\(\int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))^3} \, dx\) [278]

Optimal result

Integrand size = 35, antiderivative size = 386 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))^3} \, dx=\frac {d \left (A d \left (12 c^2+16 c d+7 d^2\right )-B \left (6 c^3+12 c^2 d+13 c d^2+4 d^3\right )\right ) \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{a^2 (c-d)^4 (c+d)^2 \sqrt {c^2-d^2} f}-\frac {d \left (A \left (2 c^2-16 c d-21 d^2\right )+B \left (4 c^2+19 c d+12 d^2\right )\right ) \cos (e+f x)}{6 a^2 (c-d)^3 (c+d) f (c+d \sin (e+f x))^2}-\frac {(A c+2 B c-8 A d+5 B d) \cos (e+f x)}{3 a^2 (c-d)^2 f (1+\sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {(A-B) \cos (e+f x)}{3 (c-d) f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^2}-\frac {d \left (A \left (2 c^3-16 c^2 d-59 c d^2-32 d^3\right )+B \left (4 c^3+37 c^2 d+44 c d^2+20 d^3\right )\right ) \cos (e+f x)}{6 a^2 (c-d)^4 (c+d)^2 f (c+d \sin (e+f x))} \]

[Out]

-1/6*d*(A*(2*c^2-16*c*d-21*d^2)+B*(4*c^2+19*c*d+12*d^2))*cos(f*x+e)/a^2/(c-d)^3/(c+d)/f/(c+d*sin(f*x+e))^2-1/3
*(A*c-8*A*d+2*B*c+5*B*d)*cos(f*x+e)/a^2/(c-d)^2/f/(1+sin(f*x+e))/(c+d*sin(f*x+e))^2-1/3*(A-B)*cos(f*x+e)/(c-d)
/f/(a+a*sin(f*x+e))^2/(c+d*sin(f*x+e))^2-1/6*d*(A*(2*c^3-16*c^2*d-59*c*d^2-32*d^3)+B*(4*c^3+37*c^2*d+44*c*d^2+
20*d^3))*cos(f*x+e)/a^2/(c-d)^4/(c+d)^2/f/(c+d*sin(f*x+e))+d*(A*d*(12*c^2+16*c*d+7*d^2)-B*(6*c^3+12*c^2*d+13*c
*d^2+4*d^3))*arctan((d+c*tan(1/2*f*x+1/2*e))/(c^2-d^2)^(1/2))/a^2/(c-d)^4/(c+d)^2/f/(c^2-d^2)^(1/2)

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {3057, 2833, 12, 2739, 632, 210} \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))^3} \, dx=\frac {d \left (A d \left (12 c^2+16 c d+7 d^2\right )-B \left (6 c^3+12 c^2 d+13 c d^2+4 d^3\right )\right ) \arctan \left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{a^2 f (c-d)^4 (c+d)^2 \sqrt {c^2-d^2}}-\frac {d \left (A \left (2 c^2-16 c d-21 d^2\right )+B \left (4 c^2+19 c d+12 d^2\right )\right ) \cos (e+f x)}{6 a^2 f (c-d)^3 (c+d) (c+d \sin (e+f x))^2}-\frac {d \left (A \left (2 c^3-16 c^2 d-59 c d^2-32 d^3\right )+B \left (4 c^3+37 c^2 d+44 c d^2+20 d^3\right )\right ) \cos (e+f x)}{6 a^2 f (c-d)^4 (c+d)^2 (c+d \sin (e+f x))}-\frac {(A c-8 A d+2 B c+5 B d) \cos (e+f x)}{3 a^2 f (c-d)^2 (\sin (e+f x)+1) (c+d \sin (e+f x))^2}-\frac {(A-B) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2} \]

[In]

Int[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^3),x]

[Out]

(d*(A*d*(12*c^2 + 16*c*d + 7*d^2) - B*(6*c^3 + 12*c^2*d + 13*c*d^2 + 4*d^3))*ArcTan[(d + c*Tan[(e + f*x)/2])/S
qrt[c^2 - d^2]])/(a^2*(c - d)^4*(c + d)^2*Sqrt[c^2 - d^2]*f) - (d*(A*(2*c^2 - 16*c*d - 21*d^2) + B*(4*c^2 + 19
*c*d + 12*d^2))*Cos[e + f*x])/(6*a^2*(c - d)^3*(c + d)*f*(c + d*Sin[e + f*x])^2) - ((A*c + 2*B*c - 8*A*d + 5*B
*d)*Cos[e + f*x])/(3*a^2*(c - d)^2*f*(1 + Sin[e + f*x])*(c + d*Sin[e + f*x])^2) - ((A - B)*Cos[e + f*x])/(3*(c
- d)*f*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^2) - (d*(A*(2*c^3 - 16*c^2*d - 59*c*d^2 - 32*d^3) + B*(4*c
^3 + 37*c^2*d + 44*c*d^2 + 20*d^3))*Cos[e + f*x])/(6*a^2*(c - d)^4*(c + d)^2*f*(c + d*Sin[e + f*x]))

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 2833

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 3057

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])

