3.472 \(\int \frac{\sec ^4(c+d x)}{(a+b \sin (c+d x))^8} \, dx\)

Optimal. Leaf size=653 \[ \frac{165 a b^4 \left (112 a^4 b^2+70 a^2 b^4+32 a^6+7 b^6\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{8 d \left (a^2-b^2\right )^{19/2}}+\frac{b \left (28420 a^4 b^2+12907 a^2 b^4+9212 a^6+512 b^6\right ) \sec ^3(c+d x)}{112 d \left (a^2-b^2\right )^7 (a+b \sin (c+d x))}+\frac{13 a b \left (336 a^2 b^2+140 a^4+85 b^4\right ) \sec ^3(c+d x)}{112 d \left (a^2-b^2\right )^6 (a+b \sin (c+d x))^2}+\frac{b \left (1421 a^2 b^2+882 a^4+128 b^4\right ) \sec ^3(c+d x)}{168 d \left (a^2-b^2\right )^5 (a+b \sin (c+d x))^3}+\frac{a b \left (118 a^2+103 b^2\right ) \sec ^3(c+d x)}{56 d \left (a^2-b^2\right )^4 (a+b \sin (c+d x))^4}+\frac{b \left (13 a^2+4 b^2\right ) \sec ^3(c+d x)}{14 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))^5}+\frac{17 a b \sec ^3(c+d x)}{42 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^6}+\frac{b \sec ^3(c+d x)}{7 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^7}-\frac{\sec ^3(c+d x) \left (1155 a b \left (112 a^4 b^2+70 a^2 b^4+32 a^6+7 b^6\right )-\left (52528 a^6 b^2+142902 a^4 b^4+57665 a^2 b^6+112 a^8+2048 b^8\right ) \sin (c+d x)\right )}{336 d \left (a^2-b^2\right )^8}+\frac{\sec (c+d x) \left (\left (-6048 a^8 b^2-207332 a^6 b^4-413024 a^4 b^6-135489 a^2 b^8+224 a^{10}-4096 b^{10}\right ) \sin (c+d x)+3465 a b^3 \left (112 a^4 b^2+70 a^2 b^4+32 a^6+7 b^6\right )\right )}{336 d \left (a^2-b^2\right )^9} \]

[Out]

(165*a*b^4*(32*a^6 + 112*a^4*b^2 + 70*a^2*b^4 + 7*b^6)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(8*(a
^2 - b^2)^(19/2)*d) + (b*Sec[c + d*x]^3)/(7*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^7) + (17*a*b*Sec[c + d*x]^3)/(4
2*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^6) + (b*(13*a^2 + 4*b^2)*Sec[c + d*x]^3)/(14*(a^2 - b^2)^3*d*(a + b*Sin
[c + d*x])^5) + (a*b*(118*a^2 + 103*b^2)*Sec[c + d*x]^3)/(56*(a^2 - b^2)^4*d*(a + b*Sin[c + d*x])^4) + (b*(882
*a^4 + 1421*a^2*b^2 + 128*b^4)*Sec[c + d*x]^3)/(168*(a^2 - b^2)^5*d*(a + b*Sin[c + d*x])^3) + (13*a*b*(140*a^4
+ 336*a^2*b^2 + 85*b^4)*Sec[c + d*x]^3)/(112*(a^2 - b^2)^6*d*(a + b*Sin[c + d*x])^2) + (b*(9212*a^6 + 28420*a
^4*b^2 + 12907*a^2*b^4 + 512*b^6)*Sec[c + d*x]^3)/(112*(a^2 - b^2)^7*d*(a + b*Sin[c + d*x])) - (Sec[c + d*x]^3
*(1155*a*b*(32*a^6 + 112*a^4*b^2 + 70*a^2*b^4 + 7*b^6) - (112*a^8 + 52528*a^6*b^2 + 142902*a^4*b^4 + 57665*a^2
*b^6 + 2048*b^8)*Sin[c + d*x]))/(336*(a^2 - b^2)^8*d) + (Sec[c + d*x]*(3465*a*b^3*(32*a^6 + 112*a^4*b^2 + 70*a
^2*b^4 + 7*b^6) + (224*a^10 - 6048*a^8*b^2 - 207332*a^6*b^4 - 413024*a^4*b^6 - 135489*a^2*b^8 - 4096*b^10)*Sin
[c + d*x]))/(336*(a^2 - b^2)^9*d)

________________________________________________________________________________________

Rubi [A]  time = 2.137, antiderivative size = 653, normalized size of antiderivative = 1., number of steps used = 13, 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} \[ \frac{165 a b^4 \left (112 a^4 b^2+70 a^2 b^4+32 a^6+7 b^6\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{8 d \left (a^2-b^2\right )^{19/2}}+\frac{b \left (28420 a^4 b^2+12907 a^2 b^4+9212 a^6+512 b^6\right ) \sec ^3(c+d x)}{112 d \left (a^2-b^2\right )^7 (a+b \sin (c+d x))}+\frac{13 a b \left (336 a^2 b^2+140 a^4+85 b^4\right ) \sec ^3(c+d x)}{112 d \left (a^2-b^2\right )^6 (a+b \sin (c+d x))^2}+\frac{b \left (1421 a^2 b^2+882 a^4+128 b^4\right ) \sec ^3(c+d x)}{168 d \left (a^2-b^2\right )^5 (a+b \sin (c+d x))^3}+\frac{a b \left (118 a^2+103 b^2\right ) \sec ^3(c+d x)}{56 d \left (a^2-b^2\right )^4 (a+b \sin (c+d x))^4}+\frac{b \left (13 a^2+4 b^2\right ) \sec ^3(c+d x)}{14 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))^5}+\frac{17 a b \sec ^3(c+d x)}{42 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^6}+\frac{b \sec ^3(c+d x)}{7 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^7}-\frac{\sec ^3(c+d x) \left (1155 a b \left (112 a^4 b^2+70 a^2 b^4+32 a^6+7 b^6\right )-\left (52528 a^6 b^2+142902 a^4 b^4+57665 a^2 b^6+112 a^8+2048 b^8\right ) \sin (c+d x)\right )}{336 d \left (a^2-b^2\right )^8}+\frac{\sec (c+d x) \left (\left (-6048 a^8 b^2-207332 a^6 b^4-413024 a^4 b^6-135489 a^2 b^8+224 a^{10}-4096 b^{10}\right ) \sin (c+d x)+3465 a b^3 \left (112 a^4 b^2+70 a^2 b^4+32 a^6+7 b^6\right )\right )}{336 d \left (a^2-b^2\right )^9} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^4/(a + b*Sin[c + d*x])^8,x]

[Out]

