3.285 \(\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c+d \sin (e+f x))^3} \, dx\)

Optimal. Leaf size=508 \[ -\frac{d^2 \left (A d \left (20 c^2+30 c d+13 d^2\right )-3 B \left (8 c^2 d+4 c^3+7 c d^2+2 d^3\right )\right ) \tan ^{-1}\left (\frac{c \tan \left (\frac{1}{2} (e+f x)\right )+d}{\sqrt{c^2-d^2}}\right )}{a^3 f (c-d)^5 (c+d)^2 \sqrt{c^2-d^2}}-\frac{d \left (A \left (142 c^2 d^2-30 c^3 d+4 c^4+525 c d^3+304 d^4\right )+3 B \left (-119 c^2 d^2-20 c^3 d+2 c^4-130 c d^3-48 d^4\right )\right ) \cos (e+f x)}{30 a^3 f (c-d)^5 (c+d)^2 (c+d \sin (e+f x))}-\frac{d \left (A \left (-30 c^2 d+4 c^3+146 c d^2+195 d^3\right )+3 B \left (-20 c^2 d+2 c^3-57 c d^2-30 d^3\right )\right ) \cos (e+f x)}{30 a^3 f (c-d)^4 (c+d) (c+d \sin (e+f x))^2}-\frac{\left (A \left (2 c^2-15 c d+76 d^2\right )+3 B \left (c^2-10 c d-12 d^2\right )\right ) \cos (e+f x)}{15 f (c-d)^3 \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^2}-\frac{(2 A c-11 A d+3 B c+6 B d) \cos (e+f x)}{15 a f (c-d)^2 (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}-\frac{(A-B) \cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2} \]

[Out]

-((d^2*(A*d*(20*c^2 + 30*c*d + 13*d^2) - 3*B*(4*c^3 + 8*c^2*d + 7*c*d^2 + 2*d^3))*ArcTan[(d + c*Tan[(e + f*x)/
2])/Sqrt[c^2 - d^2]])/(a^3*(c - d)^5*(c + d)^2*Sqrt[c^2 - d^2]*f)) - (d*(3*B*(2*c^3 - 20*c^2*d - 57*c*d^2 - 30
*d^3) + A*(4*c^3 - 30*c^2*d + 146*c*d^2 + 195*d^3))*Cos[e + f*x])/(30*a^3*(c - d)^4*(c + d)*f*(c + d*Sin[e + f
*x])^2) - ((A - B)*Cos[e + f*x])/(5*(c - d)*f*(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^2) - ((2*A*c + 3*B*c
- 11*A*d + 6*B*d)*Cos[e + f*x])/(15*a*(c - d)^2*f*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^2) - ((3*B*(c^2
- 10*c*d - 12*d^2) + A*(2*c^2 - 15*c*d + 76*d^2))*Cos[e + f*x])/(15*(c - d)^3*f*(a^3 + a^3*Sin[e + f*x])*(c +
d*Sin[e + f*x])^2) - (d*(3*B*(2*c^4 - 20*c^3*d - 119*c^2*d^2 - 130*c*d^3 - 48*d^4) + A*(4*c^4 - 30*c^3*d + 14
2*c^2*d^2 + 525*c*d^3 + 304*d^4))*Cos[e + f*x])/(30*a^3*(c - d)^5*(c + d)^2*f*(c + d*Sin[e + f*x]))

________________________________________________________________________________________

Rubi [A]  time = 1.44656, antiderivative size = 508, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {2978, 2754, 12, 2660, 618, 204} \[ -\frac{d^2 \left (A d \left (20 c^2+30 c d+13 d^2\right )-3 B \left (8 c^2 d+4 c^3+7 c d^2+2 d^3\right )\right ) \tan ^{-1}\left (\frac{c \tan \left (\frac{1}{2} (e+f x)\right )+d}{\sqrt{c^2-d^2}}\right )}{a^3 f (c-d)^5 (c+d)^2 \sqrt{c^2-d^2}}-\frac{d \left (A \left (142 c^2 d^2-30 c^3 d+4 c^4+525 c d^3+304 d^4\right )+3 B \left (-119 c^2 d^2-20 c^3 d+2 c^4-130 c d^3-48 d^4\right )\right ) \cos (e+f x)}{30 a^3 f (c-d)^5 (c+d)^2 (c+d \sin (e+f x))}-\frac{d \left (A \left (-30 c^2 d+4 c^3+146 c d^2+195 d^3\right )+3 B \left (-20 c^2 d+2 c^3-57 c d^2-30 d^3\right )\right ) \cos (e+f x)}{30 a^3 f (c-d)^4 (c+d) (c+d \sin (e+f x))^2}-\frac{\left (A \left (2 c^2-15 c d+76 d^2\right )+3 B \left (c^2-10 c d-12 d^2\right )\right ) \cos (e+f x)}{15 f (c-d)^3 \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^2}-\frac{(2 A c-11 A d+3 B c+6 B d) \cos (e+f x)}{15 a f (c-d)^2 (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}-\frac{(A-B) \cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((d^2*(A*d*(20*c^2 + 30*c*d + 13*d^2) - 3*B*(4*c^3 + 8*c^2*d + 7*c*d^2 + 2*d^3))*ArcTan[(d + c*Tan[(e + f*x)/
2])/Sqrt[c^2 - d^2]])/(a^3*(c - d)^5*(c + d)^2*Sqrt[c^2 - d^2]*f)) - (d*(3*B*(2*c^3 - 20*c^2*d - 57*c*d^2 - 30
*d^3) + A*(4*c^3 - 30*c^2*d + 146*c*d^2 + 195*d^3))*Cos[e + f*x])/(30*a^3*(c - d)^4*(c + d)*f*(c + d*Sin[e + f
*x])^2) - ((A - B)*Cos[e + f*x])/(5*(c - d)*f*(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^2) - ((2*A*c + 3*B*c
- 11*A*d + 6*B*d)*Cos[e + f*x])/(15*a*(c - d)^2*f*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^2) - ((3*B*(c^2
- 10*c*d - 12*d^2) + A*(2*c^2 - 15*c*d + 76*d^2))*Cos[e + f*x])/(15*(c - d)^3*f*(a^3 + a^3*Sin[e + f*x])*(c +
d*Sin[e + f*x])^2) - (d*(3*B*(2*c^4 - 20*c^3*d - 119*c^2*d^2 - 130*c*d^3 - 48*d^4) + A*(4*c^4 - 30*c^3*d + 14
2*c^2*d^2 + 525*c*d^3 + 304*d^4))*Cos[e + f*x])/(30*a^3*(c - d)^5*(c + d)^2*f*(c + d*Sin[e + f*x]))

