∫ sec2 (c+dx)__ (a+b sin(c+dx))8 dx [471]

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

-9/8*a*b^2*(64*a^6+336*a^4*b^2+280*a^2*b^4+35*b^6)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/(a^2-b^2)^

(17/2)/d+1/7*b*sec(d*x+c)/(a^2-b^2)/d/(a+b*sin(d*x+c))^7+5/14*a*b*sec(d*x+c)/(a^2-b^2)^2/d/(a+b*sin(d*x+c))^6+

1/70*b*(49*a^2+16*b^2)*sec(d*x+c)/(a^2-b^2)^3/d/(a+b*sin(d*x+c))^5+13/280*a*b*(28*a^2+27*b^2)*sec(d*x+c)/(a^2-

b^2)^4/d/(a+b*sin(d*x+c))^4+1/280*b*(700*a^4+1317*a^2*b^2+128*b^4)*sec(d*x+c)/(a^2-b^2)^5/d/(a+b*sin(d*x+c))^3

+11/560*a*b*(280*a^4+844*a^2*b^2+241*b^4)*sec(d*x+c)/(a^2-b^2)^6/d/(a+b*sin(d*x+c))^2+1/560*b*(9800*a^6+41484*

a^4*b^2+22767*a^2*b^4+1024*b^6)*sec(d*x+c)/(a^2-b^2)^7/d/(a+b*sin(d*x+c))-1/560*sec(d*x+c)*(315*a*b*(64*a^6+33

6*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+2048*b^8)*sin(d*x+c))/(a^2-b

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Rubi [A]
time = 1.15, antiderivative size = 529, normalized size of antiderivative = 1.00, 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 = {2773, 2943, 2945, 12, 2739, 632, 210} \begin {gather*} \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 {11 a b \left (280 a^4+844 a^2 b^2+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 (700 a^4+1317 a^2 b^2+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 {9 a b^2 \left (64 a^6+336 a^4 b^2+280 a^2 b^4+35 b^6\right ) \text {ArcTan}\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 (9800 a^6+41484 a^4 b^2+22767 a^2 b^4+1024 b^6\right ) \sec (c+d x)}{560 d \left (a^2-b^2\right )^7 (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 d \left (a^2-b^2\right )^8} \end {gather*}

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

(-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)

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

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])

], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

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]

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] &&

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] /;

FreeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*p]

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)*S

imp[(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]

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]

\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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 4.34, size = 494, normalized size = 0.93 \begin {gather*} -\frac {\frac {630 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 )}{\left (a^2-b^2\right )^{17/2}}+\frac {80 b^3 \cos (c+d x)}{\left (a^2-b^2\right )^2 (a+b \sin (c+d x))^7}+\frac {360 a b^3 \cos (c+d x)}{\left (a^2-b^2\right )^3 (a+b \sin (c+d x))^6}+\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 {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 {2 b^3 \left (2616 a^4+3207 a^2 b^2+232 b^4\right ) \cos (c+d x)}{\left (a^2-b^2\right )^6 (a+b \sin (c+d x))^3}+\frac {a b^3 \left (11112 a^4+23066 a^2 b^2+5057 b^4\right ) \cos (c+d x)}{\left (a^2-b^2\right )^7 (a+b \sin (c+d x))^2}+\frac {b^3 \left (26792 a^6+86434 a^4 b^2+38831 a^2 b^4+1488 b^6\right ) \cos (c+d x)}{\left (a^2-b^2\right )^8 (a+b \sin (c+d x))}-\frac {560 \sec (c+d x) \left (-8 a b \left (a^6+7 a^4 b^2+7 a^2 b^4+b^6\right )+\left (a^8+28 a^6 b^2+70 a^4 b^4+28 a^2 b^6+b^8\right ) \sin (c+d x)\right )}{\left (a^2-b^2\right )^8}}{560 d} \end {gather*}

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

-1/560*((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*(2

616*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 + 2

3066*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 + 38831*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 -

