3.5.72 \(\int \frac {\sec ^4(c+d x)}{(a+b \sin (c+d x))^8} \, dx\) [472]

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

[Out]

165/8*a*b^4*(32*a^6+112*a^4*b^2+70*a^2*b^4+7*b^6)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/(a^2-b^2)^(
19/2)/d+1/7*b*sec(d*x+c)^3/(a^2-b^2)/d/(a+b*sin(d*x+c))^7+17/42*a*b*sec(d*x+c)^3/(a^2-b^2)^2/d/(a+b*sin(d*x+c)
)^6+1/14*b*(13*a^2+4*b^2)*sec(d*x+c)^3/(a^2-b^2)^3/d/(a+b*sin(d*x+c))^5+1/56*a*b*(118*a^2+103*b^2)*sec(d*x+c)^
3/(a^2-b^2)^4/d/(a+b*sin(d*x+c))^4+1/168*b*(882*a^4+1421*a^2*b^2+128*b^4)*sec(d*x+c)^3/(a^2-b^2)^5/d/(a+b*sin(
d*x+c))^3+13/112*a*b*(140*a^4+336*a^2*b^2+85*b^4)*sec(d*x+c)^3/(a^2-b^2)^6/d/(a+b*sin(d*x+c))^2+1/112*b*(9212*
a^6+28420*a^4*b^2+12907*a^2*b^4+512*b^6)*sec(d*x+c)^3/(a^2-b^2)^7/d/(a+b*sin(d*x+c))-1/336*sec(d*x+c)^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(d*
x+c))/(a^2-b^2)^8/d+1/336*sec(d*x+c)*(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(d*x+c))/(a^2-b^2)^9/d

________________________________________________________________________________________

Rubi [A]
time = 1.38, antiderivative size = 653, normalized size of antiderivative = 1.00, 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 = {2773, 2943, 2945, 12, 2739, 632, 210} \begin {gather*} \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 {13 a b \left (140 a^4+336 a^2 b^2+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 (882 a^4+1421 a^2 b^2+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 {165 a b^4 \left (32 a^6+112 a^4 b^2+70 a^2 b^4+7 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 )^{19/2}}+\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 d \left (a^2-b^2\right )^7 (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 d \left (a^2-b^2\right )^8}+\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 d \left (a^2-b^2\right )^9} \end {gather*}

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 12

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

Rule 210

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])

Rule 632

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 2739

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 2773

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]

Rule 2943

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]

Rule 2945

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]

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 ) \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 )^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 ) \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 )^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*}

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Mathematica [A]
time = 5.25, size = 597, normalized size = 0.91 \begin {gather*} \frac {\frac {6930 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 )}{\left (a^2-b^2\right )^{19/2}}+\frac {48 b^5 \cos (c+d x)}{\left (a^2-b^2\right )^3 (a+b \sin (c+d x))^7}+\frac {328 a b^5 \cos (c+d x)}{\left (a^2-b^2\right )^4 (a+b \sin (c+d x))^6}+\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 {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 {2 b^5 \left (6058 a^4+5273 a^2 b^2+296 b^4\right ) \cos (c+d x)}{\left (a^2-b^2\right )^7 (a+b \sin (c+d x))^3}+\frac {a b^5 \left (33284 a^4+48820 a^2 b^2+8287 b^4\right ) \cos (c+d x)}{\left (a^2-b^2\right )^8 (a+b \sin (c+d x))^2}+\frac {b^5 \left (103844 a^6+234272 a^4 b^2+81057 a^2 b^4+2528 b^6\right ) \cos (c+d x)}{\left (a^2-b^2\right )^9 (a+b \sin (c+d x))}+\frac {112 \sec ^3(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}+\frac {224 \sec (c+d x) \left (12 \left (15 a^7 b^3+63 a^5 b^5+45 a^3 b^7+5 a b^9\right )+\left (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}\right ) \sin (c+d x)\right )}{\left (a^2-b^2\right )^9}}{336 d} \end {gather*}

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)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1379\) vs. \(2(628)=1256\).
time = 4.30, size = 1380, normalized size = 2.11

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1380\)
default	\(\text {Expression too large to display}\)	\(1380\)
risch	\(\text {Expression too large to display}\)	\(3064\)

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,method=_RETURNVERBOSE)

[Out]