Rule 2978

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

Rule 2754

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[((
b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(f*(m + 1)*(a^2 - b^2)), x] + Dist[1/((m + 1)*(a^2 - b^2
)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*(m + 2)*Sin[e + f*x], x], x], x] /
; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -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{A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c+d \sin (e+f x))^3} \, dx &=-\frac{(A-B) \cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^2}-\frac{\int \frac{-a (2 A c+3 B c-7 A d+2 B d)-4 a (A-B) d \sin (e+f x)}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))^3} \, dx}{5 a^2 (c-d)}\\ &=-\frac{(A-B) \cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^2}-\frac{(2 A c+3 B c-11 A d+6 B d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^2}+\frac{\int \frac{a^2 \left (3 B \left (c^2-7 c d-6 d^2\right )+A \left (2 c^2-9 c d+43 d^2\right )\right )+3 a^2 d (A (2 c-11 d)+3 B (c+2 d)) \sin (e+f x)}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^3} \, dx}{15 a^4 (c-d)^2}\\ &=-\frac{(A-B) \cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^2}-\frac{(2 A c+3 B c-11 A d+6 B d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^2}-\frac{\left (3 B \left (c^2-10 c d-12 d^2\right )+A \left (2 c^2-15 c d+76 d^2\right )\right ) \cos (e+f x)}{15 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^2}-\frac{\int \frac{-3 a^3 d^2 (2 A c+33 B c-65 A d+30 B d)-2 a^3 d \left (3 B \left (c^2-10 c d-12 d^2\right )+A \left (2 c^2-15 c d+76 d^2\right )\right ) \sin (e+f x)}{(c+d \sin (e+f x))^3} \, dx}{15 a^6 (c-d)^3}\\ &=-\frac{d \left (3 B \left (2 c^3-20 c^2 d-57 c d^2-30 d^3\right )+A \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right )\right ) \cos (e+f x)}{30 a^3 (c-d)^4 (c+d) f (c+d \sin (e+f x))^2}-\frac{(A-B) \cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^2}-\frac{(2 A c+3 B c-11 A d+6 B d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^2}-\frac{\left (3 B \left (c^2-10 c d-12 d^2\right )+A \left (2 c^2-15 c d+76 d^2\right )\right ) \cos (e+f x)}{15 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^2}+\frac{\int \frac{2 a^3 d^2 \left (2 A c^2+93 B c^2-165 A c d+150 B c d-152 A d^2+72 B d^2\right )+a^3 d \left (3 B \left (2 c^3-20 c^2 d-57 c d^2-30 d^3\right )+A \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right )\right ) \sin (e+f x)}{(c+d \sin (e+f x))^2} \, dx}{30 a^6 (c-d)^4 (c+d)}\\ &=-\frac{d \left (3 B \left (2 c^3-20 c^2 d-57 c d^2-30 d^3\right )+A \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right )\right ) \cos (e+f x)}{30 a^3 (c-d)^4 (c+d) f (c+d \sin (e+f x))^2}-\frac{(A-B) \cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^2}-\frac{(2 A c+3 B c-11 A d+6 B d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^2}-\frac{\left (3 B \left (c^2-10 c d-12 d^2\right )+A \left (2 c^2-15 c d+76 d^2\right )\right ) \cos (e+f x)}{15 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^2}-\frac{d \left (3 B \left (2 c^4-20 c^3 d-119 c^2 d^2-130 c d^3-48 d^4\right )+A \left (4 c^4-30 c^3 d+142 c^2 d^2+525 c d^3+304 d^4\right )\right ) \cos (e+f x)}{30 a^3 (c-d)^5 (c+d)^2 f (c+d \sin (e+f x))}-\frac{\int \frac{15 a^3 d^2 \left (A d \left (20 c^2+30 c d+13 d^2\right )-3 B \left (4 c^3+8 c^2 d+7 c d^2+2 d^3\right )\right )}{c+d \sin (e+f x)} \, dx}{30 a^6 (c-d)^5 (c+d)^2}\\ &=-\frac{d \left (3 B \left (2 c^3-20 c^2 d-57 c d^2-30 d^3\right )+A \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right )\right ) \cos (e+f x)}{30 a^3 (c-d)^4 (c+d) f (c+d \sin (e+f x))^2}-\frac{(A-B) \cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^2}-\frac{(2 A c+3 B c-11 A d+6 B d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^2}-\frac{\left (3 B \left (c^2-10 c d-12 d^2\right )+A \left (2 c^2-15 c d+76 d^2\right )\right ) \cos (e+f x)}{15 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^2}-\frac{d \left (3 B \left (2 c^4-20 c^3 d-119 c^2 d^2-130 c d^3-48 d^4\right )+A \left (4 c^4-30 c^3 d+142 c^2 d^2+525 c d^3+304 d^4\right )\right ) \cos (e+f x)}{30 a^3 (c-d)^5 (c+d)^2 f (c+d \sin (e+f x))}-\frac{\left (d^2 \left (A d \left (20 c^2+30 c d+13 d^2\right )-3 B \left (4 c^3+8 c^2 d+7 c d^2+2 d^3\right )\right )\right ) \int \frac{1}{c+d \sin (e+f x)} \, dx}{2 a^3 (c-d)^5 (c+d)^2}\\ &=-\frac{d \left (3 B \left (2 c^3-20 c^2 d-57 c d^2-30 d^3\right )+A \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right )\right ) \cos (e+f x)}{30 a^3 (c-d)^4 (c+d) f (c+d \sin (e+f x))^2}-\frac{(A-B) \cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^2}-\frac{(2 A c+3 B c-11 A d+6 B d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^2}-\frac{\left (3 B \left (c^2-10 c d-12 d^2\right )+A \left (2 c^2-15 c d+76 d^2\right )\right ) \cos (e+f x)}{15 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^2}-\frac{d \left (3 B \left (2 c^4-20 c^3 d-119 c^2 d^2-130 c d^3-48 d^4\right )+A \left (4 c^4-30 c^3 d+142 c^2 d^2+525 c d^3+304 d^4\right )\right ) \cos (e+f x)}{30 a^3 (c-d)^5 (c+d)^2 f (c+d \sin (e+f x))}-\frac{\left (d^2 \left (A d \left (20 c^2+30 c d+13 d^2\right )-3 B \left (4 c^3+8 c^2 d+7 c d^2+2 d^3\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 )}{a^3 (c-d)^5 (c+d)^2 f}\\ &=-\frac{d \left (3 B \left (2 c^3-20 c^2 d-57 c d^2-30 d^3\right )+A \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right )\right ) \cos (e+f x)}{30 a^3 (c-d)^4 (c+d) f (c+d \sin (e+f x))^2}-\frac{(A-B) \cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^2}-\frac{(2 A c+3 B c-11 A d+6 B d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^2}-\frac{\left (3 B \left (c^2-10 c d-12 d^2\right )+A \left (2 c^2-15 c d+76 d^2\right )\right ) \cos (e+f x)}{15 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^2}-\frac{d \left (3 B \left (2 c^4-20 c^3 d-119 c^2 d^2-130 c d^3-48 d^4\right )+A \left (4 c^4-30 c^3 d+142 c^2 d^2+525 c d^3+304 d^4\right )\right ) \cos (e+f x)}{30 a^3 (c-d)^5 (c+d)^2 f (c+d \sin (e+f x))}+\frac{\left (2 d^2 \left (A d \left (20 c^2+30 c d+13 d^2\right )-3 B \left (4 c^3+8 c^2 d+7 c d^2+2 d^3\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 )}{a^3 (c-d)^5 (c+d)^2 f}\\ &=-\frac{d^2 \left (A d \left (20 c^2+30 c d+13 d^2\right )-3 B \left (4 c^3+8 c^2 d+7 c d^2+2 d^3\right )\right ) \tan ^{-1}\left (\frac{d+c \tan \left (\frac{1}{2} (e+f x)\right )}{\sqrt{c^2-d^2}}\right )}{a^3 (c-d)^5 (c+d)^2 \sqrt{c^2-d^2} f}-\frac{d \left (3 B \left (2 c^3-20 c^2 d-57 c d^2-30 d^3\right )+A \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right )\right ) \cos (e+f x)}{30 a^3 (c-d)^4 (c+d) f (c+d \sin (e+f x))^2}-\frac{(A-B) \cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^2}-\frac{(2 A c+3 B c-11 A d+6 B d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^2}-\frac{\left (3 B \left (c^2-10 c d-12 d^2\right )+A \left (2 c^2-15 c d+76 d^2\right )\right ) \cos (e+f x)}{15 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^2}-\frac{d \left (3 B \left (2 c^4-20 c^3 d-119 c^2 d^2-130 c d^3-48 d^4\right )+A \left (4 c^4-30 c^3 d+142 c^2 d^2+525 c d^3+304 d^4\right )\right ) \cos (e+f x)}{30 a^3 (c-d)^5 (c+d)^2 f (c+d \sin (e+f x))}\\ \end{align*}