Rubi steps \begin{align*} \text {integral}& = -\frac {(A-B) \cos (e+f x)}{3 (c-d) f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^2}-\frac {\int \frac {-a (A (c-5 d)+2 B (c+d))-3 a (A-B) d \sin (e+f x)}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^3} \, dx}{3 a^2 (c-d)} \\ & = -\frac {(A c+2 B c-8 A d+5 B d) \cos (e+f x)}{3 a^2 (c-d)^2 f (1+\sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {(A-B) \cos (e+f x)}{3 (c-d) f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^2}+\frac {\int \frac {-3 a^2 d (3 B c-7 A d+4 B d)+2 a^2 d (A c+2 B c-8 A d+5 B d) \sin (e+f x)}{(c+d \sin (e+f x))^3} \, dx}{3 a^4 (c-d)^2} \\ & = -\frac {d \left (A \left (2 c^2-16 c d-21 d^2\right )+B \left (4 c^2+19 c d+12 d^2\right )\right ) \cos (e+f x)}{6 a^2 (c-d)^3 (c+d) f (c+d \sin (e+f x))^2}-\frac {(A c+2 B c-8 A d+5 B d) \cos (e+f x)}{3 a^2 (c-d)^2 f (1+\sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {(A-B) \cos (e+f x)}{3 (c-d) f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^2}-\frac {\int \frac {-2 a^2 d \left (A d (19 c+16 d)-B \left (9 c^2+16 c d+10 d^2\right )\right )-a^2 d \left (2 A c^2+4 B c^2-16 A c d+19 B c d-21 A d^2+12 B d^2\right ) \sin (e+f x)}{(c+d \sin (e+f x))^2} \, dx}{6 a^4 (c-d)^3 (c+d)} \\ & = -\frac {d \left (A \left (2 c^2-16 c d-21 d^2\right )+B \left (4 c^2+19 c d+12 d^2\right )\right ) \cos (e+f x)}{6 a^2 (c-d)^3 (c+d) f (c+d \sin (e+f x))^2}-\frac {(A c+2 B c-8 A d+5 B d) \cos (e+f x)}{3 a^2 (c-d)^2 f (1+\sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {(A-B) \cos (e+f x)}{3 (c-d) f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^2}-\frac {d \left (A \left (2 c^3-16 c^2 d-59 c d^2-32 d^3\right )+B \left (4 c^3+37 c^2 d+44 c d^2+20 d^3\right )\right ) \cos (e+f x)}{6 a^2 (c-d)^4 (c+d)^2 f (c+d \sin (e+f x))}+\frac {\int \frac {3 a^2 d \left (A d \left (12 c^2+16 c d+7 d^2\right )-B \left (6 c^3+12 c^2 d+13 c d^2+4 d^3\right )\right )}{c+d \sin (e+f x)} \, dx}{6 a^4 (c-d)^4 (c+d)^2} \\ & = -\frac {d \left (A \left (2 c^2-16 c d-21 d^2\right )+B \left (4 c^2+19 c d+12 d^2\right )\right ) \cos (e+f x)}{6 a^2 (c-d)^3 (c+d) f (c+d \sin (e+f x))^2}-\frac {(A c+2 B c-8 A d+5 B d) \cos (e+f x)}{3 a^2 (c-d)^2 f (1+\sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {(A-B) \cos (e+f x)}{3 (c-d) f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^2}-\frac {d \left (A \left (2 c^3-16 c^2 d-59 c d^2-32 d^3\right )+B \left (4 c^3+37 c^2 d+44 c d^2+20 d^3\right )\right ) \cos (e+f x)}{6 a^2 (c-d)^4 (c+d)^2 f (c+d \sin (e+f x))}+\frac {\left (d \left (A d \left (12 c^2+16 c d+7 d^2\right )-B \left (6 c^3+12 c^2 d+13 c d^2+4 d^3\right )\right )\right ) \int \frac {1}{c+d \sin (e+f x)} \, dx}{2 a^2 (c-d)^4 (c+d)^2} \\ & = -\frac {d \left (A \left (2 c^2-16 c d-21 d^2\right )+B \left (4 c^2+19 c d+12 d^2\right )\right ) \cos (e+f x)}{6 a^2 (c-d)^3 (c+d) f (c+d \sin (e+f x))^2}-\frac {(A c+2 B c-8 A d+5 B d) \cos (e+f x)}{3 a^2 (c-d)^2 f (1+\sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {(A-B) \cos (e+f x)}{3 (c-d) f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^2}-\frac {d \left (A \left (2 c^3-16 c^2 d-59 c d^2-32 d^3\right )+B \left (4 c^3+37 c^2 d+44 c d^2+20 d^3\right )\right ) \cos (e+f x)}{6 a^2 (c-d)^4 (c+d)^2 f (c+d \sin (e+f x))}+\frac {\left (d \left (A d \left (12 c^2+16 c d+7 d^2\right )-B \left (6 c^3+12 c^2 d+13 c d^2+4 d^3\right )\right )\right ) \text {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^2 (c-d)^4 (c+d)^2 f} \\ & = -\frac {d \left (A \left (2 c^2-16 c d-21 d^2\right )+B \left (4 c^2+19 c d+12 d^2\right )\right ) \cos (e+f x)}{6 a^2 (c-d)^3 (c+d) f (c+d \sin (e+f x))^2}-\frac {(A c+2 B c-8 A d+5 B d) \cos (e+f x)}{3 a^2 (c-d)^2 f (1+\sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {(A-B) \cos (e+f x)}{3 (c-d) f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^2}-\frac {d \left (A \left (2 c^3-16 c^2 d-59 c d^2-32 d^3\right )+B \left (4 c^3+37 c^2 d+44 c d^2+20 d^3\right )\right ) \cos (e+f x)}{6 a^2 (c-d)^4 (c+d)^2 f (c+d \sin (e+f x))}-\frac {\left (2 d \left (A d \left (12 c^2+16 c d+7 d^2\right )-B \left (6 c^3+12 c^2 d+13 c d^2+4 d^3\right )\right )\right ) \text {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^2 (c-d)^4 (c+d)^2 f} \\ & = \frac {d \left (A d \left (12 c^2+16 c d+7 d^2\right )-B \left (6 c^3+12 c^2 d+13 c d^2+4 d^3\right )\right ) \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{a^2 (c-d)^4 (c+d)^2 \sqrt {c^2-d^2} f}-\frac {d \left (A \left (2 c^2-16 c d-21 d^2\right )+B \left (4 c^2+19 c d+12 d^2\right )\right ) \cos (e+f x)}{6 a^2 (c-d)^3 (c+d) f (c+d \sin (e+f x))^2}-\frac {(A c+2 B c-8 A d+5 B d) \cos (e+f x)}{3 a^2 (c-d)^2 f (1+\sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {(A-B) \cos (e+f x)}{3 (c-d) f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^2}-\frac {d \left (A \left (2 c^3-16 c^2 d-59 c d^2-32 d^3\right )+B \left (4 c^3+37 c^2 d+44 c d^2+20 d^3\right )\right ) \cos (e+f x)}{6 a^2 (c-d)^4 (c+d)^2 f (c+d \sin (e+f x))} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1257\) vs. \(2(386)=772\).

