\(\int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c+d \sin (e+f x))^2} \, dx\) [263]

Optimal result

Integrand size = 35, antiderivative size = 283 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c+d \sin (e+f x))^2} \, dx=-\frac {a^3 \left (2 A (2 c-3 d) d-B \left (6 c^2-12 c d+7 d^2\right )\right ) x}{2 d^4}+\frac {2 a^3 (c-d)^2 \left (A d (2 c+3 d)-B \left (3 c^2+3 c d-d^2\right )\right ) \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{d^4 (c+d) \sqrt {c^2-d^2} f}-\frac {a^3 \left (4 A c d-B \left (6 c^2-3 c d-5 d^2\right )\right ) \cos (e+f x)}{2 d^3 (c+d) f}+\frac {(2 A d-B (3 c+d)) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{2 d^2 (c+d) f}+\frac {a (B c-A d) \cos (e+f x) (a+a \sin (e+f x))^2}{d (c+d) f (c+d \sin (e+f x))} \]

[Out]

-1/2*a^3*(2*A*(2*c-3*d)*d-B*(6*c^2-12*c*d+7*d^2))*x/d^4-1/2*a^3*(4*A*c*d-B*(6*c^2-3*c*d-5*d^2))*cos(f*x+e)/d^3
/(c+d)/f+1/2*(2*A*d-B*(3*c+d))*cos(f*x+e)*(a^3+a^3*sin(f*x+e))/d^2/(c+d)/f+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*a^3*(c-d)^2*(A*d*(2*c+3*d)-B*(3*c^2+3*c*d-d^2))*arctan((d+c*tan(1/2*f*x+1
/2*e))/(c^2-d^2)^(1/2))/d^4/(c+d)/f/(c^2-d^2)^(1/2)

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {3054, 3055, 3047, 3102, 2814, 2739, 632, 210} \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c+d \sin (e+f x))^2} \, dx=\frac {2 a^3 (c-d)^2 \left (A d (2 c+3 d)-B \left (3 c^2+3 c d-d^2\right )\right ) \arctan \left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{d^4 f (c+d) \sqrt {c^2-d^2}}-\frac {a^3 x \left (2 A d (2 c-3 d)-B \left (6 c^2-12 c d+7 d^2\right )\right )}{2 d^4}-\frac {a^3 \left (4 A c d-B \left (6 c^2-3 c d-5 d^2\right )\right ) \cos (e+f x)}{2 d^3 f (c+d)}+\frac {(2 A d-B (3 c+d)) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{2 d^2 f (c+d)}+\frac {a (B c-A d) \cos (e+f x) (a \sin (e+f x)+a)^2}{d f (c+d) (c+d \sin (e+f x))} \]

[In]

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

[Out]

-1/2*(a^3*(2*A*(2*c - 3*d)*d - B*(6*c^2 - 12*c*d + 7*d^2))*x)/d^4 + (2*a^3*(c - d)^2*(A*d*(2*c + 3*d) - B*(3*c
^2 + 3*c*d - d^2))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(d^4*(c + d)*Sqrt[c^2 - d^2]*f) - (a^3*(4
*A*c*d - B*(6*c^2 - 3*c*d - 5*d^2))*Cos[e + f*x])/(2*d^3*(c + d)*f) + ((2*A*d - B*(3*c + d))*Cos[e + f*x]*(a^3
+ a^3*Sin[e + f*x]))/(2*d^2*(c + d)*f) + (a*(B*c - A*d)*Cos[e + f*x]*(a + a*Sin[e + f*x])^2)/(d*(c + d)*f*(c
+ d*Sin[e + f*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 2814

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 3047

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 3054

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
*Sin[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 3055