Mathematica [A]  time = 4.57133, size = 548, normalized size = 1.08 \[ \frac{\left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right ) \left (\frac{15 d^3 \left (B \left (7 c^2+6 c d+2 d^2\right )-3 A d (3 c+2 d)\right ) \cos (e+f x) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^5}{(c+d)^2 (c+d \sin (e+f x))}+4 \left (A \left (2 c^2-19 c d+107 d^2\right )+3 B \left (c^2-12 c d-19 d^2\right )\right ) \sin \left (\frac{1}{2} (e+f x)\right ) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^4+\frac{30 d^2 \left (3 B \left (8 c^2 d+4 c^3+7 c d^2+2 d^3\right )-A d \left (20 c^2+30 c d+13 d^2\right )\right ) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^5 \tan ^{-1}\left (\frac{c \tan \left (\frac{1}{2} (e+f x)\right )+d}{\sqrt{c^2-d^2}}\right )}{(c+d)^2 \sqrt{c^2-d^2}}+\frac{15 d^3 (c-d) (B c-A d) \cos (e+f x) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^5}{(c+d) (c+d \sin (e+f x))^2}+12 (A-B) (c-d)^2 \sin \left (\frac{1}{2} (e+f x)\right )-2 (c-d) (A (2 c-17 d)+3 B (c+4 d)) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3+4 (c-d) (A (2 c-17 d)+3 B (c+4 d)) \sin \left (\frac{1}{2} (e+f x)\right ) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^2+6 (B-A) (c-d)^2 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )\right )}{30 a^3 f (c-d)^5 (\sin (e+f x)+1)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(12*(A - B)*(c - d)^2*Sin[(e + f*x)/2] + 6*(-A + B)*(c - d)^2*(Cos[(e +
f*x)/2] + Sin[(e + f*x)/2]) + 4*(c - d)*(A*(2*c - 17*d) + 3*B*(c + 4*d))*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] +
Sin[(e + f*x)/2])^2 - 2*(c - d)*(A*(2*c - 17*d) + 3*B*(c + 4*d))*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + 4*
(3*B*(c^2 - 12*c*d - 19*d^2) + A*(2*c^2 - 19*c*d + 107*d^2))*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x
)/2])^4 + (30*d^2*(-(A*d*(20*c^2 + 30*c*d + 13*d^2)) + 3*B*(4*c^3 + 8*c^2*d + 7*c*d^2 + 2*d^3))*ArcTan[(d + c*
Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)/((c + d)^2*Sqrt[c^2 - d^2]) + (15*
(c - d)*d^3*(B*c - A*d)*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)/((c + d)*(c + d*Sin[e + f*x])^2)
+ (15*d^3*(-3*A*d*(3*c + 2*d) + B*(7*c^2 + 6*c*d + 2*d^2))*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])
^5)/((c + d)^2*(c + d*Sin[e + f*x]))))/(30*a^3*(c - d)^5*f*(1 + Sin[e + f*x])^3)

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Maple [B]  time = 0.193, size = 2918, normalized size = 5.7 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^3,x)

[Out]