(165*a*b^4*(32*a^6 + 112*a^4*b^2 + 70*a^2*b^4 + 7*b^6)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(8*(a
^2 - b^2)^(19/2)*d) + (b*Sec[c + d*x]^3)/(7*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^7) + (17*a*b*Sec[c + d*x]^3)/(4
2*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^6) + (b*(13*a^2 + 4*b^2)*Sec[c + d*x]^3)/(14*(a^2 - b^2)^3*d*(a + b*Sin
[c + d*x])^5) + (a*b*(118*a^2 + 103*b^2)*Sec[c + d*x]^3)/(56*(a^2 - b^2)^4*d*(a + b*Sin[c + d*x])^4) + (b*(882
*a^4 + 1421*a^2*b^2 + 128*b^4)*Sec[c + d*x]^3)/(168*(a^2 - b^2)^5*d*(a + b*Sin[c + d*x])^3) + (13*a*b*(140*a^4
+ 336*a^2*b^2 + 85*b^4)*Sec[c + d*x]^3)/(112*(a^2 - b^2)^6*d*(a + b*Sin[c + d*x])^2) + (b*(9212*a^6 + 28420*a
^4*b^2 + 12907*a^2*b^4 + 512*b^6)*Sec[c + d*x]^3)/(112*(a^2 - b^2)^7*d*(a + b*Sin[c + d*x])) - (Sec[c + d*x]^3
*(1155*a*b*(32*a^6 + 112*a^4*b^2 + 70*a^2*b^4 + 7*b^6) - (112*a^8 + 52528*a^6*b^2 + 142902*a^4*b^4 + 57665*a^2
*b^6 + 2048*b^8)*Sin[c + d*x]))/(336*(a^2 - b^2)^8*d) + (Sec[c + d*x]*(3465*a*b^3*(32*a^6 + 112*a^4*b^2 + 70*a
^2*b^4 + 7*b^6) + (224*a^10 - 6048*a^8*b^2 - 207332*a^6*b^4 - 413024*a^4*b^6 - 135489*a^2*b^8 - 4096*b^10)*Sin
[c + d*x]))/(336*(a^2 - b^2)^9*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 ^4(c+d x)}{(a+b \sin (c+d x))^8} \, dx &=\frac{b \sec ^3(c+d x)}{7 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^7}-\frac{\int \frac{\sec ^4(c+d x) (-7 a+10 b \sin (c+d x))}{(a+b \sin (c+d x))^7} \, dx}{7 \left (a^2-b^2\right )}\\ &=\frac{b \sec ^3(c+d x)}{7 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^7}+\frac{17 a b \sec ^3(c+d x)}{42 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^6}+\frac{\int \frac{\sec ^4(c+d x) \left (6 \left (7 a^2+10 b^2\right )-153 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 ^3(c+d x)}{7 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^7}+\frac{17 a b \sec ^3(c+d x)}{42 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^6}+\frac{b \left (13 a^2+4 b^2\right ) \sec ^3(c+d x)}{14 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^5}-\frac{\int \frac{\sec ^4(c+d x) \left (-15 a \left (14 a^2+71 b^2\right )+120 b \left (13 a^2+4 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 ^3(c+d x)}{7 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^7}+\frac{17 a b \sec ^3(c+d x)}{42 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^6}+\frac{b \left (13 a^2+4 b^2\right ) \sec ^3(c+d x)}{14 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^5}+\frac{a b \left (118 a^2+103 b^2\right ) \sec ^3(c+d x)}{56 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^4}+\frac{\int \frac{\sec ^4(c+d x) \left (60 \left (14 a^4+175 a^2 b^2+32 b^4\right )-105 a b \left (118 a^2+103 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 ^3(c+d x)}{7 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^7}+\frac{17 a b \sec ^3(c+d x)}{42 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^6}+\frac{b \left (13 a^2+4 b^2\right ) \sec ^3(c+d x)}{14 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^5}+\frac{a b \left (118 a^2+103 b^2\right ) \sec ^3(c+d x)}{56 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^4}+\frac{b \left (882 a^4+1421 a^2 b^2+128 b^4\right ) \sec ^3(c+d x)}{168 \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))^3}-\frac{\int \frac{\sec ^4(c+d x) \left (-45 a \left (56 a^4+1526 a^2 b^2+849 b^4\right )+90 b \left (882 a^4+1421 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 ^3(c+d x)}{7 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^7}+\frac{17 a b \sec ^3(c+d x)}{42 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^6}+\frac{b \left (13 a^2+4 b^2\right ) \sec ^3(c+d x)}{14 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^5}+\frac{a b \left (118 a^2+103 b^2\right ) \sec ^3(c+d x)}{56 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^4}+\frac{b \left (882 a^4+1421 a^2 b^2+128 b^4\right ) \sec ^3(c+d x)}{168 \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))^3}+\frac{13 a b \left (140 a^4+336 a^2 b^2+85 b^4\right ) \sec ^3(c+d x)}{112 \left (a^2-b^2\right )^6 d (a+b \sin (c+d x))^2}+\frac{\int \frac{\sec ^4(c+d x) \left (90 \left (56 a^6+3290 a^4 b^2+3691 a^2 b^4+256 b^6\right )-2925 a b \left (140 a^4+336 a^2 b^2+85 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 ^3(c+d x)}{7 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^7}+\frac{17 a b \sec ^3(c+d x)}{42 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^6}+\frac{b \left (13 a^2+4 b^2\right ) \sec ^3(c+d x)}{14 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^5}+\frac{a b \left (118 a^2+103 b^2\right ) \sec ^3(c+d x)}{56 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^4}+\frac{b \left (882 a^4+1421 a^2 b^2+128 b^4\right ) \sec ^3(c+d x)}{168 \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))^3}+\frac{13 a b \left (140 a^4+336 a^2 b^2+85 b^4\right ) \sec ^3(c+d x)}{112 \left (a^2-b^2\right )^6 d (a+b \sin (c+d x))^2}+\frac{b \left (9212 a^6+28420 a^4 b^2+12907 a^2 b^4+512 b^6\right ) \sec ^3(c+d x)}{112 \left (a^2-b^2\right )^7 d (a+b \sin (c+d x))}-\frac{\int \frac{\sec ^4(c+d x) \left (-45 a \left (112 a^6+15680 a^4 b^2+29222 a^2 b^4+6037 b^6\right )+180 b \left (9212 a^6+28420 a^4 b^2+12907 a^2 b^4+512 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 ^3(c+d x)}{7 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^7}+\frac{17 a b \sec ^3(c+d x)}{42 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^6}+\frac{b \left (13 a^2+4 b^2\right ) \sec ^3(c+d x)}{14 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^5}+\frac{a b \left (118 a^2+103 b^2\right ) \sec ^3(c+d x)}{56 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^4}+\frac{b \left (882 a^4+1421 a^2 b^2+128 b^4\right ) \sec ^3(c+d x)}{168 \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))^3}+\frac{13 a b \left (140 a^4+336 a^2 b^2+85 b^4\right ) \sec ^3(c+d x)}{112 \left (a^2-b^2\right )^6 d (a+b \sin (c+d x))^2}+\frac{b \left (9212 a^6+28420 a^4 b^2+12907 a^2 b^4+512 b^6\right ) \sec ^3(c+d x)}{112 \left (a^2-b^2\right )^7 d (a+b \sin (c+d x))}-\frac{\sec ^3(c+d x) \left (1155 a b \left (32 a^6+112 a^4 b^2+70 a^2 b^4+7 b^6\right )-\left (112 a^8+52528 a^6 b^2+142902 a^4 b^4+57665 a^2 b^6+2048 b^8\right ) \sin (c+d x)\right )}{336 \left (a^2-b^2\right )^8 d}+\frac{\int \frac{\sec ^2(c+d x) \left (45 \left (224 a^9-5824 a^7 b^2-102276 a^5 b^4-127220 a^3 b^6-20159 a b^8\right )+90 b \left (112 a^8+52528 a^6 b^2+142902 a^4 b^4+57665 a^2 b^6+2048 b^8\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{15120 \left (a^2-b^2\right )^8}\\ &=\frac{b \sec ^3(c+d x)}{7 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^7}+\frac{17 a b \sec ^3(c+d x)}{42 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^6}+\frac{b \left (13 a^2+4 