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

Rule 3102

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*} \text {integral}& = \frac {a (B c-A d) \cos (e+f x) (a+a \sin (e+f x))^2}{d (c+d) f (c+d \sin (e+f x))}+\frac {\int \frac {(a+a \sin (e+f x))^2 (-a (B (2 c-d)-3 A d)-a (2 A d-B (3 c+d)) \sin (e+f x))}{c+d \sin (e+f x)} \, dx}{d (c+d)} \\ & = \frac {(2 A d-B (3 c+d)) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{2 d^2 (c+d) f}+\frac {a (B c-A d) \cos (e+f x) (a+a \sin (e+f x))^2}{d (c+d) f (c+d \sin (e+f x))}+\frac {\int \frac {(a+a \sin (e+f x)) \left (-a^2 \left (2 A (c-3 d) d-B \left (3 c^2-3 c d+2 d^2\right )\right )+a^2 \left (4 A c d-B \left (6 c^2-3 c d-5 d^2\right )\right ) \sin (e+f x)\right )}{c+d \sin (e+f x)} \, dx}{2 d^2 (c+d)} \\ & = \frac {(2 A d-B (3 c+d)) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{2 d^2 (c+d) f}+\frac {a (B c-A d) \cos (e+f x) (a+a \sin (e+f x))^2}{d (c+d) f (c+d \sin (e+f x))}+\frac {\int \frac {-a^3 \left (2 A (c-3 d) d-B \left (3 c^2-3 c d+2 d^2\right )\right )+\left (a^3 \left (4 A c d-B \left (6 c^2-3 c d-5 d^2\right )\right )-a^3 \left (2 A (c-3 d) d-B \left (3 c^2-3 c d+2 d^2\right )\right )\right ) \sin (e+f x)+a^3 \left (4 A c d-B \left (6 c^2-3 c d-5 d^2\right )\right ) \sin ^2(e+f x)}{c+d \sin (e+f x)} \, dx}{2 d^2 (c+d)} \\ & = -\frac {a^3 \left (4 A c d-B \left (6 c^2-3 c d-5 d^2\right )\right ) \cos (e+f x)}{2 d^3 (c+d) f}+\frac {(2 A d-B (3 c+d)) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{2 d^2 (c+d) f}+\frac {a (B c-A d) \cos (e+f x) (a+a \sin (e+f x))^2}{d (c+d) f (c+d \sin (e+f x))}+\frac {\int \frac {-a^3 d \left (2 A (c-3 d) d-B \left (3 c^2-3 c d+2 d^2\right )\right )-a^3 (c+d) \left (2 A (2 c-3 d) d-B \left (6 c^2-12 c d+7 d^2\right )\right ) \sin (e+f x)}{c+d \sin (e+f x)} \, dx}{2 d^3 (c+d)} \\ & = -\frac {a^3 \left (2 A (2 c-3 d) d-B \left (6 c^2-12 c d+7 d^2\right )\right ) x}{2 d^4}-\frac {a^3 \left (4 A c d-B \left (6 c^2-3 c d-5 d^2\right )\right ) \cos (e+f x)}{2 d^3 (c+d) f}+\frac {(2 A d-B (3 c+d)) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{2 d^2 (c+d) f}+\frac {a (B c-A d) \cos (e+f x) (a+a \sin (e+f x))^2}{d (c+d) f (c+d \sin (e+f x))}+\frac {\left (a^3 (c-d)^2 \left (A d (2 c+3 d)-B \left (3 c^2+3 c d-d^2\right )\right )\right ) \int \frac {1}{c+d \sin (e+f x)} \, dx}{d^4 (c+d)} \\ & = -\frac {a^3 \left (2 A (2 c-3 d) d-B \left (6 c^2-12 c d+7 d^2\right )\right ) x}{2 d^4}-\frac {a^3 \left (4 A c d-B \left (6 c^2-3 c d-5 d^2\right )\right ) \cos (e+f x)}{2 d^3 (c+d) f}+\frac {(2 A d-B (3 c+d)) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{2 d^2 (c+d) f}+\frac {a (B c-A d) \cos (e+f x) (a+a \sin (e+f x))^2}{d (c+d) f (c+d \sin (e+f x))}+\frac {\left (2 a^3 (c-d)^2 \left (A d (2 c+3 d)-B \left (3 c^2+3 c d-d^2\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 )}{d^4 (c+d) f} \\ & = -\frac {a^3 \left (2 A (2 c-3 d) d-B \left (6 c^2-12 c d+7 d^2\right )\right ) x}{2 d^4}-\frac {a^3 \left (4 A c d-B \left (6 c^2-3 c d-5 d^2\right )\right ) \cos (e+f x)}{2 d^3 (c+d) f}+\frac {(2 A d-B (3 c+d)) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{2 d^2 (c+d) f}+\frac {a (B c-A d) \cos (e+f x) (a+a \sin (e+f x))^2}{d (c+d) f (c+d \sin (e+f x))}-\frac {\left (4 a^3 (c-d)^2 \left (A d (2 c+3 d)-B \left (3 c^2+3 c d-d^2\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 )}{d^4 (c+d) f} \\ & = -\frac {a^3 \left (2 A (2 c-3 d) d-B \left (6 c^2-12 c d+7 d^2\right )\right ) x}{2 d^4}+\frac {2 a^3 (c-d)^2 \left (A d (2 c+3 d)-B \left (3 c^2+3 c d-d^2\right )\right ) \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{d^4 (c+d) \sqrt {c^2-d^2} f}-\frac {a^3 \left (4 A c d-B \left (6 c^2-3 c d-5 d^2\right )\right ) \cos (e+f x)}{2 d^3 (c+d) f}+\frac {(2 A d-B (3 c+d)) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{2 d^2 (c+d) f}+\frac {a (B c-A d) \cos (e+f x) (a+a \sin (e+f x))^2}{d (c+d) f (c+d \sin (e+f x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.95 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.86 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c+d \sin (e+f x))^2} \, dx=\frac {a^3 (1+\sin (e+f x))^3 \left (2 \left (2 A d (-2 c+3 d)+B \left (6 c^2-12 c d+7 d^2\right )\right ) (e+f x)-\frac {8 (c-d)^2 \left (-A d (2 c+3 d)+B \left (3 c^2+3 c d-d^2\right )\right ) \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{(c+d) \sqrt {c^2-d^2}}-4 d (-2 B c+A d+3 B d) \cos (e+f x)+\frac {4 (c-d)^2 d (B c-A d) \cos (e+f x)}{(c+d) (c+d \sin (e+f x))}-B d^2 \sin (2 (e+f x))\right )}{4 d^4 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.67 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.43

method	result	size
derivativedivides	\(\frac {2 a^{3} \left (-\frac {\frac {-\frac {B \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) d^{2}}{2}+\left (A \,d^{2}-2 c d B +3 d^{2} B \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {B \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) d^{2}}{2}+A \,d^{2}-2 c d B +3 d^{2} B}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}+\frac {\left (4 A c d -6 A \,d^{2}-6 B \,c^{2}+12 c d B -7 d^{2} B \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}}{d^{4}}+\frac {\frac {-\frac {d^{2} \left (c^{2} d A -2 d^{2} c A +A \,d^{3}-B \,c^{3}+2 c^{2} d B -d^{2} c B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (c +d \right ) c}-\frac {d \left (c^{2} d A -2 d^{2} c A +A \,d^{3}-B \,c^{3}+2 c^{2} d B -d^{2} c B \right )}{c +d}}{\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c +2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c}+\frac {\left (2 A \,c^{3} d -A \,c^{2} d^{2}-4 A c \,d^{3}+3 A \,d^{4}-3 B \,c^{4}+3 B \,c^{3} d +4 B \,c^{2} d^{2}-5 B c \,d^{3}+B \,d^{4}\right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{\left (c +d \right ) \sqrt {c^{2}-d^{2}}}}{d^{4}}\right )}{f}\)	\(406\)
default	\(\frac {2 a^{3} \left (-\frac {\frac {-\frac {B \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) d^{2}}{2}+\left (A \,d^{2}-2 c d B +3 d^{2} B \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {B \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) d^{2}}{2}+A \,d^{2}-2 c d B +3 d^{2} B}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}+\frac {\left (4 A c d -6 A \,d^{2}-6 B \,c^{2}+12 c d B -7 d^{2} B \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}}{d^{4}}+\frac {\frac {-\frac {d^{2} \left (c^{2} d A -2 d^{2} c A +A \,d^{3}-B \,c^{3}+2 c^{2} d B -d^{2} c B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (c +d \right ) c}-\frac {d \left (c^{2} d A -2 d^{2} c A +A \,d^{3}-B \,c^{3}+2 c^{2} d B -d^{2} c B \right )}{c +d}}{\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c +2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c}+\frac {\left (2 A \,c^{3} d -A \,c^{2} d^{2}-4 A c \,d^{3}+3 A \,d^{4}-3 B \,c^{4}+3 B \,c^{3} d +4 B \,c^{2} d^{2}-5 B c \,d^{3}+B \,d^{4}\right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{\left (c +d \right ) \sqrt {c^{2}-d^{2}}}}{d^{4}}\right )}{f}\)	\(406\)
risch	\(\text {Expression too large to display}\)	\(1083\)

[In]

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

[Out]

