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

Optimal. Leaf size=305 \[ -\frac {\left (A d (c+4 d)-B \left (3 c^2+4 c d-2 d^2\right )\right ) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{2 d^2 f (c+d)^2 (c+d \sin (e+f x))}-\frac {a^3 (c-d) \left (A d \left (2 c^2+6 c d+7 d^2\right )-3 B \left (2 c^3+4 c^2 d+c d^2-2 d^3\right )\right ) \tan ^{-1}\left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{d^4 f (c+d)^2 \sqrt {c^2-d^2}}-\frac {a^3 x (-A d+3 B c-3 B d)}{d^4}-\frac {a^3 (3 B c (2 c+3 d)-A d (2 c+5 d)) \cos (e+f x)}{2 d^3 f (c+d)^2}+\frac {a (B c-A d) \cos (e+f x) (a \sin (e+f x)+a)^2}{2 d f (c+d) (c+d \sin (e+f x))^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.93, antiderivative size = 305, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2975, 2968, 3023, 2735, 2660, 618, 204} \[ -\frac {a^3 (c-d) \left (A d \left (2 c^2+6 c d+7 d^2\right )-3 B \left (4 c^2 d+2 c^3+c d^2-2 d^3\right )\right ) \tan ^{-1}\left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{d^4 f (c+d)^2 \sqrt {c^2-d^2}}-\frac {\left (A d (c+4 d)-B \left (3 c^2+4 c d-2 d^2\right )\right ) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{2 d^2 f (c+d)^2 (c+d \sin (e+f x))}-\frac {a^3 (3 B c (2 c+3 d)-A d (2 c+5 d)) \cos (e+f x)}{2 d^3 f (c+d)^2}-\frac {a^3 x (-A d+3 B c-3 B d)}{d^4}+\frac {a (B c-A d) \cos (e+f x) (a \sin (e+f x)+a)^2}{2 d f (c+d) (c+d \sin (e+f x))^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a^3*(3*B*c - A*d - 3*B*d)*x)/d^4) - (a^3*(c - d)*(A*d*(2*c^2 + 6*c*d + 7*d^2) - 3*B*(2*c^3 + 4*c^2*d + c*d^
2 - 2*d^3))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(d^4*(c + d)^2*Sqrt[c^2 - d^2]*f) - (a^3*(3*B*c*
(2*c + 3*d) - A*d*(2*c + 5*d))*Cos[e + f*x])/(2*d^3*(c + d)^2*f) + (a*(B*c - A*d)*Cos[e + f*x]*(a + a*Sin[e +
f*x])^2)/(2*d*(c + d)*f*(c + d*Sin[e + f*x])^2) - ((A*d*(c + 4*d) - B*(3*c^2 + 4*c*d - 2*d^2))*Cos[e + f*x]*(a
^3 + a^3*Sin[e + f*x]))/(2*d^2*(c + d)^2*f*(c + d*Sin[e + f*x]))

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])

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 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 2735

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

Rule 2968

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

Rule 2975

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^2*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*S
in[e + f*x])^(n + 1))/(d*f*(n + 1)*(b*c + a*d)), x] - Dist[b/(d*(n + 1)*(b*c + a*d)), Int[(a + b*Sin[e + f*x])
^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[a*A*d*(m - n - 2) - B*(a*c*(m - 1) + b*d*(n + 1)) - (A*b*d*(m + n +
1) - B*(b*c*m - a*d*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d
, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] && LtQ[n, -1] && IntegerQ[2*m] && (IntegerQ[2*n]
|| EqQ[c, 0])

Rule 3023

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[1/(b*
(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x]
, x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rubi steps

\begin {align*} \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c+d \sin (e+f x))^3} \, dx &=\frac {a (B c-A d) \cos (e+f x) (a+a \sin (e+f x))^2}{2 d (c+d) f (c+d \sin (e+f x))^2}+\frac {\int \frac {(a+a \sin (e+f x))^2 (-2 a (B (c-d)-2 A d)+a (3 B c-A d+2 B d) \sin (e+f x))}{(c+d \sin (e+f x))^2} \, dx}{2 d (c+d)}\\ &=\frac {a (B c-A d) \cos (e+f x) (a+a \sin (e+f x))^2}{2 d (c+d) f (c+d \sin (e+f x))^2}-\frac {\left (A d (c+4 d)-B \left (3 c^2+4 c d-2 d^2\right )\right ) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{2 d^2 (c+d)^2 f (c+d \sin (e+f x))}+\frac {\int \frac {(a+a \sin (e+f x)) \left (a^2 \left (A d (c+7 d)-3 B \left (c^2+c d-2 d^2\right )\right )+a^2 (3 B c (2 c+3 d)-A d (2 c+5 d)) \sin (e+f x)\right )}{c+d \sin (e+f x)} \, dx}{2 d^2 (c+d)^2}\\ &=\frac {a (B c-A d) \cos (e+f x) (a+a \sin (e+f x))^2}{2 d (c+d) f (c+d \sin (e+f x))^2}-\frac {\left (A d (c+4 d)-B \left (3 c^2+4 c d-2 d^2\right )\right ) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{2 d^2 (c+d)^2 f (c+d \sin (e+f x))}+\frac {\int \frac {a^3 \left (A d (c+7 d)-3 B \left (c^2+c d-2 d^2\right )\right )+\left (a^3 (3 B c (2 c+3 d)-A d (2 c+5 d))+a^3 \left (A d (c+7 d)-3 B \left (c^2+c d-2 d^2\right )\right )\right ) \sin (e+f x)+a^3 (3 B c (2 c+3 d)-A d (2 c+5 d)) \sin ^2(e+f x)}{c+d \sin (e+f x)} \, dx}{2 d^2 (c+d)^2}\\ &=-\frac {a^3 (3 B c (2 c+3 d)-A d (2 c+5 d)) \cos (e+f x)}{2 d^3 (c+d)^2 f}+\frac {a (B c-A d) \cos (e+f x) (a+a \sin (e+f x))^2}{2 d (c+d) f (c+d \sin (e+f x))^2}-\frac {\left (A d (c+4 d)-B \left (3 c^2+4 c d-2 d^2\right )\right ) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{2 d^2 (c+d)^2 f (c+d \sin (e+f x))}+\frac {\int \frac {a^3 d \left (A d (c+7 d)-3 B \left (c^2+c d-2 d^2\right )\right )-2 a^3 (c+d)^2 (3 B (c-d)-A d) \sin (e+f x)}{c+d \sin (e+f x)} \, dx}{2 d^3 (c+d)^2}\\ &=-\frac {a^3 (3 B c-A d-3 B d) x}{d^4}-\frac {a^3 (3 B c (2 c+3 d)-A d (2 c+5 d)) \cos (e+f x)}{2 d^3 (c+d)^2 f}+\frac {a (B c-A d) \cos (e+f x) (a+a \sin (e+f x))^2}{2 d (c+d) f (c+d \sin (e+f x))^2}-\frac {\left (A d (c+4 d)-B \left (3 c^2+4 c d-2 d^2\right )\right ) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{2 d^2 (c+d)^2 f (c+d \sin (e+f x))}-\frac {\left (a^3 (c-d) \left (A d \left (2 c^2+6 c d+7 d^2\right )-3 B \left (2 c^3+4 c^2 d+c d^2-2 d^3\right )\right )\right ) \int \frac {1}{c+d \sin (e+f x)} \, dx}{2 d^4 (c+d)^2}\\ &=-\frac {a^3 (3 B c-A d-3 B d) x}{d^4}-\frac {a^3 (3 B c (2 c+3 d)-A d (2 c+5 d)) \cos (e+f x)}{2 d^3 (c+d)^2 f}+\frac {a (B c-A d) \cos (e+f x) (a+a \sin (e+f x))^2}{2 d (c+d) f (c+d \sin (e+f x))^2}-\frac {\left (A d (c+4 d)-B \left (3 c^2+4 c d-2 d^2\right )\right ) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{2 d^2 (c+d)^2 f (c+d \sin (e+f x))}-\frac {\left (a^3 (c-d) \left (A d \left (2 c^2+6 c d+7 d^2\right )-3 B \left (2 c^3+4 c^2 d+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 )}{d^4 (c+d)^2 f}\\ &=-\frac {a^3 (3 B c-A d-3 B d) x}{d^4}-\frac {a^3 (3 B c (2 c+3 d)-A d (2 c+5 d)) \cos (e+f x)}{2 d^3 (c+d)^2 f}+\frac {a (B c-A d) \cos (e+f x) (a+a \sin (e+f x))^2}{2 d (c+d) f (c+d \sin (e+f x))^2}-\frac {\left (A d (c+4 d)-B \left (3 c^2+4 c d-2 d^2\right )\right ) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{2 d^2 (c+d)^2 f (c+d \sin (e+f x))}+\frac {\left (2 a^3 (c-d) \left (A d \left (2 c^2+6 c d+7 d^2\right )-3 B \left (2 c^3+4 c^2 d+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 )}{d^4 (c+d)^2 f}\\ &=-\frac {a^3 (3 B c-A d-3 B d) x}{d^4}-\frac {a^3 (c-d) \left (A d \left (2 c^2+6 c d+7 d^2\right )-3 B \left (2 c^3+4 c^2 d+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 )}{d^4 (c+d)^2 \sqrt {c^2-d^2} f}-\frac {a^3 (3 B c (2 c+3 d)-A d (2 c+5 d)) \cos (e+f x)}{2 d^3 (c+d)^2 f}+\frac {a (B c-A d) \cos (e+f x) (a+a \sin (e+f x))^2}{2 d (c+d) f (c+d \sin (e+f x))^2}-\frac {\left (A d (c+4 d)-B \left (3 c^2+4 c d-2 d^2\right )\right ) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{2 d^2 (c+d)^2 f (c+d \sin (e+f x))}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B]  time = 3.14, size = 830, normalized size = 2.72 \[ \frac {a^3 (\sin (e+f x)+1)^3 \left (\frac {4 (c-d) \left (3 B \left (2 c^3+4 d c^2+d^2 c-2 d^3\right )-A d \left (2 c^2+6 d c+7 d^2\right )\right ) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{\sqrt {c^2-d^2}}+\frac {-12 B e c^5-12 B f x c^5+4 A d e c^4-12 B d e c^4+4 A d f x c^4-12 B d f x c^4-24 B d e \sin (e+f x) c^4-24 B d f x \sin (e+f x) c^4+8 A d^2 e c^3+6 B d^2 e c^3+8 A d^2 f x c^3+6 B d^2 f x c^3+8 A d^2 e \sin (e+f x) c^3-24 B d^2 e \sin (e+f x) c^3+8 A d^2 f x \sin (e+f x) c^3-24 B d^2 f x \sin (e+f x) c^3-9 B d^2 \sin (2 (e+f x)) c^3+6 A d^3 e c^2+6 B d^3 e c^2+6 A d^3 f x c^2+6 B d^3 f x c^2+B d^3 \cos (3 (e+f x)) c^2+16 A d^3 e \sin (e+f x) c^2+24 B d^3 e \sin (e+f x) c^2+16 A d^3 f x \sin (e+f x) c^2+24 B d^3 f x \sin (e+f x) c^2+3 A d^3 \sin (2 (e+f x)) c^2-9 B d^3 \sin (2 (e+f x)) c^2+4 A d^4 e c+6 B d^4 e c+4 A d^4 f x c+6 B d^4 f x c+2 B d^4 \cos (3 (e+f x)) c+8 A d^4 e \sin (e+f x) c+24 B d^4 e \sin (e+f x) c+8 A d^4 f x \sin (e+f x) c+24 B d^4 f x \sin (e+f x) c+3 A d^4 \sin (2 (e+f x)) c+4 B d^4 \sin (2 (e+f x)) c+2 A d^5 e+6 B d^5 e+2 A d^5 f x+6 B d^5 f x-d \left (2 A d \left (-2 c^3-4 d c^2+5 d^2 c+d^3\right )+B \left (12 c^4+12 d c^3-9 d^2 c^2+4 d^3 c+d^4\right )\right ) \cos (e+f x)-2 d^2 (c+d)^2 (-3 B c+A d+3 B d) (e+f x) \cos (2 (e+f x))+B d^5 \cos (3 (e+f x))-6 A d^5 \sin (2 (e+f x))-2 B d^5 \sin (2 (e+f x))}{(c+d \sin (e+f x))^2}\right )}{4 d^4 (c+d)^2 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*(1 + Sin[e + f*x])^3*((4*(c - d)*(-(A*d*(2*c^2 + 6*c*d + 7*d^2)) + 3*B*(2*c^3 + 4*c^2*d + c*d^2 - 2*d^3))
*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/Sqrt[c^2 - d^2] + (-12*B*c^5*e + 4*A*c^4*d*e - 12*B*c^4*d*e
+ 8*A*c^3*d^2*e + 6*B*c^3*d^2*e + 6*A*c^2*d^3*e + 6*B*c^2*d^3*e + 4*A*c*d^4*e + 6*B*c*d^4*e + 2*A*d^5*e + 6*B
*d^5*e - 12*B*c^5*f*x + 4*A*c^4*d*f*x - 12*B*c^4*d*f*x + 8*A*c^3*d^2*f*x + 6*B*c^3*d^2*f*x + 6*A*c^2*d^3*f*x +
6*B*c^2*d^3*f*x + 4*A*c*d^4*f*x + 6*B*c*d^4*f*x + 2*A*d^5*f*x + 6*B*d^5*f*x - d*(2*A*d*(-2*c^3 - 4*c^2*d + 5*
c*d^2 + d^3) + B*(12*c^4 + 12*c^3*d - 9*c^2*d^2 + 4*c*d^3 + d^4))*Cos[e + f*x] - 2*d^2*(c + d)^2*(-3*B*c + A*d
+ 3*B*d)*(e + f*x)*Cos[2*(e + f*x)] + B*c^2*d^3*Cos[3*(e + f*x)] + 2*B*c*d^4*Cos[3*(e + f*x)] + B*d^5*Cos[3*(
e + f*x)] - 24*B*c^4*d*e*Sin[e + f*x] + 8*A*c^3*d^2*e*Sin[e + f*x] - 24*B*c^3*d^2*e*Sin[e + f*x] + 16*A*c^2*d^
3*e*Sin[e + f*x] + 24*B*c^2*d^3*e*Sin[e + f*x] + 8*A*c*d^4*e*Sin[e + f*x] + 24*B*c*d^4*e*Sin[e + f*x] - 24*B*c
^4*d*f*x*Sin[e + f*x] + 8*A*c^3*d^2*f*x*Sin[e + f*x] - 24*B*c^3*d^2*f*x*Sin[e + f*x] + 16*A*c^2*d^3*f*x*Sin[e
+ f*x] + 24*B*c^2*d^3*f*x*Sin[e + f*x] + 8*A*c*d^4*f*x*Sin[e + f*x] + 24*B*c*d^4*f*x*Sin[e + f*x] - 9*B*c^3*d^
2*Sin[2*(e + f*x)] + 3*A*c^2*d^3*Sin[2*(e + f*x)] - 9*B*c^2*d^3*Sin[2*(e + f*x)] + 3*A*c*d^4*Sin[2*(e + f*x)]
+ 4*B*c*d^4*Sin[2*(e + f*x)] - 6*A*d^5*Sin[2*(e + f*x)] - 2*B*d^5*Sin[2*(e + f*x)])/(c + d*Sin[e + f*x])^2))/(
4*d^4*(c + d)^2*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6)

