3.722 \(\int \frac{1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))^3} \, dx\)

Optimal. Leaf size=669 \[ -\frac{b^3 \left (-a^2 b^2 \left (2 c^2-29 d^2\right )+10 a^3 b c d-20 a^4 d^2-4 a b^3 c d-b^4 \left (c^2+12 d^2\right )\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (e+f x)\right )+b}{\sqrt{a^2-b^2}}\right )}{f \left (a^2-b^2\right )^{5/2} (b c-a d)^5}-\frac{d^3 \left (a^2 d^2 \left (2 c^2+d^2\right )-a b \left (10 c^3 d-4 c d^3\right )+b^2 \left (-29 c^2 d^2+20 c^4+12 d^4\right )\right ) \tan ^{-1}\left (\frac{c \tan \left (\frac{1}{2} (e+f x)\right )+d}{\sqrt{c^2-d^2}}\right )}{f \left (c^2-d^2\right )^{5/2} (b c-a d)^5}+\frac{3 d \left (-a^2 b^3 d \left (-12 c^2 d^2+3 c^4+7 d^4\right )-2 a^3 b^2 c d^4-a^4 b \left (3 c^2 d^3-2 d^5\right )+a^5 c d^4+a b^4 c \left (-2 c^2 d^2+c^4+2 d^4\right )+b^5 d \left (-7 c^2 d^2+2 c^4+4 d^4\right )\right ) \cos (e+f x)}{2 f \left (a^2-b^2\right )^2 \left (c^2-d^2\right )^2 (b c-a d)^4 (c+d \sin (e+f x))}-\frac{d \left (2 a^2 b^2 d \left (4 c^2-5 d^2\right )+a^4 d^3-3 a b^3 c \left (c^2-d^2\right )-b^4 d \left (5 c^2-6 d^2\right )\right ) \cos (e+f x)}{2 f \left (a^2-b^2\right )^2 \left (c^2-d^2\right ) (b c-a d)^3 (c+d \sin (e+f x))^2}+\frac{b^2 \left (-7 a^2 d+3 a b c+4 b^2 d\right ) \cos (e+f x)}{2 f \left (a^2-b^2\right )^2 (b c-a d)^2 (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}+\frac{b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2} \]

[Out]

-((b^3*(10*a^3*b*c*d - 4*a*b^3*c*d - 20*a^4*d^2 - a^2*b^2*(2*c^2 - 29*d^2) - b^4*(c^2 + 12*d^2))*ArcTan[(b + a
*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*(b*c - a*d)^5*f)) - (d^3*(a^2*d^2*(2*c^2 + d^2) - a*b*
(10*c^3*d - 4*c*d^3) + b^2*(20*c^4 - 29*c^2*d^2 + 12*d^4))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(
(b*c - a*d)^5*(c^2 - d^2)^(5/2)*f) - (d*(a^4*d^3 - b^4*d*(5*c^2 - 6*d^2) + 2*a^2*b^2*d*(4*c^2 - 5*d^2) - 3*a*b
^3*c*(c^2 - d^2))*Cos[e + f*x])/(2*(a^2 - b^2)^2*(b*c - a*d)^3*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^2) + (b^2*Co
s[e + f*x])/(2*(a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^2) + (b^2*(3*a*b*c - 7*a^
2*d + 4*b^2*d)*Cos[e + f*x])/(2*(a^2 - b^2)^2*(b*c - a*d)^2*f*(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^2) + (
3*d*(a^5*c*d^4 - 2*a^3*b^2*c*d^4 + a*b^4*c*(c^4 - 2*c^2*d^2 + 2*d^4) + b^5*d*(2*c^4 - 7*c^2*d^2 + 4*d^4) - a^2
*b^3*d*(3*c^4 - 12*c^2*d^2 + 7*d^4) - a^4*b*(3*c^2*d^3 - 2*d^5))*Cos[e + f*x])/(2*(a^2 - b^2)^2*(b*c - a*d)^4*
(c^2 - d^2)^2*f*(c + d*Sin[e + f*x]))