2/f*a^3*(-1/d^4*((-1/2*B*tan(1/2*f*x+1/2*e)^3*d^2+(A*d^2-2*B*c*d+3*B*d^2)*tan(1/2*f*x+1/2*e)^2+1/2*B*tan(1/2*f
*x+1/2*e)*d^2+A*d^2-2*c*d*B+3*d^2*B)/(1+tan(1/2*f*x+1/2*e)^2)^2+1/2*(4*A*c*d-6*A*d^2-6*B*c^2+12*B*c*d-7*B*d^2)
*arctan(tan(1/2*f*x+1/2*e)))+1/d^4*((-d^2*(A*c^2*d-2*A*c*d^2+A*d^3-B*c^3+2*B*c^2*d-B*c*d^2)/(c+d)/c*tan(1/2*f*
x+1/2*e)-d*(A*c^2*d-2*A*c*d^2+A*d^3-B*c^3+2*B*c^2*d-B*c*d^2)/(c+d))/(tan(1/2*f*x+1/2*e)^2*c+2*d*tan(1/2*f*x+1/
2*e)+c)+(2*A*c^3*d-A*c^2*d^2-4*A*c*d^3+3*A*d^4-3*B*c^4+3*B*c^3*d+4*B*c^2*d^2-5*B*c*d^3+B*d^4)/(c+d)/(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 [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 1027, normalized size of antiderivative = 3.63 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c+d \sin (e+f x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c+d*sin(f*x+e))^2,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. 571 vs. \(2 (271) = 542\).

Time = 0.31 (sec) , antiderivative size = 571, normalized size of antiderivative = 2.02 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c+d \sin (e+f x))^2} \, dx=-\frac {\frac {4 \, {\left (3 \, B a^{3} c^{4} - 2 \, A a^{3} c^{3} d - 3 \, B a^{3} c^{3} d + A a^{3} c^{2} d^{2} - 4 \, B a^{3} c^{2} d^{2} + 4 \, A a^{3} c d^{3} + 5 \, B a^{3} c d^{3} - 3 \, A a^{3} d^{4} - B a^{3} d^{4}\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 (c d^{4} + d^{5}\right )} \sqrt {c^{2} - d^{2}}} - \frac {4 \, {\left (B a^{3} c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - A a^{3} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 2 \, B a^{3} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2 \, A a^{3} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + B a^{3} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - A a^{3} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + B a^{3} c^{4} - A a^{3} c^{3} d - 2 \, B a^{3} c^{3} d + 2 \, A a^{3} c^{2} d^{2} + B a^{3} c^{2} d^{2} - A a^{3} c d^{3}\right )}}{{\left (c^{2} d^{3} + c d^{4}\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 )}} - \frac {{\left (6 \, B a^{3} c^{2} - 4 \, A a^{3} c d - 12 \, B a^{3} c d + 6 \, A a^{3} d^{2} + 7 \, B a^{3} d^{2}\right )} {\left (f x + e\right )}}{d^{4}} - \frac {2 \, {\left (B a^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 4 \, B a^{3} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 2 \, A a^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 6 \, B a^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - B a^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 4 \, B a^{3} c - 2 \, A a^{3} d - 6 \, B a^{3} d\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}^{2} d^{3}}}{2 \, f} \]

[In]

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

[Out]

-1/2*(4*(3*B*a^3*c^4 - 2*A*a^3*c^3*d - 3*B*a^3*c^3*d + A*a^3*c^2*d^2 - 4*B*a^3*c^2*d^2 + 4*A*a^3*c*d^3 + 5*B*a
^3*c*d^3 - 3*A*a^3*d^4 - 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*d^4 + d^5)*sqrt(c^2 - d^2)) - 4*(B*a^3*c^3*d*tan(1/2*f*x + 1/2*e) - A*a^3*c^2*d^2*t
an(1/2*f*x + 1/2*e) - 2*B*a^3*c^2*d^2*tan(1/2*f*x + 1/2*e) + 2*A*a^3*c*d^3*tan(1/2*f*x + 1/2*e) + B*a^3*c*d^3*
tan(1/2*f*x + 1/2*e) - A*a^3*d^4*tan(1/2*f*x + 1/2*e) + B*a^3*c^4 - A*a^3*c^3*d - 2*B*a^3*c^3*d + 2*A*a^3*c^2*
d^2 + B*a^3*c^2*d^2 - A*a^3*c*d^3)/((c^2*d^3 + c*d^4)*(c*tan(1/2*f*x + 1/2*e)^2 + 2*d*tan(1/2*f*x + 1/2*e) + c
)) - (6*B*a^3*c^2 - 4*A*a^3*c*d - 12*B*a^3*c*d + 6*A*a^3*d^2 + 7*B*a^3*d^2)*(f*x + e)/d^4 - 2*(B*a^3*d*tan(1/2
*f*x + 1/2*e)^3 + 4*B*a^3*c*tan(1/2*f*x + 1/2*e)^2 - 2*A*a^3*d*tan(1/2*f*x + 1/2*e)^2 - 6*B*a^3*d*tan(1/2*f*x
+ 1/2*e)^2 - B*a^3*d*tan(1/2*f*x + 1/2*e) + 4*B*a^3*c - 2*A*a^3*d - 6*B*a^3*d)/((tan(1/2*f*x + 1/2*e)^2 + 1)^2
*d^3))/f

Mupad [B] (verification not implemented)

Time = 24.89 (sec) , antiderivative size = 11993, normalized size of antiderivative = 42.38 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c+d \sin (e+f x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

