\(\int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^3} \, dx\) [271]

Optimal result

Integrand size = 35, antiderivative size = 283 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^3} \, dx=-\frac {\left (3 A d \left (2 c^2+2 c d+d^2\right )-B \left (2 c^3+4 c^2 d+7 c d^2+2 d^3\right )\right ) \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{a (c-d) \left (c^2-d^2\right )^{5/2} f}-\frac {d (2 A c-3 B c+3 A d-2 B d) \cos (e+f x)}{2 a (c-d)^2 (c+d) f (c+d \sin (e+f x))^2}-\frac {(A-B) \cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {d \left (2 A c^2-5 B c^2+9 A c d-6 B c d+4 A d^2-4 B d^2\right ) \cos (e+f x)}{2 a (c-d)^3 (c+d)^2 f (c+d \sin (e+f x))} \]

[Out]

-(3*A*d*(2*c^2+2*c*d+d^2)-B*(2*c^3+4*c^2*d+7*c*d^2+2*d^3))*arctan((d+c*tan(1/2*f*x+1/2*e))/(c^2-d^2)^(1/2))/a/
(c-d)/(c^2-d^2)^(5/2)/f-1/2*d*(2*A*c+3*A*d-3*B*c-2*B*d)*cos(f*x+e)/a/(c-d)^2/(c+d)/f/(c+d*sin(f*x+e))^2-(A-B)*
cos(f*x+e)/(c-d)/f/(a+a*sin(f*x+e))/(c+d*sin(f*x+e))^2-1/2*d*(2*A*c^2+9*A*c*d+4*A*d^2-5*B*c^2-6*B*c*d-4*B*d^2)
*cos(f*x+e)/a/(c-d)^3/(c+d)^2/f/(c+d*sin(f*x+e))

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {3057, 2833, 12, 2739, 632, 210} \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^3} \, dx=-\frac {\left (3 A d \left (2 c^2+2 c d+d^2\right )-B \left (2 c^3+4 c^2 d+7 c d^2+2 d^3\right )\right ) \arctan \left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{a f (c-d) \left (c^2-d^2\right )^{5/2}}-\frac {d \left (2 A c^2+9 A c d+4 A d^2-5 B c^2-6 B c d-4 B d^2\right ) \cos (e+f x)}{2 a f (c-d)^3 (c+d)^2 (c+d \sin (e+f x))}-\frac {d (2 A c+3 A d-3 B c-2 B d) \cos (e+f x)}{2 a f (c-d)^2 (c+d) (c+d \sin (e+f x))^2}-\frac {(A-B) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^2} \]

[In]

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

[Out]