________________________________________________________________________________________

fricas [B]  time = 0.61, size = 1670, normalized size = 5.48 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*(4*(3*B*a^3*c^3*d^2 - (A - 3*B)*a^3*c^2*d^3 - (2*A + 3*B)*a^3*c*d^4 - (A + 3*B)*a^3*d^5)*f*x*cos(f*x + e
)^2 + 4*(B*a^3*c^2*d^3 + 2*B*a^3*c*d^4 + B*a^3*d^5)*cos(f*x + e)^3 - 4*(3*B*a^3*c^5 - (A - 3*B)*a^3*c^4*d - 2*
A*a^3*c^3*d^2 - 2*A*a^3*c^2*d^3 - (2*A + 3*B)*a^3*c*d^4 - (A + 3*B)*a^3*d^5)*f*x - (6*B*a^3*c^5 - 2*(A - 6*B)*
a^3*c^4*d - 3*(2*A - 3*B)*a^3*c^3*d^2 - 3*(3*A - 2*B)*a^3*c^2*d^3 - 3*(2*A - B)*a^3*c*d^4 - (7*A + 6*B)*a^3*d^
5 - (6*B*a^3*c^3*d^2 - 2*(A - 6*B)*a^3*c^2*d^3 - 3*(2*A - B)*a^3*c*d^4 - (7*A + 6*B)*a^3*d^5)*cos(f*x + e)^2 +
2*(6*B*a^3*c^4*d - 2*(A - 6*B)*a^3*c^3*d^2 - 3*(2*A - B)*a^3*c^2*d^3 - (7*A + 6*B)*a^3*c*d^4)*sin(f*x + e))*s
qrt(-(c - d)/(c + d))*log(((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 + 2*((c^2 + c*d)*cos(
f*x + e)*sin(f*x + e) + (c*d + d^2)*cos(f*x + e))*sqrt(-(c - d)/(c + d)))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x
+ e) - c^2 - d^2)) - 2*(6*B*a^3*c^4*d - 2*(A - 3*B)*a^3*c^3*d^2 - (4*A + 3*B)*a^3*c^2*d^3 + 5*(A + B)*a^3*c*d^
4 + (A + 2*B)*a^3*d^5)*cos(f*x + e) - 2*(4*(3*B*a^3*c^4*d - (A - 3*B)*a^3*c^3*d^2 - (2*A + 3*B)*a^3*c^2*d^3 -
(A + 3*B)*a^3*c*d^4)*f*x + (9*B*a^3*c^3*d^2 - 3*(A - 3*B)*a^3*c^2*d^3 - (3*A + 4*B)*a^3*c*d^4 + 2*(3*A + B)*a^
3*d^5)*cos(f*x + e))*sin(f*x + e))/((c^2*d^6 + 2*c*d^7 + d^8)*f*cos(f*x + e)^2 - 2*(c^3*d^5 + 2*c^2*d^6 + c*d^
7)*f*sin(f*x + e) - (c^4*d^4 + 2*c^3*d^5 + 2*c^2*d^6 + 2*c*d^7 + d^8)*f), -1/2*(2*(3*B*a^3*c^3*d^2 - (A - 3*B)
*a^3*c^2*d^3 - (2*A + 3*B)*a^3*c*d^4 - (A + 3*B)*a^3*d^5)*f*x*cos(f*x + e)^2 + 2*(B*a^3*c^2*d^3 + 2*B*a^3*c*d^
4 + B*a^3*d^5)*cos(f*x + e)^3 - 2*(3*B*a^3*c^5 - (A - 3*B)*a^3*c^4*d - 2*A*a^3*c^3*d^2 - 2*A*a^3*c^2*d^3 - (2*
A + 3*B)*a^3*c*d^4 - (A + 3*B)*a^3*d^5)*f*x - (6*B*a^3*c^5 - 2*(A - 6*B)*a^3*c^4*d - 3*(2*A - 3*B)*a^3*c^3*d^2
- 3*(3*A - 2*B)*a^3*c^2*d^3 - 3*(2*A - B)*a^3*c*d^4 - (7*A + 6*B)*a^3*d^5 - (6*B*a^3*c^3*d^2 - 2*(A - 6*B)*a^
3*c^2*d^3 - 3*(2*A - B)*a^3*c*d^4 - (7*A + 6*B)*a^3*d^5)*cos(f*x + e)^2 + 2*(6*B*a^3*c^4*d - 2*(A - 6*B)*a^3*c
^3*d^2 - 3*(2*A - B)*a^3*c^2*d^3 - (7*A + 6*B)*a^3*c*d^4)*sin(f*x + e))*sqrt((c - d)/(c + d))*arctan(-(c*sin(f
*x + e) + d)*sqrt((c - d)/(c + d))/((c - d)*cos(f*x + e))) - (6*B*a^3*c^4*d - 2*(A - 3*B)*a^3*c^3*d^2 - (4*A +
3*B)*a^3*c^2*d^3 + 5*(A + B)*a^3*c*d^4 + (A + 2*B)*a^3*d^5)*cos(f*x + e) - (4*(3*B*a^3*c^4*d - (A - 3*B)*a^3*
c^3*d^2 - (2*A + 3*B)*a^3*c^2*d^3 - (A + 3*B)*a^3*c*d^4)*f*x + (9*B*a^3*c^3*d^2 - 3*(A - 3*B)*a^3*c^2*d^3 - (3
*A + 4*B)*a^3*c*d^4 + 2*(3*A + B)*a^3*d^5)*cos(f*x + e))*sin(f*x + e))/((c^2*d^6 + 2*c*d^7 + d^8)*f*cos(f*x +
e)^2 - 2*(c^3*d^5 + 2*c^2*d^6 + c*d^7)*f*sin(f*x + e) - (c^4*d^4 + 2*c^3*d^5 + 2*c^2*d^6 + 2*c*d^7 + d^8)*f)]