b^2\right ) \sec ^3(c+d x)}{14 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^5}+\frac{a b \left (118 a^2+103 b^2\right ) \sec ^3(c+d x)}{56 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^4}+\frac{b \left (882 a^4+1421 a^2 b^2+128 b^4\right ) \sec ^3(c+d x)}{168 \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))^3}+\frac{13 a b \left (140 a^4+336 a^2 b^2+85 b^4\right ) \sec ^3(c+d x)}{112 \left (a^2-b^2\right )^6 d (a+b \sin (c+d x))^2}+\frac{b \left (9212 a^6+28420 a^4 b^2+12907 a^2 b^4+512 b^6\right ) \sec ^3(c+d x)}{112 \left (a^2-b^2\right )^7 d (a+b \sin (c+d x))}-\frac{\sec ^3(c+d x) \left (1155 a b \left (32 a^6+112 a^4 b^2+70 a^2 b^4+7 b^6\right )-\left (112 a^8+52528 a^6 b^2+142902 a^4 b^4+57665 a^2 b^6+2048 b^8\right ) \sin (c+d x)\right )}{336 \left (a^2-b^2\right )^8 d}+\frac{\sec (c+d x) \left (3465 a b^3 \left (32 a^6+112 a^4 b^2+70 a^2 b^4+7 b^6\right )+\left (224 a^{10}-6048 a^8 b^2-207332 a^6 b^4-413024 a^4 b^6-135489 a^2 b^8-4096 b^{10}\right ) \sin (c+d x)\right )}{336 \left (a^2-b^2\right )^9 d}-\frac{\int -\frac{155925 a b^4 \left (32 a^6+112 a^4 b^2+70 a^2 b^4+7 b^6\right )}{a+b \sin (c+d x)} \, dx}{15120 \left (a^2-b^2\right )^9}\\ &=\frac{b \sec ^3(c+d x)}{7 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^7}+\frac{17 a b \sec ^3(c+d x)}{42 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^6}+\frac{b \left (13 a^2+4 b^2\right ) \sec ^3(c+d x)}{14 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^5}+\frac{a b \left (118 a^2+103 b^2\right ) \sec ^3(c+d x)}{56 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^4}+\frac{b \left (882 a^4+1421 a^2 b^2+128 b^4\right ) \sec ^3(c+d x)}{168 \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))^3}+\frac{13 a b \left (140 a^4+336 a^2 b^2+85 b^4\right ) \sec ^3(c+d x)}{112 \left (a^2-b^2\right )^6 d (a+b \sin (c+d x))^2}+\frac{b \left (9212 a^6+28420 a^4 b^2+12907 a^2 b^4+512 b^6\right ) \sec ^3(c+d x)}{112 \left (a^2-b^2\right )^7 d (a+b \sin (c+d x))}-\frac{\sec ^3(c+d x) \left (1155 a b \left (32 a^6+112 a^4 b^2+70 a^2 b^4+7 b^6\right )-\left (112 a^8+52528 a^6 b^2+142902 a^4 b^4+57665 a^2 b^6+2048 b^8\right ) \sin (c+d x)\right )}{336 \left (a^2-b^2\right )^8 d}+\frac{\sec (c+d x) \left (3465 a b^3 \left (32 a^6+112 a^4 b^2+70 a^2 b^4+7 b^6\right )+\left (224 a^{10}-6048 a^8 b^2-207332 a^6 b^4-413024 a^4 b^6-135489 a^2 b^8-4096 b^{10}\right ) \sin (c+d x)\right )}{336 \left (a^2-b^2\right )^9 d}+\frac{\left (165 a b^4 \left (32 a^6+112 a^4 b^2+70 a^2 b^4+7 b^6\right )\right ) \int \frac{1}{a+b \sin (c+d x)} \, dx}{16 \left (a^2-b^2\right )^9}\\ &=\frac{b \sec ^3(c+d x)}{7 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^7}+\frac{17 a b \sec ^3(c+d x)}{42 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^6}+\frac{b \left (13 a^2+4 b^2\right ) \sec ^3(c+d x)}{14 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^5}+\frac{a b \left (118 a^2+103 b^2\right ) \sec ^3(c+d x)}{56 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^4}+\frac{b \left (882 a^4+1421 a^2 b^2+128 b^4\right ) \sec ^3(c+d x)}{168 \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))^3}+\frac{13 a b \left (140 a^4+336 a^2 b^2+85 b^4\right ) \sec ^3(c+d x)}{112 \left (a^2-b^2\right )^6 d (a+b \sin (c+d x))^2}+\frac{b \left (9212 a^6+28420 a^4 b^2+12907 a^2 b^4+512 b^6\right ) \sec ^3(c+d x)}{112 \left (a^2-b^2\right )^7 d (a+b \sin (c+d x))}-\frac{\sec ^3(c+d x) \left (1155 a b \left (32 a^6+112 a^4 b^2+70 a^2 b^4+7 b^6\right )-\left (112 a^8+52528 a^6 b^2+142902 a^4 b^4+57665 a^2 b^6+2048 b^8\right ) \sin (c+d x)\right )}{336 \left (a^2-b^2\right )^8 d}+\frac{\sec (c+d x) \left (3465 a b^3 \left (32 a^6+112 a^4 b^2+70 a^2 b^4+7 b^6\right )+\left (224 a^{10}-6048 a^8 b^2-207332 a^6 b^4-413024 a^4 b^6-135489 a^2 b^8-4096 b^{10}\right ) \sin (c+d x)\right )}{336 \left (a^2-b^2\right )^9 d}+\frac{\left (165 a b^4 \left (32 a^6+112 a^4 b^2+70 a^2 b^4+7 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 )^9 d}\\ &=\frac{b \sec ^3(c+d x)}{7 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^7}+\frac{17 a b \sec ^3(c+d x)}{42 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^6}+\frac{b \left (13 a^2+4 b^2\right ) \sec ^3(c+d x)}{14 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^5}+\frac{a b \left (118 a^2+103 b^2\right ) \sec ^3(c+d x)}{56 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^4}+\frac{b \left (882 a^4+1421 a^2 b^2+128 b^4\right ) \sec ^3(c+d x)}{168 \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))^3}+\frac{13 a b \left (140 a^4+336 a^2 b^2+85 b^4\right ) \sec ^3(c+d x)}{112 \left (a^2-b^2\right )^6 d (a+b \sin (c+d x))^2}+\frac{b \left (9212 a^6+28420 a^4 b^2+12907 a^2 b^4+512 b^6\right ) \sec ^3(c+d x)}{112 \left (a^2-b^2\right )^7 d (a+b \sin (c+d x))}-\frac{\sec ^3(c+d x) \left (1155 a b \left (32 a^6+112 a^4 b^2+70 a^2 b^4+7 b^6\right )-\left (112 a^8+52528 a^6 b^2+142902 a^4 b^4+57665 a^2 b^6+2048 b^8\right ) \sin (c+d x)\right )}{336 \left (a^2-b^2\right )^8 d}+\frac{\sec (c+d x) \left (3465 a b^3 \left (32 a^6+112 a^4 b^2+70 a^2 b^4+7 b^6\right )+\left (224 a^{10}-6048 a^8 b^2-207332 a^6 b^4-413024 a^4 b^6-135489 a^2 b^8-4096 b^{10}\right ) \sin (c+d x)\right )}{336 \left (a^2-b^2\right )^9 d}-\frac{\left (165 a b^4 \left (32 a^6+112 a^4 b^2+70 a^2 b^4+7 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 )^9 d}\\ &=\frac{165 a b^4 \left (32 a^6+112 a^4 b^2+70 a^2 b^4+7 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 )^{19/2} d}+\frac{b \sec ^3(c+d x)}{7 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^7}+\frac{17 a b \sec ^3(c+d x)}{42 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^6}+\frac{b \left (13 a^2+4 b^2\right ) \sec ^3(c+d x)}{14 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^5}+\frac{a b \left (118 a^2+103 b^2\right ) \sec ^3(c+d x)}{56 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^4}+\frac{b \left (882 a^4+1421 a^2 b^2+128 b^4\right ) \sec ^3(c+d x)}{168 \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))^3}+\frac{13 a b \left (140 a^4+336 a^2 b^2+85 b^4\right ) \sec ^3(c+d x)}{112 \left (a^2-b^2\right )^6 d (a+b \sin (c+d x))^2}+\frac{b \left (9212 a^6+28420 a^4 b^2+12907 a^2 b^4+512 b^6\right ) \sec ^3(c+d x)}{112 \left (a^2-b^2\right )^7 d (a+b \sin (c+d x))}-\frac{\sec ^3(c+d x) \left (1155 a b \left (32 a^6+112 a^4 b^2+70 a^2 b^4+7 b^6\right )-\left (112 a^8+52528 a^6 b^2+142902 a^4 b^4+57665 a^2 b^6+2048 b^8\right ) \sin (c+d x)\right )}{336 \left (a^2-b^2\right )^8 d}+\frac{\sec (c+d x) \left (3465 a b^3 \left (32 a^6+112 a^4 b^2+70 a^2 b^4+7 b^6\right )+\left (224 a^{10}-6048 a^8 b^2-207332 a^6 b^4-413024 a^4 b^6-135489 a^2 b^8-4096 b^{10}\right ) \sin (c+d x)\right )}{336 \left (a^2-b^2\right )^9 d}\\ \end{align*}