________________________________________________________________________________________

Rubi [A]  time = 3.30899, antiderivative size = 669, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {2802, 3055, 3001, 2660, 618, 204} \[ -\frac{b^3 \left (-a^2 b^2 \left (2 c^2-29 d^2\right )+10 a^3 b c d-20 a^4 d^2-4 a b^3 c d-b^4 \left (c^2+12 d^2\right )\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (e+f x)\right )+b}{\sqrt{a^2-b^2}}\right )}{f \left (a^2-b^2\right )^{5/2} (b c-a d)^5}-\frac{d^3 \left (a^2 d^2 \left (2 c^2+d^2\right )-a b \left (10 c^3 d-4 c d^3\right )+b^2 \left (-29 c^2 d^2+20 c^4+12 d^4\right )\right ) \tan ^{-1}\left (\frac{c \tan \left (\frac{1}{2} (e+f x)\right )+d}{\sqrt{c^2-d^2}}\right )}{f \left (c^2-d^2\right )^{5/2} (b c-a d)^5}+\frac{3 d \left (-a^2 b^3 d \left (-12 c^2 d^2+3 c^4+7 d^4\right )-2 a^3 b^2 c d^4-a^4 b \left (3 c^2 d^3-2 d^5\right )+a^5 c d^4+a b^4 c \left (-2 c^2 d^2+c^4+2 d^4\right )+b^5 d \left (-7 c^2 d^2+2 c^4+4 d^4\right )\right ) \cos (e+f x)}{2 f \left (a^2-b^2\right )^2 \left (c^2-d^2\right )^2 (b c-a d)^4 (c+d \sin (e+f x))}-\frac{d \left (2 a^2 b^2 d \left (4 c^2-5 d^2\right )+a^4 d^3-3 a b^3 c \left (c^2-d^2\right )-b^4 d \left (5 c^2-6 d^2\right )\right ) \cos (e+f x)}{2 f \left (a^2-b^2\right )^2 \left (c^2-d^2\right ) (b c-a d)^3 (c+d \sin (e+f x))^2}+\frac{b^2 \left (-7 a^2 d+3 a b c+4 b^2 d\right ) \cos (e+f x)}{2 f \left (a^2-b^2\right )^2 (b c-a d)^2 (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}+\frac{b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^3),x]

[Out]

-((b^3*(10*a^3*b*c*d - 4*a*b^3*c*d - 20*a^4*d^2 - a^2*b^2*(2*c^2 - 29*d^2) - b^4*(c^2 + 12*d^2))*ArcTan[(b + a
*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*(b*c - a*d)^5*f)) - (d^3*(a^2*d^2*(2*c^2 + d^2) - a*b*
(10*c^3*d - 4*c*d^3) + b^2*(20*c^4 - 29*c^2*d^2 + 12*d^4))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(
(b*c - a*d)^5*(c^2 - d^2)^(5/2)*f) - (d*(a^4*d^3 - b^4*d*(5*c^2 - 6*d^2) + 2*a^2*b^2*d*(4*c^2 - 5*d^2) - 3*a*b
^3*c*(c^2 - d^2))*Cos[e + f*x])/(2*(a^2 - b^2)^2*(b*c - a*d)^3*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^2) + (b^2*Co
s[e + f*x])/(2*(a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^2) + (b^2*(3*a*b*c - 7*a^
2*d + 4*b^2*d)*Cos[e + f*x])/(2*(a^2 - b^2)^2*(b*c - a*d)^2*f*(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^2) + (
3*d*(a^5*c*d^4 - 2*a^3*b^2*c*d^4 + a*b^4*c*(c^4 - 2*c^2*d^2 + 2*d^4) + b^5*d*(2*c^4 - 7*c^2*d^2 + 4*d^4) - a^2
*b^3*d*(3*c^4 - 12*c^2*d^2 + 7*d^4) - a^4*b*(3*c^2*d^3 - 2*d^5))*Cos[e + f*x])/(2*(a^2 - b^2)^2*(b*c - a*d)^4*
(c^2 - d^2)^2*f*(c + d*Sin[e + f*x]))