12/f/a^3/(c-d)^5/(tan(1/2*f*x+1/2*e)+1)*B*d^2-20/f/a^3/(c-d)^5/(tan(1/2*f*x+1/2*e)+1)*A*d^2-10/f/a^3/(c-d)^4/(
tan(1/2*f*x+1/2*e)+1)^2*A*d-2/f/a^3/(c-d)^4/(tan(1/2*f*x+1/2*e)+1)^2*B*c+8/f/a^3/(c-d)^4/(tan(1/2*f*x+1/2*e)+1
)^2*B*d-16/3/f/a^3/(c-d)^4/(tan(1/2*f*x+1/2*e)+1)^3*A*c+28/3/f/a^3/(c-d)^4/(tan(1/2*f*x+1/2*e)+1)^3*A*d+4/f/a^
3/(c-d)^4/(tan(1/2*f*x+1/2*e)+1)^3*B*c-8/f/a^3/(c-d)^4/(tan(1/2*f*x+1/2*e)+1)^3*B*d-2/f/a^3/(c-d)^5/(tan(1/2*f
*x+1/2*e)+1)*A*c^2+4/f/a^3/(c-d)^4/(tan(1/2*f*x+1/2*e)+1)^2*A*c+6/f/a^3*d^4/(c-d)^5/(c*tan(1/2*f*x+1/2*e)^2+2*
tan(1/2*f*x+1/2*e)*d+c)^2/(c^2+2*c*d+d^2)*c^2*tan(1/2*f*x+1/2*e)^2*B+17/f/a^3*d^5/(c-d)^5/(c*tan(1/2*f*x+1/2*e
)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^2+2*c*d+d^2)*c*tan(1/2*f*x+1/2*e)^2*B+2/f/a^3*d^7/(c-d)^5/(c*tan(1/2*f*x+1/
2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^2+2*c*d+d^2)/c*tan(1/2*f*x+1/2*e)^2*B-29/f/a^3*d^5/(c-d)^5/(c*tan(1/2*f*
x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2*c/(c^2+2*c*d+d^2)*tan(1/2*f*x+1/2*e)*A+23/f/a^3*d^4/(c-d)^5/(c*tan(1/2*
f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2*c^2/(c^2+2*c*d+d^2)*tan(1/2*f*x+1/2*e)*B+12/f/a^3*d^2/(c-d)^5/(c^2+2*
c*d+d^2)/(c^2-d^2)^(1/2)*arctan(1/2*(2*c*tan(1/2*f*x+1/2*e)+2*d)/(c^2-d^2)^(1/2))*B*c^3+2/f/a^3*d^7/(c-d)^5/(c
*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2/c/(c^2+2*c*d+d^2)*tan(1/2*f*x+1/2*e)^3*A+9/f/a^3*d^4/(c-d)^5
/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2*c^2/(c^2+2*c*d+d^2)*tan(1/2*f*x+1/2*e)^3*B+18/f/a^3*d^5/(
c-d)^5/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2*c/(c^2+2*c*d+d^2)*tan(1/2*f*x+1/2*e)*B-11/f/a^3*d^5
/(c-d)^5/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2*c/(c^2+2*c*d+d^2)*tan(1/2*f*x+1/2*e)^3*A-20/f/a^3
*d^3/(c-d)^5/(c^2+2*c*d+d^2)/(c^2-d^2)^(1/2)*arctan(1/2*(2*c*tan(1/2*f*x+1/2*e)+2*d)/(c^2-d^2)^(1/2))*A*c^2-30
/f/a^3*d^4/(c-d)^5/(c^2+2*c*d+d^2)/(c^2-d^2)^(1/2)*arctan(1/2*(2*c*tan(1/2*f*x+1/2*e)+2*d)/(c^2-d^2)^(1/2))*A*
c+24/f/a^3*d^3/(c-d)^5/(c^2+2*c*d+d^2)/(c^2-d^2)^(1/2)*arctan(1/2*(2*c*tan(1/2*f*x+1/2*e)+2*d)/(c^2-d^2)^(1/2)
)*B*c^2+6/f/a^3*d^4/(c-d)^5/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^2+2*c*d+d^2)*B*c^2+10/f/a^3
/(c-d)^5/(tan(1/2*f*x+1/2*e)+1)*A*c*d+1/f/a^3*d^6/(c-d)^5/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2/
(c^2+2*c*d+d^2)*A+6/f/a^3*d^5/(c-d)^5/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2*c/(c^2+2*c*d+d^2)*ta
n(1/2*f*x+1/2*e)^3*B-10/f/a^3*d^4/(c-d)^5/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^2+2*c*d+d^2)*
c^2*tan(1/2*f*x+1/2*e)^2*A-6/f/a^3*d^5/(c-d)^5/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^2+2*c*d+
d^2)*c*tan(1/2*f*x+1/2*e)^2*A+2/f/a^3*d^7/(c-d)^5/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2/c/(c^2+2
*c*d+d^2)*tan(1/2*f*x+1/2*e)*A+21/f/a^3*d^4/(c-d)^5/(c^2+2*c*d+d^2)/(c^2-d^2)^(1/2)*arctan(1/2*(2*c*tan(1/2*f*
x+1/2*e)+2*d)/(c^2-d^2)^(1/2))*B*c-12/f/a^3*d^7/(c-d)^5/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c
^2+2*c*d+d^2)/c*tan(1/2*f*x+1/2*e)^2*A-19/f/a^3*d^6/(c-d)^5/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^
2/(c^2+2*c*d+d^2)*tan(1/2*f*x+1/2*e)^2*A+8/f/a^3*d^3/(c-d)^5/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)
^2/(c^2+2*c*d+d^2)*B*c^3-13/f/a^3*d^5/(c-d)^5/(c^2+2*c*d+d^2)/(c^2-d^2)^(1/2)*arctan(1/2*(2*c*tan(1/2*f*x+1/2*
e)+2*d)/(c^2-d^2)^(1/2))*A+6/f/a^3*d^5/(c-d)^5/(c^2+2*c*d+d^2)/(c^2-d^2)^(1/2)*arctan(1/2*(2*c*tan(1/2*f*x+1/2
*e)+2*d)/(c^2-d^2)^(1/2))*B+4/f/a^3*d^6/(c-d)^5/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^2+2*c*d
+d^2)*tan(1/2*f*x+1/2*e)*B+1/f/a^3*d^5/(c-d)^5/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^2+2*c*d+
d^2)*B*c+12/f/a^3*d^6/(c-d)^5/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^2+2*c*d+d^2)*tan(1/2*f*x+
1/2*e)^2*B-18/f/a^3*d^6/(c-d)^5/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^2+2*c*d+d^2)*tan(1/2*f*
x+1/2*e)*A-6/f/a^3*d^6/(c-d)^5/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^2+2*c*d+d^2)*tan(1/2*f*x
+1/2*e)^3*A-10/f/a^3*d^4/(c-d)^5/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^2+2*c*d+d^2)*A*c^2-6/f
/a^3*d^5/(c-d)^5/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^2+2*c*d+d^2)*A*c-4/f/a^3/(c-d)^3/(tan(
1/2*f*x+1/2*e)+1)^4*B-8/5/f/a^3/(c-d)^3/(tan(1/2*f*x+1/2*e)+1)^5*A+8/5/f/a^3/(c-d)^3/(tan(1/2*f*x+1/2*e)+1)^5*
B+4/f/a^3/(c-d)^3/(tan(1/2*f*x+1/2*e)+1)^4*A+2/f/a^3*d^8/(c-d)^5/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*
d+c)^2/(c^2+2*c*d+d^2)/c^2*tan(1/2*f*x+1/2*e)^2*A+8/f/a^3*d^3/(c-d)^5/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/
2*e)*d+c)^2/(c^2+2*c*d+d^2)*c^3*tan(1/2*f*x+1/2*e)^2*B

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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((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 5.11804, size = 16342, normalized size = 32.17 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^3,x, algorithm="fricas")

[Out]