Time = 10.90 (sec) , antiderivative size = 1257, normalized size of antiderivative = 3.26 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))^3} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (-\frac {48 d \left (-A d \left (12 c^2+16 c d+7 d^2\right )+B \left (6 c^3+12 c^2 d+13 c d^2+4 d^3\right )\right ) \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3}{\sqrt {c^2-d^2}}+\frac {\left (-A d \left (96 c^4+524 c^3 d+776 c^2 d^2+487 c d^3+112 d^4\right )+B \left (48 c^5+240 c^4 d+536 c^3 d^2+701 c^2 d^3+400 c d^4+70 d^5\right )\right ) \cos \left (\frac {1}{2} (e+f x)\right )-\left (A \left (16 c^5-80 c^4 d-536 c^3 d^2-1028 c^2 d^3-695 c d^4-134 d^5\right )+B \left (32 c^5+224 c^4 d+728 c^3 d^2+893 c^2 d^3+482 c d^4+98 d^5\right )\right ) \cos \left (\frac {3}{2} (e+f x)\right )+24 B c^3 d^2 \cos \left (\frac {5}{2} (e+f x)\right )-12 A c^2 d^3 \cos \left (\frac {5}{2} (e+f x)\right )+21 B c^2 d^3 \cos \left (\frac {5}{2} (e+f x)\right )-15 A c d^4 \cos \left (\frac {5}{2} (e+f x)\right )-18 B c d^4 \cos \left (\frac {5}{2} (e+f x)\right )+6 A d^5 \cos \left (\frac {5}{2} (e+f x)\right )-6 B d^5 \cos \left (\frac {5}{2} (e+f x)\right )+4 A c^3 d^2 \cos \left (\frac {7}{2} (e+f x)\right )+8 B c^3 d^2 \cos \left (\frac {7}{2} (e+f x)\right )-32 A c^2 d^3 \cos \left (\frac {7}{2} (e+f x)\right )+59 B c^2 d^3 \cos \left (\frac {7}{2} (e+f x)\right )-97 A c d^4 \cos \left (\frac {7}{2} (e+f x)\right )+76 B c d^4 \cos \left (\frac {7}{2} (e+f x)\right )-52 A d^5 \cos \left (\frac {7}{2} (e+f x)\right )+34 B d^5 \cos \left (\frac {7}{2} (e+f x)\right )+48 A c^5 \sin \left (\frac {1}{2} (e+f x)\right )+48 B c^5 \sin \left (\frac {1}{2} (e+f x)\right )-224 A c^4 d \sin \left (\frac {1}{2} (e+f x)\right )+416 B c^4 d \sin \left (\frac {1}{2} (e+f x)\right )-872 A c^3 d^2 \sin \left (\frac {1}{2} (e+f x)\right )+992 B c^3 d^2 \sin \left (\frac {1}{2} (e+f x)\right )-1144 A c^2 d^3 \sin \left (\frac {1}{2} (e+f x)\right )+967 B c^2 d^3 \sin \left (\frac {1}{2} (e+f x)\right )-685 A c d^4 \sin \left (\frac {1}{2} (e+f x)\right )+496 B c d^4 \sin \left (\frac {1}{2} (e+f x)\right )-168 A d^5 \sin \left (\frac {1}{2} (e+f x)\right )+126 B d^5 \sin \left (\frac {1}{2} (e+f x)\right )+48 B c^4 d \sin \left (\frac {3}{2} (e+f x)\right )-132 A c^3 d^2 \sin \left (\frac {3}{2} (e+f x)\right )+96 B c^3 d^2 \sin \left (\frac {3}{2} (e+f x)\right )-204 A c^2 d^3 \sin \left (\frac {3}{2} (e+f x)\right )+207 B c^2 d^3 \sin \left (\frac {3}{2} (e+f x)\right )-165 A c d^4 \sin \left (\frac {3}{2} (e+f x)\right )+174 B c d^4 \sin \left (\frac {3}{2} (e+f x)\right )-66 A d^5 \sin \left (\frac {3}{2} (e+f x)\right )+42 B d^5 \sin \left (\frac {3}{2} (e+f x)\right )-16 A c^4 d \sin \left (\frac {5}{2} (e+f x)\right )-32 B c^4 d \sin \left (\frac {5}{2} (e+f x)\right )+116 A c^3 d^2 \sin \left (\frac {5}{2} (e+f x)\right )-224 B c^3 d^2 \sin \left (\frac {5}{2} (e+f x)\right )+412 A c^2 d^3 \sin \left (\frac {5}{2} (e+f x)\right )-409 B c^2 d^3 \sin \left (\frac {5}{2} (e+f x)\right )+403 A c d^4 \sin \left (\frac {5}{2} (e+f x)\right )-286 B c d^4 \sin \left (\frac {5}{2} (e+f x)\right )+114 A d^5 \sin \left (\frac {5}{2} (e+f x)\right )-78 B d^5 \sin \left (\frac {5}{2} (e+f x)\right )+15 B c^2 d^3 \sin \left (\frac {7}{2} (e+f x)\right )-21 A c d^4 \sin \left (\frac {7}{2} (e+f x)\right )+12 B c d^4 \sin \left (\frac {7}{2} (e+f x)\right )-12 A d^5 \sin \left (\frac {7}{2} (e+f x)\right )+6 B d^5 \sin \left (\frac {7}{2} (e+f x)\right )}{(c+d \sin (e+f x))^2}\right )}{48 a^2 (c-d)^4 (c+d)^2 f (1+\sin (e+f x))^2} \]

[In]

Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^3),x]

[Out]

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*((-48*d*(-(A*d*(12*c^2 + 16*c*d + 7*d^2)) + B*(6*c^3 + 12*c^2*d + 13*c*
d^2 + 4*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])^3)/Sqrt[c
^2 - d^2] + ((-(A*d*(96*c^4 + 524*c^3*d + 776*c^2*d^2 + 487*c*d^3 + 112*d^4)) + B*(48*c^5 + 240*c^4*d + 536*c^
3*d^2 + 701*c^2*d^3 + 400*c*d^4 + 70*d^5))*Cos[(e + f*x)/2] - (A*(16*c^5 - 80*c^4*d - 536*c^3*d^2 - 1028*c^2*d
^3 - 695*c*d^4 - 134*d^5) + B*(32*c^5 + 224*c^4*d + 728*c^3*d^2 + 893*c^2*d^3 + 482*c*d^4 + 98*d^5))*Cos[(3*(e
+ f*x))/2] + 24*B*c^3*d^2*Cos[(5*(e + f*x))/2] - 12*A*c^2*d^3*Cos[(5*(e + f*x))/2] + 21*B*c^2*d^3*Cos[(5*(e +
f*x))/2] - 15*A*c*d^4*Cos[(5*(e + f*x))/2] - 18*B*c*d^4*Cos[(5*(e + f*x))/2] + 6*A*d^5*Cos[(5*(e + f*x))/2] -
6*B*d^5*Cos[(5*(e + f*x))/2] + 4*A*c^3*d^2*Cos[(7*(e + f*x))/2] + 8*B*c^3*d^2*Cos[(7*(e + f*x))/2] - 32*A*c^2
*d^3*Cos[(7*(e + f*x))/2] + 59*B*c^2*d^3*Cos[(7*(e + f*x))/2] - 97*A*c*d^4*Cos[(7*(e + f*x))/2] + 76*B*c*d^4*C
os[(7*(e + f*x))/2] - 52*A*d^5*Cos[(7*(e + f*x))/2] + 34*B*d^5*Cos[(7*(e + f*x))/2] + 48*A*c^5*Sin[(e + f*x)/2
] + 48*B*c^5*Sin[(e + f*x)/2] - 224*A*c^4*d*Sin[(e + f*x)/2] + 416*B*c^4*d*Sin[(e + f*x)/2] - 872*A*c^3*d^2*Si
n[(e + f*x)/2] + 992*B*c^3*d^2*Sin[(e + f*x)/2] - 1144*A*c^2*d^3*Sin[(e + f*x)/2] + 967*B*c^2*d^3*Sin[(e + f*x
)/2] - 685*A*c*d^4*Sin[(e + f*x)/2] + 496*B*c*d^4*Sin[(e + f*x)/2] - 168*A*d^5*Sin[(e + f*x)/2] + 126*B*d^5*Si
n[(e + f*x)/2] + 48*B*c^4*d*Sin[(3*(e + f*x))/2] - 132*A*c^3*d^2*Sin[(3*(e + f*x))/2] + 96*B*c^3*d^2*Sin[(3*(e
+ f*x))/2] - 204*A*c^2*d^3*Sin[(3*(e + f*x))/2] + 207*B*c^2*d^3*Sin[(3*(e + f*x))/2] - 165*A*c*d^4*Sin[(3*(e
+ f*x))/2] + 174*B*c*d^4*Sin[(3*(e + f*x))/2] - 66*A*d^5*Sin[(3*(e + f*x))/2] + 42*B*d^5*Sin[(3*(e + f*x))/2]
- 16*A*c^4*d*Sin[(5*(e + f*x))/2] - 32*B*c^4*d*Sin[(5*(e + f*x))/2] + 116*A*c^3*d^2*Sin[(5*(e + f*x))/2] - 224
*B*c^3*d^2*Sin[(5*(e + f*x))/2] + 412*A*c^2*d^3*Sin[(5*(e + f*x))/2] - 409*B*c^2*d^3*Sin[(5*(e + f*x))/2] + 40
3*A*c*d^4*Sin[(5*(e + f*x))/2] - 286*B*c*d^4*Sin[(5*(e + f*x))/2] + 114*A*d^5*Sin[(5*(e + f*x))/2] - 78*B*d^5*
Sin[(5*(e + f*x))/2] + 15*B*c^2*d^3*Sin[(7*(e + f*x))/2] - 21*A*c*d^4*Sin[(7*(e + f*x))/2] + 12*B*c*d^4*Sin[(7
*(e + f*x))/2] - 12*A*d^5*Sin[(7*(e + f*x))/2] + 6*B*d^5*Sin[(7*(e + f*x))/2])/(c + d*Sin[e + f*x])^2))/(48*a^
2*(c - d)^4*(c + d)^2*f*(1 + Sin[e + f*x])^2)