- ((2*(A*a^3*d^3 - 3*B*a^3*c^3 - A*a^3*c*d^2 + 2*A*a^3*c^2*d + 2*B*a^3*c*d^2 + 3*B*a^3*c^2*d))/(d^3*(c + d)) +
(2*tan(e/2 + (f*x)/2)^4*(A*a^3*d^3 - 3*B*a^3*c^3 - B*a^3*d^3 - A*a^3*c*d^2 + 2*A*a^3*c^2*d + B*a^3*c*d^2 + 3*
B*a^3*c^2*d))/(d^3*(c + d)) + (2*tan(e/2 + (f*x)/2)^2*(2*A*a^3*d^3 - 6*B*a^3*c^3 + B*a^3*d^3 - 2*A*a^3*c*d^2 +
4*A*a^3*c^2*d + 5*B*a^3*c*d^2 + 6*B*a^3*c^2*d))/(d^3*(c + d)) + (4*tan(e/2 + (f*x)/2)^3*(A*a^3*d^3 - 3*B*a^3*
c^3 - A*a^3*c*d^2 + 2*A*a^3*c^2*d + 2*B*a^3*c*d^2 + 3*B*a^3*c^2*d))/(c*d^2*(c + d)) + (tan(e/2 + (f*x)/2)^5*(2
*A*a^3*d^3 - 3*B*a^3*c^3 - 4*A*a^3*c*d^2 + 2*A*a^3*c^2*d - 2*B*a^3*c*d^2 + 3*B*a^3*c^2*d))/(c*d^2*(c + d)) + (
tan(e/2 + (f*x)/2)*(2*A*a^3*d^3 - 9*B*a^3*c^3 + 6*A*a^3*c^2*d + 10*B*a^3*c*d^2 + 9*B*a^3*c^2*d))/(c*d^2*(c + d
)))/(f*(c + 2*d*tan(e/2 + (f*x)/2) + 3*c*tan(e/2 + (f*x)/2)^2 + 3*c*tan(e/2 + (f*x)/2)^4 + c*tan(e/2 + (f*x)/2
)^6 + 4*d*tan(e/2 + (f*x)/2)^3 + 2*d*tan(e/2 + (f*x)/2)^5)) - (atan(((((8*(36*A^2*a^6*c^2*d^9 + 24*A^2*a^6*c^3
*d^8 - 44*A^2*a^6*c^4*d^7 - 16*A^2*a^6*c^5*d^6 + 16*A^2*a^6*c^6*d^5 + 49*B^2*a^6*c^2*d^9 - 70*B^2*a^6*c^3*d^8
- 59*B^2*a^6*c^4*d^7 + 144*B^2*a^6*c^5*d^6 - 24*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 + 84
*A*B*a^6*c^2*d^9 - 32*A*B*a^6*c^3*d^8 - 148*A*B*a^6*c^4*d^7 + 88*A*B*a^6*c^5*d^6 + 72*A*B*a^6*c^6*d^5 - 48*A*B
*a^6*c^7*d^4))/(2*c*d^9 + d^10 + c^2*d^8) + (8*tan(e/2 + (f*x)/2)*(144*A^2*a^6*c^2*d^10 - 164*A^2*a^6*c^3*d^9
- 136*A^2*a^6*c^4*d^8 + 136*A^2*a^6*c^5*d^7 + 32*A^2*a^6*c^6*d^6 - 32*A^2*a^6*c^7*d^5 - 100*B^2*a^6*c^2*d^10 -
299*B^2*a^6*c^3*d^9 + 494*B^2*a^6*c^4*d^8 + 91*B^2*a^6*c^5*d^7 - 504*B^2*a^6*c^6*d^6 + 156*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 + 36*A^2*a^6*c*d^11 + 94*B^2*a^6*c*d^11 + 88*A*B*a^6*c^2*d^10 - 628*A
*B*a^6*c^3*d^9 + 208*A*B*a^6*c^4*d^8 + 572*A*B*a^6*c^5*d^7 - 320*A*B*a^6*c^6*d^6 - 144*A*B*a^6*c^7*d^5 + 96*A*
B*a^6*c^8*d^4 + 144*A*B*a^6*c*d^11))/(2*c*d^10 + d^11 + c^2*d^9) + (((((8*(4*c^2*d^13 + 8*c^3*d^12 + 4*c^4*d^1
1))/(2*c*d^9 + d^10 + c^2*d^8) + (8*tan(e/2 + (f*x)/2)*(12*c*d^15 + 24*c^2*d^14 + 4*c^3*d^13 - 16*c^4*d^12 - 8
*c^5*d^11))/(2*c*d^10 + d^11 + c^2*d^9))*(B*a^3*c^2*3i + a^3*d^2*(3*A + (7*B)/2)*1i - (a^3*d*(4*A*c + 12*B*c)*
1i)/2))/d^4 - (8*(12*A*a^3*c*d^12 + 14*B*a^3*c*d^12 + 4*A*a^3*c^2*d^11 - 12*A*a^3*c^3*d^10 - 4*A*a^3*c^4*d^9 -
20*B*a^3*c^3*d^10 + 6*B*a^3*c^5*d^8))/(2*c*d^9 + d^10 + c^2*d^8) + (8*tan(e/2 + (f*x)/2)*(24*A*a^3*c*d^13 + 8
*B*a^3*c*d^13 - 8*A*a^3*c^2*d^12 - 40*A*a^3*c^3*d^11 + 8*A*a^3*c^4*d^10 + 16*A*a^3*c^5*d^9 - 32*B*a^3*c^2*d^12
- 8*B*a^3*c^3*d^11 + 56*B*a^3*c^4*d^10 - 24*B*a^3*c^6*d^8))/(2*c*d^10 + d^11 + c^2*d^9))*(B*a^3*c^2*3i + a^3*
d^2*(3*A + (7*B)/2)*1i - (a^3*d*(4*A*c + 12*B*c)*1i)/2))/d^4)*(B*a^3*c^2*3i + a^3*d^2*(3*A + (7*B)/2)*1i - (a^
3*d*(4*A*c + 12*B*c)*1i)/2)*1i)/d^4 + (((8*(36*A^2*a^6*c^2*d^9 + 24*A^2*a^6*c^3*d^8 - 44*A^2*a^6*c^4*d^7 - 16*
A^2*a^6*c^5*d^6 + 16*A^2*a^6*c^6*d^5 + 49*B^2*a^6*c^2*d^9 - 70*B^2*a^6*c^3*d^8 - 59*B^2*a^6*c^4*d^7 + 144*B^2*
a^6*c^5*d^6 - 24*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 + 84*A*B*a^6*c^2*d^9 - 32*A*B*a^6*c
^3*d^8 - 148*A*B*a^6*c^4*d^7 + 88*A*B*a^6*c^5*d^6 + 72*A*B*a^6*c^6*d^5 - 48*A*B*a^6*c^7*d^4))/(2*c*d^9 + d^10