[-1/60*(12*(A - B)*c^8 - 24*(A - B)*c^7*d - 24*(A - B)*c^6*d^2 + 72*(A - B)*c^5*d^3 - 72*(A - B)*c^3*d^5 + 24*
(A - B)*c^2*d^6 + 24*(A - B)*c*d^7 - 12*(A - B)*d^8 + 2*(2*(2*A + 3*B)*c^6*d^2 - 30*(A + 2*B)*c^5*d^3 + 3*(46*
A - 121*B)*c^4*d^4 + 15*(37*A - 22*B)*c^3*d^5 + 3*(54*A + 71*B)*c^2*d^6 - 15*(35*A - 26*B)*c*d^7 - 16*(19*A -
9*B)*d^8)*cos(f*x + e)^5 - 2*(4*(2*A + 3*B)*c^7*d - 4*(13*A + 27*B)*c^6*d^2 + 18*(12*A - 37*B)*c^5*d^3 + 6*(18
1*A - 171*B)*c^4*d^4 + 3*(328*A - 33*B)*c^3*d^5 - 9*(69*A - 104*B)*c^2*d^6 - (1208*A - 753*B)*c*d^7 - (413*A -
198*B)*d^8)*cos(f*x + e)^4 - 2*(2*(2*A + 3*B)*c^8 - 6*(A + 4*B)*c^7*d - 20*(A + 21*B)*c^6*d^2 + 6*(128*A - 29
3*B)*c^5*d^3 + 3*(892*A - 827*B)*c^4*d^4 + 3*(769*A - 49*B)*c^3*d^5 - (1573*A - 2373*B)*c^2*d^6 - 3*(1023*A -
643*B)*c*d^7 - (1087*A - 522*B)*d^8)*cos(f*x + e)^3 + 4*(2*(2*A + 3*B)*c^8 - 5*(4*A + 3*B)*c^7*d + (19*A - 174
*B)*c^6*d^2 + 15*(22*A - 35*B)*c^5*d^3 + 3*(233*A - 173*B)*c^4*d^4 + 15*(23*A + 10*B)*c^3*d^5 - (526*A - 591*B
)*c^2*d^6 - 5*(131*A - 78*B)*c*d^7 - 4*(49*A - 24*B)*d^8)*cos(f*x + e)^2 - 15*(48*B*c^5*d^2 - 16*(5*A - 12*B)*
c^4*d^3 - 4*(70*A - 81*B)*c^3*d^4 - 12*(31*A - 24*B)*c^2*d^5 - 4*(56*A - 33*B)*c*d^6 - 4*(13*A - 6*B)*d^7 + (1
2*B*c^3*d^4 - 4*(5*A - 6*B)*c^2*d^5 - 3*(10*A - 7*B)*c*d^6 - (13*A - 6*B)*d^7)*cos(f*x + e)^5 + (24*B*c^4*d^3
- 4*(10*A - 21*B)*c^3*d^4 - 6*(20*A - 19*B)*c^2*d^5 - (116*A - 75*B)*c*d^6 - 3*(13*A - 6*B)*d^7)*cos(f*x + e)^
4 - (12*B*c^5*d^2 - 4*(5*A - 18*B)*c^4*d^3 - (110*A - 153*B)*c^3*d^4 - (193*A - 162*B)*c^2*d^5 - (142*A - 87*B
)*c*d^6 - 3*(13*A - 6*B)*d^7)*cos(f*x + e)^3 - (36*B*c^5*d^2 - 12*(5*A - 16*B)*c^4*d^3 - (290*A - 387*B)*c^3*d
^4 - (479*A - 396*B)*c^2*d^5 - (340*A - 207*B)*c*d^6 - 7*(13*A - 6*B)*d^7)*cos(f*x + e)^2 + 2*(12*B*c^5*d^2 -
4*(5*A - 12*B)*c^4*d^3 - (70*A - 81*B)*c^3*d^4 - 3*(31*A - 24*B)*c^2*d^5 - (56*A - 33*B)*c*d^6 - (13*A - 6*B)*
d^7)*cos(f*x + e) + (48*B*c^5*d^2 - 16*(5*A - 12*B)*c^4*d^3 - 4*(70*A - 81*B)*c^3*d^4 - 12*(31*A - 24*B)*c^2*d
^5 - 4*(56*A - 33*B)*c*d^6 - 4*(13*A - 6*B)*d^7 + (12*B*c^3*d^4 - 4*(5*A - 6*B)*c^2*d^5 - 3*(10*A - 7*B)*c*d^6
- (13*A - 6*B)*d^7)*cos(f*x + e)^4 - 2*(12*B*c^4*d^3 - 4*(5*A - 9*B)*c^3*d^4 - 5*(10*A - 9*B)*c^2*d^5 - (43*A
- 27*B)*c*d^6 - (13*A - 6*B)*d^7)*cos(f*x + e)^3 - (12*B*c^5*d^2 - 4*(5*A - 24*B)*c^4*d^3 - 75*(2*A - 3*B)*c^
3*d^4 - (293*A - 252*B)*c^2*d^5 - 3*(76*A - 47*B)*c*d^6 - 5*(13*A - 6*B)*d^7)*cos(f*x + e)^2 + 2*(12*B*c^5*d^2
- 4*(5*A - 12*B)*c^4*d^3 - (70*A - 81*B)*c^3*d^4 - 3*(31*A - 24*B)*c^2*d^5 - (56*A - 33*B)*c*d^6 - (13*A - 6*
B)*d^7)*cos(f*x + e))*sin(f*x + e))*sqrt(-c^2 + d^2)*log(-((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) -
c^2 - d^2 - 2*(c*cos(f*x + e)*sin(f*x + e) + d*cos(f*x + e))*sqrt(-c^2 + d^2))/(d^2*cos(f*x + e)^2 - 2*c*d*si
n(f*x + e) - c^2 - d^2)) + 12*((3*A + 2*B)*c^8 - (11*A + 9*B)*c^7*d + (9*A - 109*B)*c^6*d^2 + (213*A - 353*B)*
c^5*d^3 + 5*(95*A - 71*B)*c^4*d^4 + (237*A + 103*B)*c^3*d^5 - (359*A - 399*B)*c^2*d^6 - (439*A - 259*B)*c*d^7
- (128*A - 63*B)*d^8)*cos(f*x + e) - 2*(6*(A - B)*c^8 - 12*(A - B)*c^7*d - 12*(A - B)*c^6*d^2 + 36*(A - B)*c^5
*d^3 - 36*(A - B)*c^3*d^5 + 12*(A - B)*c^2*d^6 + 12*(A - B)*c*d^7 - 6*(A - B)*d^8 + (2*(2*A + 3*B)*c^6*d^2 - 3
0*(A + 2*B)*c^5*d^3 + 3*(46*A - 121*B)*c^4*d^4 + 15*(37*A - 22*B)*c^3*d^5 + 3*(54*A + 71*B)*c^2*d^6 - 15*(35*A
- 26*B)*c*d^7 - 16*(19*A - 9*B)*d^8)*cos(f*x + e)^4 + (4*(2*A + 3*B)*c^7*d - 6*(8*A + 17*B)*c^6*d^2 + 6*(31*A
- 121*B)*c^5*d^3 + 3*(408*A - 463*B)*c^4*d^4 + 3*(513*A - 143*B)*c^3*d^5 - 3*(153*A - 383*B)*c^2*d^6 - (1733*
A - 1143*B)*c*d^7 - 3*(239*A - 114*B)*d^8)*cos(f*x + e)^3 - 2*((2*A + 3*B)*c^8 - (7*A + 18*B)*c^7*d + (14*A -
159*B)*c^6*d^2 + 3*(97*A - 172*B)*c^5*d^3 + 6*(121*A - 91*B)*c^4*d^4 + 3*(128*A + 47*B)*c^3*d^5 - (557*A - 612
*B)*c^2*d^6 - (668*A - 393*B)*c*d^7 - 5*(37*A - 18*B)*d^8)*cos(f*x + e)^2 - 6*((2*A + 3*B)*c^8 - (9*A + 11*B)*
c^7*d + (11*A - 111*B)*c^6*d^2 + (207*A - 347*B)*c^5*d^3 + 5*(95*A - 71*B)*c^4*d^4 + (243*A + 97*B)*c^3*d^5 -
(361*A - 401*B)*c^2*d^6 - 9*(49*A - 29*B)*c*d^7 - (127*A - 62*B)*d^8)*cos(f*x + e))*sin(f*x + e))/((a^3*c^9*d^