1/d*(-1/3/(a+b)^8/(tan(1/2*d*x+1/2*c)-1)^3-1/2/(a+b)^8/(tan(1/2*d*x+1/2*c)-1)^2-(a+5*b)/(a+b)^9/(tan(1/2*d*x+1
/2*c)-1)-1/3/(a-b)^8/(tan(1/2*d*x+1/2*c)+1)^3+1/2/(a-b)^8/(tan(1/2*d*x+1/2*c)+1)^2-(a-5*b)/(a-b)^9/(tan(1/2*d*
x+1/2*c)+1)+2*b^4/(a-b)^9/(a+b)^9*((1/16*b^2*(11088*a^12+6798*a^10*b^2+3091*a^8*b^4-1344*a^6*b^6+576*a^4*b^8-1
44*a^2*b^10+16*b^12)/a*tan(1/2*d*x+1/2*c)^13+1/16*b*(7392*a^14+132528*a^12*b^2+100518*a^10*b^4+25991*a^8*b^6-8
064*a^6*b^8+3456*a^4*b^10-864*a^2*b^12+96*b^14)/a^2*tan(1/2*d*x+1/2*c)^12+1/24/a^3*b^2*(221760*a^14+1107612*a^
12*b^2+885544*a^10*b^4+155169*a^8*b^6-32256*a^6*b^8+15264*a^4*b^10-4096*a^2*b^12+480*b^14)*tan(1/2*d*x+1/2*c)^
11+1/24/a^4*b*(66528*a^16+1574496*a^14*b^2+3884100*a^12*b^4+2736860*a^10*b^6+381885*a^8*b^8-48960*a^6*b^10+266
40*a^4*b^12-7760*a^2*b^14+960*b^16)*tan(1/2*d*x+1/2*c)^10+1/48/a^5*b^2*(1718640*a^16+11886930*a^14*b^2+1849182
5*a^12*b^4+9856770*a^10*b^6+1146588*a^8*b^8-59760*a^6*b^10+46960*a^4*b^12-16512*a^2*b^14+2304*b^16)*tan(1/2*d*
x+1/2*c)^9+1/48/a^6*b*(332640*a^18+8651280*a^16*b^2+27807890*a^14*b^4+29152473*a^12*b^6+10622738*a^10*b^8+9175
92*a^8*b^10+48960*a^6*b^12+5440*a^4*b^14-7808*a^2*b^16+1536*b^18)*tan(1/2*d*x+1/2*c)^8+1/84/a^7*b^2*(5433120*a
^18+40230960*a^16*b^2+73645726*a^14*b^4+49633899*a^12*b^6+11312812*a^10*b^8+549276*a^8*b^10+136320*a^6*b^12-34
432*a^4*b^14+1280*a^2*b^16+768*b^18)*tan(1/2*d*x+1/2*c)^7+1/12/a^6*b*(110880*a^18+2766720*a^16*b^2+8967200*a^1
4*b^4+9794970*a^12*b^6+3768737*a^10*b^8+417528*a^8*b^10+18420*a^6*b^12+1360*a^4*b^14-1952*a^2*b^16+384*b^18)*t
an(1/2*d*x+1/2*c)^6+1/48/a^5*b^2*(2938320*a^16+19492110*a^14*b^2+32820667*a^12*b^4+19077284*a^10*b^6+3198648*a
^8*b^8-27040*a^6*b^10+46960*a^4*b^12-16512*a^2*b^14+2304*b^16)*tan(1/2*d*x+1/2*c)^5+1/48/a^4*b*(332640*a^16+68
19120*a^14*b^2+17886198*a^12*b^4+15000695*a^10*b^6+3081220*a^8*b^8-87552*a^6*b^10+55024*a^4*b^12-15520*a^2*b^1
4+1920*b^16)*tan(1/2*d*x+1/2*c)^4+1/24/a^3*b^2*(709632*a^14+3305412*a^12*b^2+3779732*a^10*b^4+836821*a^8*b^6-2
8824*a^6*b^8+15592*a^4*b^10-4096*a^2*b^12+480*b^14)*tan(1/2*d*x+1/2*c)^3+1/24*b*(66528*a^14+841632*a^12*b^2+11
82940*a^10*b^4+263232*a^8*b^6-8399*a^6*b^8+4624*a^4*b^10-1224*a^2*b^12+144*b^14)/a^2*tan(1/2*d*x+1/2*c)^2+1/48
*b^2*(277200*a^12+415734*a^10*b^2+92059*a^8*b^4-3078*a^6*b^6+1612*a^4*b^8-416*a^2*b^10+48*b^12)/a*tan(1/2*d*x+
1/2*c)+1/336*(155232*a^12+218064*a^10*b^2+50666*a^8*b^4-3555*a^6*b^6+1670*a^4*b^8-424*a^2*b^10+48*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+165/16*a*(32*a^6+112*a^4*b^2+70*a^2*b^4+7*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))))

________________________________________________________________________________________

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

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 >> 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?`
for more de

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2208 vs. \(2 (628) = 1256\).
time = 0.90, size = 4500, normalized size = 6.89 \begin {gather*} \text {Too large to display} \end {gather*}

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*(...

________________________________________________________________________________________

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

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

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 3047 vs. \(2 (628) = 1256\).
time = 6.09, size = 3047, normalized size = 4.67 \begin {gather*} \text {Too large to display} \end {gather*}

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...

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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(c + d*x)^4*(a + b*sin(c + d*x))^8),x)

[Out]

\text{Hanged}

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