+ c^2*d^8) + (8*tan(e/2 + (f*x)/2)*(144*A^2*a^6*c^2*d^10 - 164*A^2*a^6*c^3*d^9 - 136*A^2*a^6*c^4*d^8 + 136*A^2
*a^6*c^5*d^7 + 32*A^2*a^6*c^6*d^6 - 32*A^2*a^6*c^7*d^5 - 100*B^2*a^6*c^2*d^10 - 299*B^2*a^6*c^3*d^9 + 494*B^2*
a^6*c^4*d^8 + 91*B^2*a^6*c^5*d^7 - 504*B^2*a^6*c^6*d^6 + 156*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 + 36*A^2*a^6*c*d^11 + 94*B^2*a^6*c*d^11 + 88*A*B*a^6*c^2*d^10 - 628*A*B*a^6*c^3*d^9 + 208*A*B*a^6*c^
4*d^8 + 572*A*B*a^6*c^5*d^7 - 320*A*B*a^6*c^6*d^6 - 144*A*B*a^6*c^7*d^5 + 96*A*B*a^6*c^8*d^4 + 144*A*B*a^6*c*d
^11))/(2*c*d^10 + d^11 + c^2*d^9) + (((8*(12*A*a^3*c*d^12 + 14*B*a^3*c*d^12 + 4*A*a^3*c^2*d^11 - 12*A*a^3*c^3*
d^10 - 4*A*a^3*c^4*d^9 - 20*B*a^3*c^3*d^10 + 6*B*a^3*c^5*d^8))/(2*c*d^9 + d^10 + c^2*d^8) + (((8*(4*c^2*d^13 +
8*c^3*d^12 + 4*c^4*d^11))/(2*c*d^9 + d^10 + c^2*d^8) + (8*tan(e/2 + (f*x)/2)*(12*c*d^15 + 24*c^2*d^14 + 4*c^3
*d^13 - 16*c^4*d^12 - 8*c^5*d^11))/(2*c*d^10 + d^11 + c^2*d^9))*(B*a^3*c^2*3i + a^3*d^2*(3*A + (7*B)/2)*1i - (
a^3*d*(4*A*c + 12*B*c)*1i)/2))/d^4 - (8*tan(e/2 + (f*x)/2)*(24*A*a^3*c*d^13 + 8*B*a^3*c*d^13 - 8*A*a^3*c^2*d^1
2 - 40*A*a^3*c^3*d^11 + 8*A*a^3*c^4*d^10 + 16*A*a^3*c^5*d^9 - 32*B*a^3*c^2*d^12 - 8*B*a^3*c^3*d^11 + 56*B*a^3*
c^4*d^10 - 24*B*a^3*c^6*d^8))/(2*c*d^10 + d^11 + c^2*d^9))*(B*a^3*c^2*3i + a^3*d^2*(3*A + (7*B)/2)*1i - (a^3*d
*(4*A*c + 12*B*c)*1i)/2))/d^4)*(B*a^3*c^2*3i + a^3*d^2*(3*A + (7*B)/2)*1i - (a^3*d*(4*A*c + 12*B*c)*1i)/2)*1i)
/d^4)/((16*(132*A^3*a^9*c^3*d^6 - 252*A^3*a^9*c^2*d^7 - 54*B^3*a^9*c^9 + 76*A^3*a^9*c^4*d^5 - 80*A^3*a^9*c^5*d
^4 + 16*A^3*a^9*c^6*d^3 - 115*B^3*a^9*c^2*d^7 + 350*B^3*a^9*c^3*d^6 - 537*B^3*a^9*c^4*d^5 + 387*B^3*a^9*c^5*d^
4 + 36*B^3*a^9*c^6*d^3 - 297*B^3*a^9*c^7*d^2 + 108*A^3*a^9*c*d^8 + 14*B^3*a^9*c*d^8 + 216*B^3*a^9*c^8*d + 96*A
*B^2*a^9*c*d^8 + 108*A*B^2*a^9*c^8*d + 198*A^2*B*a^9*c*d^8 - 573*A*B^2*a^9*c^2*d^7 + 1239*A*B^2*a^9*c^3*d^6 -
1125*A*B^2*a^9*c^4*d^5 + 93*A*B^2*a^9*c^5*d^4 + 630*A*B^2*a^9*c^6*d^3 - 468*A*B^2*a^9*c^7*d^2 - 768*A^2*B*a^9*
c^2*d^7 + 996*A^2*B*a^9*c^3*d^6 - 288*A^2*B*a^9*c^4*d^5 - 402*A^2*B*a^9*c^5*d^4 + 336*A^2*B*a^9*c^6*d^3 - 72*A
^2*B*a^9*c^7*d^2))/(2*c*d^9 + d^10 + c^2*d^8) + (16*tan(e/2 + (f*x)/2)*(520*A^3*a^9*c^4*d^6 - 360*A^3*a^9*c^2*
d^8 - 168*A^3*a^9*c^3*d^7 - 216*B^3*a^9*c^10 - 112*A^3*a^9*c^5*d^5 - 160*A^3*a^9*c^6*d^4 + 64*A^3*a^9*c^7*d^3
- 728*B^3*a^9*c^2*d^8 + 1702*B^3*a^9*c^3*d^7 - 1090*B^3*a^9*c^4*d^6 - 1584*B^3*a^9*c^5*d^5 + 2898*B^3*a^9*c^6*
d^4 - 1080*B^3*a^9*c^7*d^3 - 864*B^3*a^9*c^8*d^2 + 216*A^3*a^9*c*d^9 + 98*B^3*a^9*c*d^9 + 864*B^3*a^9*c^9*d +
462*A*B^2*a^9*c*d^9 + 432*A*B^2*a^9*c^9*d + 576*A^2*B*a^9*c*d^9 - 2178*A*B^2*a^9*c^2*d^8 + 2982*A*B^2*a^9*c^3*
d^7 + 594*A*B^2*a^9*c^4*d^6 - 4668*A*B^2*a^9*c^5*d^5 + 3096*A*B^2*a^9*c^6*d^4 + 792*A*B^2*a^9*c^7*d^3 - 1512*A
*B^2*a^9*c^8*d^2 - 1752*A^2*B*a^9*c^2*d^8 + 912*A^2*B*a^9*c^3*d^7 + 2016*A^2*B*a^9*c^4*d^6 - 2352*A^2*B*a^9*c^
5*d^5 + 24*A^2*B*a^9*c^6*d^4 + 864*A^2*B*a^9*c^7*d^3 - 288*A^2*B*a^9*c^8*d^2))/(2*c*d^10 + d^11 + c^2*d^9) + (
((8*(36*A^2*a^6*c^2*d^9 + 24*A^2*a^6*c^3*d^8 - 44*A^2*a^6*c^4*d^7 - 16*A^2*a^6*c^5*d^6 + 16*A^2*a^6*c^6*d^5 +