________________________________________________________________________________________

giac [B]  time = 0.30, size = 986, normalized size = 3.23 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((6*B*a^3*c^4 - 2*A*a^3*c^3*d + 6*B*a^3*c^3*d - 4*A*a^3*c^2*d^2 - 9*B*a^3*c^2*d^2 - A*a^3*c*d^3 - 9*B*a^3*c*d^
3 + 7*A*a^3*d^4 + 6*B*a^3*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)))/((c^2*d^4 + 2*c*d^5 + d^6)*sqrt(c^2 - d^2)) - 2*B*a^3/((tan(1/2*f*x + 1/2*e)^2 + 1)*d^3) - (
3*B*a^3*c^5*d*tan(1/2*f*x + 1/2*e)^3 - A*a^3*c^4*d^2*tan(1/2*f*x + 1/2*e)^3 + 3*B*a^3*c^4*d^2*tan(1/2*f*x + 1/
2*e)^3 - 5*A*a^3*c^3*d^3*tan(1/2*f*x + 1/2*e)^3 - 6*B*a^3*c^3*d^3*tan(1/2*f*x + 1/2*e)^3 + 4*A*a^3*c^2*d^4*tan
(1/2*f*x + 1/2*e)^3 + 2*A*a^3*c*d^5*tan(1/2*f*x + 1/2*e)^3 + 4*B*a^3*c^6*tan(1/2*f*x + 1/2*e)^2 - 2*A*a^3*c^5*
d*tan(1/2*f*x + 1/2*e)^2 + 2*B*a^3*c^5*d*tan(1/2*f*x + 1/2*e)^2 - 4*A*a^3*c^4*d^2*tan(1/2*f*x + 1/2*e)^2 + B*a
^3*c^4*d^2*tan(1/2*f*x + 1/2*e)^2 + A*a^3*c^3*d^3*tan(1/2*f*x + 1/2*e)^2 + 5*B*a^3*c^3*d^3*tan(1/2*f*x + 1/2*e
)^2 - 7*A*a^3*c^2*d^4*tan(1/2*f*x + 1/2*e)^2 - 14*B*a^3*c^2*d^4*tan(1/2*f*x + 1/2*e)^2 + 10*A*a^3*c*d^5*tan(1/
2*f*x + 1/2*e)^2 + 2*B*a^3*c*d^5*tan(1/2*f*x + 1/2*e)^2 + 2*A*a^3*d^6*tan(1/2*f*x + 1/2*e)^2 + 13*B*a^3*c^5*d*
tan(1/2*f*x + 1/2*e) - 7*A*a^3*c^4*d^2*tan(1/2*f*x + 1/2*e) + 5*B*a^3*c^4*d^2*tan(1/2*f*x + 1/2*e) - 11*A*a^3*
c^3*d^3*tan(1/2*f*x + 1/2*e) - 22*B*a^3*c^3*d^3*tan(1/2*f*x + 1/2*e) + 16*A*a^3*c^2*d^4*tan(1/2*f*x + 1/2*e) +
4*B*a^3*c^2*d^4*tan(1/2*f*x + 1/2*e) + 2*A*a^3*c*d^5*tan(1/2*f*x + 1/2*e) + 4*B*a^3*c^6 - 2*A*a^3*c^5*d + 2*B
*a^3*c^5*d - 4*A*a^3*c^4*d^2 - 7*B*a^3*c^4*d^2 + 5*A*a^3*c^3*d^3 + B*a^3*c^3*d^3 + A*a^3*c^2*d^4)/((c^4*d^3 +
2*c^3*d^4 + 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) - (3*B*a^3*c - A*a^3*d - 3*B
*a^3*d)*(f*x + e)/d^4)/f

________________________________________________________________________________________

maple [B]  time = 0.56, size = 2906, normalized size = 9.53 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*a^3/f/d/(tan(1/2*f*x+1/2*e)^2*c+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)^3*B+2*a^
3/f/d^2/(tan(1/2*f*x+1/2*e)^2*c+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*A+4*a^3/f
/d/(tan(1/2*f*x+1/2*e)^2*c+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-10*a^3/f*d^2
/(tan(1/2*f*x+1/2*e)^2*c+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-9*a^3/f/d/(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-4*a^3/f/d^2/(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^2-2*a^3/f*d^3/(tan(1/2*f*x+1
/2*e)^2*c+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-4*a^3/f/d^3/(tan(1/2*f*x+1/2*
e)^2*c+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^2+2*c*d+d^2)*c^4*tan(1/2*f*x+1/2*e)^2*B-13*a^3/f/d^2/(tan(1/2*f*x+1/2*e)
^2*c+2*tan(1/2*f*x+1/2*e)*d+c)^2*c^3/(c^2+2*c*d+d^2)*tan(1/2*f*x+1/2*e)*B+6*a^3/f/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))*B*c^3+a^3/f/d/(tan(1/2*f*x+1/2*e)^2*c+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)^3*A-2*a^3/f*d^2/(tan(1/2*f*x+1/2*e)^2*c+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)^3*A-9*a^3/f/d^2/(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^2-2*a^3/f/d^2/(tan(1/2*f*x+1/2*e)^2*c+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-3*a^3/f/d^2/(tan(1/2*f*x+1/2*e)^2*c+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)^3*B-2*a^3/f/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*c^3-5*a^3/f/d/(tan(1/2*f*x+1/2*e)^2*c+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-a^3/f/d/(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+6*a^3/f/d^4/(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))*c^4*B-a^3/f/d/(tan(1/2*f*x+1/2*e)^2*c+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^2+2*c*d+d^2)*c^2*ta
n(1/2*f*x+1/2*e)^2*B-2*a^3/f*d^2/(tan(1/2*f*x+1/2*e)^2*c+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+7*a^3/f/d/(tan(1/2*f*x+1/2*e)^2*c+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)*A-2*a^3/f*d^2/(tan(1/2*f*x+1/2*e)^2*c+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-a^3/f*d/(tan(1/2*f*x+1/2*e)^2*c+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^2+2*c*d+d^2)*A-6*a^3/f/d^4*B*arctan(tan(1/2*f
*x+1/2*e))*c-a^3/f/(tan(1/2*f*x+1/2*e)^2*c+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+22*a^3/f/(tan(1/2*f*x+1/2*e)^2*c+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-5*a^3/f/
(tan(1/2*f*x+1/2*e)^2*c+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+11*a^3/f/(tan(1/2
*f*x+1/2*e)^2*c+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+5*a^3/f/(tan(1/2*f*x+1/2*e)
^2*c+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)^3*A+6*a^3/f/(tan(1/2*f*x+1/2*e)^2*c+2*ta
n(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)^3*B-2*a^3/f/d^2/(tan(1/2*f*x+1/2*e)^2*c+2*tan(1/2
*f*x+1/2*e)*d+c)^2/(c^2+2*c*d+d^2)*B*c^3+7*a^3/f/d/(tan(1/2*f*x+1/2*e)^2*c+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^2+2*
c*d+d^2)*B*c^2+2*a^3/f/d^2/(tan(1/2*f*x+1/2*e)^2*c+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^2+2*c*d+d^2)*A*c^3-4*a^3/f*d
/(tan(1/2*f*x+1/2*e)^2*c+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+7*a^3/f*d/(tan(1/2
*f*x+1/2*e)^2*c+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+14*a^3/f*d/(tan(1/2*f*x+1/2
*e)^2*c+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-16*a^3/f*d/(tan(1/2*f*x+1/2*e)^2*c+
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-4*a^3/f*d/(tan(1/2*f*x+1/2*e)^2*c+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+4*a^3/f/d/(tan(1/2*f*x+1/2*e)^2*c+2*tan(1/2*f*x+1/2*e)*d
+c)^2/(c^2+2*c*d+d^2)*A*c^2-4*a^3/f/d^3/(tan(1/2*f*x+1/2*e)^2*c+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^2+2*c*d+d^2)*c^
4*B-a^3/f/(tan(1/2*f*x+1/2*e)^2*c+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^2+2*c*d+d^2)*B*c-5*a^3/f/(tan(1/2*f*x+1/2*e)^
2*c+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^2+2*c*d+d^2)*A*c+7*a^3/f/(c^2+2*c*d+d^2)/(c^2-d^2)^(1/2)*arctan(1/2*(2*c*ta
n(1/2*f*x+1/2*e)+2*d)/(c^2-d^2)^(1/2))*A+6*a^3/f/(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-2*a^3/f/d^3*B/(1+tan(1/2*f*x+1/2*e)^2)+2*a^3/f/d^3*A*arctan(tan(1/2*f*x+1/2*e))+
6*a^3/f/d^3*B*arctan(tan(1/2*f*x+1/2*e))

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^3*(A+B*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 details)Is 4*d^2-4*c^2 positive or negative?