2 - 3*a^3*c^8*d^3 + 8*a^3*c^6*d^5 - 6*a^3*c^5*d^6 - 6*a^3*c^4*d^7 + 8*a^3*c^3*d^8 - 3*a^3*c*d^10 + a^3*d^11)*f
*cos(f*x + e)^5 + (2*a^3*c^10*d - 3*a^3*c^9*d^2 - 9*a^3*c^8*d^3 + 16*a^3*c^7*d^4 + 12*a^3*c^6*d^5 - 30*a^3*c^5
*d^6 - 2*a^3*c^4*d^7 + 24*a^3*c^3*d^8 - 6*a^3*c^2*d^9 - 7*a^3*c*d^10 + 3*a^3*d^11)*f*cos(f*x + e)^4 - (a^3*c^1
1 + a^3*c^10*d - 9*a^3*c^9*d^2 - a^3*c^8*d^3 + 26*a^3*c^7*d^4 - 6*a^3*c^6*d^5 - 34*a^3*c^5*d^6 + 14*a^3*c^4*d^
7 + 21*a^3*c^3*d^8 - 11*a^3*c^2*d^9 - 5*a^3*c*d^10 + 3*a^3*d^11)*f*cos(f*x + e)^3 - (3*a^3*c^11 + a^3*c^10*d -
23*a^3*c^9*d^2 + 3*a^3*c^8*d^3 + 62*a^3*c^7*d^4 - 22*a^3*c^6*d^5 - 78*a^3*c^5*d^6 + 38*a^3*c^4*d^7 + 47*a^3*c
^3*d^8 - 27*a^3*c^2*d^9 - 11*a^3*c*d^10 + 7*a^3*d^11)*f*cos(f*x + e)^2 + 2*(a^3*c^11 - a^3*c^10*d - 5*a^3*c^9*
d^2 + 5*a^3*c^8*d^3 + 10*a^3*c^7*d^4 - 10*a^3*c^6*d^5 - 10*a^3*c^5*d^6 + 10*a^3*c^4*d^7 + 5*a^3*c^3*d^8 - 5*a^
3*c^2*d^9 - a^3*c*d^10 + a^3*d^11)*f*cos(f*x + e) + 4*(a^3*c^11 - a^3*c^10*d - 5*a^3*c^9*d^2 + 5*a^3*c^8*d^3 +
10*a^3*c^7*d^4 - 10*a^3*c^6*d^5 - 10*a^3*c^5*d^6 + 10*a^3*c^4*d^7 + 5*a^3*c^3*d^8 - 5*a^3*c^2*d^9 - a^3*c*d^1
0 + a^3*d^11)*f + ((a^3*c^9*d^2 - 3*a^3*c^8*d^3 + 8*a^3*c^6*d^5 - 6*a^3*c^5*d^6 - 6*a^3*c^4*d^7 + 8*a^3*c^3*d^
8 - 3*a^3*c*d^10 + a^3*d^11)*f*cos(f*x + e)^4 - 2*(a^3*c^10*d - 2*a^3*c^9*d^2 - 3*a^3*c^8*d^3 + 8*a^3*c^7*d^4
+ 2*a^3*c^6*d^5 - 12*a^3*c^5*d^6 + 2*a^3*c^4*d^7 + 8*a^3*c^3*d^8 - 3*a^3*c^2*d^9 - 2*a^3*c*d^10 + a^3*d^11)*f*
cos(f*x + e)^3 - (a^3*c^11 + 3*a^3*c^10*d - 13*a^3*c^9*d^2 - 7*a^3*c^8*d^3 + 42*a^3*c^7*d^4 - 2*a^3*c^6*d^5 -
58*a^3*c^5*d^6 + 18*a^3*c^4*d^7 + 37*a^3*c^3*d^8 - 17*a^3*c^2*d^9 - 9*a^3*c*d^10 + 5*a^3*d^11)*f*cos(f*x + e)^
2 + 2*(a^3*c^11 - a^3*c^10*d - 5*a^3*c^9*d^2 + 5*a^3*c^8*d^3 + 10*a^3*c^7*d^4 - 10*a^3*c^6*d^5 - 10*a^3*c^5*d^
6 + 10*a^3*c^4*d^7 + 5*a^3*c^3*d^8 - 5*a^3*c^2*d^9 - a^3*c*d^10 + a^3*d^11)*f*cos(f*x + e) + 4*(a^3*c^11 - a^3
*c^10*d - 5*a^3*c^9*d^2 + 5*a^3*c^8*d^3 + 10*a^3*c^7*d^4 - 10*a^3*c^6*d^5 - 10*a^3*c^5*d^6 + 10*a^3*c^4*d^7 +
5*a^3*c^3*d^8 - 5*a^3*c^2*d^9 - a^3*c*d^10 + a^3*d^11)*f)*sin(f*x + e)), -1/30*(6*(A - B)*c^8 - 12*(A - B)*c^7
*d - 12*(A - B)*c^6*d^2 + 36*(A - B)*c^5*d^3 - 36*(A - B)*c^3*d^5 + 12*(A - B)*c^2*d^6 + 12*(A - B)*c*d^7 - 6*
(A - B)*d^8 + (2*(2*A + 3*B)*c^6*d^2 - 30*(A + 2*B)*c^5*d^3 + 3*(46*A - 121*B)*c^4*d^4 + 15*(37*A - 22*B)*c^3*
d^5 + 3*(54*A + 71*B)*c^2*d^6 - 15*(35*A - 26*B)*c*d^7 - 16*(19*A - 9*B)*d^8)*cos(f*x + e)^5 - (4*(2*A + 3*B)*
c^7*d - 4*(13*A + 27*B)*c^6*d^2 + 18*(12*A - 37*B)*c^5*d^3 + 6*(181*A - 171*B)*c^4*d^4 + 3*(328*A - 33*B)*c^3*
d^5 - 9*(69*A - 104*B)*c^2*d^6 - (1208*A - 753*B)*c*d^7 - (413*A - 198*B)*d^8)*cos(f*x + e)^4 - (2*(2*A + 3*B)
*c^8 - 6*(A + 4*B)*c^7*d - 20*(A + 21*B)*c^6*d^2 + 6*(128*A - 293*B)*c^5*d^3 + 3*(892*A - 827*B)*c^4*d^4 + 3*(
769*A - 49*B)*c^3*d^5 - (1573*A - 2373*B)*c^2*d^6 - 3*(1023*A - 643*B)*c*d^7 - (1087*A - 522*B)*d^8)*cos(f*x +
e)^3 + 2*(2*(2*A + 3*B)*c^8 - 5*(4*A + 3*B)*c^7*d + (19*A - 174*B)*c^6*d^2 + 15*(22*A - 35*B)*c^5*d^3 + 3*(23
3*A - 173*B)*c^4*d^4 + 15*(23*A + 10*B)*c^3*d^5 - (526*A - 591*B)*c^2*d^6 - 5*(131*A - 78*B)*c*d^7 - 4*(49*A -
24*B)*d^8)*cos(f*x + e)^2 + 15*(48*B*c^5*d^2 - 16*(5*A - 12*B)*c^4*d^3 - 4*(70*A - 81*B)*c^3*d^4 - 12*(31*A -
24*B)*c^2*d^5 - 4*(56*A - 33*B)*c*d^6 - 4*(13*A - 6*B)*d^7 + (12*B*c^3*d^4 - 4*(5*A - 6*B)*c^2*d^5 - 3*(10*A
- 7*B)*c*d^6 - (13*A - 6*B)*d^7)*cos(f*x + e)^5 + (24*B*c^4*d^3 - 4*(10*A - 21*B)*c^3*d^4 - 6*(20*A - 19*B)*c^
2*d^5 - (116*A - 75*B)*c*d^6 - 3*(13*A - 6*B)*d^7)*cos(f*x + e)^4 - (12*B*c^5*d^2 - 4*(5*A - 18*B)*c^4*d^3 - (
110*A - 153*B)*c^3*d^4 - (193*A - 162*B)*c^2*d^5 - (142*A - 87*B)*c*d^6 - 3*(13*A - 6*B)*d^7)*cos(f*x + e)^3 -
(36*B*c^5*d^2 - 12*(5*A - 16*B)*c^4*d^3 - (290*A - 387*B)*c^3*d^4 - (479*A - 396*B)*c^2*d^5 - (340*A - 207*B)
*c*d^6 - 7*(13*A - 6*B)*d^7)*cos(f*x + e)^2 + 2*(12*B*c^5*d^2 - 4*(5*A - 12*B)*c^4*d^3 - (70*A - 81*B)*c^3*d^4
- 3*(31*A - 24*B)*c^2*d^5 - (56*A - 33*B)*c*d^6 - (13*A - 6*B)*d^7)*cos(f*x + e) + (48*B*c^5*d^2 - 16*(5*A -