Maple [A] (verified)

Time = 4.52 (sec) , antiderivative size = 547, normalized size of antiderivative = 1.42

method	result	size
derivativedivides	\(\frac {-\frac {2 \left (-2 B +2 A \right )}{3 \left (c -d \right )^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {2 B -2 A}{\left (c -d \right )^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 \left (A c -4 d A +3 d B \right )}{\left (c -d \right )^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}+\frac {2 d \left (\frac {\frac {d^{2} \left (9 c^{2} d A +4 d^{2} c A -2 A \,d^{3}-7 B \,c^{3}-4 c^{2} d B \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 c \left (c^{2}+2 c d +d^{2}\right )}+\frac {d \left (8 A \,c^{4} d +4 A \,c^{3} d^{2}+15 A \,c^{2} d^{3}+8 A c \,d^{4}-2 A \,d^{5}-6 B \,c^{5}-4 B \,c^{4} d -13 B \,c^{3} d^{2}-8 B \,c^{2} d^{3}-2 B c \,d^{4}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c^{2}+2 c d +d^{2}\right ) c^{2}}+\frac {d^{2} \left (23 c^{2} d A +12 d^{2} c A -2 A \,d^{3}-17 B \,c^{3}-12 c^{2} d B -4 d^{2} c B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 c \left (c^{2}+2 c d +d^{2}\right )}+\frac {d \left (8 c^{2} d A +4 d^{2} c A -A \,d^{3}-6 B \,c^{3}-4 c^{2} d B -d^{2} c B \right )}{2 c^{2}+4 c d +2 d^{2}}}{{\left (\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c +2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c \right )}^{2}}+\frac {\left (12 c^{2} d A +16 d^{2} c A +7 A \,d^{3}-6 B \,c^{3}-12 c^{2} d B -13 d^{2} c B -4 d^{3} B \right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{2 \left (c^{2}+2 c d +d^{2}\right ) \sqrt {c^{2}-d^{2}}}\right )}{\left (c -d \right )^{4}}}{a^{2} f}\)	\(547\)
default	\(\frac {-\frac {2 \left (-2 B +2 A \right )}{3 \left (c -d \right )^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {2 B -2 A}{\left (c -d \right )^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 \left (A c -4 d A +3 d B \right )}{\left (c -d \right )^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}+\frac {2 d \left (\frac {\frac {d^{2} \left (9 c^{2} d A +4 d^{2} c A -2 A \,d^{3}-7 B \,c^{3}-4 c^{2} d B \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 c \left (c^{2}+2 c d +d^{2}\right )}+\frac {d \left (8 A \,c^{4} d +4 A \,c^{3} d^{2}+15 A \,c^{2} d^{3}+8 A c \,d^{4}-2 A \,d^{5}-6 B \,c^{5}-4 B \,c^{4} d -13 B \,c^{3} d^{2}-8 B \,c^{2} d^{3}-2 B c \,d^{4}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c^{2}+2 c d +d^{2}\right ) c^{2}}+\frac {d^{2} \left (23 c^{2} d A +12 d^{2} c A -2 A \,d^{3}-17 B \,c^{3}-12 c^{2} d B -4 d^{2} c B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 c \left (c^{2}+2 c d +d^{2}\right )}+\frac {d \left (8 c^{2} d A +4 d^{2} c A -A \,d^{3}-6 B \,c^{3}-4 c^{2} d B -d^{2} c B \right )}{2 c^{2}+4 c d +2 d^{2}}}{{\left (\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c +2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c \right )}^{2}}+\frac {\left (12 c^{2} d A +16 d^{2} c A +7 A \,d^{3}-6 B \,c^{3}-12 c^{2} d B -13 d^{2} c B -4 d^{3} B \right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{2 \left (c^{2}+2 c d +d^{2}\right ) \sqrt {c^{2}-d^{2}}}\right )}{\left (c -d \right )^{4}}}{a^{2} f}\)	\(547\)
risch	\(\text {Expression too large to display}\)	\(2375\)

[In]

int((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^2/(c+d*sin(f*x+e))^3,x,method=_RETURNVERBOSE)

[Out]

2/f/a^2*(-1/3*(-2*B+2*A)/(c-d)^3/(tan(1/2*f*x+1/2*e)+1)^3-1/2*(2*B-2*A)/(c-d)^3/(tan(1/2*f*x+1/2*e)+1)^2-(A*c-
4*A*d+3*B*d)/(c-d)^4/(tan(1/2*f*x+1/2*e)+1)+d/(c-d)^4*((1/2*d^2*(9*A*c^2*d+4*A*c*d^2-2*A*d^3-7*B*c^3-4*B*c^2*d
)/c/(c^2+2*c*d+d^2)*tan(1/2*f*x+1/2*e)^3+1/2*d*(8*A*c^4*d+4*A*c^3*d^2+15*A*c^2*d^3+8*A*c*d^4-2*A*d^5-6*B*c^5-4
*B*c^4*d-13*B*c^3*d^2-8*B*c^2*d^3-2*B*c*d^4)/(c^2+2*c*d+d^2)/c^2*tan(1/2*f*x+1/2*e)^2+1/2*d^2*(23*A*c^2*d+12*A
*c*d^2-2*A*d^3-17*B*c^3-12*B*c^2*d-4*B*c*d^2)/c/(c^2+2*c*d+d^2)*tan(1/2*f*x+1/2*e)+1/2*d*(8*A*c^2*d+4*A*c*d^2-
A*d^3-6*B*c^3-4*B*c^2*d-B*c*d^2)/(c^2+2*c*d+d^2))/(tan(1/2*f*x+1/2*e)^2*c+2*d*tan(1/2*f*x+1/2*e)+c)^2+1/2*(12*
A*c^2*d+16*A*c*d^2+7*A*d^3-6*B*c^3-12*B*c^2*d-13*B*c*d^2-4*B*d^3)/(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))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2456 vs. \(2 (373) = 746\).