-(((3*A*d*(2*c^2 + 2*c*d + d^2) - B*(2*c^3 + 4*c^2*d + 7*c*d^2 + 2*d^3))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[
c^2 - d^2]])/(a*(c - d)*(c^2 - d^2)^(5/2)*f)) - (d*(2*A*c - 3*B*c + 3*A*d - 2*B*d)*Cos[e + f*x])/(2*a*(c - d)^
2*(c + d)*f*(c + d*Sin[e + f*x])^2) - ((A - B)*Cos[e + f*x])/((c - d)*f*(a + a*Sin[e + f*x])*(c + d*Sin[e + f*
x])^2) - (d*(2*A*c^2 - 5*B*c^2 + 9*A*c*d - 6*B*c*d + 4*A*d^2 - 4*B*d^2)*Cos[e + f*x])/(2*a*(c - d)^3*(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)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {\int \frac {a (3 A d-B (c+2 d))-2 a (A-B) d \sin (e+f x)}{(c+d \sin (e+f x))^3} \, dx}{a^2 (c-d)} \\ & = -\frac {d (2 A c-3 B c+3 A d-2 B d) \cos (e+f x)}{2 a (c-d)^2 (c+d) f (c+d \sin (e+f x))^2}-\frac {(A-B) \cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^2}+\frac {\int \frac {-2 a \left (2 (A-B) d^2+c (3 A d-B (c+2 d))\right )+a d (2 A c-3 B c+3 A d-2 B d) \sin (e+f x)}{(c+d \sin (e+f x))^2} \, dx}{2 a^2 (c-d)^2 (c+d)} \\ & = -\frac {d (2 A c-3 B c+3 A d-2 B d) \cos (e+f x)}{2 a (c-d)^2 (c+d) f (c+d \sin (e+f x))^2}-\frac {(A-B) \cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {d \left (2 A c^2-5 B c^2+9 A c d-6 B c d+4 A d^2-4 B d^2\right ) \cos (e+f x)}{2 a (c-d)^3 (c+d)^2 f (c+d \sin (e+f x))}-\frac {\int \frac {a \left (3 A d \left (2 c^2+2 c d+d^2\right )-B \left (2 c^3+4 c^2 d+7 c d^2+2 d^3\right )\right )}{c+d \sin (e+f x)} \, dx}{2 a^2 (c-d)^3 (c+d)^2} \\ & = -\frac {d (2 A c-3 B c+3 A d-2 B d) \cos (e+f x)}{2 a (c-d)^2 (c+d) f (c+d \sin (e+f x))^2}-\frac {(A-B) \cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {d \left (2 A c^2-5 B c^2+9 A c d-6 B c d+4 A d^2-4 B d^2\right ) \cos (e+f x)}{2 a (c-d)^3 (c+d)^2 f (c+d \sin (e+f x))}-\frac {\left (3 A d \left (2 c^2+2 c d+d^2\right )-B \left (2 c^3+4 c^2 d+7 c d^2+2 d^3\right )\right ) \int \frac {1}{c+d \sin (e+f x)} \, dx}{2 a (c-d)^3 (c+d)^2} \\ & = -\frac {d (2 A c-3 B c+3 A d-2 B d) \cos (e+f x)}{2 a (c-d)^2 (c+d) f (c+d \sin (e+f x))^2}-\frac {(A-B) \cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {d \left (2 A c^2-5 B c^2+9 A c d-6 B c d+4 A d^2-4 B d^2\right ) \cos (e+f x)}{2 a (c-d)^3 (c+d)^2 f (c+d \sin (e+f x))}-\frac {\left (3 A d \left (2 c^2+2 c d+d^2\right )-B \left (2 c^3+4 c^2 d+7 c d^2+2 d^3\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 (c-d)^3 (c+d)^2 f} \\ & = -\frac {d (2 A c-3 B c+3 A d-2 B d) \cos (e+f x)}{2 a (c-d)^2 (c+d) f (c+d \sin (e+f x))^2}-\frac {(A-B) \cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {d \left (2 A c^2-5 B c^2+9 A c d-6 B c d+4 A d^2-4 B d^2\right ) \cos (e+f x)}{2 a (c-d)^3 (c+d)^2 f (c+d \sin (e+f x))}+\frac {\left (2 \left (3 A d \left (2 c^2+2 c d+d^2\right )-B \left (2 c^3+4 c^2 d+7 c d^2+2 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 (c-d)^3 (c+d)^2 f} \\ & = -\frac {\left (3 A d \left (2 c^2+2 c d+d^2\right )-B \left (2 c^3+4 c^2 d+7 c d^2+2 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 (c-d)^3 (c+d)^2 \sqrt {c^2-d^2} f}-\frac {d (2 A c-3 B c+3 A d-2 B d) \cos (e+f x)}{2 a (c-d)^2 (c+d) f (c+d \sin (e+f x))^2}-\frac {(A-B) \cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {d \left (2 A c^2-5 B c^2+9 A c d-6 B c d+4 A d^2-4 B d^2\right ) \cos (e+f x)}{2 a (c-d)^3 (c+d)^2 f (c+d \sin (e+f x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.28 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.11 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^3} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (4 (A-B) \sin \left (\frac {1}{2} (e+f x)\right )+\frac {2 \left (-3 A d \left (2 c^2+2 c d+d^2\right )+B \left (2 c^3+4 c^2 d+7 c d^2+2 d^3\right )\right ) \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )}{(c+d)^2 \sqrt {c^2-d^2}}+\frac {(c-d) d (B c-A d) \cos (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )}{(c+d) (c+d \sin (e+f x))^2}+\frac {d \left (-A d (5 c+2 d)+B \left (3 c^2+2 c d+2 d^2\right )\right ) \cos (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )}{(c+d)^2 (c+d \sin (e+f x))}\right )}{2 a (c-d)^3 f (1+\sin (e+f x))} \]

[In]

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

[Out]

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(4*(A - B)*Sin[(e + f*x)/2] + (2*(-3*A*d*(2*c^2 + 2*c*d + d^2) + B*(2*c
^3 + 4*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]))/((c + d)^2*Sqrt[c^2 - d^2]) + ((c - d)*d*(B*c - A*d)*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)
/2]))/((c + d)*(c + d*Sin[e + f*x])^2) + (d*(-(A*d*(5*c + 2*d)) + B*(3*c^2 + 2*c*d + 2*d^2))*Cos[e + f*x]*(Cos
[(e + f*x)/2] + Sin[(e + f*x)/2]))/((c + d)^2*(c + d*Sin[e + f*x]))))/(2*a*(c - d)^3*f*(1 + Sin[e + f*x]))

Maple [A] (verified)