49*B^2*a^6*c^2*d^9 - 70*B^2*a^6*c^3*d^8 - 59*B^2*a^6*c^4*d^7 + 144*B^2*a^6*c^5*d^6 - 24*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 + 84*A*B*a^6*c^2*d^9 - 32*A*B*a^6*c^3*d^8 - 148*A*B*a^6*c^4*d^7 + 88*A*B*a
^6*c^5*d^6 + 72*A*B*a^6*c^6*d^5 - 48*A*B*a^6*c^7*d^4))/(2*c*d^9 + d^10 + c^2*d^8) + (8*tan(e/2 + (f*x)/2)*(144
*A^2*a^6*c^2*d^10 - 164*A^2*a^6*c^3*d^9 - 136*A^2*a^6*c^4*d^8 + 136*A^2*a^6*c^5*d^7 + 32*A^2*a^6*c^6*d^6 - 32*
A^2*a^6*c^7*d^5 - 100*B^2*a^6*c^2*d^10 - 299*B^2*a^6*c^3*d^9 + 494*B^2*a^6*c^4*d^8 + 91*B^2*a^6*c^5*d^7 - 504*
B^2*a^6*c^6*d^6 + 156*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 + 36*A^2*a^6*c*d^11 + 94*B^2*
a^6*c*d^11 + 88*A*B*a^6*c^2*d^10 - 628*A*B*a^6*c^3*d^9 + 208*A*B*a^6*c^4*d^8 + 572*A*B*a^6*c^5*d^7 - 320*A*B*a
^6*c^6*d^6 - 144*A*B*a^6*c^7*d^5 + 96*A*B*a^6*c^8*d^4 + 144*A*B*a^6*c*d^11))/(2*c*d^10 + d^11 + c^2*d^9) + (((
((8*(4*c^2*d^13 + 8*c^3*d^12 + 4*c^4*d^11))/(2*c*d^9 + d^10 + c^2*d^8) + (8*tan(e/2 + (f*x)/2)*(12*c*d^15 + 24
*c^2*d^14 + 4*c^3*d^13 - 16*c^4*d^12 - 8*c^5*d^11))/(2*c*d^10 + d^11 + c^2*d^9))*(B*a^3*c^2*3i + a^3*d^2*(3*A
+ (7*B)/2)*1i - (a^3*d*(4*A*c + 12*B*c)*1i)/2))/d^4 - (8*(12*A*a^3*c*d^12 + 14*B*a^3*c*d^12 + 4*A*a^3*c^2*d^11
- 12*A*a^3*c^3*d^10 - 4*A*a^3*c^4*d^9 - 20*B*a^3*c^3*d^10 + 6*B*a^3*c^5*d^8))/(2*c*d^9 + d^10 + c^2*d^8) + (8
*tan(e/2 + (f*x)/2)*(24*A*a^3*c*d^13 + 8*B*a^3*c*d^13 - 8*A*a^3*c^2*d^12 - 40*A*a^3*c^3*d^11 + 8*A*a^3*c^4*d^1
0 + 16*A*a^3*c^5*d^9 - 32*B*a^3*c^2*d^12 - 8*B*a^3*c^3*d^11 + 56*B*a^3*c^4*d^10 - 24*B*a^3*c^6*d^8))/(2*c*d^10
+ d^11 + c^2*d^9))*(B*a^3*c^2*3i + a^3*d^2*(3*A + (7*B)/2)*1i - (a^3*d*(4*A*c + 12*B*c)*1i)/2))/d^4)*(B*a^3*c
^2*3i + a^3*d^2*(3*A + (7*B)/2)*1i - (a^3*d*(4*A*c + 12*B*c)*1i)/2))/d^4 - (((8*(36*A^2*a^6*c^2*d^9 + 24*A^2*a
^6*c^3*d^8 - 44*A^2*a^6*c^4*d^7 - 16*A^2*a^6*c^5*d^6 + 16*A^2*a^6*c^6*d^5 + 49*B^2*a^6*c^2*d^9 - 70*B^2*a^6*c^
3*d^8 - 59*B^2*a^6*c^4*d^7 + 144*B^2*a^6*c^5*d^6 - 24*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 + 84*A*B*a^6*c^2*d^9 - 32*A*B*a^6*c^3*d^8 - 148*A*B*a^6*c^4*d^7 + 88*A*B*a^6*c^5*d^6 + 72*A*B*a^6*c^6*d^5 -
48*A*B*a^6*c^7*d^4))/(2*c*d^9 + d^10 + c^2*d^8) + (8*tan(e/2 + (f*x)/2)*(144*A^2*a^6*c^2*d^10 - 164*A^2*a^6*c^
3*d^9 - 136*A^2*a^6*c^4*d^8 + 136*A^2*a^6*c^5*d^7 + 32*A^2*a^6*c^6*d^6 - 32*A^2*a^6*c^7*d^5 - 100*B^2*a^6*c^2*
d^10 - 299*B^2*a^6*c^3*d^9 + 494*B^2*a^6*c^4*d^8 + 91*B^2*a^6*c^5*d^7 - 504*B^2*a^6*c^6*d^6 + 156*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 + 36*A^2*a^6*c*d^11 + 94*B^2*a^6*c*d^11 + 88*A*B*a^6*c^2*d^10 -
628*A*B*a^6*c^3*d^9 + 208*A*B*a^6*c^4*d^8 + 572*A*B*a^6*c^5*d^7 - 320*A*B*a^6*c^6*d^6 - 144*A*B*a^6*c^7*d^5 +
96*A*B*a^6*c^8*d^4 + 144*A*B*a^6*c*d^11))/(2*c*d^10 + d^11 + c^2*d^9) + (((8*(12*A*a^3*c*d^12 + 14*B*a^3*c*d^
12 + 4*A*a^3*c^2*d^11 - 12*A*a^3*c^3*d^10 - 4*A*a^3*c^4*d^9 - 20*B*a^3*c^3*d^10 + 6*B*a^3*c^5*d^8))/(2*c*d^9 +
d^10 + c^2*d^8) + (((8*(4*c^2*d^13 + 8*c^3*d^12 + 4*c^4*d^11))/(2*c*d^9 + d^10 + c^2*d^8) + (8*tan(e/2 + (f*x
)/2)*(12*c*d^15 + 24*c^2*d^14 + 4*c^3*d^13 - 16*c^4*d^12 - 8*c^5*d^11))/(2*c*d^10 + d^11 + c^2*d^9))*(B*a^3*c^
2*3i + a^3*d^2*(3*A + (7*B)/2)*1i - (a^3*d*(4*A*c + 12*B*c)*1i)/2))/d^4 - (8*tan(e/2 + (f*x)/2)*(24*A*a^3*c*d^