Rule 2802

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -S
imp[(b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 -
b^2)), x] + Dist[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n
*Simp[a*(b*c - a*d)*(m + 1) + b^2*d*(m + n + 2) - (b^2*c + b*(b*c - a*d)*(m + 1))*Sin[e + f*x] - b^2*d*(m + n
+ 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] &
& NeQ[c^2 - d^2, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*n] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !
(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) || EqQ[a, 0])))

Rule 3055

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((A*b^2 - a*b*B + a^2*C)*Cos[e +
f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2)), x] + Dis
t[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b
*c - a*d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*
c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A*b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /;
FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && Lt
Q[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&
!IntegerQ[m]) || EqQ[a, 0])))

Rule 3001

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)])), x_Symbol] :> Dist[(A*b - a*B)/(b*c - a*d), Int[1/(a + b*Sin[e + f*x]), x], x] + Dist[(B*c - A
*d)/(b*c - a*d), Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0]
&& NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

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{1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))^3} \, dx &=\frac{b^2 \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2}-\frac{\int \frac{-2 \left (a b c-a^2 d+2 b^2 d\right )+b (b c-2 a d) \sin (e+f x)+3 b^2 d \sin ^2(e+f x)}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))^3} \, dx}{2 \left (a^2-b^2\right ) (b c-a d)}\\ &=\frac{b^2 \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2}+\frac{b^2 \left (3 a b c-7 a^2 d+4 b^2 d\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}+\frac{\int \frac{-4 a^3 b c d+7 a b^3 c d+2 a^4 d^2+2 a^2 b^2 \left (c^2-10 d^2\right )+b^4 \left (c^2+12 d^2\right )+b d \left (3 b^3 c-4 a^3 d+a b^2 d\right ) \sin (e+f x)-2 b^2 d \left (3 a b c-7 a^2 d+4 b^2 d\right ) \sin ^2(e+f x)}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^3} \, dx}{2 \left (a^2-b^2\right )^2 (b c-a d)^2}\\ &=-\frac{d \left (a^4 d^3-b^4 d \left (5 c^2-6 d^2\right )+2 a^2 b^2 d \left (4 c^2-5 d^2\right )-3 a b^3 c \left (c^2-d^2\right )\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^3 \left (c^2-d^2\right ) f (c+d \sin (e+f x))^2}+\frac{b^2 \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2}+\frac{b^2 \left (3 a b c-7 a^2 d+4 b^2 d\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}+\frac{\int \frac{-2 \left (2 a^5 c d^3+2 a^3 b^2 c d \left (3 c^2-5 d^2\right )-2 a b^4 c d \left (3 c^2-4 d^2\right )-6 a^4 b d^2 \left (c^2-d^2\right )-b^5 \left (c^4+11 c^2 d^2-12 d^4\right )-a^2 b^3 \left (2 c^4-23 c^2 d^2+21 d^4\right )\right )-2 d \left (2 a^4 b c d^2-a^5 d^3-2 b^5 c \left (c^2-2 d^2\right )+2 a^3 b^2 d \left (3 c^2-2 d^2\right )-a b^4 d \left (3 c^2-2 d^2\right )-a^2 b^3 c \left (c^2+3 d^2\right )\right ) \sin (e+f x)+2 b d \left (a^4 d^3-b^4 d \left (5 c^2-6 d^2\right )+2 a^2 b^2 d \left (4 c^2-5 d^2\right )-3 a b^3 c \left (c^2-d^2\right )\right ) \sin ^2(e+f x)}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^2} \, dx}{4 \left (a^2-b^2\right )^2 (b c-a d)^3 \left (c^2-d^2\right )}\\ &=-\frac{d \left (a^4 d^3-b^4 d \left (5 c^2-6 d^2\right )+2 a^2 b^2 d \left (4 c^2-5 d^2\right )-3 a b^3 c \left (c^2-d^2\right )\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^3 \left (c^2-d^2\right ) f (c+d \sin (e+f x))^2}+\frac{b^2 \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2}+\frac{b^2 \left (3 a b c-7 a^2 d+4 b^2 d\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}+\frac{3 d \left (a^5 c d^4-2 a^3 b^2 c d^4+a b^4 c \left (c^4-2 c^2 d^2+2 d^4\right )+b^5 d \left (2 c^4-7 c^2 d^2+4 d^4\right )-a^2 b^3 d \left (3 c^4-12 c^2 d^2+7 d^4\right )-a^4 b \left (3 c^2 d^3-2 d^5\right )\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^4 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))}+\frac{\int \frac{2 \left (b^6 \left (c^2-d^2\right )^2 \left (c^2+12 d^2\right )+2 a^4 b^2 d^2 \left (6 c^4-14 c^2 d^2+5 d^4\right )-2 a^3 b^3 c d \left (4 c^4-16 c^2 d^2+9 d^4\right )+a b^5 c d \left (5 c^4-18 c^2 d^2+10 d^4\right )-a^5 b \left (8 c^3 d^3-5 c d^5\right )+a^2 b^4 \left (2 c^6-28 c^4 d^2+52 c^2 d^4-23 d^6\right )+a^6 \left (2 c^2 d^4+d^6\right )\right )+2 b d (b c+a d) \left (2 a^2 b^2 c^4+b^4 c^4-10 a^3 b c^3 d+4 a b^3 c^3 d+2 a^4 c^2 d^2+8 a^2 b^2 c^2 d^2-10 b^4 c^2 d^2+4 a^3 b c d^3+2 a b^3 c d^3+a^4 d^4-10 a^2 b^2 d^4+6 b^4 d^4\right ) \sin (e+f x)}{(a+b \sin (e+f x)) (c+d \sin (e+f x))} \, dx}{4 \left (a^2-b^2\right )^2 (b c-a d)^4 \left (c^2-d^2\right )^2}\\ &=-\frac{d \left (a^4 d^3-b^4 d \left (5 c^2-6 d^2\right )+2 a^2 b^2 d \left (4 c^2-5 d^2\right )-3 a b^3 c \left (c^2-d^2\right )\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^3 \left (c^2-d^2\right ) f (c+d \sin (e+f x))^2}+\frac{b^2 \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2}+\frac{b^2 \left (3 a b c-7 a^2 d+4 b^2 d\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}+\frac{3 d \left (a^5 c d^4-2 a^3 b^2 c d^4+a b^4 c \left (c^4-2 c^2 d^2+2 d^4\right )+b^5 d \left (2 c^4-7 c^2 d^2+4 d^4\right )-a^2 b^3 d \left (3 c^4-12 c^2 d^2+7 d^4\right )-a^4 b \left (3 c^2 d^3-2 d^5\right )\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^4 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))}-\frac{\left (b^3 \left (10 a^3 b c d-4 a b^3 c d-20 a^4 d^2-a^2 b^2 \left (2 c^2-29 d^2\right )-b^4 \left (c^2+12 d^2\right )\right )\right ) \int \frac{1}{a+b \sin (e+f x)} \, dx}{2 \left (a^2-b^2\right )^2 (b c-a d)^5}-\frac{\left (d^3 \left (a^2 d^2 \left (2 c^2+d^2\right )-a b \left (10 c^3 d-4 c d^3\right )+b^2 \left (20 c^4-29 c^2 d^2+12 d^4\right )\right )\right ) \int \frac{1}{c+d \sin (e+f x)} \, dx}{2 (b c-a d)^5 \left (c^2-d^2\right )^2}\\ &=-\frac{d \left (a^4 d^3-b^4 d \left (5 c^2-6 d^2\right )+2 a^2 b^2 d \left (4 c^2-5 d^2\right )-3 a b^3 c \left (c^2-d^2\right )\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^3 \left (c^2-d^2\right ) f (c+d \sin (e+f x))^2}+\frac{b^2 \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2}+\frac{b^2 \left (3 a b c-7 a^2 d+4 b^2 d\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}+\frac{3 d \left (a^5 c d^4-2 a^3 b^2 c d^4+a b^4 c \left (c^4-2 c^2 d^2+2 d^4\right )+b^5 d \left (2 c^4-7 c^2 d^2+4 d^4\right )-a^2 b^3 d \left (3 c^4-12 