________________________________________________________________________________________

mupad [B]  time = 25.41, size = 13891, normalized size = 45.54 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- ((A*a^3*d^4 + 6*B*a^3*c^4 + 5*A*a^3*c*d^3 - 2*A*a^3*c^3*d + B*a^3*c*d^3 + 6*B*a^3*c^3*d - 4*A*a^3*c^2*d^2 -
5*B*a^3*c^2*d^2)/(d^3*(c + d)^2) + (4*tan(e/2 + (f*x)/2)^3*(A*a^3*d^4 + 6*B*a^3*c^4 + 5*A*a^3*c*d^3 - 2*A*a^3*
c^3*d + B*a^3*c*d^3 + 6*B*a^3*c^3*d - 4*A*a^3*c^2*d^2 - 5*B*a^3*c^2*d^2))/(c*d^2*(c + d)^2) + (tan(e/2 + (f*x)
/2)^5*(2*A*a^3*d^4 + 3*B*a^3*c^4 + 4*A*a^3*c*d^3 - A*a^3*c^3*d + 3*B*a^3*c^3*d - 5*A*a^3*c^2*d^2 - 6*B*a^3*c^2
*d^2))/(c*d^2*(c + d)^2) + (2*tan(e/2 + (f*x)/2)^2*(A*a^3*d^6 + 6*B*a^3*c^6 + 5*A*a^3*c*d^5 - 2*A*a^3*c^5*d +
B*a^3*c*d^5 + 6*B*a^3*c^5*d - 3*A*a^3*c^2*d^4 + 3*A*a^3*c^3*d^3 - 4*A*a^3*c^4*d^2 - 3*B*a^3*c^2*d^4 + 11*B*a^3
*c^3*d^3 + 3*B*a^3*c^4*d^2))/(c^2*d^3*(c + d)^2) + (tan(e/2 + (f*x)/2)^4*(2*A*a^3*d^6 + 6*B*a^3*c^6 + 10*A*a^3
*c*d^5 - 2*A*a^3*c^5*d + 2*B*a^3*c*d^5 + 6*B*a^3*c^5*d - 7*A*a^3*c^2*d^4 + A*a^3*c^3*d^3 - 4*A*a^3*c^4*d^2 - 1
4*B*a^3*c^2*d^4 + 5*B*a^3*c^3*d^3 + 3*B*a^3*c^4*d^2))/(c^2*d^3*(c + d)^2) + (tan(e/2 + (f*x)/2)*(2*A*a^3*d^4 +
21*B*a^3*c^4 + 16*A*a^3*c*d^3 - 7*A*a^3*c^3*d + 4*B*a^3*c*d^3 + 21*B*a^3*c^3*d - 11*A*a^3*c^2*d^2 - 14*B*a^3*
c^2*d^2))/(c*d^2*(c + d)^2))/(f*(tan(e/2 + (f*x)/2)^2*(3*c^2 + 4*d^2) + tan(e/2 + (f*x)/2)^4*(3*c^2 + 4*d^2) +
c^2*tan(e/2 + (f*x)/2)^6 + c^2 + 8*c*d*tan(e/2 + (f*x)/2)^3 + 4*c*d*tan(e/2 + (f*x)/2)^5 + 4*c*d*tan(e/2 + (f
*x)/2))) - (atan((((B*a^3*c*3i - a^3*d*(A + 3*B)*1i)*((8*(4*A^2*a^6*c^2*d^9 + 16*A^2*a^6*c^3*d^8 + 24*A^2*a^6*
c^4*d^7 + 16*A^2*a^6*c^5*d^6 + 4*A^2*a^6*c^6*d^5 + 36*B^2*a^6*c^2*d^9 + 72*B^2*a^6*c^3*d^8 - 36*B^2*a^6*c^4*d^
7 - 144*B^2*a^6*c^5*d^6 - 36*B^2*a^6*c^6*d^5 + 72*B^2*a^6*c^7*d^4 + 36*B^2*a^6*c^8*d^3 + 24*A*B*a^6*c^2*d^9 +
72*A*B*a^6*c^3*d^8 + 48*A*B*a^6*c^4*d^7 - 48*A*B*a^6*c^5*d^6 - 72*A*B*a^6*c^6*d^5 - 24*A*B*a^6*c^7*d^4))/(4*c*
d^11 + d^12 + 6*c^2*d^10 + 4*c^3*d^9 + c^4*d^8) + (8*tan(e/2 + (f*x)/2)*(46*A^2*a^6*c^2*d^10 + 99*A^2*a^6*c^3*
d^9 + 36*A^2*a^6*c^4*d^8 - 36*A^2*a^6*c^5*d^7 - 32*A^2*a^6*c^6*d^6 - 8*A^2*a^6*c^7*d^5 + 252*B^2*a^6*c^2*d^10
- 81*B^2*a^6*c^3*d^9 - 594*B^2*a^6*c^4*d^8 - 81*B^2*a^6*c^5*d^7 + 504*B^2*a^6*c^6*d^6 + 180*B^2*a^6*c^7*d^5 -
144*B^2*a^6*c^8*d^4 - 72*B^2*a^6*c^9*d^3 - 41*A^2*a^6*c*d^11 + 36*B^2*a^6*c*d^11 + 282*A*B*a^6*c^2*d^10 + 228*
A*B*a^6*c^3*d^9 - 318*A*B*a^6*c^4*d^8 - 372*A*B*a^6*c^5*d^7 + 24*A*B*a^6*c^6*d^6 + 144*A*B*a^6*c^7*d^5 + 48*A*
B*a^6*c^8*d^4 - 36*A*B*a^6*c*d^11))/(4*c*d^12 + d^13 + 6*c^2*d^11 + 4*c^3*d^10 + c^4*d^9) + ((B*a^3*c*3i - a^3
*d*(A + 3*B)*1i)*((8*tan(e/2 + (f*x)/2)*(28*A*a^3*c*d^14 + 24*B*a^3*c*d^14 + 52*A*a^3*c^2*d^13 + 4*A*a^3*c^3*d
^12 - 44*A*a^3*c^4*d^11 - 32*A*a^3*c^5*d^10 - 8*A*a^3*c^6*d^9 + 12*B*a^3*c^2*d^13 - 84*B*a^3*c^3*d^12 - 84*B*a
^3*c^4*d^11 + 36*B*a^3*c^5*d^10 + 72*B*a^3*c^6*d^9 + 24*B*a^3*c^7*d^8))/(4*c*d^12 + d^13 + 6*c^2*d^11 + 4*c^3*
d^10 + c^4*d^9) - (8*(4*A*a^3*c*d^13 + 12*B*a^3*c*d^13 + 2*A*a^3*c^2*d^12 - 6*A*a^3*c^3*d^11 - 2*A*a^3*c^4*d^1
0 + 2*A*a^3*c^5*d^9 + 24*B*a^3*c^2*d^12 + 6*B*a^3*c^3*d^11 - 18*B*a^3*c^4*d^10 - 18*B*a^3*c^5*d^9 - 6*B*a^3*c^
6*d^8))/(4*c*d^11 + d^12 + 6*c^2*d^10 + 4*c^3*d^9 + c^4*d^8) + (((8*(4*c^2*d^15 + 16*c^3*d^14 + 24*c^4*d^13 +
16*c^5*d^12 + 4*c^6*d^11))/(4*c*d^11 + d^12 + 6*c^2*d^10 + 4*c^3*d^9 + c^4*d^8) + (8*tan(e/2 + (f*x)/2)*(12*c*
d^17 + 48*c^2*d^16 + 64*c^3*d^15 + 16*c^4*d^14 - 36*c^5*d^13 - 32*c^6*d^12 - 8*c^7*d^11))/(4*c*d^12 + d^13 + 6
*c^2*d^11 + 4*c^3*d^10 + c^4*d^9))*(B*a^3*c*3i - a^3*d*(A + 3*B)*1i))/d^4))/d^4)*1i)/d^4 + ((B*a^3*c*3i - a^3*
d*(A + 3*B)*1i)*((8*(4*A^2*a^6*c^2*d^9 + 16*A^2*a^6*c^3*d^8 + 24*A^2*a^6*c^4*d^7 + 16*A^2*a^6*c^5*d^6 + 4*A^2*
a^6*c^6*d^5 + 36*B^2*a^6*c^2*d^9 + 72*B^2*a^6*c^3*d^8 - 36*B^2*a^6*c^4*d^7 - 144*B^2*a^6*c^5*d^6 - 36*B^2*a^6*
c^6*d^5 + 72*B^2*a^6*c^7*d^4 + 36*B^2*a^6*c^8*d^3 + 24*A*B*a^6*c^2*d^9 + 72*A*B*a^6*c^3*d^8 + 48*A*B*a^6*c^4*d
^7 - 48*A*B*a^6*c^5*d^6 - 72*A*B*a^6*c^6*d^5 - 24*A*B*a^6*c^7*d^4))/(4*c*d^11 + d^12 + 6*c^2*d^10 + 4*c^3*d^9
+ c^4*d^8) + (8*tan(e/2 + (f*x)/2)*(46*A^2*a^6*c^2*d^10 + 99*A^2*a^6*c^3*d^9 + 36*A^2*a^6*c^4*d^8 - 36*A^2*a^6
*c^5*d^7 - 32*A^2*a^6*c^6*d^6 - 8*A^2*a^6*c^7*d^5 + 252*B^2*a^6*c^2*d^10 - 81*B^2*a^6*c^3*d^9 - 594*B^2*a^6*c^
4*d^8 - 81*B^2*a^6*c^5*d^7 + 504*B^2*a^6*c^6*d^6 + 180*B^2*a^6*c^7*d^5 - 144*B^2*a^6*c^8*d^4 - 72*B^2*a^6*c^9*
d^3 - 41*A^2*a^6*c*d^11 + 36*B^2*a^6*c*d^11 + 282*A*B*a^6*c^2*d^10 + 228*A*B*a^6*c^3*d^9 - 318*A*B*a^6*c^4*d^8