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1285\) vs. \(2(506)=1012\).
time = 2.86, size = 1286, normalized size = 2.43


method	result	size

derivativedivides	\(\text {Expression too large to display}\)	\(1286\)
default	\(\text {Expression too large to display}\)	\(1286\)
risch	\(\text {Expression too large to display}\)	\(2333\)

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

1/d*(-1/(a+b)^8/(tan(1/2*d*x+1/2*c)-1)-1/(a-b)^8/(tan(1/2*d*x+1/2*c)+1)-2*b^2/(a-b)^8/(a+b)^8*((1/16*b^2*(2352

*a^12+1176*a^10*b^2+1419*a^8*b^4-896*a^6*b^6+448*a^4*b^8-128*a^2*b^10+16*b^12)/a*tan(1/2*d*x+1/2*c)^13+1/16*b*

(1344*a^14+27216*a^12*b^2+21240*a^10*b^4+10591*a^8*b^6-5376*a^6*b^8+2688*a^4*b^10-768*a^2*b^12+96*b^14)/a^2*ta

n(1/2*d*x+1/2*c)^12+1/8/a^3*b^2*(14112*a^14+74592*a^12*b^2+68466*a^10*b^4+18179*a^8*b^6-7168*a^6*b^8+3968*a^4*

b^10-1216*a^2*b^12+160*b^14)*tan(1/2*d*x+1/2*c)^11+1/8/a^4*b*(4032*a^16+100800*a^14*b^2+267072*a^12*b^4+222502

*a^10*b^6+38423*a^8*b^8-10304*a^6*b^10+6784*a^4*b^12-2288*a^2*b^14+320*b^16)*tan(1/2*d*x+1/2*c)^10+1/80/a^5*b^

2*(529200*a^16+3849720*a^14*b^2+6666195*a^12*b^4+4159190*a^10*b^6+487396*a^8*b^8-43648*a^6*b^10+55568*a^4*b^12

-23936*a^2*b^14+3840*b^16)*tan(1/2*d*x+1/2*c)^9+1/80/a^6*b*(100800*a^18+2682960*a^16*b^2+9329400*a^14*b^4+1109

6155*a^12*b^6+4640510*a^10*b^8+305704*a^8*b^10+83968*a^6*b^12-1088*a^4*b^14-10624*a^2*b^16+2560*b^18)*tan(1/2*

d*x+1/2*c)^8+1/140/a^7*b^2*(1646400*a^18+12759600*a^16*b^2+26124840*a^14*b^4+20046285*a^12*b^6+5007436*a^10*b^

8+165284*a^8*b^10+170752*a^6*b^12-54400*a^4*b^14+3328*a^2*b^16+1280*b^18)*tan(1/2*d*x+1/2*c)^7+1/20/a^6*b*(336

00*a^18+843360*a^16*b^2+2993040*a^14*b^4+3713960*a^12*b^6+1637615*a^10*b^8+166096*a^8*b^10+24732*a^6*b^12-272*

a^4*b^14-2656*a^2*b^16+640*b^18)*tan(1/2*d*x+1/2*c)^6+1/80/a^5*b^2*(882000*a^16+6146280*a^14*b^2+11822205*a^12

*b^4+7892620*a^10*b^6+1476776*a^8*b^8-25008*a^6*b^10+55568*a^4*b^12-23936*a^2*b^14+3840*b^16)*tan(1/2*d*x+1/2*

c)^5+1/80/a^4*b*(100800*a^16+2061360*a^14*b^2+6111816*a^12*b^4+5950817*a^10*b^6+1509628*a^8*b^8-100576*a^6*b^1

0+69360*a^4*b^12-22880*a^2*b^14+3200*b^16)*tan(1/2*d*x+1/2*c)^4+1/40/a^3*b^2*(211680*a^14+1061760*a^12*b^2+146

4342*a^10*b^4+418379*a^8*b^6-34264*a^6*b^8+20088*a^4*b^10-6080*a^2*b^12+800*b^14)*tan(1/2*d*x+1/2*c)^3+1/40*b*