Time = 2.49 (sec) , antiderivative size = 482, normalized size of antiderivative = 1.70

method	result	size
derivativedivides	\(\frac {-\frac {2 \left (\frac {\frac {d^{2} \left (7 c^{2} d A +2 d^{2} c A -2 A \,d^{3}-5 B \,c^{3}-2 c^{2} d B \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 c \left (c^{2}+2 c d +d^{2}\right )}+\frac {d \left (6 A \,c^{4} d +2 A \,c^{3} d^{2}+11 A \,c^{2} d^{3}+4 A c \,d^{4}-2 A \,d^{5}-4 B \,c^{5}-2 B \,c^{4} d -9 B \,c^{3} d^{2}-4 B \,c^{2} d^{3}-2 B c \,d^{4}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c^{2}+2 c d +d^{2}\right ) c^{2}}+\frac {d^{2} \left (17 c^{2} d A +6 d^{2} c A -2 A \,d^{3}-11 B \,c^{3}-6 c^{2} d B -4 d^{2} c B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 c \left (c^{2}+2 c d +d^{2}\right )}+\frac {d \left (6 c^{2} d A +2 d^{2} c A -A \,d^{3}-4 B \,c^{3}-2 c^{2} d B -d^{2} c B \right )}{2 c^{2}+4 c d +2 d^{2}}}{{\left (\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c +2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c \right )}^{2}}+\frac {\left (6 c^{2} d A +6 d^{2} c A +3 A \,d^{3}-2 B \,c^{3}-4 c^{2} d B -7 d^{2} c B -2 d^{3} B \right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{2 \left (c^{2}+2 c d +d^{2}\right ) \sqrt {c^{2}-d^{2}}}\right )}{\left (c -d \right )^{3}}-\frac {2 \left (A -B \right )}{\left (c -d \right )^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}}{a f}\)	\(482\)
default	\(\frac {-\frac {2 \left (\frac {\frac {d^{2} \left (7 c^{2} d A +2 d^{2} c A -2 A \,d^{3}-5 B \,c^{3}-2 c^{2} d B \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 c \left (c^{2}+2 c d +d^{2}\right )}+\frac {d \left (6 A \,c^{4} d +2 A \,c^{3} d^{2}+11 A \,c^{2} d^{3}+4 A c \,d^{4}-2 A \,d^{5}-4 B \,c^{5}-2 B \,c^{4} d -9 B \,c^{3} d^{2}-4 B \,c^{2} d^{3}-2 B c \,d^{4}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c^{2}+2 c d +d^{2}\right ) c^{2}}+\frac {d^{2} \left (17 c^{2} d A +6 d^{2} c A -2 A \,d^{3}-11 B \,c^{3}-6 c^{2} d B -4 d^{2} c B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 c \left (c^{2}+2 c d +d^{2}\right )}+\frac {d \left (6 c^{2} d A +2 d^{2} c A -A \,d^{3}-4 B \,c^{3}-2 c^{2} d B -d^{2} c B \right )}{2 c^{2}+4 c d +2 d^{2}}}{{\left (\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c +2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c \right )}^{2}}+\frac {\left (6 c^{2} d A +6 d^{2} c A +3 A \,d^{3}-2 B \,c^{3}-4 c^{2} d B -7 d^{2} c B -2 d^{3} B \right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{2 \left (c^{2}+2 c d +d^{2}\right ) \sqrt {c^{2}-d^{2}}}\right )}{\left (c -d \right )^{3}}-\frac {2 \left (A -B \right )}{\left (c -d \right )^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}}{a f}\)	\(482\)
risch	\(\text {Expression too large to display}\)	\(1881\)

[In]

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

[Out]

2/f/a*(-1/(c-d)^3*((1/2*d^2*(7*A*c^2*d+2*A*c*d^2-2*A*d^3-5*B*c^3-2*B*c^2*d)/c/(c^2+2*c*d+d^2)*tan(1/2*f*x+1/2*
e)^3+1/2*d*(6*A*c^4*d+2*A*c^3*d^2+11*A*c^2*d^3+4*A*c*d^4-2*A*d^5-4*B*c^5-2*B*c^4*d-9*B*c^3*d^2-4*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*(17*A*c^2*d+6*A*c*d^2-2*A*d^3-11*B*c^3-6*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*(6*A*c^2*d+2*A*c*d^2-A*d^3-4*B*c^3-2*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*(6*A*c^2*d+6*A*c*d^2+3*A*d^3-2*B*c^3-4*B*c^
2*d-7*B*c*d^2-2*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)
))-(A-B)/(c-d)^3/(tan(1/2*f*x+1/2*e)+1))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1608 vs. \(2 (274) = 548\).