- 372*A*B*a^6*c^5*d^7 + 24*A*B*a^6*c^6*d^6 + 144*A*B*a^6*c^7*d^5 + 48*A*B*a^6*c^8*d^4 - 36*A*B*a^6*c*d^11))/(
4*c*d^12 + d^13 + 6*c^2*d^11 + 4*c^3*d^10 + c^4*d^9) + ((B*a^3*c*3i - a^3*d*(A + 3*B)*1i)*((8*(4*A*a^3*c*d^13
+ 12*B*a^3*c*d^13 + 2*A*a^3*c^2*d^12 - 6*A*a^3*c^3*d^11 - 2*A*a^3*c^4*d^10 + 2*A*a^3*c^5*d^9 + 24*B*a^3*c^2*d^
12 + 6*B*a^3*c^3*d^11 - 18*B*a^3*c^4*d^10 - 18*B*a^3*c^5*d^9 - 6*B*a^3*c^6*d^8))/(4*c*d^11 + d^12 + 6*c^2*d^10
+ 4*c^3*d^9 + c^4*d^8) - (8*tan(e/2 + (f*x)/2)*(28*A*a^3*c*d^14 + 24*B*a^3*c*d^14 + 52*A*a^3*c^2*d^13 + 4*A*a
^3*c^3*d^12 - 44*A*a^3*c^4*d^11 - 32*A*a^3*c^5*d^10 - 8*A*a^3*c^6*d^9 + 12*B*a^3*c^2*d^13 - 84*B*a^3*c^3*d^12
- 84*B*a^3*c^4*d^11 + 36*B*a^3*c^5*d^10 + 72*B*a^3*c^6*d^9 + 24*B*a^3*c^7*d^8))/(4*c*d^12 + d^13 + 6*c^2*d^11
+ 4*c^3*d^10 + c^4*d^9) + (((8*(4*c^2*d^15 + 16*c^3*d^14 + 24*c^4*d^13 + 16*c^5*d^12 + 4*c^6*d^11))/(4*c*d^11
+ d^12 + 6*c^2*d^10 + 4*c^3*d^9 + c^4*d^8) + (8*tan(e/2 + (f*x)/2)*(12*c*d^17 + 48*c^2*d^16 + 64*c^3*d^15 + 16
*c^4*d^14 - 36*c^5*d^13 - 32*c^6*d^12 - 8*c^7*d^11))/(4*c*d^12 + d^13 + 6*c^2*d^11 + 4*c^3*d^10 + c^4*d^9))*(B
*a^3*c*3i - a^3*d*(A + 3*B)*1i))/d^4))/d^4)*1i)/d^4)/((16*(54*B^3*a^9*c^8 - 29*A^3*a^9*c^3*d^5 - 18*A^3*a^9*c^
4*d^4 - 2*A^3*a^9*c^5*d^3 - 324*B^3*a^9*c^2*d^6 + 81*B^3*a^9*c^3*d^5 + 405*B^3*a^9*c^4*d^4 - 135*B^3*a^9*c^5*d
^3 - 243*B^3*a^9*c^6*d^2 + 49*A^3*a^9*c*d^7 + 108*B^3*a^9*c*d^7 + 54*B^3*a^9*c^7*d + 288*A*B^2*a^9*c*d^7 - 54*
A*B^2*a^9*c^7*d + 231*A^2*B*a^9*c*d^7 - 576*A*B^2*a^9*c^2*d^6 - 135*A*B^2*a^9*c^3*d^5 + 540*A*B^2*a^9*c^4*d^4
+ 135*A*B^2*a^9*c^5*d^3 - 198*A*B^2*a^9*c^6*d^2 - 231*A^2*B*a^9*c^2*d^6 - 201*A^2*B*a^9*c^3*d^5 + 69*A^2*B*a^9
*c^4*d^4 + 114*A^2*B*a^9*c^5*d^3 + 18*A^2*B*a^9*c^6*d^2))/(4*c*d^11 + d^12 + 6*c^2*d^10 + 4*c^3*d^9 + c^4*d^8)
+ (16*tan(e/2 + (f*x)/2)*(216*B^3*a^9*c^9 + 52*A^3*a^9*c^2*d^7 + 4*A^3*a^9*c^3*d^6 - 44*A^3*a^9*c^4*d^5 - 32*
A^3*a^9*c^5*d^4 - 8*A^3*a^9*c^6*d^3 - 324*B^3*a^9*c^2*d^7 - 756*B^3*a^9*c^3*d^6 + 864*B^3*a^9*c^4*d^5 + 1080*B
^3*a^9*c^5*d^4 - 756*B^3*a^9*c^6*d^3 - 756*B^3*a^9*c^7*d^2 + 28*A^3*a^9*c*d^8 + 216*B^3*a^9*c*d^8 + 216*B^3*a^
9*c^8*d + 396*A*B^2*a^9*c*d^8 - 216*A*B^2*a^9*c^8*d + 192*A^2*B*a^9*c*d^8 - 108*A*B^2*a^9*c^2*d^7 - 1224*A*B^2
*a^9*c^3*d^6 + 1260*A*B^2*a^9*c^5*d^4 + 324*A*B^2*a^9*c^6*d^3 - 432*A*B^2*a^9*c^7*d^2 + 156*A^2*B*a^9*c^2*d^7
- 372*A^2*B*a^9*c^3*d^6 - 372*A^2*B*a^9*c^4*d^5 + 108*A^2*B*a^9*c^5*d^4 + 216*A^2*B*a^9*c^6*d^3 + 72*A^2*B*a^9
*c^7*d^2))/(4*c*d^12 + d^13 + 6*c^2*d^11 + 4*c^3*d^10 + c^4*d^9) + ((B*a^3*c*3i - a^3*d*(A + 3*B)*1i)*((8*(4*A
^2*a^6*c^2*d^9 + 16*A^2*a^6*c^3*d^8 + 24*A^2*a^6*c^4*d^7 + 16*A^2*a^6*c^5*d^6 + 4*A^2*a^6*c^6*d^5 + 36*B^2*a^6
*c^2*d^9 + 72*B^2*a^6*c^3*d^8 - 36*B^2*a^6*c^4*d^7 - 144*B^2*a^6*c^5*d^6 - 36*B^2*a^6*c^6*d^5 + 72*B^2*a^6*c^7
*d^4 + 36*B^2*a^6*c^8*d^3 + 24*A*B*a^6*c^2*d^9 + 72*A*B*a^6*c^3*d^8 + 48*A*B*a^6*c^4*d^7 - 48*A*B*a^6*c^5*d^6
- 72*A*B*a^6*c^6*d^5 - 24*A*B*a^6*c^7*d^4))/(4*c*d^11 + d^12 + 6*c^2*d^10 + 4*c^3*d^9 + c^4*d^8) + (8*tan(e/2
+ (f*x)/2)*(46*A^2*a^6*c^2*d^10 + 99*A^2*a^6*c^3*d^9 + 36*A^2*a^6*c^4*d^8 - 36*A^2*a^6*c^5*d^7 - 32*A^2*a^6*c^
6*d^6 - 8*A^2*a^6*c^7*d^5 + 252*B^2*a^6*c^2*d^10 - 81*B^2*a^6*c^3*d^9 - 594*B^2*a^6*c^4*d^8 - 81*B^2*a^6*c^5*d
^7 + 504*B^2*a^6*c^6*d^6 + 180*B^2*a^6*c^7*d^5 - 144*B^2*a^6*c^8*d^4 - 72*B^2*a^6*c^9*d^3 - 41*A^2*a^6*c*d^11
+ 36*B^2*a^6*c*d^11 + 282*A*B*a^6*c^2*d^10 + 228*A*B*a^6*c^3*d^9 - 318*A*B*a^6*c^4*d^8 - 372*A*B*a^6*c^5*d^7 +
24*A*B*a^6*c^6*d^6 + 144*A*B*a^6*c^7*d^5 + 48*A*B*a^6*c^8*d^4 - 36*A*B*a^6*c*d^11))/(4*c*d^12 + d^13 + 6*c^2*
d^11 + 4*c^3*d^10 + c^4*d^9) + ((B*a^3*c*3i - a^3*d*(A + 3*B)*1i)*((8*tan(e/2 + (f*x)/2)*(28*A*a^3*c*d^14 + 24
*B*a^3*c*d^14 + 52*A*a^3*c^2*d^13 + 4*A*a^3*c^3*d^12 - 44*A*a^3*c^4*d^11 - 32*A*a^3*c^5*d^10 - 8*A*a^3*c^6*d^9
+ 12*B*a^3*c^2*d^13 - 84*B*a^3*c^3*d^12 - 84*B*a^3*c^4*d^11 + 36*B*a^3*c^5*d^10 + 72*B*a^3*c^6*d^9 + 24*B*a^3
*c^7*d^8))/(4*c*d^12 + d^13 + 6*c^2*d^11 + 4*c^3*d^10 + c^4*d^9) - (8*(4*A*a^3*c*d^13 + 12*B*a^3*c*d^13 + 2*A*
a^3*c^2*d^12 - 6*A*a^3*c^3*d^11 - 2*A*a^3*c^4*d^10 + 2*A*a^3*c^5*d^9 + 24*B*a^3*c^2*d^12 + 6*B*a^3*c^3*d^11 -
18*B*a^3*c^4*d^10 - 18*B*a^3*c^5*d^9 - 6*B*a^3*c^6*d^8))/(4*c*d^11 + d^12 + 6*c^2*d^10 + 4*c^3*d^9 + c^4*d^8)
+ (((8*(4*c^2*d^15 + 16*c^3*d^14 + 24*c^4*d^13 + 16*c^5*d^12 + 4*c^6*d^11))/(4*c*d^11 + d^12 + 6*c^2*d^10 + 4*
c^3*d^9 + c^4*d^8) + (8*tan(e/2 + (f*x)/2)*(12*c*d^17 + 48*c^2*d^16 + 64*c^3*d^15 + 16*c^4*d^14 - 36*c^5*d^13
- 32*c^6*d^12 - 8*c^7*d^11))/(4*c*d^12 + d^13 + 6*c^2*d^11 + 4*c^3*d^10 + c^4*d^9))*(B*a^3*c*3i - a^3*d*(A + 3