c^2 d^2+7 d^4\right )-a^4 b \left (3 c^2 d^3-2 d^5\right )\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^4 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))}-\frac{\left (b^3 \left (10 a^3 b c d-4 a b^3 c d-20 a^4 d^2-a^2 b^2 \left (2 c^2-29 d^2\right )-b^4 \left (c^2+12 d^2\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac{1}{2} (e+f x)\right )\right )}{\left (a^2-b^2\right )^2 (b c-a d)^5 f}-\frac{\left (d^3 \left (a^2 d^2 \left (2 c^2+d^2\right )-a b \left (10 c^3 d-4 c d^3\right )+b^2 \left (20 c^4-29 c^2 d^2+12 d^4\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{c+2 d x+c x^2} \, dx,x,\tan \left (\frac{1}{2} (e+f x)\right )\right )}{(b c-a d)^5 \left (c^2-d^2\right )^2 f}\\ &=-\frac{d \left (a^4 d^3-b^4 d \left (5 c^2-6 d^2\right )+2 a^2 b^2 d \left (4 c^2-5 d^2\right )-3 a b^3 c \left (c^2-d^2\right )\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^3 \left (c^2-d^2\right ) f (c+d \sin (e+f x))^2}+\frac{b^2 \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2}+\frac{b^2 \left (3 a b c-7 a^2 d+4 b^2 d\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}+\frac{3 d \left (a^5 c d^4-2 a^3 b^2 c d^4+a b^4 c \left (c^4-2 c^2 d^2+2 d^4\right )+b^5 d \left (2 c^4-7 c^2 d^2+4 d^4\right )-a^2 b^3 d \left (3 c^4-12 c^2 d^2+7 d^4\right )-a^4 b \left (3 c^2 d^3-2 d^5\right )\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^4 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))}+\frac{\left (2 b^3 \left (10 a^3 b c d-4 a b^3 c d-20 a^4 d^2-a^2 b^2 \left (2 c^2-29 d^2\right )-b^4 \left (c^2+12 d^2\right )\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} (e+f x)\right )\right )}{\left (a^2-b^2\right )^2 (b c-a d)^5 f}+\frac{\left (2 d^3 \left (a^2 d^2 \left (2 c^2+d^2\right )-a b \left (10 c^3 d-4 c d^3\right )+b^2 \left (20 c^4-29 c^2 d^2+12 d^4\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \left (c^2-d^2\right )-x^2} \, dx,x,2 d+2 c \tan \left (\frac{1}{2} (e+f x)\right )\right )}{(b c-a d)^5 \left (c^2-d^2\right )^2 f}\\ &=-\frac{b^3 \left (10 a^3 b c d-4 a b^3 c d-20 a^4 d^2-a^2 b^2 \left (2 c^2-29 d^2\right )-b^4 \left (c^2+12 d^2\right )\right ) \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (e+f x)\right )}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2} (b c-a d)^5 f}-\frac{d^3 \left (a^2 d^2 \left (2 c^2+d^2\right )-a b \left (10 c^3 d-4 c d^3\right )+b^2 \left (20 c^4-29 c^2 d^2+12 d^4\right )\right ) \tan ^{-1}\left (\frac{d+c \tan \left (\frac{1}{2} (e+f x)\right )}{\sqrt{c^2-d^2}}\right )}{(b c-a d)^5 \left (c^2-d^2\right )^{5/2} f}-\frac{d \left (a^4 d^3-b^4 d \left (5 c^2-6 d^2\right )+2 a^2 b^2 d \left (4 c^2-5 d^2\right )-3 a b^3 c \left (c^2-d^2\right )\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^3 \left (c^2-d^2\right ) f (c+d \sin (e+f x))^2}+\frac{b^2 \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2}+\frac{b^2 \left (3 a b c-7 a^2 d+4 b^2 d\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}+\frac{3 d \left (a^5 c d^4-2 a^3 b^2 c d^4+a b^4 c \left (c^4-2 c^2 d^2+2 d^4\right )+b^5 d \left (2 c^4-7 c^2 d^2+4 d^4\right )-a^2 b^3 d \left (3 c^4-12 c^2 d^2+7 d^4\right )-a^4 b \left (3 c^2 d^3-2 d^5\right )\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^4 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))}\\ \end{align*}