Time = 0.37 (sec) , antiderivative size = 3303, normalized size of antiderivative = 11.67 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/4*(4*(A - B)*c^6 - 12*(A - B)*c^4*d^2 + 12*(A - B)*c^2*d^4 - 4*(A - B)*d^6 - 2*((2*A - 5*B)*c^4*d^2 + 3*(3*
A - 2*B)*c^3*d^3 + (2*A + B)*c^2*d^4 - 3*(3*A - 2*B)*c*d^5 - 4*(A - B)*d^6)*cos(f*x + e)^3 + 2*(4*(A - 2*B)*c^
5*d + 4*(3*A - 2*B)*c^4*d^2 - (2*A - 7*B)*c^3*d^3 - 5*(3*A - 2*B)*c^2*d^4 - (2*A - B)*c*d^5 + (3*A - 2*B)*d^6)
*cos(f*x + e)^2 - (2*B*c^5 - 2*(3*A - 4*B)*c^4*d - (18*A - 17*B)*c^3*d^2 - (21*A - 20*B)*c^2*d^3 - (12*A - 11*
B)*c*d^4 - (3*A - 2*B)*d^5 - (2*B*c^3*d^2 - 2*(3*A - 2*B)*c^2*d^3 - (6*A - 7*B)*c*d^4 - (3*A - 2*B)*d^5)*cos(f
*x + e)^3 - (4*B*c^4*d - 2*(6*A - 5*B)*c^3*d^2 - 18*(A - B)*c^2*d^3 - (12*A - 11*B)*c*d^4 - (3*A - 2*B)*d^5)*c
os(f*x + e)^2 + (2*B*c^5 - 2*(3*A - 2*B)*c^4*d - 3*(2*A - 3*B)*c^3*d^2 - 3*(3*A - 2*B)*c^2*d^3 - (6*A - 7*B)*c
*d^4 - (3*A - 2*B)*d^5)*cos(f*x + e) + (2*B*c^5 - 2*(3*A - 4*B)*c^4*d - (18*A - 17*B)*c^3*d^2 - (21*A - 20*B)*
c^2*d^3 - (12*A - 11*B)*c*d^4 - (3*A - 2*B)*d^5 - (2*B*c^3*d^2 - 2*(3*A - 2*B)*c^2*d^3 - (6*A - 7*B)*c*d^4 - (
3*A - 2*B)*d^5)*cos(f*x + e)^2 + 2*(2*B*c^4*d - 2*(3*A - 2*B)*c^3*d^2 - (6*A - 7*B)*c^2*d^3 - (3*A - 2*B)*c*d^
4)*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)) + 2*(2*(A - B)*c^6 + 4*(A - 2*B)*c^5*d + (8*A - 7*B)*c^4*d^2 + (7*A + B)*c^3*d^3 - (7*A -
5*B)*c^2*d^4 - (11*A - 7*B)*c*d^5 - (3*A - 4*B)*d^6)*cos(f*x + e) - 2*(2*(A - B)*c^6 - 6*(A - B)*c^4*d^2 + 6*(
A - B)*c^2*d^4 - 2*(A - B)*d^6 - ((2*A - 5*B)*c^4*d^2 + 3*(3*A - 2*B)*c^3*d^3 + (2*A + B)*c^2*d^4 - 3*(3*A - 2
*B)*c*d^5 - 4*(A - B)*d^6)*cos(f*x + e)^2 - (4*(A - 2*B)*c^5*d + (14*A - 13*B)*c^4*d^2 + (7*A + B)*c^3*d^3 - (
13*A - 11*B)*c^2*d^4 - (11*A - 7*B)*c*d^5 - (A - 2*B)*d^6)*cos(f*x + e))*sin(f*x + e))/((a*c^7*d^2 - a*c^6*d^3
- 3*a*c^5*d^4 + 3*a*c^4*d^5 + 3*a*c^3*d^6 - 3*a*c^2*d^7 - a*c*d^8 + a*d^9)*f*cos(f*x + e)^3 + (2*a*c^8*d - a*
c^7*d^2 - 7*a*c^6*d^3 + 3*a*c^5*d^4 + 9*a*c^4*d^5 - 3*a*c^3*d^6 - 5*a*c^2*d^7 + a*c*d^8 + a*d^9)*f*cos(f*x + e
)^2 - (a*c^9 - a*c^8*d - 2*a*c^7*d^2 + 2*a*c^6*d^3 + 2*a*c^3*d^6 - 2*a*c^2*d^7 - a*c*d^8 + a*d^9)*f*cos(f*x +
e) - (a*c^9 + a*c^8*d - 4*a*c^7*d^2 - 4*a*c^6*d^3 + 6*a*c^5*d^4 + 6*a*c^4*d^5 - 4*a*c^3*d^6 - 4*a*c^2*d^7 + a*
c*d^8 + a*d^9)*f + ((a*c^7*d^2 - a*c^6*d^3 - 3*a*c^5*d^4 + 3*a*c^4*d^5 + 3*a*c^3*d^6 - 3*a*c^2*d^7 - a*c*d^8 +
a*d^9)*f*cos(f*x + e)^2 - 2*(a*c^8*d - a*c^7*d^2 - 3*a*c^6*d^3 + 3*a*c^5*d^4 + 3*a*c^4*d^5 - 3*a*c^3*d^6 - a*
c^2*d^7 + a*c*d^8)*f*cos(f*x + e) - (a*c^9 + a*c^8*d - 4*a*c^7*d^2 - 4*a*c^6*d^3 + 6*a*c^5*d^4 + 6*a*c^4*d^5 -