*B)*1i))/d^4))/d^4))/d^4 - ((B*a^3*c*3i - a^3*d*(A + 3*B)*1i)*((8*(4*A^2*a^6*c^2*d^9 + 16*A^2*a^6*c^3*d^8 + 24
*A^2*a^6*c^4*d^7 + 16*A^2*a^6*c^5*d^6 + 4*A^2*a^6*c^6*d^5 + 36*B^2*a^6*c^2*d^9 + 72*B^2*a^6*c^3*d^8 - 36*B^2*a
^6*c^4*d^7 - 144*B^2*a^6*c^5*d^6 - 36*B^2*a^6*c^6*d^5 + 72*B^2*a^6*c^7*d^4 + 36*B^2*a^6*c^8*d^3 + 24*A*B*a^6*c
^2*d^9 + 72*A*B*a^6*c^3*d^8 + 48*A*B*a^6*c^4*d^7 - 48*A*B*a^6*c^5*d^6 - 72*A*B*a^6*c^6*d^5 - 24*A*B*a^6*c^7*d^
4))/(4*c*d^11 + d^12 + 6*c^2*d^10 + 4*c^3*d^9 + c^4*d^8) + (8*tan(e/2 + (f*x)/2)*(46*A^2*a^6*c^2*d^10 + 99*A^2
*a^6*c^3*d^9 + 36*A^2*a^6*c^4*d^8 - 36*A^2*a^6*c^5*d^7 - 32*A^2*a^6*c^6*d^6 - 8*A^2*a^6*c^7*d^5 + 252*B^2*a^6*
c^2*d^10 - 81*B^2*a^6*c^3*d^9 - 594*B^2*a^6*c^4*d^8 - 81*B^2*a^6*c^5*d^7 + 504*B^2*a^6*c^6*d^6 + 180*B^2*a^6*c
^7*d^5 - 144*B^2*a^6*c^8*d^4 - 72*B^2*a^6*c^9*d^3 - 41*A^2*a^6*c*d^11 + 36*B^2*a^6*c*d^11 + 282*A*B*a^6*c^2*d^
10 + 228*A*B*a^6*c^3*d^9 - 318*A*B*a^6*c^4*d^8 - 372*A*B*a^6*c^5*d^7 + 24*A*B*a^6*c^6*d^6 + 144*A*B*a^6*c^7*d^
5 + 48*A*B*a^6*c^8*d^4 - 36*A*B*a^6*c*d^11))/(4*c*d^12 + d^13 + 6*c^2*d^11 + 4*c^3*d^10 + c^4*d^9) + ((B*a^3*c
*3i - a^3*d*(A + 3*B)*1i)*((8*(4*A*a^3*c*d^13 + 12*B*a^3*c*d^13 + 2*A*a^3*c^2*d^12 - 6*A*a^3*c^3*d^11 - 2*A*a^
3*c^4*d^10 + 2*A*a^3*c^5*d^9 + 24*B*a^3*c^2*d^12 + 6*B*a^3*c^3*d^11 - 18*B*a^3*c^4*d^10 - 18*B*a^3*c^5*d^9 - 6
*B*a^3*c^6*d^8))/(4*c*d^11 + d^12 + 6*c^2*d^10 + 4*c^3*d^9 + c^4*d^8) - (8*tan(e/2 + (f*x)/2)*(28*A*a^3*c*d^14
+ 24*B*a^3*c*d^14 + 52*A*a^3*c^2*d^13 + 4*A*a^3*c^3*d^12 - 44*A*a^3*c^4*d^11 - 32*A*a^3*c^5*d^10 - 8*A*a^3*c^
6*d^9 + 12*B*a^3*c^2*d^13 - 84*B*a^3*c^3*d^12 - 84*B*a^3*c^4*d^11 + 36*B*a^3*c^5*d^10 + 72*B*a^3*c^6*d^9 + 24*
B*a^3*c^7*d^8))/(4*c*d^12 + d^13 + 6*c^2*d^11 + 4*c^3*d^10 + c^4*d^9) + (((8*(4*c^2*d^15 + 16*c^3*d^14 + 24*c^
4*d^13 + 16*c^5*d^12 + 4*c^6*d^11))/(4*c*d^11 + d^12 + 6*c^2*d^10 + 4*c^3*d^9 + c^4*d^8) + (8*tan(e/2 + (f*x)/
2)*(12*c*d^17 + 48*c^2*d^16 + 64*c^3*d^15 + 16*c^4*d^14 - 36*c^5*d^13 - 32*c^6*d^12 - 8*c^7*d^11))/(4*c*d^12 +
d^13 + 6*c^2*d^11 + 4*c^3*d^10 + c^4*d^9))*(B*a^3*c*3i - a^3*d*(A + 3*B)*1i))/d^4))/d^4))/d^4))*(B*a^3*c*3i -
a^3*d*(A + 3*B)*1i)*2i)/(d^4*f) - (a^3*atan(((a^3*(-(c + d)^5*(c - d))^(1/2)*((8*(4*A^2*a^6*c^2*d^9 + 16*A^2*
a^6*c^3*d^8 + 24*A^2*a^6*c^4*d^7 + 16*A^2*a^6*c^5*d^6 + 4*A^2*a^6*c^6*d^5 + 36*B^2*a^6*c^2*d^9 + 72*B^2*a^6*c^
3*d^8 - 36*B^2*a^6*c^4*d^7 - 144*B^2*a^6*c^5*d^6 - 36*B^2*a^6*c^6*d^5 + 72*B^2*a^6*c^7*d^4 + 36*B^2*a^6*c^8*d^
3 + 24*A*B*a^6*c^2*d^9 + 72*A*B*a^6*c^3*d^8 + 48*A*B*a^6*c^4*d^7 - 48*A*B*a^6*c^5*d^6 - 72*A*B*a^6*c^6*d^5 - 2
4*A*B*a^6*c^7*d^4))/(4*c*d^11 + d^12 + 6*c^2*d^10 + 4*c^3*d^9 + c^4*d^8) + (8*tan(e/2 + (f*x)/2)*(46*A^2*a^6*c
^2*d^10 + 99*A^2*a^6*c^3*d^9 + 36*A^2*a^6*c^4*d^8 - 36*A^2*a^6*c^5*d^7 - 32*A^2*a^6*c^6*d^6 - 8*A^2*a^6*c^7*d^
5 + 252*B^2*a^6*c^2*d^10 - 81*B^2*a^6*c^3*d^9 - 594*B^2*a^6*c^4*d^8 - 81*B^2*a^6*c^5*d^7 + 504*B^2*a^6*c^6*d^6
+ 180*B^2*a^6*c^7*d^5 - 144*B^2*a^6*c^8*d^4 - 72*B^2*a^6*c^9*d^3 - 41*A^2*a^6*c*d^11 + 36*B^2*a^6*c*d^11 + 28
2*A*B*a^6*c^2*d^10 + 228*A*B*a^6*c^3*d^9 - 318*A*B*a^6*c^4*d^8 - 372*A*B*a^6*c^5*d^7 + 24*A*B*a^6*c^6*d^6 + 14
4*A*B*a^6*c^7*d^5 + 48*A*B*a^6*c^8*d^4 - 36*A*B*a^6*c*d^11))/(4*c*d^12 + d^13 + 6*c^2*d^11 + 4*c^3*d^10 + c^4*
d^9) + (a^3*(-(c + d)^5*(c - d))^(1/2)*((8*tan(e/2 + (f*x)/2)*(28*A*a^3*c*d^14 + 24*B*a^3*c*d^14 + 52*A*a^3*c^
2*d^13 + 4*A*a^3*c^3*d^12 - 44*A*a^3*c^4*d^11 - 32*A*a^3*c^5*d^10 - 8*A*a^3*c^6*d^9 + 12*B*a^3*c^2*d^13 - 84*B
*a^3*c^3*d^12 - 84*B*a^3*c^4*d^11 + 36*B*a^3*c^5*d^10 + 72*B*a^3*c^6*d^9 + 24*B*a^3*c^7*d^8))/(4*c*d^12 + d^13
+ 6*c^2*d^11 + 4*c^3*d^10 + c^4*d^9) - (8*(4*A*a^3*c*d^13 + 12*B*a^3*c*d^13 + 2*A*a^3*c^2*d^12 - 6*A*a^3*c^3*
d^11 - 2*A*a^3*c^4*d^10 + 2*A*a^3*c^5*d^9 + 24*B*a^3*c^2*d^12 + 6*B*a^3*c^3*d^11 - 18*B*a^3*c^4*d^10 - 18*B*a^
3*c^5*d^9 - 6*B*a^3*c^6*d^8))/(4*c*d^11 + d^12 + 6*c^2*d^10 + 4*c^3*d^9 + c^4*d^8) + (a^3*(-(c + d)^5*(c - d))
^(1/2)*((8*(4*c^2*d^15 + 16*c^3*d^14 + 24*c^4*d^13 + 16*c^5*d^12 + 4*c^6*d^11))/(4*c*d^11 + d^12 + 6*c^2*d^10
+ 4*c^3*d^9 + c^4*d^8) + (8*tan(e/2 + (f*x)/2)*(12*c*d^17 + 48*c^2*d^16 + 64*c^3*d^15 + 16*c^4*d^14 - 36*c^5*d
^13 - 32*c^6*d^12 - 8*c^7*d^11))/(4*c*d^12 + d^13 + 6*c^2*d^11 + 4*c^3*d^10 + c^4*d^9))*(7*A*d^3 - 6*B*c^3 + 6
*B*d^3 + 6*A*c*d^2 + 2*A*c^2*d - 3*B*c*d^2 - 12*B*c^2*d))/(2*(5*c*d^8 + d^9 + 10*c^2*d^7 + 10*c^3*d^6 + 5*c^4*
d^5 + c^5*d^4)))*(7*A*d^3 - 6*B*c^3 + 6*B*d^3 + 6*A*c*d^2 + 2*A*c^2*d - 3*B*c*d^2 - 12*B*c^2*d))/(2*(5*c*d^8 +
d^9 + 10*c^2*d^7 + 10*c^3*d^6 + 5*c^4*d^5 + c^5*d^4)))*(7*A*d^3 - 6*B*c^3 + 6*B*d^3 + 6*A*c*d^2 + 2*A*c^2*d -