Mathematica [A]  time = 5.45203, size = 597, normalized size = 0.91 \[ \frac{\frac{6930 a b^4 \left (112 a^4 b^2+70 a^2 b^4+32 a^6+7 b^6\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{19/2}}+\frac{b^5 \left (234272 a^4 b^2+81057 a^2 b^4+103844 a^6+2528 b^6\right ) \cos (c+d x)}{\left (a^2-b^2\right )^9 (a+b \sin (c+d x))}+\frac{a b^5 \left (48820 a^2 b^2+33284 a^4+8287 b^4\right ) \cos (c+d x)}{\left (a^2-b^2\right )^8 (a+b \sin (c+d x))^2}+\frac{2 b^5 \left (5273 a^2 b^2+6058 a^4+296 b^4\right ) \cos (c+d x)}{\left (a^2-b^2\right )^7 (a+b \sin (c+d x))^3}+\frac{2 a b^5 \left (2138 a^2+925 b^2\right ) \cos (c+d x)}{\left (a^2-b^2\right )^6 (a+b \sin (c+d x))^4}+\frac{8 b^5 \left (167 a^2+24 b^2\right ) \cos (c+d x)}{\left (a^2-b^2\right )^5 (a+b \sin (c+d x))^5}+\frac{328 a b^5 \cos (c+d x)}{\left (a^2-b^2\right )^4 (a+b \sin (c+d x))^6}+\frac{48 b^5 \cos (c+d x)}{\left (a^2-b^2\right )^3 (a+b \sin (c+d x))^7}+\frac{112 \sec ^3(c+d x) \left (\left (28 a^6 b^2+70 a^4 b^4+28 a^2 b^6+a^8+b^8\right ) \sin (c+d x)-8 a b \left (7 a^4 b^2+7 a^2 b^4+a^6+b^6\right )\right )}{\left (a^2-b^2\right )^8}+\frac{224 \sec (c+d x) \left (\left (-27 a^8 b^2-462 a^6 b^4-798 a^4 b^6-243 a^2 b^8+a^{10}-7 b^{10}\right ) \sin (c+d x)+12 \left (45 a^3 b^7+63 a^5 b^5+15 a^7 b^3+5 a b^9\right )\right )}{\left (a^2-b^2\right )^9}}{336 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^4/(a + b*Sin[c + d*x])^8,x]