4*a*c^3*d^6 - 4*a*c^2*d^7 + a*c*d^8 + a*d^9)*f)*sin(f*x + e)), 1/2*(2*(A - B)*c^6 - 6*(A - B)*c^4*d^2 + 6*(A
- B)*c^2*d^4 - 2*(A - B)*d^6 - ((2*A - 5*B)*c^4*d^2 + 3*(3*A - 2*B)*c^3*d^3 + (2*A + B)*c^2*d^4 - 3*(3*A - 2*B
)*c*d^5 - 4*(A - B)*d^6)*cos(f*x + e)^3 + (4*(A - 2*B)*c^5*d + 4*(3*A - 2*B)*c^4*d^2 - (2*A - 7*B)*c^3*d^3 - 5
*(3*A - 2*B)*c^2*d^4 - (2*A - B)*c*d^5 + (3*A - 2*B)*d^6)*cos(f*x + e)^2 + (2*B*c^5 - 2*(3*A - 4*B)*c^4*d - (1
8*A - 17*B)*c^3*d^2 - (21*A - 20*B)*c^2*d^3 - (12*A - 11*B)*c*d^4 - (3*A - 2*B)*d^5 - (2*B*c^3*d^2 - 2*(3*A -
2*B)*c^2*d^3 - (6*A - 7*B)*c*d^4 - (3*A - 2*B)*d^5)*cos(f*x + e)^3 - (4*B*c^4*d - 2*(6*A - 5*B)*c^3*d^2 - 18*(
A - B)*c^2*d^3 - (12*A - 11*B)*c*d^4 - (3*A - 2*B)*d^5)*cos(f*x + e)^2 + (2*B*c^5 - 2*(3*A - 2*B)*c^4*d - 3*(2
*A - 3*B)*c^3*d^2 - 3*(3*A - 2*B)*c^2*d^3 - (6*A - 7*B)*c*d^4 - (3*A - 2*B)*d^5)*cos(f*x + e) + (2*B*c^5 - 2*(
3*A - 4*B)*c^4*d - (18*A - 17*B)*c^3*d^2 - (21*A - 20*B)*c^2*d^3 - (12*A - 11*B)*c*d^4 - (3*A - 2*B)*d^5 - (2*
B*c^3*d^2 - 2*(3*A - 2*B)*c^2*d^3 - (6*A - 7*B)*c*d^4 - (3*A - 2*B)*d^5)*cos(f*x + e)^2 + 2*(2*B*c^4*d - 2*(3*
A - 2*B)*c^3*d^2 - (6*A - 7*B)*c^2*d^3 - (3*A - 2*B)*c*d^4)*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*(A - B)*c^6 + 4*(A - 2*B)*c^5*d + (8*A - 7*B)*c^4*
d^2 + (7*A + B)*c^3*d^3 - (7*A - 5*B)*c^2*d^4 - (11*A - 7*B)*c*d^5 - (3*A - 4*B)*d^6)*cos(f*x + e) - (2*(A - B
)*c^6 - 6*(A - B)*c^4*d^2 + 6*(A - B)*c^2*d^4 - 2*(A - B)*d^6 - ((2*A - 5*B)*c^4*d^2 + 3*(3*A - 2*B)*c^3*d^3 +
(2*A + B)*c^2*d^4 - 3*(3*A - 2*B)*c*d^5 - 4*(A - B)*d^6)*cos(f*x + e)^2 - (4*(A - 2*B)*c^5*d + (14*A - 13*B)*
c^4*d^2 + (7*A + B)*c^3*d^3 - (13*A - 11*B)*c^2*d^4 - (11*A - 7*B)*c*d^5 - (A - 2*B)*d^6)*cos(f*x + e))*sin(f*
x + e))/((a*c^7*d^2 - a*c^6*d^3 - 3*a*c^5*d^4 + 3*a*c^4*d^5 + 3*a*c^3*d^6 - 3*a*c^2*d^7 - a*c*d^8 + a*d^9)*f*c
os(f*x + e)^3 + (2*a*c^8*d - a*c^7*d^2 - 7*a*c^6*d^3 + 3*a*c^5*d^4 + 9*a*c^4*d^5 - 3*a*c^3*d^6 - 5*a*c^2*d^7 +
a*c*d^8 + a*d^9)*f*cos(f*x + e)^2 - (a*c^9 - a*c^8*d - 2*a*c^7*d^2 + 2*a*c^6*d^3 + 2*a*c^3*d^6 - 2*a*c^2*d^7
- a*c*d^8 + a*d^9)*f*cos(f*x + e) - (a*c^9 + a*c^8*d - 4*a*c^7*d^2 - 4*a*c^6*d^3 + 6*a*c^5*d^4 + 6*a*c^4*d^5 -
4*a*c^3*d^6 - 4*a*c^2*d^7 + a*c*d^8 + a*d^9)*f + ((a*c^7*d^2 - a*c^6*d^3 - 3*a*c^5*d^4 + 3*a*c^4*d^5 + 3*a*c^
3*d^6 - 3*a*c^2*d^7 - a*c*d^8 + a*d^9)*f*cos(f*x + e)^2 - 2*(a*c^8*d - a*c^7*d^2 - 3*a*c^6*d^3 + 3*a*c^5*d^4 +
3*a*c^4*d^5 - 3*a*c^3*d^6 - a*c^2*d^7 + a*c*d^8)*f*cos(f*x + e) - (a*c^9 + a*c^8*d - 4*a*c^7*d^2 - 4*a*c^6*d^
3 + 6*a*c^5*d^4 + 6*a*c^4*d^5 - 4*a*c^3*d^6 - 4*a*c^2*d^7 + a*c*d^8 + a*d^9)*f)*sin(f*x + e))]