(20160*a^14+255360*a^12*b^2+454176*a^10*b^4+131714*a^8*b^6-10433*a^6*b^8+6080*a^4*b^10-1832*a^2*b^12+240*b^14)

/a^2*tan(1/2*d*x+1/2*c)^2+1/80*b^2*(82320*a^12+158760*a^10*b^2+46329*a^8*b^4-3842*a^6*b^6+2132*a^4*b^8-624*a^2

*b^10+80*b^12)/a*tan(1/2*d*x+1/2*c)+1/560*(47040*a^12+82320*a^10*b^2+26712*a^8*b^4-4161*a^6*b^6+2186*a^4*b^8-6

32*a^2*b^10+80*b^12)*b)/(a*tan(1/2*d*x+1/2*c)^2+2*b*tan(1/2*d*x+1/2*c)+a)^7+9/16*a*(64*a^6+336*a^4*b^2+280*a^2

*b^4+35*b^6)/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2))))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a

ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?`

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1899 vs. \(2 (506) = 1012\).
time = 0.70, size = 3882, normalized size = 7.34 \begin {gather*} \text {Too large to display} \end {gather*}

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

[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*...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2610 vs. \(2 (506) = 1012\).
time = 6.57, size = 2610, normalized size = 4.93 \begin {gather*} \text {Too large to display} \end {gather*}

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

-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*ta...

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Mupad [B]
time = 53.32, size = 2500, normalized size = 4.73 \begin {gather*} \text {Too large to display} \end {gather*}

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

- ((4480*a^14*b + 80*b^15 - 632*a^2*b^13 + 2186*a^4*b^11 - 4161*a^6*b^9 + 31192*a^8*b^7 + 113680*a^10*b^5 + 78

400*a^12*b^3)/(280*(a^16 + b^16 - 8*a^2*b^14 + 28*a^4*b^12 - 56*a^6*b^10 + 70*a^8*b^8 - 56*a^10*b^6 + 28*a^12*

b^4 - 8*a^14*b^2)) - (9*tan(c/2 + (d*x)/2)^8*(560*b^15 + 10360*a^2*b^13 + 59766*a^4*b^11 + 117497*a^6*b^9 + 91

112*a^8*b^7 + 25200*a^10*b^5 + 2240*a^12*b^3))/(8*(a^16 + b^16 - 8*a^2*b^14 + 28*a^4*b^12 - 56*a^6*b^10 + 70*a

^8*b^8 - 56*a^10*b^6 + 28*a^12*b^4 - 8*a^14*b^2)) + (tan(c/2 + (d*x)/2)*(80*b^16 - 80*a^16 - 624*a^2*b^14 + 21

32*a^4*b^12 - 3842*a^6*b^10 + 55209*a^8*b^8 + 219240*a^10*b^6 + 139440*a^12*b^4 + 6720*a^14*b^2))/(40*a*(a^16

+ b^16 - 8*a^2*b^14 + 28*a^4*b^12 - 56*a^6*b^10 + 70*a^8*b^8 - 56*a^10*b^6 + 28*a^12*b^4 - 8*a^14*b^2)) + (tan

(c/2 + (d*x)/2)^7*(5120*b^22 - 19600*a^22 - 13568*a^2*b^20 - 50048*a^4*b^18 + 294032*a^6*b^16 + 1158752*a^8*b^

14 + 11762072*a^10*b^12 + 34250720*a^12*b^10 + 32332965*a^14*b^8 + 15431080*a^16*b^6 + 1234800*a^18*b^4 + 2352

00*a^20*b^2))/(280*a^7*(a^16 + b^16 - 8*a^2*b^14 + 28*a^4*b^12 - 56*a^6*b^10 + 70*a^8*b^8 - 56*a^10*b^6 + 28*a

^12*b^4 - 8*a^14*b^2)) - (tan(c/2 + (d*x)/2)^9*(19600*a^22 + 5120*b^22 - 13568*a^2*b^20 - 50048*a^4*b^18 + 294