12*B)*c^4*d^3 - 4*(70*A - 81*B)*c^3*d^4 - 12*(31*A - 24*B)*c^2*d^5 - 4*(56*A - 33*B)*c*d^6 - 4*(13*A - 6*B)*d^
7 + (12*B*c^3*d^4 - 4*(5*A - 6*B)*c^2*d^5 - 3*(10*A - 7*B)*c*d^6 - (13*A - 6*B)*d^7)*cos(f*x + e)^4 - 2*(12*B*
c^4*d^3 - 4*(5*A - 9*B)*c^3*d^4 - 5*(10*A - 9*B)*c^2*d^5 - (43*A - 27*B)*c*d^6 - (13*A - 6*B)*d^7)*cos(f*x + e
)^3 - (12*B*c^5*d^2 - 4*(5*A - 24*B)*c^4*d^3 - 75*(2*A - 3*B)*c^3*d^4 - (293*A - 252*B)*c^2*d^5 - 3*(76*A - 47
*B)*c*d^6 - 5*(13*A - 6*B)*d^7)*cos(f*x + e)^2 + 2*(12*B*c^5*d^2 - 4*(5*A - 12*B)*c^4*d^3 - (70*A - 81*B)*c^3*
d^4 - 3*(31*A - 24*B)*c^2*d^5 - (56*A - 33*B)*c*d^6 - (13*A - 6*B)*d^7)*cos(f*x + e))*sin(f*x + e))*sqrt(c^2 -
d^2)*arctan(-(c*sin(f*x + e) + d)/(sqrt(c^2 - d^2)*cos(f*x + e))) + 6*((3*A + 2*B)*c^8 - (11*A + 9*B)*c^7*d +
(9*A - 109*B)*c^6*d^2 + (213*A - 353*B)*c^5*d^3 + 5*(95*A - 71*B)*c^4*d^4 + (237*A + 103*B)*c^3*d^5 - (359*A
- 399*B)*c^2*d^6 - (439*A - 259*B)*c*d^7 - (128*A - 63*B)*d^8)*cos(f*x + e) - (6*(A - B)*c^8 - 12*(A - B)*c^7*
d - 12*(A - B)*c^6*d^2 + 36*(A - B)*c^5*d^3 - 36*(A - B)*c^3*d^5 + 12*(A - B)*c^2*d^6 + 12*(A - B)*c*d^7 - 6*(
A - B)*d^8 + (2*(2*A + 3*B)*c^6*d^2 - 30*(A + 2*B)*c^5*d^3 + 3*(46*A - 121*B)*c^4*d^4 + 15*(37*A - 22*B)*c^3*d
^5 + 3*(54*A + 71*B)*c^2*d^6 - 15*(35*A - 26*B)*c*d^7 - 16*(19*A - 9*B)*d^8)*cos(f*x + e)^4 + (4*(2*A + 3*B)*c
^7*d - 6*(8*A + 17*B)*c^6*d^2 + 6*(31*A - 121*B)*c^5*d^3 + 3*(408*A - 463*B)*c^4*d^4 + 3*(513*A - 143*B)*c^3*d
^5 - 3*(153*A - 383*B)*c^2*d^6 - (1733*A - 1143*B)*c*d^7 - 3*(239*A - 114*B)*d^8)*cos(f*x + e)^3 - 2*((2*A + 3
*B)*c^8 - (7*A + 18*B)*c^7*d + (14*A - 159*B)*c^6*d^2 + 3*(97*A - 172*B)*c^5*d^3 + 6*(121*A - 91*B)*c^4*d^4 +
3*(128*A + 47*B)*c^3*d^5 - (557*A - 612*B)*c^2*d^6 - (668*A - 393*B)*c*d^7 - 5*(37*A - 18*B)*d^8)*cos(f*x + e)
^2 - 6*((2*A + 3*B)*c^8 - (9*A + 11*B)*c^7*d + (11*A - 111*B)*c^6*d^2 + (207*A - 347*B)*c^5*d^3 + 5*(95*A - 71
*B)*c^4*d^4 + (243*A + 97*B)*c^3*d^5 - (361*A - 401*B)*c^2*d^6 - 9*(49*A - 29*B)*c*d^7 - (127*A - 62*B)*d^8)*c
os(f*x + e))*sin(f*x + e))/((a^3*c^9*d^2 - 3*a^3*c^8*d^3 + 8*a^3*c^6*d^5 - 6*a^3*c^5*d^6 - 6*a^3*c^4*d^7 + 8*a
^3*c^3*d^8 - 3*a^3*c*d^10 + a^3*d^11)*f*cos(f*x + e)^5 + (2*a^3*c^10*d - 3*a^3*c^9*d^2 - 9*a^3*c^8*d^3 + 16*a^
3*c^7*d^4 + 12*a^3*c^6*d^5 - 30*a^3*c^5*d^6 - 2*a^3*c^4*d^7 + 24*a^3*c^3*d^8 - 6*a^3*c^2*d^9 - 7*a^3*c*d^10 +
3*a^3*d^11)*f*cos(f*x + e)^4 - (a^3*c^11 + a^3*c^10*d - 9*a^3*c^9*d^2 - a^3*c^8*d^3 + 26*a^3*c^7*d^4 - 6*a^3*c
^6*d^5 - 34*a^3*c^5*d^6 + 14*a^3*c^4*d^7 + 21*a^3*c^3*d^8 - 11*a^3*c^2*d^9 - 5*a^3*c*d^10 + 3*a^3*d^11)*f*cos(
f*x + e)^3 - (3*a^3*c^11 + a^3*c^10*d - 23*a^3*c^9*d^2 + 3*a^3*c^8*d^3 + 62*a^3*c^7*d^4 - 22*a^3*c^6*d^5 - 78*
a^3*c^5*d^6 + 38*a^3*c^4*d^7 + 47*a^3*c^3*d^8 - 27*a^3*c^2*d^9 - 11*a^3*c*d^10 + 7*a^3*d^11)*f*cos(f*x + e)^2
+ 2*(a^3*c^11 - a^3*c^10*d - 5*a^3*c^9*d^2 + 5*a^3*c^8*d^3 + 10*a^3*c^7*d^4 - 10*a^3*c^6*d^5 - 10*a^3*c^5*d^6
+ 10*a^3*c^4*d^7 + 5*a^3*c^3*d^8 - 5*a^3*c^2*d^9 - a^3*c*d^10 + a^3*d^11)*f*cos(f*x + e) + 4*(a^3*c^11 - a^3*c
^10*d - 5*a^3*c^9*d^2 + 5*a^3*c^8*d^3 + 10*a^3*c^7*d^4 - 10*a^3*c^6*d^5 - 10*a^3*c^5*d^6 + 10*a^3*c^4*d^7 + 5*
a^3*c^3*d^8 - 5*a^3*c^2*d^9 - a^3*c*d^10 + a^3*d^11)*f + ((a^3*c^9*d^2 - 3*a^3*c^8*d^3 + 8*a^3*c^6*d^5 - 6*a^3
*c^5*d^6 - 6*a^3*c^4*d^7 + 8*a^3*c^3*d^8 - 3*a^3*c*d^10 + a^3*d^11)*f*cos(f*x + e)^4 - 2*(a^3*c^10*d - 2*a^3*c
^9*d^2 - 3*a^3*c^8*d^3 + 8*a^3*c^7*d^4 + 2*a^3*c^6*d^5 - 12*a^3*c^5*d^6 + 2*a^3*c^4*d^7 + 8*a^3*c^3*d^8 - 3*a^
3*c^2*d^9 - 2*a^3*c*d^10 + a^3*d^11)*f*cos(f*x + e)^3 - (a^3*c^11 + 3*a^3*c^10*d - 13*a^3*c^9*d^2 - 7*a^3*c^8*
d^3 + 42*a^3*c^7*d^4 - 2*a^3*c^6*d^5 - 58*a^3*c^5*d^6 + 18*a^3*c^4*d^7 + 37*a^3*c^3*d^8 - 17*a^3*c^2*d^9 - 9*a
^3*c*d^10 + 5*a^3*d^11)*f*cos(f*x + e)^2 + 2*(a^3*c^11 - a^3*c^10*d - 5*a^3*c^9*d^2 + 5*a^3*c^8*d^3 + 10*a^3*c
^7*d^4 - 10*a^3*c^6*d^5 - 10*a^3*c^5*d^6 + 10*a^3*c^4*d^7 + 5*a^3*c^3*d^8 - 5*a^3*c^2*d^9 - a^3*c*d^10 + a^3*d
^11)*f*cos(f*x + e) + 4*(a^3*c^11 - a^3*c^10*d - 5*a^3*c^9*d^2 + 5*a^3*c^8*d^3 + 10*a^3*c^7*d^4 - 10*a^3*c^6*d
^5 - 10*a^3*c^5*d^6 + 10*a^3*c^4*d^7 + 5*a^3*c^3*d^8 - 5*a^3*c^2*d^9 - a^3*c*d^10 + a^3*d^11)*f)*sin(f*x + e))
]