Time = 0.45 (sec) , antiderivative size = 4997, normalized size of antiderivative = 12.95 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))^3} \, dx=\text {Too large to display} \]

[In]

integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^2/(c+d*sin(f*x+e))^3,x, algorithm="fricas")

[Out]

[-1/12*(4*(A - B)*c^7 - 4*(A - B)*c^6*d - 12*(A - B)*c^5*d^2 + 12*(A - B)*c^4*d^3 + 12*(A - B)*c^3*d^4 - 12*(A
- B)*c^2*d^5 - 4*(A - B)*c*d^6 + 4*(A - B)*d^7 - 2*(2*(A + 2*B)*c^5*d^2 - (16*A - 37*B)*c^4*d^3 - (61*A - 40*
B)*c^3*d^4 - (16*A + 17*B)*c^2*d^5 + (59*A - 44*B)*c*d^6 + 4*(8*A - 5*B)*d^7)*cos(f*x + e)^4 - 2*(4*(A + 2*B)*
c^6*d - 4*(7*A - 16*B)*c^5*d^2 - 118*(A - B)*c^4*d^3 - (106*A - 25*B)*c^3*d^4 + (71*A - 98*B)*c^2*d^5 + (134*A
- 89*B)*c*d^6 + (43*A - 28*B)*d^7)*cos(f*x + e)^3 + 2*(2*(A + 2*B)*c^7 - 6*(2*A - 3*B)*c^6*d - 12*(3*A - 4*B)
*c^5*d^2 - 3*(18*A - 17*B)*c^4*d^3 - 3*(13*A + B)*c^3*d^4 + 3*(13*A - 17*B)*c^2*d^5 + (73*A - 49*B)*c*d^6 + 9*
(3*A - 2*B)*d^7)*cos(f*x + e)^2 + 3*(12*B*c^5*d - 24*(A - 2*B)*c^4*d^2 - 2*(40*A - 43*B)*c^3*d^3 - 6*(17*A - 1
4*B)*c^2*d^4 - 6*(10*A - 7*B)*c*d^5 - 2*(7*A - 4*B)*d^6 + (6*B*c^3*d^3 - 12*(A - B)*c^2*d^4 - (16*A - 13*B)*c*
d^5 - (7*A - 4*B)*d^6)*cos(f*x + e)^4 - (12*B*c^4*d^2 - 6*(4*A - 5*B)*c^3*d^3 - 2*(22*A - 19*B)*c^2*d^4 - 3*(1
0*A - 7*B)*c*d^5 - (7*A - 4*B)*d^6)*cos(f*x + e)^3 - (6*B*c^5*d - 12*(A - 3*B)*c^4*d^2 - (64*A - 79*B)*c^3*d^3
- (107*A - 92*B)*c^2*d^4 - (76*A - 55*B)*c*d^5 - 3*(7*A - 4*B)*d^6)*cos(f*x + e)^2 + (6*B*c^5*d - 12*(A - 2*B
)*c^4*d^2 - (40*A - 43*B)*c^3*d^3 - 3*(17*A - 14*B)*c^2*d^4 - 3*(10*A - 7*B)*c*d^5 - (7*A - 4*B)*d^6)*cos(f*x
+ e) + (12*B*c^5*d - 24*(A - 2*B)*c^4*d^2 - 2*(40*A - 43*B)*c^3*d^3 - 6*(17*A - 14*B)*c^2*d^4 - 6*(10*A - 7*B)
*c*d^5 - 2*(7*A - 4*B)*d^6 - (6*B*c^3*d^3 - 12*(A - B)*c^2*d^4 - (16*A - 13*B)*c*d^5 - (7*A - 4*B)*d^6)*cos(f*
x + e)^3 - 2*(6*B*c^4*d^2 - 6*(2*A - 3*B)*c^3*d^3 - (28*A - 25*B)*c^2*d^4 - (23*A - 17*B)*c*d^5 - (7*A - 4*B)*
d^6)*cos(f*x + e)^2 + (6*B*c^5*d - 12*(A - 2*B)*c^4*d^2 - (40*A - 43*B)*c^3*d^3 - 3*(17*A - 14*B)*c^2*d^4 - 3*
(10*A - 7*B)*c*d^5 - (7*A - 4*B)*d^6)*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*sin(f*x + e) - c^2 - d^2)) + 4*((2*A + B)*c^7 - (5*A - 14*B)*c^6*d - 3*(12*A - 19*
B)*c^5*d^2 - 3*(25*A - 21*B)*c^4*d^3 - 3*(13*A + 4*B)*c^3*d^4 + 3*(20*A - 21*B)*c^2*d^5 + (73*A - 46*B)*c*d^6
+ 2*(10*A - 7*B)*d^7)*cos(f*x + e) - 2*(2*(A - B)*c^7 - 2*(A - B)*c^6*d - 6*(A - B)*c^5*d^2 + 6*(A - B)*c^4*d^
3 + 6*(A - B)*c^3*d^4 - 6*(A - B)*c^2*d^5 - 2*(A - B)*c*d^6 + 2*(A - B)*d^7 + (2*(A + 2*B)*c^5*d^2 - (16*A - 3
7*B)*c^4*d^3 - (61*A - 40*B)*c^3*d^4 - (16*A + 17*B)*c^2*d^5 + (59*A - 44*B)*c*d^6 + 4*(8*A - 5*B)*d^7)*cos(f*
x + e)^3 - (4*(A + 2*B)*c^6*d - 30*(A - 2*B)*c^5*d^2 - 3*(34*A - 27*B)*c^4*d^3 - 15*(3*A + B)*c^3*d^4 + 3*(29*
A - 27*B)*c^2*d^5 + 15*(5*A - 3*B)*c*d^6 + (11*A - 8*B)*d^7)*cos(f*x + e)^2 - 2*((A + 2*B)*c^7 - (4*A - 13*B)*
c^6*d - 3*(11*A - 18*B)*c^5*d^2 - 6*(13*A - 11*B)*c^4*d^3 - 3*(14*A + 3*B)*c^3*d^4 + 3*(21*A - 22*B)*c^2*d^5 +
(74*A - 47*B)*c*d^6 + (19*A - 13*B)*d^7)*cos(f*x + e))*sin(f*x + e))/((a^2*c^8*d^2 - 2*a^2*c^7*d^3 - 2*a^2*c^
6*d^4 + 6*a^2*c^5*d^5 - 6*a^2*c^3*d^7 + 2*a^2*c^2*d^8 + 2*a^2*c*d^9 - a^2*d^10)*f*cos(f*x + e)^4 - (2*a^2*c^9*
d - 3*a^2*c^8*d^2 - 6*a^2*c^7*d^3 + 10*a^2*c^6*d^4 + 6*a^2*c^5*d^5 - 12*a^2*c^4*d^6 - 2*a^2*c^3*d^7 + 6*a^2*c^
2*d^8 - a^2*d^10)*f*cos(f*x + e)^3 - (a^2*c^10 + 2*a^2*c^9*d - 7*a^2*c^8*d^2 - 8*a^2*c^7*d^3 + 18*a^2*c^6*d^4
+ 12*a^2*c^5*d^5 - 22*a^2*c^4*d^6 - 8*a^2*c^3*d^7 + 13*a^2*c^2*d^8 + 2*a^2*c*d^9 - 3*a^2*d^10)*f*cos(f*x + e)^
2 + (a^2*c^10 - 5*a^2*c^8*d^2 + 10*a^2*c^6*d^4 - 10*a^2*c^4*d^6 + 5*a^2*c^2*d^8 - a^2*d^10)*f*cos(f*x + e) + 2
*(a^2*c^10 - 5*a^2*c^8*d^2 + 10*a^2*c^6*d^4 - 10*a^2*c^4*d^6 + 5*a^2*c^2*d^8 - a^2*d^10)*f - ((a^2*c^8*d^2 - 2
*a^2*c^7*d^3 - 2*a^2*c^6*d^4 + 6*a^2*c^5*d^5 - 6*a^2*c^3*d^7 + 2*a^2*c^2*d^8 + 2*a^2*c*d^9 - a^2*d^10)*f*cos(f
*x + e)^3 + 2*(a^2*c^9*d - a^2*c^8*d^2 - 4*a^2*c^7*d^3 + 4*a^2*c^6*d^4 + 6*a^2*c^5*d^5 - 6*a^2*c^4*d^6 - 4*a^2
*c^3*d^7 + 4*a^2*c^2*d^8 + a^2*c*d^9 - a^2*d^10)*f*cos(f*x + e)^2 - (a^2*c^10 - 5*a^2*c^8*d^2 + 10*a^2*c^6*d^4