032*a^6*b^16 + 1217552*a^8*b^14 + 21572852*a^10*b^12 + 69353690*a^12*b^10 + 86769515*a^14*b^8 + 39441080*a^16*

b^6 + 6762000*a^18*b^4 + 78400*a^20*b^2))/(280*a^7*(a^16 + b^16 - 8*a^2*b^14 + 28*a^4*b^12 - 56*a^6*b^10 + 70*

a^8*b^8 - 56*a^10*b^6 + 28*a^12*b^4 - 8*a^14*b^2)) - (3*tan(c/2 + (d*x)/2)^11*(560*a^20 + 1280*b^20 - 8512*a^2

*b^18 + 22576*a^4*b^16 - 27776*a^6*b^14 + 201292*a^8*b^12 + 1695400*a^10*b^10 + 2917285*a^12*b^8 + 1708840*a^1

4*b^6 + 311920*a^16*b^4 + 8960*a^18*b^2))/(40*a^5*(a^16 + b^16 - 8*a^2*b^14 + 28*a^4*b^12 - 56*a^6*b^10 + 70*a

^8*b^8 - 56*a^10*b^6 + 28*a^12*b^4 - 8*a^14*b^2)) + (3*tan(c/2 + (d*x)/2)^5*(1280*b^20 - 560*a^20 - 8512*a^2*b

^18 + 22576*a^4*b^16 - 21728*a^6*b^14 + 643528*a^8*b^12 + 3165074*a^10*b^10 + 4325867*a^12*b^8 + 2252600*a^14*

b^6 + 337680*a^16*b^4 + 17920*a^18*b^2))/(40*a^5*(a^16 + b^16 - 8*a^2*b^14 + 28*a^4*b^12 - 56*a^6*b^10 + 70*a^

8*b^8 - 56*a^10*b^6 + 28*a^12*b^4 - 8*a^14*b^2)) - (tan(c/2 + (d*x)/2)^13*(112*a^18 + 320*b^18 - 2448*a^2*b^16

+ 8064*a^4*b^14 - 14784*a^6*b^12 + 38598*a^8*b^10 + 171465*a^10*b^8 + 232680*a^12*b^6 + 58800*a^14*b^4 + 2688

*a^16*b^2))/(8*a^3*(a^16 + b^16 - 8*a^2*b^14 + 28*a^4*b^12 - 56*a^6*b^10 + 70*a^8*b^8 - 56*a^10*b^6 + 28*a^12*

b^4 - 8*a^14*b^2)) + (tan(c/2 + (d*x)/2)^3*(320*b^18 - 112*a^18 - 2448*a^2*b^16 + 8160*a^4*b^14 - 14132*a^6*b^

12 + 202616*a^8*b^10 + 800359*a^10*b^8 + 621880*a^12*b^6 + 133840*a^14*b^4 + 6272*a^16*b^2))/(8*a^3*(a^16 + b^

16 - 8*a^2*b^14 + 28*a^4*b^12 - 56*a^6*b^10 + 70*a^8*b^8 - 56*a^10*b^6 + 28*a^12*b^4 - 8*a^14*b^2)) - (tan(c/2

+ (d*x)/2)^15*(16*a^16 + 16*b^16 - 128*a^2*b^14 + 448*a^4*b^12 - 896*a^6*b^10 + 1435*a^8*b^8 + 1624*a^10*b^6

+ 3472*a^12*b^4 + 448*a^14*b^2))/(8*a*(a^16 + b^16 - 8*a^2*b^14 + 28*a^4*b^12 - 56*a^6*b^10 + 70*a^8*b^8 - 56*

a^10*b^6 + 28*a^12*b^4 - 8*a^14*b^2)) + (tan(c/2 + (d*x)/2)^6*(5600*a^20*b + 2560*b^21 - 13824*a^2*b^19 + 2179