Mathematica [B]  time = 8.3794, size = 1815, normalized size = 2.71 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^3),x]

[Out]

-((b^3*(2*a^2*b^2*c^2 + b^4*c^2 - 10*a^3*b*c*d + 4*a*b^3*c*d + 20*a^4*d^2 - 29*a^2*b^2*d^2 + 12*b^4*d^2)*ArcTa
n[(Sec[(e + f*x)/2]*(b*Cos[(e + f*x)/2] + a*Sin[(e + f*x)/2]))/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*(-(b*c) +
a*d)^5*f)) - (d^3*(20*b^2*c^4 - 10*a*b*c^3*d + 2*a^2*c^2*d^2 - 29*b^2*c^2*d^2 + 4*a*b*c*d^3 + a^2*d^4 + 12*b^2
*d^4)*ArcTan[(Sec[(e + f*x)/2]*(d*Cos[(e + f*x)/2] + c*Sin[(e + f*x)/2]))/Sqrt[c^2 - d^2]])/((b*c - a*d)^5*(c^
2 - d^2)^(5/2)*f) + (32*a^2*b^5*c^7*Cos[e + f*x] - 8*b^7*c^7*Cos[e + f*x] - 80*a^3*b^4*c^6*d*Cos[e + f*x] + 68
*a*b^6*c^6*d*Cos[e + f*x] - 92*a^2*b^5*c^5*d^2*Cos[e + f*x] + 38*b^7*c^5*d^2*Cos[e + f*x] + 140*a^3*b^4*c^4*d^
3*Cos[e + f*x] - 122*a*b^6*c^4*d^3*Cos[e + f*x] - 80*a^6*b*c^3*d^4*Cos[e + f*x] + 140*a^4*b^3*c^3*d^4*Cos[e +
f*x] + 48*a^2*b^5*c^3*d^4*Cos[e + f*x] - 72*b^7*c^3*d^4*Cos[e + f*x] + 32*a^7*c^2*d^5*Cos[e + f*x] - 92*a^5*b^
2*c^2*d^5*Cos[e + f*x] + 48*a^3*b^4*c^2*d^5*Cos[e + f*x] + 12*a*b^6*c^2*d^5*Cos[e + f*x] + 68*a^6*b*c*d^6*Cos[
e + f*x] - 122*a^4*b^3*c*d^6*Cos[e + f*x] + 12*a^2*b^5*c*d^6*Cos[e + f*x] + 36*b^7*c*d^6*Cos[e + f*x] - 8*a^7*
d^7*Cos[e + f*x] + 38*a^5*b^2*d^7*Cos[e + f*x] - 72*a^3*b^4*d^7*Cos[e + f*x] + 36*a*b^6*d^7*Cos[e + f*x] - 12*
a*b^6*c^6*d*Cos[3*(e + f*x)] + 28*a^2*b^5*c^5*d^2*Cos[3*(e + f*x)] - 22*b^7*c^5*d^2*Cos[3*(e + f*x)] + 20*a^3*
b^4*c^4*d^3*Cos[3*(e + f*x)] + 10*a*b^6*c^4*d^3*Cos[3*(e + f*x)] + 20*a^4*b^3*c^3*d^4*Cos[3*(e + f*x)] - 96*a^
2*b^5*c^3*d^4*Cos[3*(e + f*x)] + 64*b^7*c^3*d^4*Cos[3*(e + f*x)] + 28*a^5*b^2*c^2*d^5*Cos[3*(e + f*x)] - 96*a^
3*b^4*c^2*d^5*Cos[3*(e + f*x)] + 44*a*b^6*c^2*d^5*Cos[3*(e + f*x)] - 12*a^6*b*c*d^6*Cos[3*(e + f*x)] + 10*a^4*
b^3*c*d^6*Cos[3*(e + f*x)] + 44*a^2*b^5*c*d^6*Cos[3*(e + f*x)] - 36*b^7*c*d^6*Cos[3*(e + f*x)] - 22*a^5*b^2*d^
7*Cos[3*(e + f*x)] + 64*a^3*b^4*d^7*Cos[3*(e + f*x)] - 36*a*b^6*d^7*Cos[3*(e + f*x)] + 12*a*b^6*c^7*Sin[2*(e +