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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((A+B*sin(f*x+e))/(a+a*sin(f*x+e))**3/(c+d*sin(f*x+e))**3,x)

[Out]

Timed out

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Giac [B]  time = 1.55749, size = 1717, normalized size = 3.38 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^3,x, algorithm="giac")

[Out]

1/15*(15*(12*B*c^3*d^2 - 20*A*c^2*d^3 + 24*B*c^2*d^3 - 30*A*c*d^4 + 21*B*c*d^4 - 13*A*d^5 + 6*B*d^5)*(pi*floor
(1/2*(f*x + e)/pi + 1/2)*sgn(c) + arctan((c*tan(1/2*f*x + 1/2*e) + d)/sqrt(c^2 - d^2)))/((a^3*c^7 - 3*a^3*c^6*
d + a^3*c^5*d^2 + 5*a^3*c^4*d^3 - 5*a^3*c^3*d^4 - a^3*c^2*d^5 + 3*a^3*c*d^6 - a^3*d^7)*sqrt(c^2 - d^2)) + 15*(
9*B*c^4*d^4*tan(1/2*f*x + 1/2*e)^3 - 11*A*c^3*d^5*tan(1/2*f*x + 1/2*e)^3 + 6*B*c^3*d^5*tan(1/2*f*x + 1/2*e)^3
- 6*A*c^2*d^6*tan(1/2*f*x + 1/2*e)^3 + 2*A*c*d^7*tan(1/2*f*x + 1/2*e)^3 + 8*B*c^5*d^3*tan(1/2*f*x + 1/2*e)^2 -
10*A*c^4*d^4*tan(1/2*f*x + 1/2*e)^2 + 6*B*c^4*d^4*tan(1/2*f*x + 1/2*e)^2 - 6*A*c^3*d^5*tan(1/2*f*x + 1/2*e)^2
+ 17*B*c^3*d^5*tan(1/2*f*x + 1/2*e)^2 - 19*A*c^2*d^6*tan(1/2*f*x + 1/2*e)^2 + 12*B*c^2*d^6*tan(1/2*f*x + 1/2*
e)^2 - 12*A*c*d^7*tan(1/2*f*x + 1/2*e)^2 + 2*B*c*d^7*tan(1/2*f*x + 1/2*e)^2 + 2*A*d^8*tan(1/2*f*x + 1/2*e)^2 +
23*B*c^4*d^4*tan(1/2*f*x + 1/2*e) - 29*A*c^3*d^5*tan(1/2*f*x + 1/2*e) + 18*B*c^3*d^5*tan(1/2*f*x + 1/2*e) - 1
8*A*c^2*d^6*tan(1/2*f*x + 1/2*e) + 4*B*c^2*d^6*tan(1/2*f*x + 1/2*e) + 2*A*c*d^7*tan(1/2*f*x + 1/2*e) + 8*B*c^5
*d^3 - 10*A*c^4*d^4 + 6*B*c^4*d^4 - 6*A*c^3*d^5 + B*c^3*d^5 + A*c^2*d^6)/((a^3*c^9 - 3*a^3*c^8*d + a^3*c^7*d^2
+ 5*a^3*c^6*d^3 - 5*a^3*c^5*d^4 - a^3*c^4*d^5 + 3*a^3*c^3*d^6 - a^3*c^2*d^7)*(c*tan(1/2*f*x + 1/2*e)^2 + 2*d*
tan(1/2*f*x + 1/2*e) + c)^2) - 2*(15*A*c^2*tan(1/2*f*x + 1/2*e)^4 - 75*A*c*d*tan(1/2*f*x + 1/2*e)^4 + 150*A*d^
2*tan(1/2*f*x + 1/2*e)^4 - 90*B*d^2*tan(1/2*f*x + 1/2*e)^4 + 30*A*c^2*tan(1/2*f*x + 1/2*e)^3 + 15*B*c^2*tan(1/
2*f*x + 1/2*e)^3 - 195*A*c*d*tan(1/2*f*x + 1/2*e)^3 - 75*B*c*d*tan(1/2*f*x + 1/2*e)^3 + 525*A*d^2*tan(1/2*f*x
+ 1/2*e)^3 - 300*B*d^2*tan(1/2*f*x + 1/2*e)^3 + 40*A*c^2*tan(1/2*f*x + 1/2*e)^2 + 15*B*c^2*tan(1/2*f*x + 1/2*e
)^2 - 245*A*c*d*tan(1/2*f*x + 1/2*e)^2 - 135*B*c*d*tan(1/2*f*x + 1/2*e)^2 + 745*A*d^2*tan(1/2*f*x + 1/2*e)^2 -
420*B*d^2*tan(1/2*f*x + 1/2*e)^2 + 20*A*c^2*tan(1/2*f*x + 1/2*e) + 15*B*c^2*tan(1/2*f*x + 1/2*e) - 145*A*c*d*
tan(1/2*f*x + 1/2*e) - 105*B*c*d*tan(1/2*f*x + 1/2*e) + 485*A*d^2*tan(1/2*f*x + 1/2*e) - 270*B*d^2*tan(1/2*f*x
+ 1/2*e) + 7*A*c^2 + 3*B*c^2 - 44*A*c*d - 21*B*c*d + 127*A*d^2 - 72*B*d^2)/((a^3*c^5 - 5*a^3*c^4*d + 10*a^3*c
^3*d^2 - 10*a^3*c^2*d^3 + 5*a^3*c*d^4 - a^3*d^5)*(tan(1/2*f*x + 1/2*e) + 1)^5))/f