- 10*a^2*c^4*d^6 + 5*a^2*c^2*d^8 - a^2*d^10)*f*cos(f*x + e) - 2*(a^2*c^10 - 5*a^2*c^8*d^2 + 10*a^2*c^6*d^4 -
10*a^2*c^4*d^6 + 5*a^2*c^2*d^8 - a^2*d^10)*f)*sin(f*x + e)), -1/6*(2*(A - B)*c^7 - 2*(A - B)*c^6*d - 6*(A - B)
*c^5*d^2 + 6*(A - B)*c^4*d^3 + 6*(A - B)*c^3*d^4 - 6*(A - B)*c^2*d^5 - 2*(A - B)*c*d^6 + 2*(A - B)*d^7 - (2*(A
+ 2*B)*c^5*d^2 - (16*A - 37*B)*c^4*d^3 - (61*A - 40*B)*c^3*d^4 - (16*A + 17*B)*c^2*d^5 + (59*A - 44*B)*c*d^6
+ 4*(8*A - 5*B)*d^7)*cos(f*x + e)^4 - (4*(A + 2*B)*c^6*d - 4*(7*A - 16*B)*c^5*d^2 - 118*(A - B)*c^4*d^3 - (106
*A - 25*B)*c^3*d^4 + (71*A - 98*B)*c^2*d^5 + (134*A - 89*B)*c*d^6 + (43*A - 28*B)*d^7)*cos(f*x + e)^3 + (2*(A
+ 2*B)*c^7 - 6*(2*A - 3*B)*c^6*d - 12*(3*A - 4*B)*c^5*d^2 - 3*(18*A - 17*B)*c^4*d^3 - 3*(13*A + B)*c^3*d^4 + 3
*(13*A - 17*B)*c^2*d^5 + (73*A - 49*B)*c*d^6 + 9*(3*A - 2*B)*d^7)*cos(f*x + e)^2 - 3*(12*B*c^5*d - 24*(A - 2*B
)*c^4*d^2 - 2*(40*A - 43*B)*c^3*d^3 - 6*(17*A - 14*B)*c^2*d^4 - 6*(10*A - 7*B)*c*d^5 - 2*(7*A - 4*B)*d^6 + (6*
B*c^3*d^3 - 12*(A - B)*c^2*d^4 - (16*A - 13*B)*c*d^5 - (7*A - 4*B)*d^6)*cos(f*x + e)^4 - (12*B*c^4*d^2 - 6*(4*
A - 5*B)*c^3*d^3 - 2*(22*A - 19*B)*c^2*d^4 - 3*(10*A - 7*B)*c*d^5 - (7*A - 4*B)*d^6)*cos(f*x + e)^3 - (6*B*c^5
*d - 12*(A - 3*B)*c^4*d^2 - (64*A - 79*B)*c^3*d^3 - (107*A - 92*B)*c^2*d^4 - (76*A - 55*B)*c*d^5 - 3*(7*A - 4*
B)*d^6)*cos(f*x + e)^2 + (6*B*c^5*d - 12*(A - 2*B)*c^4*d^2 - (40*A - 43*B)*c^3*d^3 - 3*(17*A - 14*B)*c^2*d^4 -
3*(10*A - 7*B)*c*d^5 - (7*A - 4*B)*d^6)*cos(f*x + e) + (12*B*c^5*d - 24*(A - 2*B)*c^4*d^2 - 2*(40*A - 43*B)*c
^3*d^3 - 6*(17*A - 14*B)*c^2*d^4 - 6*(10*A - 7*B)*c*d^5 - 2*(7*A - 4*B)*d^6 - (6*B*c^3*d^3 - 12*(A - B)*c^2*d^
4 - (16*A - 13*B)*c*d^5 - (7*A - 4*B)*d^6)*cos(f*x + e)^3 - 2*(6*B*c^4*d^2 - 6*(2*A - 3*B)*c^3*d^3 - (28*A - 2
5*B)*c^2*d^4 - (23*A - 17*B)*c*d^5 - (7*A - 4*B)*d^6)*cos(f*x + e)^2 + (6*B*c^5*d - 12*(A - 2*B)*c^4*d^2 - (40
*A - 43*B)*c^3*d^3 - 3*(17*A - 14*B)*c^2*d^4 - 3*(10*A - 7*B)*c*d^5 - (7*A - 4*B)*d^6)*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))) + 2*((2*A + B)*c^7 - (5*A -
14*B)*c^6*d - 3*(12*A - 19*B)*c^5*d^2 - 3*(25*A - 21*B)*c^4*d^3 - 3*(13*A + 4*B)*c^3*d^4 + 3*(20*A - 21*B)*c^2
*d^5 + (73*A - 46*B)*c*d^6 + 2*(10*A - 7*B)*d^7)*cos(f*x + e) - (2*(A - B)*c^7 - 2*(A - B)*c^6*d - 6*(A - B)*c
^5*d^2 + 6*(A - B)*c^4*d^3 + 6*(A - B)*c^3*d^4 - 6*(A - B)*c^2*d^5 - 2*(A - B)*c*d^6 + 2*(A - B)*d^7 + (2*(A +
2*B)*c^5*d^2 - (16*A - 37*B)*c^4*d^3 - (61*A - 40*B)*c^3*d^4 - (16*A + 17*B)*c^2*d^5 + (59*A - 44*B)*c*d^6 +
4*(8*A - 5*B)*d^7)*cos(f*x + e)^3 - (4*(A + 2*B)*c^6*d - 30*(A - 2*B)*c^5*d^2 - 3*(34*A - 27*B)*c^4*d^3 - 15*(
3*A + B)*c^3*d^4 + 3*(29*A - 27*B)*c^2*d^5 + 15*(5*A - 3*B)*c*d^6 + (11*A - 8*B)*d^7)*cos(f*x + e)^2 - 2*((A +
2*B)*c^7 - (4*A - 13*B)*c^6*d - 3*(11*A - 18*B)*c^5*d^2 - 6*(13*A - 11*B)*c^4*d^3 - 3*(14*A + 3*B)*c^3*d^4 +
3*(21*A - 22*B)*c^2*d^5 + (74*A - 47*B)*c*d^6 + (19*A - 13*B)*d^7)*cos(f*x + e))*sin(f*x + e))/((a^2*c^8*d^2 -
2*a^2*c^7*d^3 - 2*a^2*c^6*d^4 + 6*a^2*c^5*d^5 - 6*a^2*c^3*d^7 + 2*a^2*c^2*d^8 + 2*a^2*c*d^9 - a^2*d^10)*f*cos
(f*x + e)^4 - (2*a^2*c^9*d - 3*a^2*c^8*d^2 - 6*a^2*c^7*d^3 + 10*a^2*c^6*d^4 + 6*a^2*c^5*d^5 - 12*a^2*c^4*d^6 -
2*a^2*c^3*d^7 + 6*a^2*c^2*d^8 - a^2*d^10)*f*cos(f*x + e)^3 - (a^2*c^10 + 2*a^2*c^9*d - 7*a^2*c^8*d^2 - 8*a^2*
c^7*d^3 + 18*a^2*c^6*d^4 + 12*a^2*c^5*d^5 - 22*a^2*c^4*d^6 - 8*a^2*c^3*d^7 + 13*a^2*c^2*d^8 + 2*a^2*c*d^9 - 3*
a^2*d^10)*f*cos(f*x + e)^2 + (a^2*c^10 - 5*a^2*c^8*d^2 + 10*a^2*c^6*d^4 - 10*a^2*c^4*d^6 + 5*a^2*c^2*d^8 - a^2
*d^10)*f*cos(f*x + e) + 2*(a^2*c^10 - 5*a^2*c^8*d^2 + 10*a^2*c^6*d^4 - 10*a^2*c^4*d^6 + 5*a^2*c^2*d^8 - a^2*d^
10)*f - ((a^2*c^8*d^2 - 2*a^2*c^7*d^3 - 2*a^2*c^6*d^4 + 6*a^2*c^5*d^5 - 6*a^2*c^3*d^7 + 2*a^2*c^2*d^8 + 2*a^2*
c*d^9 - a^2*d^10)*f*cos(f*x + e)^3 + 2*(a^2*c^9*d - a^2*c^8*d^2 - 4*a^2*c^7*d^3 + 4*a^2*c^6*d^4 + 6*a^2*c^5*d^
5 - 6*a^2*c^4*d^6 - 4*a^2*c^3*d^7 + 4*a^2*c^2*d^8 + a^2*c*d^9 - a^2*d^10)*f*cos(f*x + e)^2 - (a^2*c^10 - 5*a^2
*c^8*d^2 + 10*a^2*c^6*d^4 - 10*a^2*c^4*d^6 + 5*a^2*c^2*d^8 - a^2*d^10)*f*cos(f*x + e) - 2*(a^2*c^10 - 5*a^2*c^
8*d^2 + 10*a^2*c^6*d^4 - 10*a^2*c^4*d^6 + 5*a^2*c^2*d^8 - a^2*d^10)*f)*sin(f*x + e))]