f*x)] - 4*a^2*b^5*c^6*d*Sin[2*(e + f*x)] + 16*b^7*c^6*d*Sin[2*(e + f*x)] - 80*a^3*b^4*c^5*d^2*Sin[2*(e + f*x)
] + 38*a*b^6*c^5*d^2*Sin[2*(e + f*x)] - 10*a^2*b^5*c^4*d^3*Sin[2*(e + f*x)] - 20*b^7*c^4*d^3*Sin[2*(e + f*x)]
- 80*a^5*b^2*c^3*d^4*Sin[2*(e + f*x)] + 320*a^3*b^4*c^3*d^4*Sin[2*(e + f*x)] - 192*a*b^6*c^3*d^4*Sin[2*(e + f*
x)] - 4*a^6*b*c^2*d^5*Sin[2*(e + f*x)] - 10*a^4*b^3*c^2*d^5*Sin[2*(e + f*x)] + 64*a^2*b^5*c^2*d^5*Sin[2*(e + f
*x)] - 26*b^7*c^2*d^5*Sin[2*(e + f*x)] + 12*a^7*c*d^6*Sin[2*(e + f*x)] + 38*a^5*b^2*c*d^6*Sin[2*(e + f*x)] - 1
92*a^3*b^4*c*d^6*Sin[2*(e + f*x)] + 124*a*b^6*c*d^6*Sin[2*(e + f*x)] + 16*a^6*b*d^7*Sin[2*(e + f*x)] - 20*a^4*
b^3*d^7*Sin[2*(e + f*x)] - 26*a^2*b^5*d^7*Sin[2*(e + f*x)] + 24*b^7*d^7*Sin[2*(e + f*x)] - 3*a*b^6*c^5*d^2*Sin
[4*(e + f*x)] + 9*a^2*b^5*c^4*d^3*Sin[4*(e + f*x)] - 6*b^7*c^4*d^3*Sin[4*(e + f*x)] + 6*a*b^6*c^3*d^4*Sin[4*(e
+ f*x)] + 9*a^4*b^3*c^2*d^5*Sin[4*(e + f*x)] - 36*a^2*b^5*c^2*d^5*Sin[4*(e + f*x)] + 21*b^7*c^2*d^5*Sin[4*(e
+ f*x)] - 3*a^5*b^2*c*d^6*Sin[4*(e + f*x)] + 6*a^3*b^4*c*d^6*Sin[4*(e + f*x)] - 6*a*b^6*c*d^6*Sin[4*(e + f*x)]
- 6*a^4*b^3*d^7*Sin[4*(e + f*x)] + 21*a^2*b^5*d^7*Sin[4*(e + f*x)] - 12*b^7*d^7*Sin[4*(e + f*x)])/(16*(a^2 -
b^2)^2*(-(b*c) + a*d)^4*(c^2 - d^2)^2*f*(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^2)

________________________________________________________________________________________

Maple [B]  time = 0.234, size = 7348, normalized size = 11. \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^3,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(1/(a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [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(1/(a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^3,x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

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(1/(a+b*sin(f*x+e))**3/(c+d*sin(f*x+e))**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [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(1/(a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^3,x, algorithm="giac")

[Out]

Timed out