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))/(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. 727 vs. \(2 (274) = 548\).

Time = 0.38 (sec) , antiderivative size = 727, normalized size of antiderivative = 2.57 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^3} \, dx=\frac {\frac {{\left (2 \, B c^{3} - 6 \, A c^{2} d + 4 \, B c^{2} d - 6 \, A c d^{2} + 7 \, B c d^{2} - 3 \, A d^{3} + 2 \, B d^{3}\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (c\right ) + \arctan \left (\frac {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + d}{\sqrt {c^{2} - d^{2}}}\right )\right )}}{{\left (a c^{5} - a c^{4} d - 2 \, a c^{3} d^{2} + 2 \, a c^{2} d^{3} + a c d^{4} - a d^{5}\right )} \sqrt {c^{2} - d^{2}}} - \frac {2 \, {\left (A - B\right )}}{{\left (a c^{3} - 3 \, a c^{2} d + 3 \, a c d^{2} - a d^{3}\right )} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}} + \frac {5 \, B c^{4} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 7 \, A c^{3} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2 \, B c^{3} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 2 \, A c^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2 \, A c d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 4 \, B c^{5} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 6 \, A c^{4} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, B c^{4} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 2 \, A c^{3} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 9 \, B c^{3} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 11 \, A c^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 4 \, B c^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 4 \, A c d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, B c d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, A d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 11 \, B c^{4} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 17 \, A c^{3} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 6 \, B c^{3} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 6 \, A c^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 4 \, B c^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2 \, A c d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 4 \, B c^{5} d - 6 \, A c^{4} d^{2} + 2 \, B c^{4} d^{2} - 2 \, A c^{3} d^{3} + B c^{3} d^{3} + A c^{2} d^{4}}{{\left (a c^{7} - a c^{6} d - 2 \, a c^{5} d^{2} + 2 \, a c^{4} d^{3} + a c^{3} d^{4} - a c^{2} d^{5}\right )} {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + c\right )}^{2}}}{f} \]

[In]

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

[Out]

((2*B*c^3 - 6*A*c^2*d + 4*B*c^2*d - 6*A*c*d^2 + 7*B*c*d^2 - 3*A*d^3 + 2*B*d^3)*(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*c^5 - a*c^4*d - 2*a*c^3*d^2 + 2*a*c^2*d^
3 + a*c*d^4 - a*d^5)*sqrt(c^2 - d^2)) - 2*(A - B)/((a*c^3 - 3*a*c^2*d + 3*a*c*d^2 - a*d^3)*(tan(1/2*f*x + 1/2*
e) + 1)) + (5*B*c^4*d^2*tan(1/2*f*x + 1/2*e)^3 - 7*A*c^3*d^3*tan(1/2*f*x + 1/2*e)^3 + 2*B*c^3*d^3*tan(1/2*f*x
+ 1/2*e)^3 - 2*A*c^2*d^4*tan(1/2*f*x + 1/2*e)^3 + 2*A*c*d^5*tan(1/2*f*x + 1/2*e)^3 + 4*B*c^5*d*tan(1/2*f*x + 1
/2*e)^2 - 6*A*c^4*d^2*tan(1/2*f*x + 1/2*e)^2 + 2*B*c^4*d^2*tan(1/2*f*x + 1/2*e)^2 - 2*A*c^3*d^3*tan(1/2*f*x +
1/2*e)^2 + 9*B*c^3*d^3*tan(1/2*f*x + 1/2*e)^2 - 11*A*c^2*d^4*tan(1/2*f*x + 1/2*e)^2 + 4*B*c^2*d^4*tan(1/2*f*x
+ 1/2*e)^2 - 4*A*c*d^5*tan(1/2*f*x + 1/2*e)^2 + 2*B*c*d^5*tan(1/2*f*x + 1/2*e)^2 + 2*A*d^6*tan(1/2*f*x + 1/2*e
)^2 + 11*B*c^4*d^2*tan(1/2*f*x + 1/2*e) - 17*A*c^3*d^3*tan(1/2*f*x + 1/2*e) + 6*B*c^3*d^3*tan(1/2*f*x + 1/2*e)
- 6*A*c^2*d^4*tan(1/2*f*x + 1/2*e) + 4*B*c^2*d^4*tan(1/2*f*x + 1/2*e) + 2*A*c*d^5*tan(1/2*f*x + 1/2*e) + 4*B*
c^5*d - 6*A*c^4*d^2 + 2*B*c^4*d^2 - 2*A*c^3*d^3 + B*c^3*d^3 + A*c^2*d^4)/((a*c^7 - a*c^6*d - 2*a*c^5*d^2 + 2*a
*c^4*d^3 + a*c^3*d^4 - a*c^2*d^5)*(c*tan(1/2*f*x + 1/2*e)^2 + 2*d*tan(1/2*f*x + 1/2*e) + c)^2))/f