13 + 8*B*a^3*c*d^13 - 8*A*a^3*c^2*d^12 - 40*A*a^3*c^3*d^11 + 8*A*a^3*c^4*d^10 + 16*A*a^3*c^5*d^9 - 32*B*a^3*c^
2*d^12 - 8*B*a^3*c^3*d^11 + 56*B*a^3*c^4*d^10 - 24*B*a^3*c^6*d^8))/(2*c*d^10 + d^11 + c^2*d^9))*(B*a^3*c^2*3i
+ a^3*d^2*(3*A + (7*B)/2)*1i - (a^3*d*(4*A*c + 12*B*c)*1i)/2))/d^4)*(B*a^3*c^2*3i + a^3*d^2*(3*A + (7*B)/2)*1i
- (a^3*d*(4*A*c + 12*B*c)*1i)/2))/d^4))*(B*a^3*c^2*3i + a^3*d^2*(3*A + (7*B)/2)*1i - (a^3*d*(4*A*c + 12*B*c)*
1i)/2)*2i)/(d^4*f) - (a^3*atan(((a^3*(-(c + d)^3*(c - d)^3)^(1/2)*((8*(36*A^2*a^6*c^2*d^9 + 24*A^2*a^6*c^3*d^8
- 44*A^2*a^6*c^4*d^7 - 16*A^2*a^6*c^5*d^6 + 16*A^2*a^6*c^6*d^5 + 49*B^2*a^6*c^2*d^9 - 70*B^2*a^6*c^3*d^8 - 59
*B^2*a^6*c^4*d^7 + 144*B^2*a^6*c^5*d^6 - 24*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 + 84*A*B
*a^6*c^2*d^9 - 32*A*B*a^6*c^3*d^8 - 148*A*B*a^6*c^4*d^7 + 88*A*B*a^6*c^5*d^6 + 72*A*B*a^6*c^6*d^5 - 48*A*B*a^6
*c^7*d^4))/(2*c*d^9 + d^10 + c^2*d^8) + (8*tan(e/2 + (f*x)/2)*(144*A^2*a^6*c^2*d^10 - 164*A^2*a^6*c^3*d^9 - 13
6*A^2*a^6*c^4*d^8 + 136*A^2*a^6*c^5*d^7 + 32*A^2*a^6*c^6*d^6 - 32*A^2*a^6*c^7*d^5 - 100*B^2*a^6*c^2*d^10 - 299
*B^2*a^6*c^3*d^9 + 494*B^2*a^6*c^4*d^8 + 91*B^2*a^6*c^5*d^7 - 504*B^2*a^6*c^6*d^6 + 156*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 + 36*A^2*a^6*c*d^11 + 94*B^2*a^6*c*d^11 + 88*A*B*a^6*c^2*d^10 - 628*A*B*a
^6*c^3*d^9 + 208*A*B*a^6*c^4*d^8 + 572*A*B*a^6*c^5*d^7 - 320*A*B*a^6*c^6*d^6 - 144*A*B*a^6*c^7*d^5 + 96*A*B*a^
6*c^8*d^4 + 144*A*B*a^6*c*d^11))/(2*c*d^10 + d^11 + c^2*d^9) + (a^3*(-(c + d)^3*(c - d)^3)^(1/2)*((8*tan(e/2 +
(f*x)/2)*(24*A*a^3*c*d^13 + 8*B*a^3*c*d^13 - 8*A*a^3*c^2*d^12 - 40*A*a^3*c^3*d^11 + 8*A*a^3*c^4*d^10 + 16*A*a
^3*c^5*d^9 - 32*B*a^3*c^2*d^12 - 8*B*a^3*c^3*d^11 + 56*B*a^3*c^4*d^10 - 24*B*a^3*c^6*d^8))/(2*c*d^10 + d^11 +
c^2*d^9) - (8*(12*A*a^3*c*d^12 + 14*B*a^3*c*d^12 + 4*A*a^3*c^2*d^11 - 12*A*a^3*c^3*d^10 - 4*A*a^3*c^4*d^9 - 20
*B*a^3*c^3*d^10 + 6*B*a^3*c^5*d^8))/(2*c*d^9 + d^10 + c^2*d^8) + (a^3*((8*(4*c^2*d^13 + 8*c^3*d^12 + 4*c^4*d^1
1))/(2*c*d^9 + d^10 + c^2*d^8) + (8*tan(e/2 + (f*x)/2)*(12*c*d^15 + 24*c^2*d^14 + 4*c^3*d^13 - 16*c^4*d^12 - 8
*c^5*d^11))/(2*c*d^10 + d^11 + c^2*d^9))*(-(c + d)^3*(c - d)^3)^(1/2)*(3*A*d^2 - 3*B*c^2 + B*d^2 + 2*A*c*d - 3
*B*c*d))/(3*c*d^6 + d^7 + 3*c^2*d^5 + c^3*d^4))*(3*A*d^2 - 3*B*c^2 + B*d^2 + 2*A*c*d - 3*B*c*d))/(3*c*d^6 + d^
7 + 3*c^2*d^5 + c^3*d^4))*(3*A*d^2 - 3*B*c^2 + B*d^2 + 2*A*c*d - 3*B*c*d)*1i)/(3*c*d^6 + d^7 + 3*c^2*d^5 + c^3
*d^4) + (a^3*(-(c + d)^3*(c - d)^3)^(1/2)*((8*(36*A^2*a^6*c^2*d^9 + 24*A^2*a^6*c^3*d^8 - 44*A^2*a^6*c^4*d^7 -
16*A^2*a^6*c^5*d^6 + 16*A^2*a^6*c^6*d^5 + 49*B^2*a^6*c^2*d^9 - 70*B^2*a^6*c^3*d^8 - 59*B^2*a^6*c^4*d^7 + 144*B
^2*a^6*c^5*d^6 - 24*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 + 84*A*B*a^6*c^2*d^9 - 32*A*B*a^
6*c^3*d^8 - 148*A*B*a^6*c^4*d^7 + 88*A*B*a^6*c^5*d^6 + 72*A*B*a^6*c^6*d^5 - 48*A*B*a^6*c^7*d^4))/(2*c*d^9 + d^
10 + c^2*d^8) + (8*tan(e/2 + (f*x)/2)*(144*A^2*a^6*c^2*d^10 - 164*A^2*a^6*c^3*d^9 - 136*A^2*a^6*c^4*d^8 + 136*
A^2*a^6*c^5*d^7 + 32*A^2*a^6*c^6*d^6 - 32*A^2*a^6*c^7*d^5 - 100*B^2*a^6*c^2*d^10 - 299*B^2*a^6*c^3*d^9 + 494*B