[Out]

((6930*a*b^4*(32*a^6 + 112*a^4*b^2 + 70*a^2*b^4 + 7*b^6)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^
2 - b^2)^(19/2) + (48*b^5*Cos[c + d*x])/((a^2 - b^2)^3*(a + b*Sin[c + d*x])^7) + (328*a*b^5*Cos[c + d*x])/((a^
2 - b^2)^4*(a + b*Sin[c + d*x])^6) + (8*b^5*(167*a^2 + 24*b^2)*Cos[c + d*x])/((a^2 - b^2)^5*(a + b*Sin[c + d*x
])^5) + (2*a*b^5*(2138*a^2 + 925*b^2)*Cos[c + d*x])/((a^2 - b^2)^6*(a + b*Sin[c + d*x])^4) + (2*b^5*(6058*a^4
+ 5273*a^2*b^2 + 296*b^4)*Cos[c + d*x])/((a^2 - b^2)^7*(a + b*Sin[c + d*x])^3) + (a*b^5*(33284*a^4 + 48820*a^2
*b^2 + 8287*b^4)*Cos[c + d*x])/((a^2 - b^2)^8*(a + b*Sin[c + d*x])^2) + (b^5*(103844*a^6 + 234272*a^4*b^2 + 81
057*a^2*b^4 + 2528*b^6)*Cos[c + d*x])/((a^2 - b^2)^9*(a + b*Sin[c + d*x])) + (112*Sec[c + d*x]^3*(-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 + (224*Sec[c + d*x]*(12*(15*a^7*b^3 + 63*a^5*b^5 + 45*a^3*b^7 + 5*a*b^9) + (a^10 - 27*a^8*b^2 - 462*a^6*b^4
- 798*a^4*b^6 - 243*a^2*b^8 - 7*b^10)*Sin[c + d*x]))/(a^2 - b^2)^9)/(336*d)

________________________________________________________________________________________

Maple [B]  time = 0.354, size = 7823, normalized size = 12. \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)^4/(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)^4/(a+b*sin(d*x+c))^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 13.9975, size = 11641, normalized size = 17.83 \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)^4/(a+b*sin(d*x+c))^8,x, algorithm="fricas")

[Out]