Mupad [B] (verification not implemented)

Time = 17.95 (sec) , antiderivative size = 1076, normalized size of antiderivative = 3.80 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^3} \, dx=\frac {\frac {A\,d^4-2\,A\,c^4+2\,B\,c^4-8\,A\,c^2\,d^2+4\,B\,c^2\,d^2-2\,A\,c\,d^3-4\,A\,c^3\,d+B\,c\,d^3+8\,B\,c^3\,d}{\left (c+d\right )\,\left (c^2-d^2\right )\,\left (c^2-2\,c\,d+d^2\right )}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (2\,A\,d^6-13\,A\,c^2\,d^4-17\,A\,c^3\,d^3-22\,A\,c^4\,d^2+4\,B\,c^2\,d^4+19\,B\,c^3\,d^3+23\,B\,c^4\,d^2-2\,A\,c\,d^5-8\,A\,c^5\,d+2\,B\,c\,d^5+12\,B\,c^5\,d\right )}{c^2\,\left (c^2-2\,c\,d+d^2\right )\,\left (-c^3-c^2\,d+c\,d^2+d^3\right )}+\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (2\,A\,d^5-4\,A\,c^5+4\,B\,c^5-21\,A\,c^2\,d^3-14\,A\,c^3\,d^2+14\,B\,c^2\,d^3+17\,B\,c^3\,d^2-4\,A\,c\,d^4-4\,A\,c^4\,d+2\,B\,c\,d^4+8\,B\,c^4\,d\right )}{c^2\,\left (c^2-d^2\right )\,\left (c^2-2\,c\,d+d^2\right )}+\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (2\,A\,c^5-2\,A\,d^5-2\,B\,c^5+7\,A\,c^2\,d^3+2\,A\,c^3\,d^2-2\,B\,c^2\,d^3-7\,B\,c^3\,d^2+2\,A\,c\,d^4+4\,A\,c^4\,d-4\,B\,c^4\,d\right )}{c\,\left (c^2-2\,c\,d+d^2\right )\,\left (-c^3-c^2\,d+c\,d^2+d^3\right )}+\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (2\,A\,d^5-27\,A\,c^2\,d^3-22\,A\,c^3\,d^2+15\,B\,c^2\,d^3+29\,B\,c^3\,d^2-5\,A\,c\,d^4-8\,A\,c^4\,d+4\,B\,c\,d^4+12\,B\,c^4\,d\right )}{c\,\left (c+d\right )\,\left (c^2-d^2\right )\,\left (c^2-2\,c\,d+d^2\right )}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (2\,a\,c^2+4\,a\,c\,d+4\,a\,d^2\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (2\,a\,c^2+4\,a\,c\,d+4\,a\,d^2\right )+a\,c^2+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (a\,c^2+4\,a\,d\,c\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (a\,c^2+4\,a\,d\,c\right )+a\,c^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\right )}-\frac {\mathrm {atan}\left (\frac {\frac {\left (-2\,a\,c^5\,d+2\,a\,c^4\,d^2+4\,a\,c^3\,d^3-4\,a\,c^2\,d^4-2\,a\,c\,d^5+2\,a\,d^6\right )\,\left (2\,B\,c^3-3\,A\,d^3+2\,B\,d^3-6\,A\,c\,d^2-6\,A\,c^2\,d+7\,B\,c\,d^2+4\,B\,c^2\,d\right )}{2\,a\,{\left (c+d\right )}^{5/2}\,{\left (c-d\right )}^{7/2}}-\frac {c\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (a\,c^5-a\,c^4\,d-2\,a\,c^3\,d^2+2\,a\,c^2\,d^3+a\,c\,d^4-a\,d^5\right )\,\left (2\,B\,c^3-3\,A\,d^3+2\,B\,d^3-6\,A\,c\,d^2-6\,A\,c^2\,d+7\,B\,c\,d^2+4\,B\,c^2\,d\right )}{a\,{\left (c+d\right )}^{5/2}\,{\left (c-d\right )}^{7/2}}}{2\,B\,c^3-3\,A\,d^3+2\,B\,d^3-6\,A\,c\,d^2-6\,A\,c^2\,d+7\,B\,c\,d^2+4\,B\,c^2\,d}\right )\,\left (2\,B\,c^3-3\,A\,d^3+2\,B\,d^3-6\,A\,c\,d^2-6\,A\,c^2\,d+7\,B\,c\,d^2+4\,B\,c^2\,d\right )}{a\,f\,{\left (c+d\right )}^{5/2}\,{\left (c-d\right )}^{7/2}} \]