2*a^4*b^17 + 29568*a^6*b^15 + 997920*a^8*b^13 + 6528192*a^10*b^11 + 12687263*a^12*b^9 + 9211384*a^14*b^7 + 256

8720*a^16*b^5 + 168000*a^18*b^3))/(40*a^6*(a^16 + b^16 - 8*a^2*b^14 + 28*a^4*b^12 - 56*a^6*b^10 + 70*a^8*b^8 -

56*a^10*b^6 + 28*a^12*b^4 - 8*a^14*b^2)) - (tan(c/2 + (d*x)/2)^10*(3360*a^20*b + 2560*b^21 - 13824*a^2*b^19 +

21792*a^4*b^17 + 16128*a^6*b^15 + 462504*a^8*b^13 + 5492760*a^10*b^11 + 12382335*a^12*b^9 + 10502520*a^14*b^7

+ 3079440*a^16*b^5 + 257600*a^18*b^3))/(40*a^6*(a^16 + b^16 - 8*a^2*b^14 + 28*a^4*b^12 - 56*a^6*b^10 + 70*a^8

*b^8 - 56*a^10*b^6 + 28*a^12*b^4 - 8*a^14*b^2)) - (tan(c/2 + (d*x)/2)^12*(448*a^18*b + 640*b^19 - 4672*a^2*b^1

7 + 14336*a^4*b^15 - 23296*a^6*b^13 + 86702*a^8*b^11 + 550445*a^10*b^9 + 787976*a^12*b^7 + 312368*a^14*b^5 + 3

1808*a^16*b^3))/(8*a^4*(a^16 + b^16 - 8*a^2*b^14 + 28*a^4*b^12 - 56*a^6*b^10 + 70*a^8*b^8 - 56*a^10*b^6 + 28*a

^12*b^4 - 8*a^14*b^2)) + (tan(c/2 + (d*x)/2)^4*(6720*a^18*b + 3200*b^19 - 23360*a^2*b^17 + 73024*a^4*b^15 - 11

2736*a^6*b^13 + 1866494*a^8*b^11 + 7831069*a^10*b^9 + 7851144*a^12*b^7 + 2787120*a^14*b^5 + 212800*a^16*b^3))/

(40*a^4*(a^16 + b^16 - 8*a^2*b^14 + 28*a^4*b^12 - 56*a^6*b^10 + 70*a^8*b^8 - 56*a^10*b^6 + 28*a^12*b^4 - 8*a^1

4*b^2)) - (3*tan(c/2 + (d*x)/2)^14*(32*a^16*b + 32*b^17 - 256*a^2*b^15 + 896*a^4*b^13 - 1792*a^6*b^11 + 3605*a

^8*b^9 + 9128*a^10*b^7 + 14000*a^12*b^5 + 2240*a^14*b^3))/(8*a^2*(a^16 + b^16 - 8*a^2*b^14 + 28*a^4*b^12 - 56*

a^6*b^10 + 70*a^8*b^8 - 56*a^10*b^6 + 28*a^12*b^4 - 8*a^14*b^2)) + (3*tan(c/2 + (d*x)/2)^2*(7840*a^16*b + 1120

*b^17 - 8576*a^2*b^15 + 28584*a^4*b^13 - 49416*a^6*b^11 + 738879*a^8*b^9 + 2925944*a^10*b^7 + 1932560*a^12*b^5

+ 203840*a^14*b^3))/(280*a^2*(a^16 + b^16 - 8*a^2*b^14 + 28*a^4*b^12 - 56*a^6*b^10 + 70*a^8*b^8 - 56*a^10*b^6

+ 28*a^12*b^4 - 8*a^14*b^2)))/(d*(tan(c/2 + (d*x)/2)^5*(126*a^6*b + 672*a^2*b^5 + 840*a^4*b^3) - tan(c/2 + (d

*x)/2)^11*(126*a^6*b + 672*a^2*b^5 + 840*a^4*b^...

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3.5.71 \(\int \frac {\sec ^2(c+d x)}{(a+b \sin (c+d x))^8} \, dx\) [471]