[1/672*(224*a^18*b - 2016*a^16*b^3 + 8064*a^14*b^5 - 18816*a^12*b^7 + 28224*a^10*b^9 - 28224*a^8*b^11 + 18816*
a^6*b^13 - 8064*a^4*b^15 + 2016*a^2*b^17 - 224*b^19 - 2*(224*a^12*b^7 - 6272*a^10*b^9 - 201284*a^8*b^11 - 2056
92*a^6*b^13 + 277535*a^4*b^15 + 131393*a^2*b^17 + 4096*b^19)*cos(d*x + c)^10 + 28*(336*a^14*b^5 - 9352*a^12*b^
7 - 252014*a^10*b^9 - 230159*a^8*b^11 + 297312*a^6*b^13 + 165122*a^4*b^15 + 27731*a^2*b^17 + 1024*b^19)*cos(d*
x + c)^8 - 70*(224*a^16*b^3 - 5936*a^14*b^5 - 126448*a^12*b^7 - 243082*a^10*b^9 - 29747*a^8*b^11 + 284285*a^6*
b^13 + 109607*a^4*b^15 + 10585*a^2*b^17 + 512*b^19)*cos(d*x + c)^6 + 28*(112*a^18*b - 2296*a^16*b^3 - 35224*a^
14*b^5 - 308392*a^12*b^7 - 337750*a^10*b^9 + 149783*a^8*b^11 + 394751*a^6*b^13 + 130949*a^4*b^15 + 7427*a^2*b^
17 + 640*b^19)*cos(d*x + c)^4 - 224*(7*a^18*b - 46*a^16*b^3 + 116*a^14*b^5 - 112*a^12*b^7 - 70*a^10*b^9 + 308*
a^8*b^11 - 364*a^6*b^13 + 224*a^4*b^15 - 73*a^2*b^17 + 10*b^19)*cos(d*x + c)^2 + 3465*(7*(32*a^8*b^10 + 112*a^
6*b^12 + 70*a^4*b^14 + 7*a^2*b^16)*cos(d*x + c)^9 - 7*(160*a^10*b^8 + 656*a^8*b^10 + 686*a^6*b^12 + 245*a^4*b^
14 + 21*a^2*b^16)*cos(d*x + c)^7 + 7*(96*a^12*b^6 + 656*a^10*b^8 + 1426*a^8*b^10 + 1057*a^6*b^12 + 280*a^4*b^1
4 + 21*a^2*b^16)*cos(d*x + c)^5 - (32*a^14*b^4 + 784*a^12*b^6 + 3542*a^10*b^8 + 5621*a^8*b^10 + 3381*a^6*b^12
+ 735*a^4*b^14 + 49*a^2*b^16)*cos(d*x + c)^3 + ((32*a^7*b^11 + 112*a^5*b^13 + 70*a^3*b^15 + 7*a*b^17)*cos(d*x
+ c)^9 - 3*(224*a^9*b^9 + 816*a^7*b^11 + 602*a^5*b^13 + 119*a^3*b^15 + 7*a*b^17)*cos(d*x + c)^7 + (1120*a^11*b
^7 + 5264*a^9*b^9 + 7250*a^7*b^11 + 3521*a^5*b^13 + 504*a^3*b^15 + 21*a*b^17)*cos(d*x + c)^5 - (224*a^13*b^5 +
1904*a^11*b^7 + 5082*a^9*b^9 + 4883*a^7*b^11 + 1827*a^5*b^13 + 217*a^3*b^15 + 7*a*b^17)*cos(d*x + c)^3)*sin(d
*x + c))*sqrt(-a^2 + b^2)*log(-((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*(16*a^19 - 144*a^17*b^2 + 576*a^15*b^4 - 1344*a^13*b^6 + 2016*a^11*b^8 - 2016*a^9*b^10 + 1344*a^7*b^12 - 57
6*a^5*b^14 + 144*a^3*b^16 - 16*a*b^18 - (224*a^13*b^6 - 6272*a^11*b^8 - 185444*a^9*b^10 - 166092*a^7*b^12 + 25
6745*a^5*b^14 + 100208*a^3*b^16 + 631*a*b^18)*cos(d*x + c)^8 + 10*(112*a^15*b^4 - 3080*a^13*b^6 - 73962*a^11*b
^8 - 78323*a^9*b^10 + 60829*a^7*b^12 + 73923*a^5*b^14 + 20401*a^3*b^16 + 100*a*b^18)*cos(d*x + c)^6 - 3*(224*a
^17*b^2 - 5712*a^15*b^4 - 95648*a^13*b^6 - 254254*a^11*b^8 - 120855*a^9*b^10 + 282886*a^7*b^12 + 157892*a^5*b^
14 + 35448*a^3*b^16 + 19*a*b^18)*cos(d*x + c)^4 + 16*(2*a^19 - 35*a^17*b^2 + 208*a^15*b^4 - 644*a^13*b^6 + 120
4*a^11*b^8 - 1442*a^9*b^10 + 1120*a^7*b^12 - 548*a^5*b^14 + 154*a^3*b^16 - 19*a*b^18)*cos(d*x + c)^2)*sin(d*x
+ c))/(7*(a^21*b^6 - 10*a^19*b^8 + 45*a^17*b^10 - 120*a^15*b^12 + 210*a^13*b^14 - 252*a^11*b^16 + 210*a^9*b^18
- 120*a^7*b^20 + 45*a^5*b^22 - 10*a^3*b^24 + a*b^26)*d*cos(d*x + c)^9 - 7*(5*a^23*b^4 - 47*a^21*b^6 + 195*a^1
9*b^8 - 465*a^17*b^10 + 690*a^15*b^12 - 630*a^13*b^14 + 294*a^11*b^16 + 30*a^9*b^18 - 135*a^7*b^20 + 85*a^5*b^
22 - 25*a^3*b^24 + 3*a*b^26)*d*cos(d*x + c)^7 + 7*(3*a^25*b^2 - 20*a^23*b^4 + 38*a^21*b^6 + 60*a^19*b^8 - 435*
a^17*b^10 + 984*a^15*b^12 - 1260*a^13*b^14 + 984*a^11*b^16 - 435*a^9*b^18 + 60*a^7*b^20 + 38*a^5*b^22 - 20*a^3
*b^24 + 3*a*b^26)*d*cos(d*x + c)^5 - (a^27 + 11*a^25*b^2 - 130*a^23*b^4 + 482*a^21*b^6 - 805*a^19*b^8 + 273*a^
17*b^10 + 1428*a^15*b^12 - 3060*a^13*b^14 + 3111*a^11*b^16 - 1795*a^9*b^18 + 526*a^7*b^20 - 14*a^5*b^22 - 35*a
^3*b^24 + 7*a*b^26)*d*cos(d*x + c)^3 + ((a^20*b^7 - 10*a^18*b^9 + 45*a^16*b^11 - 120*a^14*b^13 + 210*a^12*b^15
- 252*a^10*b^17 + 210*a^8*b^19 - 120*a^6*b^21 + 45*a^4*b^23 - 10*a^2*b^25 + b^27)*d*cos(d*x + c)^9 - 3*(7*a^2
2*b^5 - 69*a^20*b^7 + 305*a^18*b^9 - 795*a^16*b^11 + 1350*a^14*b^13 - 1554*a^12*b^15 + 1218*a^10*b^17 - 630*a^
8*b^19 + 195*a^6*b^21 - 25*a^4*b^23 - 3*a^2*b^25 + b^27)*d*cos(d*x + c)^7 + (35*a^24*b^3 - 308*a^22*b^5 + 1158
*a^20*b^7 - 2340*a^18*b^9 + 2445*a^16*b^11 - 360*a^14*b^13 - 2604*a^12*b^15 + 3864*a^10*b^17 - 2835*a^8*b^19 +
1180*a^6*b^21 - 250*a^4*b^23 + 12*a^2*b^25 + 3*b^27)*d*cos(d*x + c)^5 - (7*a^26*b - 35*a^24*b^3 - 14*a^22*b^5
+ 526*a^20*b^7 - 1795*a^18*b^9 + 3111*a^16*b^11 - 3060*a^14*b^13 + 1428*a^12*b^15 + 273*a^10*b^17 - 805*a^8*b
^19 + 482*a^6*b^21 - 130*a^4*b^23 + 11*a^2*b^25 + b^27)*d*cos(d*x + c)^3)*sin(d*x + c)), 1/336*(112*a^18*b - 1