3*B*c*d^2 - 12*B*c^2*d)*1i)/(2*(5*c*d^8 + d^9 + 10*c^2*d^7 + 10*c^3*d^6 + 5*c^4*d^5 + c^5*d^4)) + (a^3*(-(c +
d)^5*(c - d))^(1/2)*((8*(4*A^2*a^6*c^2*d^9 + 16*A^2*a^6*c^3*d^8 + 24*A^2*a^6*c^4*d^7 + 16*A^2*a^6*c^5*d^6 + 4
*A^2*a^6*c^6*d^5 + 36*B^2*a^6*c^2*d^9 + 72*B^2*a^6*c^3*d^8 - 36*B^2*a^6*c^4*d^7 - 144*B^2*a^6*c^5*d^6 - 36*B^2
*a^6*c^6*d^5 + 72*B^2*a^6*c^7*d^4 + 36*B^2*a^6*c^8*d^3 + 24*A*B*a^6*c^2*d^9 + 72*A*B*a^6*c^3*d^8 + 48*A*B*a^6*
c^4*d^7 - 48*A*B*a^6*c^5*d^6 - 72*A*B*a^6*c^6*d^5 - 24*A*B*a^6*c^7*d^4))/(4*c*d^11 + d^12 + 6*c^2*d^10 + 4*c^3
*d^9 + c^4*d^8) + (8*tan(e/2 + (f*x)/2)*(46*A^2*a^6*c^2*d^10 + 99*A^2*a^6*c^3*d^9 + 36*A^2*a^6*c^4*d^8 - 36*A^
2*a^6*c^5*d^7 - 32*A^2*a^6*c^6*d^6 - 8*A^2*a^6*c^7*d^5 + 252*B^2*a^6*c^2*d^10 - 81*B^2*a^6*c^3*d^9 - 594*B^2*a
^6*c^4*d^8 - 81*B^2*a^6*c^5*d^7 + 504*B^2*a^6*c^6*d^6 + 180*B^2*a^6*c^7*d^5 - 144*B^2*a^6*c^8*d^4 - 72*B^2*a^6
*c^9*d^3 - 41*A^2*a^6*c*d^11 + 36*B^2*a^6*c*d^11 + 282*A*B*a^6*c^2*d^10 + 228*A*B*a^6*c^3*d^9 - 318*A*B*a^6*c^
4*d^8 - 372*A*B*a^6*c^5*d^7 + 24*A*B*a^6*c^6*d^6 + 144*A*B*a^6*c^7*d^5 + 48*A*B*a^6*c^8*d^4 - 36*A*B*a^6*c*d^1
1))/(4*c*d^12 + d^13 + 6*c^2*d^11 + 4*c^3*d^10 + c^4*d^9) + (a^3*(-(c + d)^5*(c - d))^(1/2)*((8*(4*A*a^3*c*d^1
3 + 12*B*a^3*c*d^13 + 2*A*a^3*c^2*d^12 - 6*A*a^3*c^3*d^11 - 2*A*a^3*c^4*d^10 + 2*A*a^3*c^5*d^9 + 24*B*a^3*c^2*
d^12 + 6*B*a^3*c^3*d^11 - 18*B*a^3*c^4*d^10 - 18*B*a^3*c^5*d^9 - 6*B*a^3*c^6*d^8))/(4*c*d^11 + d^12 + 6*c^2*d^
10 + 4*c^3*d^9 + c^4*d^8) - (8*tan(e/2 + (f*x)/2)*(28*A*a^3*c*d^14 + 24*B*a^3*c*d^14 + 52*A*a^3*c^2*d^13 + 4*A
*a^3*c^3*d^12 - 44*A*a^3*c^4*d^11 - 32*A*a^3*c^5*d^10 - 8*A*a^3*c^6*d^9 + 12*B*a^3*c^2*d^13 - 84*B*a^3*c^3*d^1
2 - 84*B*a^3*c^4*d^11 + 36*B*a^3*c^5*d^10 + 72*B*a^3*c^6*d^9 + 24*B*a^3*c^7*d^8))/(4*c*d^12 + d^13 + 6*c^2*d^1
1 + 4*c^3*d^10 + c^4*d^9) + (a^3*(-(c + d)^5*(c - d))^(1/2)*((8*(4*c^2*d^15 + 16*c^3*d^14 + 24*c^4*d^13 + 16*c
^5*d^12 + 4*c^6*d^11))/(4*c*d^11 + d^12 + 6*c^2*d^10 + 4*c^3*d^9 + c^4*d^8) + (8*tan(e/2 + (f*x)/2)*(12*c*d^17
+ 48*c^2*d^16 + 64*c^3*d^15 + 16*c^4*d^14 - 36*c^5*d^13 - 32*c^6*d^12 - 8*c^7*d^11))/(4*c*d^12 + d^13 + 6*c^2
*d^11 + 4*c^3*d^10 + c^4*d^9))*(7*A*d^3 - 6*B*c^3 + 6*B*d^3 + 6*A*c*d^2 + 2*A*c^2*d - 3*B*c*d^2 - 12*B*c^2*d))
/(2*(5*c*d^8 + d^9 + 10*c^2*d^7 + 10*c^3*d^6 + 5*c^4*d^5 + c^5*d^4)))*(7*A*d^3 - 6*B*c^3 + 6*B*d^3 + 6*A*c*d^2
+ 2*A*c^2*d - 3*B*c*d^2 - 12*B*c^2*d))/(2*(5*c*d^8 + d^9 + 10*c^2*d^7 + 10*c^3*d^6 + 5*c^4*d^5 + c^5*d^4)))*(
7*A*d^3 - 6*B*c^3 + 6*B*d^3 + 6*A*c*d^2 + 2*A*c^2*d - 3*B*c*d^2 - 12*B*c^2*d)*1i)/(2*(5*c*d^8 + d^9 + 10*c^2*d
^7 + 10*c^3*d^6 + 5*c^4*d^5 + c^5*d^4)))/((16*(54*B^3*a^9*c^8 - 29*A^3*a^9*c^3*d^5 - 18*A^3*a^9*c^4*d^4 - 2*A^
3*a^9*c^5*d^3 - 324*B^3*a^9*c^2*d^6 + 81*B^3*a^9*c^3*d^5 + 405*B^3*a^9*c^4*d^4 - 135*B^3*a^9*c^5*d^3 - 243*B^3
*a^9*c^6*d^2 + 49*A^3*a^9*c*d^7 + 108*B^3*a^9*c*d^7 + 54*B^3*a^9*c^7*d + 288*A*B^2*a^9*c*d^7 - 54*A*B^2*a^9*c^
7*d + 231*A^2*B*a^9*c*d^7 - 576*A*B^2*a^9*c^2*d^6 - 135*A*B^2*a^9*c^3*d^5 + 540*A*B^2*a^9*c^4*d^4 + 135*A*B^2*
a^9*c^5*d^3 - 198*A*B^2*a^9*c^6*d^2 - 231*A^2*B*a^9*c^2*d^6 - 201*A^2*B*a^9*c^3*d^5 + 69*A^2*B*a^9*c^4*d^4 + 1
14*A^2*B*a^9*c^5*d^3 + 18*A^2*B*a^9*c^6*d^2))/(4*c*d^11 + d^12 + 6*c^2*d^10 + 4*c^3*d^9 + c^4*d^8) + (16*tan(e
/2 + (f*x)/2)*(216*B^3*a^9*c^9 + 52*A^3*a^9*c^2*d^7 + 4*A^3*a^9*c^3*d^6 - 44*A^3*a^9*c^4*d^5 - 32*A^3*a^9*c^5*
d^4 - 8*A^3*a^9*c^6*d^3 - 324*B^3*a^9*c^2*d^7 - 756*B^3*a^9*c^3*d^6 + 864*B^3*a^9*c^4*d^5 + 1080*B^3*a^9*c^5*d
^4 - 756*B^3*a^9*c^6*d^3 - 756*B^3*a^9*c^7*d^2 + 28*A^3*a^9*c*d^8 + 216*B^3*a^9*c*d^8 + 216*B^3*a^9*c^8*d + 39
6*A*B^2*a^9*c*d^8 - 216*A*B^2*a^9*c^8*d + 192*A^2*B*a^9*c*d^8 - 108*A*B^2*a^9*c^2*d^7 - 1224*A*B^2*a^9*c^3*d^6
+ 1260*A*B^2*a^9*c^5*d^4 + 324*A*B^2*a^9*c^6*d^3 - 432*A*B^2*a^9*c^7*d^2 + 156*A^2*B*a^9*c^2*d^7 - 372*A^2*B*
a^9*c^3*d^6 - 372*A^2*B*a^9*c^4*d^5 + 108*A^2*B*a^9*c^5*d^4 + 216*A^2*B*a^9*c^6*d^3 + 72*A^2*B*a^9*c^7*d^2))/(
4*c*d^12 + d^13 + 6*c^2*d^11 + 4*c^3*d^10 + c^4*d^9) + (a^3*(-(c + d)^5*(c - d))^(1/2)*((8*(4*A^2*a^6*c^2*d^9
+ 16*A^2*a^6*c^3*d^8 + 24*A^2*a^6*c^4*d^7 + 16*A^2*a^6*c^5*d^6 + 4*A^2*a^6*c^6*d^5 + 36*B^2*a^6*c^2*d^9 + 72*B
^2*a^6*c^3*d^8 - 36*B^2*a^6*c^4*d^7 - 144*B^2*a^6*c^5*d^6 - 36*B^2*a^6*c^6*d^5 + 72*B^2*a^6*c^7*d^4 + 36*B^2*a
^6*c^8*d^3 + 24*A*B*a^6*c^2*d^9 + 72*A*B*a^6*c^3*d^8 + 48*A*B*a^6*c^4*d^7 - 48*A*B*a^6*c^5*d^6 - 72*A*B*a^6*c^
6*d^5 - 24*A*B*a^6*c^7*d^4))/(4*c*d^11 + d^12 + 6*c^2*d^10 + 4*c^3*d^9 + c^4*d^8) + (8*tan(e/2 + (f*x)/2)*(46*
A^2*a^6*c^2*d^10 + 99*A^2*a^6*c^3*d^9 + 36*A^2*a^6*c^4*d^8 - 36*A^2*a^6*c^5*d^7 - 32*A^2*a^6*c^6*d^6 - 8*A^2*a
^6*c^7*d^5 + 252*B^2*a^6*c^2*d^10 - 81*B^2*a^6*c^3*d^9 - 594*B^2*a^6*c^4*d^8 - 81*B^2*a^6*c^5*d^7 + 504*B^2*a^