[In]

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

[Out]

((A*d^4 - 2*A*c^4 + 2*B*c^4 - 8*A*c^2*d^2 + 4*B*c^2*d^2 - 2*A*c*d^3 - 4*A*c^3*d + B*c*d^3 + 8*B*c^3*d)/((c + d
)*(c^2 - d^2)*(c^2 - 2*c*d + d^2)) - (tan(e/2 + (f*x)/2)^3*(2*A*d^6 - 13*A*c^2*d^4 - 17*A*c^3*d^3 - 22*A*c^4*d
^2 + 4*B*c^2*d^4 + 19*B*c^3*d^3 + 23*B*c^4*d^2 - 2*A*c*d^5 - 8*A*c^5*d + 2*B*c*d^5 + 12*B*c^5*d))/(c^2*(c^2 -
2*c*d + d^2)*(c*d^2 - c^2*d - c^3 + d^3)) + (tan(e/2 + (f*x)/2)^2*(2*A*d^5 - 4*A*c^5 + 4*B*c^5 - 21*A*c^2*d^3
- 14*A*c^3*d^2 + 14*B*c^2*d^3 + 17*B*c^3*d^2 - 4*A*c*d^4 - 4*A*c^4*d + 2*B*c*d^4 + 8*B*c^4*d))/(c^2*(c^2 - d^2
)*(c^2 - 2*c*d + d^2)) + (tan(e/2 + (f*x)/2)^4*(2*A*c^5 - 2*A*d^5 - 2*B*c^5 + 7*A*c^2*d^3 + 2*A*c^3*d^2 - 2*B*
c^2*d^3 - 7*B*c^3*d^2 + 2*A*c*d^4 + 4*A*c^4*d - 4*B*c^4*d))/(c*(c^2 - 2*c*d + d^2)*(c*d^2 - c^2*d - c^3 + d^3)
) + (tan(e/2 + (f*x)/2)*(2*A*d^5 - 27*A*c^2*d^3 - 22*A*c^3*d^2 + 15*B*c^2*d^3 + 29*B*c^3*d^2 - 5*A*c*d^4 - 8*A
*c^4*d + 4*B*c*d^4 + 12*B*c^4*d))/(c*(c + d)*(c^2 - d^2)*(c^2 - 2*c*d + d^2)))/(f*(tan(e/2 + (f*x)/2)^2*(2*a*c
^2 + 4*a*d^2 + 4*a*c*d) + tan(e/2 + (f*x)/2)^3*(2*a*c^2 + 4*a*d^2 + 4*a*c*d) + a*c^2 + tan(e/2 + (f*x)/2)*(a*c
^2 + 4*a*c*d) + tan(e/2 + (f*x)/2)^4*(a*c^2 + 4*a*c*d) + a*c^2*tan(e/2 + (f*x)/2)^5)) - (atan((((2*a*d^6 - 4*a
*c^2*d^4 + 4*a*c^3*d^3 + 2*a*c^4*d^2 - 2*a*c*d^5 - 2*a*c^5*d)*(2*B*c^3 - 3*A*d^3 + 2*B*d^3 - 6*A*c*d^2 - 6*A*c
^2*d + 7*B*c*d^2 + 4*B*c^2*d))/(2*a*(c + d)^(5/2)*(c - d)^(7/2)) - (c*tan(e/2 + (f*x)/2)*(a*c^5 - a*d^5 + 2*a*
c^2*d^3 - 2*a*c^3*d^2 + a*c*d^4 - a*c^4*d)*(2*B*c^3 - 3*A*d^3 + 2*B*d^3 - 6*A*c*d^2 - 6*A*c^2*d + 7*B*c*d^2 +
4*B*c^2*d))/(a*(c + d)^(5/2)*(c - d)^(7/2)))/(2*B*c^3 - 3*A*d^3 + 2*B*d^3 - 6*A*c*d^2 - 6*A*c^2*d + 7*B*c*d^2
+ 4*B*c^2*d))*(2*B*c^3 - 3*A*d^3 + 2*B*d^3 - 6*A*c*d^2 - 6*A*c^2*d + 7*B*c*d^2 + 4*B*c^2*d))/(a*f*(c + d)^(5/2
)*(c - d)^(7/2))