008*a^16*b^3 + 4032*a^14*b^5 - 9408*a^12*b^7 + 14112*a^10*b^9 - 14112*a^8*b^11 + 9408*a^6*b^13 - 4032*a^4*b^15
+ 1008*a^2*b^17 - 112*b^19 - (224*a^12*b^7 - 6272*a^10*b^9 - 201284*a^8*b^11 - 205692*a^6*b^13 + 277535*a^4*b
^15 + 131393*a^2*b^17 + 4096*b^19)*cos(d*x + c)^10 + 14*(336*a^14*b^5 - 9352*a^12*b^7 - 252014*a^10*b^9 - 2301
59*a^8*b^11 + 297312*a^6*b^13 + 165122*a^4*b^15 + 27731*a^2*b^17 + 1024*b^19)*cos(d*x + c)^8 - 35*(224*a^16*b^
3 - 5936*a^14*b^5 - 126448*a^12*b^7 - 243082*a^10*b^9 - 29747*a^8*b^11 + 284285*a^6*b^13 + 109607*a^4*b^15 + 1
0585*a^2*b^17 + 512*b^19)*cos(d*x + c)^6 + 14*(112*a^18*b - 2296*a^16*b^3 - 35224*a^14*b^5 - 308392*a^12*b^7 -
337750*a^10*b^9 + 149783*a^8*b^11 + 394751*a^6*b^13 + 130949*a^4*b^15 + 7427*a^2*b^17 + 640*b^19)*cos(d*x + c
)^4 - 112*(7*a^18*b - 46*a^16*b^3 + 116*a^14*b^5 - 112*a^12*b^7 - 70*a^10*b^9 + 308*a^8*b^11 - 364*a^6*b^13 +
224*a^4*b^15 - 73*a^2*b^17 + 10*b^19)*cos(d*x + c)^2 - 3465*(7*(32*a^8*b^10 + 112*a^6*b^12 + 70*a^4*b^14 + 7*a
^2*b^16)*cos(d*x + c)^9 - 7*(160*a^10*b^8 + 656*a^8*b^10 + 686*a^6*b^12 + 245*a^4*b^14 + 21*a^2*b^16)*cos(d*x
+ c)^7 + 7*(96*a^12*b^6 + 656*a^10*b^8 + 1426*a^8*b^10 + 1057*a^6*b^12 + 280*a^4*b^14 + 21*a^2*b^16)*cos(d*x +
c)^5 - (32*a^14*b^4 + 784*a^12*b^6 + 3542*a^10*b^8 + 5621*a^8*b^10 + 3381*a^6*b^12 + 735*a^4*b^14 + 49*a^2*b^
16)*cos(d*x + c)^3 + ((32*a^7*b^11 + 112*a^5*b^13 + 70*a^3*b^15 + 7*a*b^17)*cos(d*x + c)^9 - 3*(224*a^9*b^9 +
816*a^7*b^11 + 602*a^5*b^13 + 119*a^3*b^15 + 7*a*b^17)*cos(d*x + c)^7 + (1120*a^11*b^7 + 5264*a^9*b^9 + 7250*a
^7*b^11 + 3521*a^5*b^13 + 504*a^3*b^15 + 21*a*b^17)*cos(d*x + c)^5 - (224*a^13*b^5 + 1904*a^11*b^7 + 5082*a^9*
b^9 + 4883*a^7*b^11 + 1827*a^5*b^13 + 217*a^3*b^15 + 7*a*b^17)*cos(d*x + c)^3)*sin(d*x + c))*sqrt(a^2 - b^2)*a
rctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c))) - 7*(16*a^19 - 144*a^17*b^2 + 576*a^15*b^4 - 1344*
a^13*b^6 + 2016*a^11*b^8 - 2016*a^9*b^10 + 1344*a^7*b^12 - 576*a^5*b^14 + 144*a^3*b^16 - 16*a*b^18 - (224*a^13
*b^6 - 6272*a^11*b^8 - 185444*a^9*b^10 - 166092*a^7*b^12 + 256745*a^5*b^14 + 100208*a^3*b^16 + 631*a*b^18)*cos
(d*x + c)^8 + 10*(112*a^15*b^4 - 3080*a^13*b^6 - 73962*a^11*b^8 - 78323*a^9*b^10 + 60829*a^7*b^12 + 73923*a^5*
b^14 + 20401*a^3*b^16 + 100*a*b^18)*cos(d*x + c)^6 - 3*(224*a^17*b^2 - 5712*a^15*b^4 - 95648*a^13*b^6 - 254254
*a^11*b^8 - 120855*a^9*b^10 + 282886*a^7*b^12 + 157892*a^5*b^14 + 35448*a^3*b^16 + 19*a*b^18)*cos(d*x + c)^4 +
16*(2*a^19 - 35*a^17*b^2 + 208*a^15*b^4 - 644*a^13*b^6 + 1204*a^11*b^8 - 1442*a^9*b^10 + 1120*a^7*b^12 - 548*
a^5*b^14 + 154*a^3*b^16 - 19*a*b^18)*cos(d*x + c)^2)*sin(d*x + c))/(7*(a^21*b^6 - 10*a^19*b^8 + 45*a^17*b^10 -
120*a^15*b^12 + 210*a^13*b^14 - 252*a^11*b^16 + 210*a^9*b^18 - 120*a^7*b^20 + 45*a^5*b^22 - 10*a^3*b^24 + a*b
^26)*d*cos(d*x + c)^9 - 7*(5*a^23*b^4 - 47*a^21*b^6 + 195*a^19*b^8 - 465*a^17*b^10 + 690*a^15*b^12 - 630*a^13*
b^14 + 294*a^11*b^16 + 30*a^9*b^18 - 135*a^7*b^20 + 85*a^5*b^22 - 25*a^3*b^24 + 3*a*b^26)*d*cos(d*x + c)^7 + 7
*(3*a^25*b^2 - 20*a^23*b^4 + 38*a^21*b^6 + 60*a^19*b^8 - 435*a^17*b^10 + 984*a^15*b^12 - 1260*a^13*b^14 + 984*
a^11*b^16 - 435*a^9*b^18 + 60*a^7*b^20 + 38*a^5*b^22 - 20*a^3*b^24 + 3*a*b^26)*d*cos(d*x + c)^5 - (a^27 + 11*a
^25*b^2 - 130*a^23*b^4 + 482*a^21*b^6 - 805*a^19*b^8 + 273*a^17*b^10 + 1428*a^15*b^12 - 3060*a^13*b^14 + 3111*
a^11*b^16 - 1795*a^9*b^18 + 526*a^7*b^20 - 14*a^5*b^22 - 35*a^3*b^24 + 7*a*b^26)*d*cos(d*x + c)^3 + ((a^20*b^7
- 10*a^18*b^9 + 45*a^16*b^11 - 120*a^14*b^13 + 210*a^12*b^15 - 252*a^10*b^17 + 210*a^8*b^19 - 120*a^6*b^21 +
45*a^4*b^23 - 10*a^2*b^25 + b^27)*d*cos(d*x + c)^9 - 3*(7*a^22*b^5 - 69*a^20*b^7 + 305*a^18*b^9 - 795*a^16*b^1
1 + 1350*a^14*b^13 - 1554*a^12*b^15 + 1218*a^10*b^17 - 630*a^8*b^19 + 195*a^6*b^21 - 25*a^4*b^23 - 3*a^2*b^25
+ b^27)*d*cos(d*x + c)^7 + (35*a^24*b^3 - 308*a^22*b^5 + 1158*a^20*b^7 - 2340*a^18*b^9 + 2445*a^16*b^11 - 360*
a^14*b^13 - 2604*a^12*b^15 + 3864*a^10*b^17 - 2835*a^8*b^19 + 1180*a^6*b^21 - 250*a^4*b^23 + 12*a^2*b^25 + 3*b
^27)*d*cos(d*x + c)^5 - (7*a^26*b - 35*a^24*b^3 - 14*a^22*b^5 + 526*a^20*b^7 - 1795*a^18*b^9 + 3111*a^16*b^11
- 3060*a^14*b^13 + 1428*a^12*b^15 + 273*a^10*b^17 - 805*a^8*b^19 + 482*a^6*b^21 - 130*a^4*b^23 + 11*a^2*b^25 +
b^27)*d*cos(d*x + c)^3)*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)**4/(a+b*sin(d*x+c))**8,x)