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))^3} \, dx=\text {Timed out} \]

[In]

integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))**2/(c+d*sin(f*x+e))**3,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))^3} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^2/(c+d*sin(f*x+e))^3,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*d^2-4*c^2>0)', see `assume?`
for more de

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 911 vs. \(2 (373) = 746\).

Time = 0.37 (sec) , antiderivative size = 911, normalized size of antiderivative = 2.36 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))^3} \, dx=\text {Too large to display} \]

[In]

integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^2/(c+d*sin(f*x+e))^3,x, algorithm="giac")

[Out]

-1/3*(3*(6*B*c^3*d - 12*A*c^2*d^2 + 12*B*c^2*d^2 - 16*A*c*d^3 + 13*B*c*d^3 - 7*A*d^4 + 4*B*d^4)*(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^2*c^6 - 2*a^2*c^5*d - a
^2*c^4*d^2 + 4*a^2*c^3*d^3 - a^2*c^2*d^4 - 2*a^2*c*d^5 + a^2*d^6)*sqrt(c^2 - d^2)) + 3*(7*B*c^4*d^3*tan(1/2*f*
x + 1/2*e)^3 - 9*A*c^3*d^4*tan(1/2*f*x + 1/2*e)^3 + 4*B*c^3*d^4*tan(1/2*f*x + 1/2*e)^3 - 4*A*c^2*d^5*tan(1/2*f
*x + 1/2*e)^3 + 2*A*c*d^6*tan(1/2*f*x + 1/2*e)^3 + 6*B*c^5*d^2*tan(1/2*f*x + 1/2*e)^2 - 8*A*c^4*d^3*tan(1/2*f*
x + 1/2*e)^2 + 4*B*c^4*d^3*tan(1/2*f*x + 1/2*e)^2 - 4*A*c^3*d^4*tan(1/2*f*x + 1/2*e)^2 + 13*B*c^3*d^4*tan(1/2*
f*x + 1/2*e)^2 - 15*A*c^2*d^5*tan(1/2*f*x + 1/2*e)^2 + 8*B*c^2*d^5*tan(1/2*f*x + 1/2*e)^2 - 8*A*c*d^6*tan(1/2*
f*x + 1/2*e)^2 + 2*B*c*d^6*tan(1/2*f*x + 1/2*e)^2 + 2*A*d^7*tan(1/2*f*x + 1/2*e)^2 + 17*B*c^4*d^3*tan(1/2*f*x
+ 1/2*e) - 23*A*c^3*d^4*tan(1/2*f*x + 1/2*e) + 12*B*c^3*d^4*tan(1/2*f*x + 1/2*e) - 12*A*c^2*d^5*tan(1/2*f*x +
1/2*e) + 4*B*c^2*d^5*tan(1/2*f*x + 1/2*e) + 2*A*c*d^6*tan(1/2*f*x + 1/2*e) + 6*B*c^5*d^2 - 8*A*c^4*d^3 + 4*B*c
^4*d^3 - 4*A*c^3*d^4 + B*c^3*d^4 + A*c^2*d^5)/((a^2*c^8 - 2*a^2*c^7*d - a^2*c^6*d^2 + 4*a^2*c^5*d^3 - a^2*c^4*
d^4 - 2*a^2*c^3*d^5 + a^2*c^2*d^6)*(c*tan(1/2*f*x + 1/2*e)^2 + 2*d*tan(1/2*f*x + 1/2*e) + c)^2) + 2*(3*A*c*tan
(1/2*f*x + 1/2*e)^2 - 12*A*d*tan(1/2*f*x + 1/2*e)^2 + 9*B*d*tan(1/2*f*x + 1/2*e)^2 + 3*A*c*tan(1/2*f*x + 1/2*e
) + 3*B*c*tan(1/2*f*x + 1/2*e) - 21*A*d*tan(1/2*f*x + 1/2*e) + 15*B*d*tan(1/2*f*x + 1/2*e) + 2*A*c + B*c - 11*
A*d + 8*B*d)/((a^2*c^4 - 4*a^2*c^3*d + 6*a^2*c^2*d^2 - 4*a^2*c*d^3 + a^2*d^4)*(tan(1/2*f*x + 1/2*e) + 1)^3))/f