6*c^6*d^6 + 180*B^2*a^6*c^7*d^5 - 144*B^2*a^6*c^8*d^4 - 72*B^2*a^6*c^9*d^3 - 41*A^2*a^6*c*d^11 + 36*B^2*a^6*c*
d^11 + 282*A*B*a^6*c^2*d^10 + 228*A*B*a^6*c^3*d^9 - 318*A*B*a^6*c^4*d^8 - 372*A*B*a^6*c^5*d^7 + 24*A*B*a^6*c^6
*d^6 + 144*A*B*a^6*c^7*d^5 + 48*A*B*a^6*c^8*d^4 - 36*A*B*a^6*c*d^11))/(4*c*d^12 + d^13 + 6*c^2*d^11 + 4*c^3*d^
10 + c^4*d^9) + (a^3*(-(c + d)^5*(c - d))^(1/2)*((8*tan(e/2 + (f*x)/2)*(28*A*a^3*c*d^14 + 24*B*a^3*c*d^14 + 52
*A*a^3*c^2*d^13 + 4*A*a^3*c^3*d^12 - 44*A*a^3*c^4*d^11 - 32*A*a^3*c^5*d^10 - 8*A*a^3*c^6*d^9 + 12*B*a^3*c^2*d^
13 - 84*B*a^3*c^3*d^12 - 84*B*a^3*c^4*d^11 + 36*B*a^3*c^5*d^10 + 72*B*a^3*c^6*d^9 + 24*B*a^3*c^7*d^8))/(4*c*d^
12 + d^13 + 6*c^2*d^11 + 4*c^3*d^10 + c^4*d^9) - (8*(4*A*a^3*c*d^13 + 12*B*a^3*c*d^13 + 2*A*a^3*c^2*d^12 - 6*A
*a^3*c^3*d^11 - 2*A*a^3*c^4*d^10 + 2*A*a^3*c^5*d^9 + 24*B*a^3*c^2*d^12 + 6*B*a^3*c^3*d^11 - 18*B*a^3*c^4*d^10
- 18*B*a^3*c^5*d^9 - 6*B*a^3*c^6*d^8))/(4*c*d^11 + d^12 + 6*c^2*d^10 + 4*c^3*d^9 + c^4*d^8) + (a^3*(-(c + d)^5
*(c - d))^(1/2)*((8*(4*c^2*d^15 + 16*c^3*d^14 + 24*c^4*d^13 + 16*c^5*d^12 + 4*c^6*d^11))/(4*c*d^11 + d^12 + 6*
c^2*d^10 + 4*c^3*d^9 + c^4*d^8) + (8*tan(e/2 + (f*x)/2)*(12*c*d^17 + 48*c^2*d^16 + 64*c^3*d^15 + 16*c^4*d^14 -
36*c^5*d^13 - 32*c^6*d^12 - 8*c^7*d^11))/(4*c*d^12 + d^13 + 6*c^2*d^11 + 4*c^3*d^10 + c^4*d^9))*(7*A*d^3 - 6*
B*c^3 + 6*B*d^3 + 6*A*c*d^2 + 2*A*c^2*d - 3*B*c*d^2 - 12*B*c^2*d))/(2*(5*c*d^8 + d^9 + 10*c^2*d^7 + 10*c^3*d^6
+ 5*c^4*d^5 + c^5*d^4)))*(7*A*d^3 - 6*B*c^3 + 6*B*d^3 + 6*A*c*d^2 + 2*A*c^2*d - 3*B*c*d^2 - 12*B*c^2*d))/(2*(
5*c*d^8 + d^9 + 10*c^2*d^7 + 10*c^3*d^6 + 5*c^4*d^5 + c^5*d^4)))*(7*A*d^3 - 6*B*c^3 + 6*B*d^3 + 6*A*c*d^2 + 2*
A*c^2*d - 3*B*c*d^2 - 12*B*c^2*d))/(2*(5*c*d^8 + d^9 + 10*c^2*d^7 + 10*c^3*d^6 + 5*c^4*d^5 + c^5*d^4)) - (a^3*
(-(c + d)^5*(c - d))^(1/2)*((8*(4*A^2*a^6*c^2*d^9 + 16*A^2*a^6*c^3*d^8 + 24*A^2*a^6*c^4*d^7 + 16*A^2*a^6*c^5*d
^6 + 4*A^2*a^6*c^6*d^5 + 36*B^2*a^6*c^2*d^9 + 72*B^2*a^6*c^3*d^8 - 36*B^2*a^6*c^4*d^7 - 144*B^2*a^6*c^5*d^6 -
36*B^2*a^6*c^6*d^5 + 72*B^2*a^6*c^7*d^4 + 36*B^2*a^6*c^8*d^3 + 24*A*B*a^6*c^2*d^9 + 72*A*B*a^6*c^3*d^8 + 48*A*
B*a^6*c^4*d^7 - 48*A*B*a^6*c^5*d^6 - 72*A*B*a^6*c^6*d^5 - 24*A*B*a^6*c^7*d^4))/(4*c*d^11 + d^12 + 6*c^2*d^10 +
4*c^3*d^9 + c^4*d^8) + (8*tan(e/2 + (f*x)/2)*(46*A^2*a^6*c^2*d^10 + 99*A^2*a^6*c^3*d^9 + 36*A^2*a^6*c^4*d^8 -
36*A^2*a^6*c^5*d^7 - 32*A^2*a^6*c^6*d^6 - 8*A^2*a^6*c^7*d^5 + 252*B^2*a^6*c^2*d^10 - 81*B^2*a^6*c^3*d^9 - 594
*B^2*a^6*c^4*d^8 - 81*B^2*a^6*c^5*d^7 + 504*B^2*a^6*c^6*d^6 + 180*B^2*a^6*c^7*d^5 - 144*B^2*a^6*c^8*d^4 - 72*B
^2*a^6*c^9*d^3 - 41*A^2*a^6*c*d^11 + 36*B^2*a^6*c*d^11 + 282*A*B*a^6*c^2*d^10 + 228*A*B*a^6*c^3*d^9 - 318*A*B*
a^6*c^4*d^8 - 372*A*B*a^6*c^5*d^7 + 24*A*B*a^6*c^6*d^6 + 144*A*B*a^6*c^7*d^5 + 48*A*B*a^6*c^8*d^4 - 36*A*B*a^6
*c*d^11))/(4*c*d^12 + d^13 + 6*c^2*d^11 + 4*c^3*d^10 + c^4*d^9) + (a^3*(-(c + d)^5*(c - d))^(1/2)*((8*(4*A*a^3
*c*d^13 + 12*B*a^3*c*d^13 + 2*A*a^3*c^2*d^12 - 6*A*a^3*c^3*d^11 - 2*A*a^3*c^4*d^10 + 2*A*a^3*c^5*d^9 + 24*B*a^
3*c^2*d^12 + 6*B*a^3*c^3*d^11 - 18*B*a^3*c^4*d^10 - 18*B*a^3*c^5*d^9 - 6*B*a^3*c^6*d^8))/(4*c*d^11 + d^12 + 6*
c^2*d^10 + 4*c^3*d^9 + c^4*d^8) - (8*tan(e/2 + (f*x)/2)*(28*A*a^3*c*d^14 + 24*B*a^3*c*d^14 + 52*A*a^3*c^2*d^13
+ 4*A*a^3*c^3*d^12 - 44*A*a^3*c^4*d^11 - 32*A*a^3*c^5*d^10 - 8*A*a^3*c^6*d^9 + 12*B*a^3*c^2*d^13 - 84*B*a^3*c
^3*d^12 - 84*B*a^3*c^4*d^11 + 36*B*a^3*c^5*d^10 + 72*B*a^3*c^6*d^9 + 24*B*a^3*c^7*d^8))/(4*c*d^12 + d^13 + 6*c
^2*d^11 + 4*c^3*d^10 + c^4*d^9) + (a^3*(-(c + d)^5*(c - d))^(1/2)*((8*(4*c^2*d^15 + 16*c^3*d^14 + 24*c^4*d^13
+ 16*c^5*d^12 + 4*c^6*d^11))/(4*c*d^11 + d^12 + 6*c^2*d^10 + 4*c^3*d^9 + c^4*d^8) + (8*tan(e/2 + (f*x)/2)*(12*
c*d^17 + 48*c^2*d^16 + 64*c^3*d^15 + 16*c^4*d^14 - 36*c^5*d^13 - 32*c^6*d^12 - 8*c^7*d^11))/(4*c*d^12 + d^13 +
6*c^2*d^11 + 4*c^3*d^10 + c^4*d^9))*(7*A*d^3 - 6*B*c^3 + 6*B*d^3 + 6*A*c*d^2 + 2*A*c^2*d - 3*B*c*d^2 - 12*B*c
^2*d))/(2*(5*c*d^8 + d^9 + 10*c^2*d^7 + 10*c^3*d^6 + 5*c^4*d^5 + c^5*d^4)))*(7*A*d^3 - 6*B*c^3 + 6*B*d^3 + 6*A
*c*d^2 + 2*A*c^2*d - 3*B*c*d^2 - 12*B*c^2*d))/(2*(5*c*d^8 + d^9 + 10*c^2*d^7 + 10*c^3*d^6 + 5*c^4*d^5 + c^5*d^
4)))*(7*A*d^3 - 6*B*c^3 + 6*B*d^3 + 6*A*c*d^2 + 2*A*c^2*d - 3*B*c*d^2 - 12*B*c^2*d))/(2*(5*c*d^8 + d^9 + 10*c^
2*d^7 + 10*c^3*d^6 + 5*c^4*d^5 + c^5*d^4))))*(-(c + d)^5*(c - d))^(1/2)*(7*A*d^3 - 6*B*c^3 + 6*B*d^3 + 6*A*c*d
^2 + 2*A*c^2*d - 3*B*c*d^2 - 12*B*c^2*d)*1i)/(f*(5*c*d^8 + d^9 + 10*c^2*d^7 + 10*c^3*d^6 + 5*c^4*d^5 + c^5*d^4
))

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________