[Out]

Timed out

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Giac [B]  time = 2.7795, size = 4113, normalized size = 6.3 \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)^4/(a+b*sin(d*x+c))^8,x, algorithm="giac")

[Out]

1/168*(3465*(32*a^7*b^4 + 112*a^5*b^6 + 70*a^3*b^8 + 7*a*b^10)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(a) + arct
an((a*tan(1/2*d*x + 1/2*c) + b)/sqrt(a^2 - b^2)))/((a^18 - 9*a^16*b^2 + 36*a^14*b^4 - 84*a^12*b^6 + 126*a^10*b
^8 - 126*a^8*b^10 + 84*a^6*b^12 - 36*a^4*b^14 + 9*a^2*b^16 - b^18)*sqrt(a^2 - b^2)) - 112*(3*a^10*tan(1/2*d*x
+ 1/2*c)^5 - 27*a^8*b^2*tan(1/2*d*x + 1/2*c)^5 - 882*a^6*b^4*tan(1/2*d*x + 1/2*c)^5 - 1638*a^4*b^6*tan(1/2*d*x
+ 1/2*c)^5 - 513*a^2*b^8*tan(1/2*d*x + 1/2*c)^5 - 15*b^10*tan(1/2*d*x + 1/2*c)^5 - 24*a^9*b*tan(1/2*d*x + 1/2
*c)^4 + 216*a^7*b^3*tan(1/2*d*x + 1/2*c)^4 + 1512*a^5*b^5*tan(1/2*d*x + 1/2*c)^4 + 1224*a^3*b^7*tan(1/2*d*x +
1/2*c)^4 + 144*a*b^9*tan(1/2*d*x + 1/2*c)^4 - 2*a^10*tan(1/2*d*x + 1/2*c)^3 + 162*a^8*b^2*tan(1/2*d*x + 1/2*c)
^3 + 1932*a^6*b^4*tan(1/2*d*x + 1/2*c)^3 + 3108*a^4*b^6*tan(1/2*d*x + 1/2*c)^3 + 918*a^2*b^8*tan(1/2*d*x + 1/2
*c)^3 + 26*b^10*tan(1/2*d*x + 1/2*c)^3 - 720*a^7*b^3*tan(1/2*d*x + 1/2*c)^2 - 3024*a^5*b^5*tan(1/2*d*x + 1/2*c
)^2 - 2160*a^3*b^7*tan(1/2*d*x + 1/2*c)^2 - 240*a*b^9*tan(1/2*d*x + 1/2*c)^2 + 3*a^10*tan(1/2*d*x + 1/2*c) - 2
7*a^8*b^2*tan(1/2*d*x + 1/2*c) - 882*a^6*b^4*tan(1/2*d*x + 1/2*c) - 1638*a^4*b^6*tan(1/2*d*x + 1/2*c) - 513*a^
2*b^8*tan(1/2*d*x + 1/2*c) - 15*b^10*tan(1/2*d*x + 1/2*c) - 8*a^9*b + 312*a^7*b^3 + 1512*a^5*b^5 + 1128*a^3*b^
7 + 128*a*b^9)/((a^18 - 9*a^16*b^2 + 36*a^14*b^4 - 84*a^12*b^6 + 126*a^10*b^8 - 126*a^8*b^10 + 84*a^6*b^12 - 3
6*a^4*b^14 + 9*a^2*b^16 - b^18)*(tan(1/2*d*x + 1/2*c)^2 - 1)^3) + (232848*a^18*b^6*tan(1/2*d*x + 1/2*c)^13 + 1
42758*a^16*b^8*tan(1/2*d*x + 1/2*c)^13 + 64911*a^14*b^10*tan(1/2*d*x + 1/2*c)^13 - 28224*a^12*b^12*tan(1/2*d*x
+ 1/2*c)^13 + 12096*a^10*b^14*tan(1/2*d*x + 1/2*c)^13 - 3024*a^8*b^16*tan(1/2*d*x + 1/2*c)^13 + 336*a^6*b^18*
tan(1/2*d*x + 1/2*c)^13 + 155232*a^19*b^5*tan(1/2*d*x + 1/2*c)^12 + 2783088*a^17*b^7*tan(1/2*d*x + 1/2*c)^12 +
2110878*a^15*b^9*tan(1/2*d*x + 1/2*c)^12 + 545811*a^13*b^11*tan(1/2*d*x + 1/2*c)^12 - 169344*a^11*b^13*tan(1/
2*d*x + 1/2*c)^12 + 72576*a^9*b^15*tan(1/2*d*x + 1/2*c)^12 - 18144*a^7*b^17*tan(1/2*d*x + 1/2*c)^12 + 2016*a^5
*b^19*tan(1/2*d*x + 1/2*c)^12 + 3104640*a^18*b^6*tan(1/2*d*x + 1/2*c)^11 + 15506568*a^16*b^8*tan(1/2*d*x + 1/2
*c)^11 + 12397616*a^14*b^10*tan(1/2*d*x + 1/2*c)^11 + 2172366*a^12*b^12*tan(1/2*d*x + 1/2*c)^11 - 451584*a^10*
b^14*tan(1/2*d*x + 1/2*c)^11 + 213696*a^8*b^16*tan(1/2*d*x + 1/2*c)^11 - 57344*a^6*b^18*tan(1/2*d*x + 1/2*c)^1
1 + 6720*a^4*b^20*tan(1/2*d*x + 1/2*c)^11 + 931392*a^19*b^5*tan(1/2*d*x + 1/2*c)^10 + 22042944*a^17*b^7*tan(1/
2*d*x + 1/2*c)^10 + 54377400*a^15*b^9*tan(1/2*d*x + 1/2*c)^10 + 38316040*a^13*b^11*tan(1/2*d*x + 1/2*c)^10 + 5
346390*a^11*b^13*tan(1/2*d*x + 1/2*c)^10 - 685440*a^9*b^15*tan(1/2*d*x + 1/2*c)^10 + 372960*a^7*b^17*tan(1/2*d
*x + 1/2*c)^10 - 108640*a^5*b^19*tan(1/2*d*x + 1/2*c)^10 + 13440*a^3*b^21*tan(1/2*d*x + 1/2*c)^10 + 12030480*a
^18*b^6*tan(1/2*d*x + 1/2*c)^9 + 83208510*a^16*b^8*tan(1/2*d*x + 1/2*c)^9 + 129442775*a^14*b^10*tan(1/2*d*x +
1/2*c)^9 + 68997390*a^12*b^12*tan(1/2*d*x + 1/2*c)^9 + 8026116*a^10*b^14*tan(1/2*d*x + 1/2*c)^9 - 418320*a^8*b
^16*tan(1/2*d*x + 1/2*c)^9 + 328720*a^6*b^18*tan(1/2*d*x + 1/2*c)^9 - 115584*a^4*b^20*tan(1/2*d*x + 1/2*c)^9 +
16128*a^2*b^22*tan(1/2*d*x + 1/2*c)^9 + 2328480*a^19*b^5*tan(1/2*d*x + 1/2*c)^8 + 60558960*a^17*b^7*tan(1/2*d
*x + 1/2*c)^8 + 194655230*a^15*b^9*tan(1/2*d*x + 1/2*c)^8 + 204067311*a^13*b^11*tan(1/2*d*x + 1/2*c)^8 + 74359
166*a^11*b^13*tan(1/2*d*x + 1/2*c)^8 + 6423144*a^9*b^15*tan(1/2*d*x + 1/2*c)^8 + 342720*a^7*b^17*tan(1/2*d*x +
1/2*c)^8 + 38080*a^5*b^19*tan(1/2*d*x + 1/2*c)^8 - 54656*a^3*b^21*tan(1/2*d*x + 1/2*c)^8 + 10752*a*b^23*tan(1
/2*d*x + 1/2*c)^8 + 21732480*a^18*b^6*tan(1/2*d*x + 1/2*c)^7 + 160923840*a^16*b^8*tan(1/2*d*x + 1/2*c)^7 + 294
582904*a^14*b^10*tan(1/2*d*x + 1/2*c)^7 + 198535596*a^12*b^12*tan(1/2*d*x + 1/2*c)^7 + 45251248*a^10*b^14*tan(
1/2*d*x + 1/2*c)^7 + 2197104*a^8*b^16*tan(1/2*d*x + 1/2*c)^7 + 545280*a^6*b^18*tan(1/2*d*x + 1/2*c)^7 - 137728
*a^4*b^20*tan(1/2*d*x + 1/2*c)^7 + 5120*a^2*b^22*tan(1/2*d*x + 1/2*c)^7 + 3072*b^24*tan(1/2*d*x + 1/2*c)^7 + 3
104640*a^19*b^5*tan(1/2*d*x + 1/2*c)^6 + 77468160*a^17*b^7*tan(1/2*d*x + 1/2*c)^6 + 251081600*a^15*b^9*tan(1/2
*d*x + 1/2*c)^6 + 274259160*a^13*b^11*tan(1/2*d*x + 1/2*c)^6 + 105524636*a^11*b^13*tan(1/2*d*x + 1/2*c)^6 + 11
690784*a^9*b^15*tan(1/2*d*x + 1/2*c)^6 + 515760*a^7*b^17*tan(1/2*d*x + 1/2*c)^6 + 38080*a^5*b^19*tan(1/2*d*x +
1/2*c)^6 - 54656*a^3*b^21*tan(1/2*d*x + 1/2*c)^6 + 10752*a*b^23*tan(1/2*d*x + 1/2*c)^6 + 20568240*a^18*b^6*ta
n(1/2*d*x + 1/2*c)^5 + 136444770*a^16*b^8*tan(1/2*d*x + 1/2*c)^5 + 229744669*a^14*b^10*tan(1/2*d*x + 1/2*c)^5
+ 133540988*a^12*b^12*tan(1/2*d*x + 1/2*c)^5 + 22390536*a^10*b^14*tan(1/2*d*x + 1/2*c)^5 - 189280*a^8*b^16*tan
(1/2*d*x + 1/2*c)^5 + 328720*a^6*b^18*tan(1/2*d*x + 1/2*c)^5 - 115584*a^4*b^20*tan(1/2*d*x + 1/2*c)^5 + 16128*
a^2*b^22*tan(1/2*d*x + 1/2*c)^5 + 2328480*a^19*b^5*tan(1/2*d*x + 1/2*c)^4 + 47733840*a^17*b^7*tan(1/2*d*x + 1/
2*c)^4 + 125203386*a^15*b^9*tan(1/2*d*x + 1/2*c)^4 + 105004865*a^13*b^11*tan(1/2*d*x + 1/2*c)^4 + 21568540*a^1
1*b^13*tan(1/2*d*x + 1/2*c)^4 - 612864*a^9*b^15*tan(1/2*d*x + 1/2*c)^4 + 385168*a^7*b^17*tan(1/2*d*x + 1/2*c)^
4 - 108640*a^5*b^19*tan(1/2*d*x + 1/2*c)^4 + 13440*a^3*b^21*tan(1/2*d*x + 1/2*c)^4 + 9934848*a^18*b^6*tan(1/2*
d*x + 1/2*c)^3 + 46275768*a^16*b^8*tan(1/2*d*x + 1/2*c)^3 + 52916248*a^14*b^10*tan(1/2*d*x + 1/2*c)^3 + 117154
94*a^12*b^12*tan(1/2*d*x + 1/2*c)^3 - 403536*a^10*b^14*tan(1/2*d*x + 1/2*c)^3 + 218288*a^8*b^16*tan(1/2*d*x +
1/2*c)^3 - 57344*a^6*b^18*tan(1/2*d*x + 1/2*c)^3 + 6720*a^4*b^20*tan(1/2*d*x + 1/2*c)^3 + 931392*a^19*b^5*tan(
1/2*d*x + 1/2*c)^2 + 11782848*a^17*b^7*tan(1/2*d*x + 1/2*c)^2 + 16561160*a^15*b^9*tan(1/2*d*x + 1/2*c)^2 + 368
5248*a^13*b^11*tan(1/2*d*x + 1/2*c)^2 - 117586*a^11*b^13*tan(1/2*d*x + 1/2*c)^2 + 64736*a^9*b^15*tan(1/2*d*x +
1/2*c)^2 - 17136*a^7*b^17*tan(1/2*d*x + 1/2*c)^2 + 2016*a^5*b^19*tan(1/2*d*x + 1/2*c)^2 + 1940400*a^18*b^6*ta
n(1/2*d*x + 1/2*c) + 2910138*a^16*b^8*tan(1/2*d*x + 1/2*c) + 644413*a^14*b^10*tan(1/2*d*x + 1/2*c) - 21546*a^1
2*b^12*tan(1/2*d*x + 1/2*c) + 11284*a^10*b^14*tan(1/2*d*x + 1/2*c) - 2912*a^8*b^16*tan(1/2*d*x + 1/2*c) + 336*
a^6*b^18*tan(1/2*d*x + 1/2*c) + 155232*a^19*b^5 + 218064*a^17*b^7 + 50666*a^15*b^9 - 3555*a^13*b^11 + 1670*a^1
1*b^13 - 424*a^9*b^15 + 48*a^7*b^17)/((a^25 - 9*a^23*b^2 + 36*a^21*b^4 - 84*a^19*b^6 + 126*a^17*b^8 - 126*a^15
*b^10 + 84*a^13*b^12 - 36*a^11*b^14 + 9*a^9*b^16 - a^7*b^18)*(a*tan(1/2*d*x + 1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2
*c) + a)^7))/d