Mupad [B] (verification not implemented)

Time = 17.88 (sec) , antiderivative size = 1686, normalized size of antiderivative = 4.37 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))^3} \, dx=\text {Too large to display} \]

[In]

int((A + B*sin(e + f*x))/((a + a*sin(e + f*x))^2*(c + d*sin(e + f*x))^3),x)

[Out]

(d*atan(((d*(4*a^2*c*d^6 - 2*a^2*d^7 - 2*a^2*c^6*d + 2*a^2*c^2*d^5 - 8*a^2*c^3*d^4 + 2*a^2*c^4*d^3 + 4*a^2*c^5
*d^2)*(6*B*c^3 - 7*A*d^3 + 4*B*d^3 - 16*A*c*d^2 - 12*A*c^2*d + 13*B*c*d^2 + 12*B*c^2*d))/(2*a^2*(c + d)^(5/2)*
(c - d)^(9/2)) + (c*d*tan(e/2 + (f*x)/2)*(2*a^2*c*d^5 - a^2*d^6 - a^2*c^6 + 2*a^2*c^5*d + a^2*c^2*d^4 - 4*a^2*
c^3*d^3 + a^2*c^4*d^2)*(6*B*c^3 - 7*A*d^3 + 4*B*d^3 - 16*A*c*d^2 - 12*A*c^2*d + 13*B*c*d^2 + 12*B*c^2*d))/(a^2
*(c + d)^(5/2)*(c - d)^(9/2)))/(4*B*d^4 - 7*A*d^4 - 12*A*c^2*d^2 + 12*B*c^2*d^2 - 16*A*c*d^3 + 13*B*c*d^3 + 6*
B*c^3*d))*(6*B*c^3 - 7*A*d^3 + 4*B*d^3 - 16*A*c*d^2 - 12*A*c^2*d + 13*B*c*d^2 + 12*B*c^2*d))/(a^2*f*(c + d)^(5
/2)*(c - d)^(9/2)) - ((tan(e/2 + (f*x)/2)^5*(2*A*c^6 + 2*A*d^6 + 2*B*c^6 - 23*A*c^2*d^4 - 40*A*c^3*d^3 - 38*A*
c^4*d^2 + 6*B*c^2*d^4 + 43*B*c^3*d^3 + 40*B*c^4*d^2 - 4*A*c*d^5 - 4*A*c^5*d + 2*B*c*d^5 + 12*B*c^5*d))/(c^2*(c
^5 - 3*c^4*d - 3*c*d^4 + d^5 + 2*c^2*d^3 + 2*c^3*d^2)) + (4*A*c^5 + 3*A*d^5 + 2*B*c^5 - 46*A*c^2*d^3 - 40*A*c^
3*d^2 + 28*B*c^2*d^3 + 52*B*c^3*d^2 - 12*A*c*d^4 - 14*A*c^4*d + 3*B*c*d^4 + 20*B*c^4*d)/(3*(c + d)*(c^2 - d^2)
*(3*c*d^2 - 3*c^2*d + c^3 - d^3)) + (2*tan(e/2 + (f*x)/2)^3*(6*A*c^6 + 9*A*d^6 + 6*B*c^6 - 177*A*c^2*d^4 - 212
*A*c^3*d^3 - 102*A*c^4*d^2 + 105*B*c^2*d^4 + 215*B*c^3*d^3 + 150*B*c^4*d^2 - 33*A*c*d^5 - 16*A*c^5*d + 9*B*c*d
^5 + 40*B*c^5*d))/(3*c^2*(c^2 - d^2)*(3*c*d^2 - 3*c^2*d + c^3 - d^3)) + (tan(e/2 + (f*x)/2)*(6*A*c^5 + 6*A*d^5
+ 6*B*c^5 - 160*A*c^2*d^3 - 114*A*c^3*d^2 + 97*B*c^2*d^3 + 156*B*c^3*d^2 - 33*A*c*d^4 - 20*A*c^4*d + 12*B*c*d
^4 + 44*B*c^4*d))/(3*c*(c^2 - d^2)*(3*c*d^2 - 3*c^2*d + c^3 - d^3)) + (tan(e/2 + (f*x)/2)^2*(14*A*c^7 + 6*A*d^
7 + 4*B*c^7 - 232*A*c^2*d^5 - 583*A*c^3*d^4 - 532*A*c^4*d^3 - 226*A*c^5*d^2 + 124*B*c^2*d^5 + 412*B*c^3*d^4 +
595*B*c^4*d^3 + 352*B*c^5*d^2 - 6*A*c*d^6 - 16*A*c^6*d + 6*B*c*d^6 + 82*B*c^6*d))/(3*c^2*(c + d)*(c^2 - d^2)*(
3*c*d^2 - 3*c^2*d + c^3 - d^3)) + (tan(e/2 + (f*x)/2)^4*(16*A*c^7 + 18*A*d^7 + 2*B*c^7 - 303*A*c^2*d^5 - 522*A
*c^3*d^4 - 502*A*c^4*d^3 - 220*A*c^5*d^2 + 156*B*c^2*d^5 + 453*B*c^3*d^4 + 538*B*c^4*d^3 + 328*B*c^5*d^2 - 48*
A*c*d^6 - 14*A*c^6*d + 18*B*c*d^6 + 80*B*c^6*d))/(3*c^2*(c + d)*(c^2 - d^2)*(3*c*d^2 - 3*c^2*d + c^3 - d^3)) +
(tan(e/2 + (f*x)/2)^6*(2*A*c^6 + 2*A*d^6 - 9*A*c^2*d^4 - 8*A*c^3*d^3 - 14*A*c^4*d^2 + 4*B*c^2*d^4 + 13*B*c^3*
d^3 + 12*B*c^4*d^2 - 4*A*c*d^5 - 4*A*c^5*d + 6*B*c^5*d))/(c*(c - d)*(2*c*d + c^2 + d^2)*(3*c*d^2 - 3*c^2*d + c
^3 - d^3)))/(f*(tan(e/2 + (f*x)/2)*(3*a^2*c^2 + 4*a^2*c*d) + tan(e/2 + (f*x)/2)^2*(5*a^2*c^2 + 4*a^2*d^2 + 12*
a^2*c*d) + tan(e/2 + (f*x)/2)^5*(5*a^2*c^2 + 4*a^2*d^2 + 12*a^2*c*d) + tan(e/2 + (f*x)/2)^3*(7*a^2*c^2 + 12*a^
2*d^2 + 16*a^2*c*d) + tan(e/2 + (f*x)/2)^4*(7*a^2*c^2 + 12*a^2*d^2 + 16*a^2*c*d) + tan(e/2 + (f*x)/2)^6*(3*a^2
*c^2 + 4*a^2*c*d) + a^2*c^2 + a^2*c^2*tan(e/2 + (f*x)/2)^7))