b^19*tan(1/2*d*x + 1/2*c)^10 +
3704400*a^18*b^4*tan(1/2*d*x + 1/2*c)^9 + 26948040*a^16*b^6*tan(1/2*d*x + 1/2*c)^9 + 46663365*a^14*b^8*tan(1/2
*d*x + 1/2*c)^9 + 29114330*a^12*b^10*tan(1/2*d*x + 1/2*c)^9 + 3411772*a^10*b^12*tan(1/2*d*x + 1/2*c)^9 - 30553
6*a^8*b^14*tan(1/2*d*x + 1/2*c)^9 + 388976*a^6*b^16*tan(1/2*d*x + 1/2*c)^9 - 167552*a^4*b^18*tan(1/2*d*x + 1/2
*c)^9 + 26880*a^2*b^20*tan(1/2*d*x + 1/2*c)^9 + 705600*a^19*b^3*tan(1/2*d*x + 1/2*c)^8 + 18780720*a^17*b^5*tan
(1/2*d*x + 1/2*c)^8 + 65305800*a^15*b^7*tan(1/2*d*x + 1/2*c)^8 + 77673085*a^13*b^9*tan(1/2*d*x + 1/2*c)^8 + 32
483570*a^11*b^11*tan(1/2*d*x + 1/2*c)^8 + 2139928*a^9*b^13*tan(1/2*d*x + 1/2*c)^8 + 587776*a^7*b^15*tan(1/2*d*
x + 1/2*c)^8 - 7616*a^5*b^17*tan(1/2*d*x + 1/2*c)^8 - 74368*a^3*b^19*tan(1/2*d*x + 1/2*c)^8 + 17920*a*b^21*tan
(1/2*d*x + 1/2*c)^8 + 6585600*a^18*b^4*tan(1/2*d*x + 1/2*c)^7 + 51038400*a^16*b^6*tan(1/2*d*x + 1/2*c)^7 + 104
499360*a^14*b^8*tan(1/2*d*x + 1/2*c)^7 + 80185140*a^12*b^10*tan(1/2*d*x + 1/2*c)^7 + 20029744*a^10*b^12*tan(1/
2*d*x + 1/2*c)^7 + 661136*a^8*b^14*tan(1/2*d*x + 1/2*c)^7 + 683008*a^6*b^16*tan(1/2*d*x + 1/2*c)^7 - 217600*a^
4*b^18*tan(1/2*d*x + 1/2*c)^7 + 13312*a^2*b^20*tan(1/2*d*x + 1/2*c)^7 + 5120*b^22*tan(1/2*d*x + 1/2*c)^7 + 940
800*a^19*b^3*tan(1/2*d*x + 1/2*c)^6 + 23614080*a^17*b^5*tan(1/2*d*x + 1/2*c)^6 + 83805120*a^15*b^7*tan(1/2*d*x
+ 1/2*c)^6 + 103990880*a^13*b^9*tan(1/2*d*x + 1/2*c)^6 + 45853220*a^11*b^11*tan(1/2*d*x + 1/2*c)^6 + 4650688*
a^9*b^13*tan(1/2*d*x + 1/2*c)^6 + 692496*a^7*b^15*tan(1/2*d*x + 1/2*c)^6 - 7616*a^5*b^17*tan(1/2*d*x + 1/2*c)^
6 - 74368*a^3*b^19*tan(1/2*d*x + 1/2*c)^6 + 17920*a*b^21*tan(1/2*d*x + 1/2*c)^6 + 6174000*a^18*b^4*tan(1/2*d*x
+ 1/2*c)^5 + 43023960*a^16*b^6*tan(1/2*d*x + 1/2*c)^5 + 82755435*a^14*b^8*tan(1/2*d*x + 1/2*c)^5 + 55248340*a
^12*b^10*tan(1/2*d*x + 1/2*c)^5 + 10337432*a^10*b^12*tan(1/2*d*x + 1/2*c)^5 - 175056*a^8*b^14*tan(1/2*d*x + 1/
2*c)^5 + 388976*a^6*b^16*tan(1/2*d*x + 1/2*c)^5 - 167552*a^4*b^18*tan(1/2*d*x + 1/2*c)^5 + 26880*a^2*b^20*tan(
1/2*d*x + 1/2*c)^5 + 705600*a^19*b^3*tan(1/2*d*x + 1/2*c)^4 + 14429520*a^17*b^5*tan(1/2*d*x + 1/2*c)^4 + 42782
712*a^15*b^7*tan(1/2*d*x + 1/2*c)^4 + 41655719*a^13*b^9*tan(1/2*d*x + 1/2*c)^4 + 10567396*a^11*b^11*tan(1/2*d*
x + 1/2*c)^4 - 704032*a^9*b^13*tan(1/2*d*x + 1/2*c)^4 + 485520*a^7*b^15*tan(1/2*d*x + 1/2*c)^4 - 160160*a^5*b^
17*tan(1/2*d*x + 1/2*c)^4 + 22400*a^3*b^19*tan(1/2*d*x + 1/2*c)^4 + 2963520*a^18*b^4*tan(1/2*d*x + 1/2*c)^3 +
14864640*a^16*b^6*tan(1/2*d*x + 1/2*c)^3 + 20500788*a^14*b^8*tan(1/2*d*x + 1/2*c)^3 + 5857306*a^12*b^10*tan(1/
2*d*x + 1/2*c)^3 - 479696*a^10*b^12*tan(1/2*d*x + 1/2*c)^3 + 281232*a^8*b^14*tan(1/2*d*x + 1/2*c)^3 - 85120*a^
6*b^16*tan(1/2*d*x + 1/2*c)^3 + 11200*a^4*b^18*tan(1/2*d*x + 1/2*c)^3 + 282240*a^19*b^3*tan(1/2*d*x + 1/2*c)^2
+ 3575040*a^17*b^5*tan(1/2*d*x + 1/2*c)^2 + 6358464*a^15*b^7*tan(1/2*d*x + 1/2*c)^2 + 1843996*a^13*b^9*tan(1/
2*d*x + 1/2*c)^2 - 146062*a^11*b^11*tan(1/2*d*x + 1/2*c)^2 + 85120*a^9*b^13*tan(1/2*d*x + 1/2*c)^2 - 25648*a^7
*b^15*tan(1/2*d*x + 1/2*c)^2 + 3360*a^5*b^17*tan(1/2*d*x + 1/2*c)^2 + 576240*a^18*b^4*tan(1/2*d*x + 1/2*c) + 1
111320*a^16*b^6*tan(1/2*d*x + 1/2*c) + 324303*a^14*b^8*tan(1/2*d*x + 1/2*c) - 26894*a^12*b^10*tan(1/2*d*x + 1/
2*c) + 14924*a^10*b^12*tan(1/2*d*x + 1/2*c) - 4368*a^8*b^14*tan(1/2*d*x + 1/2*c) + 560*a^6*b^16*tan(1/2*d*x +
1/2*c) + 47040*a^19*b^3 + 82320*a^17*b^5 + 26712*a^15*b^7 - 4161*a^13*b^9 + 2186*a^11*b^11 - 632*a^9*b^13 + 80
*a^7*b^15)/((a^23 - 8*a^21*b^2 + 28*a^19*b^4 - 56*a^17*b^6 + 70*a^15*b^8 - 56*a^13*b^10 + 28*a^11*b^12 - 8*a^9
*b^14 + a^7*b^16)*(a*tan(1/2*d*x + 1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c) + a)^7))/d