^2*a^6*c^4*d^8 + 91*B^2*a^6*c^5*d^7 - 504*B^2*a^6*c^6*d^6 + 156*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 + 36*A^2*a^6*c*d^11 + 94*B^2*a^6*c*d^11 + 88*A*B*a^6*c^2*d^10 - 628*A*B*a^6*c^3*d^9 + 208*A*B*a^6
*c^4*d^8 + 572*A*B*a^6*c^5*d^7 - 320*A*B*a^6*c^6*d^6 - 144*A*B*a^6*c^7*d^5 + 96*A*B*a^6*c^8*d^4 + 144*A*B*a^6*
c*d^11))/(2*c*d^10 + d^11 + c^2*d^9) + (a^3*(-(c + d)^3*(c - d)^3)^(1/2)*((8*(12*A*a^3*c*d^12 + 14*B*a^3*c*d^1
2 + 4*A*a^3*c^2*d^11 - 12*A*a^3*c^3*d^10 - 4*A*a^3*c^4*d^9 - 20*B*a^3*c^3*d^10 + 6*B*a^3*c^5*d^8))/(2*c*d^9 +
d^10 + c^2*d^8) - (8*tan(e/2 + (f*x)/2)*(24*A*a^3*c*d^13 + 8*B*a^3*c*d^13 - 8*A*a^3*c^2*d^12 - 40*A*a^3*c^3*d^
11 + 8*A*a^3*c^4*d^10 + 16*A*a^3*c^5*d^9 - 32*B*a^3*c^2*d^12 - 8*B*a^3*c^3*d^11 + 56*B*a^3*c^4*d^10 - 24*B*a^3
*c^6*d^8))/(2*c*d^10 + d^11 + c^2*d^9) + (a^3*((8*(4*c^2*d^13 + 8*c^3*d^12 + 4*c^4*d^11))/(2*c*d^9 + d^10 + c^
2*d^8) + (8*tan(e/2 + (f*x)/2)*(12*c*d^15 + 24*c^2*d^14 + 4*c^3*d^13 - 16*c^4*d^12 - 8*c^5*d^11))/(2*c*d^10 +
d^11 + c^2*d^9))*(-(c + d)^3*(c - d)^3)^(1/2)*(3*A*d^2 - 3*B*c^2 + B*d^2 + 2*A*c*d - 3*B*c*d))/(3*c*d^6 + d^7
+ 3*c^2*d^5 + c^3*d^4))*(3*A*d^2 - 3*B*c^2 + B*d^2 + 2*A*c*d - 3*B*c*d))/(3*c*d^6 + d^7 + 3*c^2*d^5 + c^3*d^4)
)*(3*A*d^2 - 3*B*c^2 + B*d^2 + 2*A*c*d - 3*B*c*d)*1i)/(3*c*d^6 + d^7 + 3*c^2*d^5 + c^3*d^4))/((16*(132*A^3*a^9
*c^3*d^6 - 252*A^3*a^9*c^2*d^7 - 54*B^3*a^9*c^9 + 76*A^3*a^9*c^4*d^5 - 80*A^3*a^9*c^5*d^4 + 16*A^3*a^9*c^6*d^3
- 115*B^3*a^9*c^2*d^7 + 350*B^3*a^9*c^3*d^6 - 537*B^3*a^9*c^4*d^5 + 387*B^3*a^9*c^5*d^4 + 36*B^3*a^9*c^6*d^3
- 297*B^3*a^9*c^7*d^2 + 108*A^3*a^9*c*d^8 + 14*B^3*a^9*c*d^8 + 216*B^3*a^9*c^8*d + 96*A*B^2*a^9*c*d^8 + 108*A*
B^2*a^9*c^8*d + 198*A^2*B*a^9*c*d^8 - 573*A*B^2*a^9*c^2*d^7 + 1239*A*B^2*a^9*c^3*d^6 - 1125*A*B^2*a^9*c^4*d^5
+ 93*A*B^2*a^9*c^5*d^4 + 630*A*B^2*a^9*c^6*d^3 - 468*A*B^2*a^9*c^7*d^2 - 768*A^2*B*a^9*c^2*d^7 + 996*A^2*B*a^9
*c^3*d^6 - 288*A^2*B*a^9*c^4*d^5 - 402*A^2*B*a^9*c^5*d^4 + 336*A^2*B*a^9*c^6*d^3 - 72*A^2*B*a^9*c^7*d^2))/(2*c
*d^9 + d^10 + c^2*d^8) + (16*tan(e/2 + (f*x)/2)*(520*A^3*a^9*c^4*d^6 - 360*A^3*a^9*c^2*d^8 - 168*A^3*a^9*c^3*d
^7 - 216*B^3*a^9*c^10 - 112*A^3*a^9*c^5*d^5 - 160*A^3*a^9*c^6*d^4 + 64*A^3*a^9*c^7*d^3 - 728*B^3*a^9*c^2*d^8 +
1702*B^3*a^9*c^3*d^7 - 1090*B^3*a^9*c^4*d^6 - 1584*B^3*a^9*c^5*d^5 + 2898*B^3*a^9*c^6*d^4 - 1080*B^3*a^9*c^7*
d^3 - 864*B^3*a^9*c^8*d^2 + 216*A^3*a^9*c*d^9 + 98*B^3*a^9*c*d^9 + 864*B^3*a^9*c^9*d + 462*A*B^2*a^9*c*d^9 + 4
32*A*B^2*a^9*c^9*d + 576*A^2*B*a^9*c*d^9 - 2178*A*B^2*a^9*c^2*d^8 + 2982*A*B^2*a^9*c^3*d^7 + 594*A*B^2*a^9*c^4
*d^6 - 4668*A*B^2*a^9*c^5*d^5 + 3096*A*B^2*a^9*c^6*d^4 + 792*A*B^2*a^9*c^7*d^3 - 1512*A*B^2*a^9*c^8*d^2 - 1752
*A^2*B*a^9*c^2*d^8 + 912*A^2*B*a^9*c^3*d^7 + 2016*A^2*B*a^9*c^4*d^6 - 2352*A^2*B*a^9*c^5*d^5 + 24*A^2*B*a^9*c^
6*d^4 + 864*A^2*B*a^9*c^7*d^3 - 288*A^2*B*a^9*c^8*d^2))/(2*c*d^10 + d^11 + c^2*d^9) + (a^3*(-(c + d)^3*(c - d)
^3)^(1/2)*((8*(36*A^2*a^6*c^2*d^9 + 24*A^2*a^6*c^3*d^8 - 44*A^2*a^6*c^4*d^7 - 16*A^2*a^6*c^5*d^6 + 16*A^2*a^6*
c^6*d^5 + 49*B^2*a^6*c^2*d^9 - 70*B^2*a^6*c^3*d^8 - 59*B^2*a^6*c^4*d^7 + 144*B^2*a^6*c^5*d^6 - 24*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 + 84*A*B*a^6*c^2*d^9 - 32*A*B*a^6*c^3*d^8 - 148*A*B*a^6*c^4*d^7
+ 88*A*B*a^6*c^5*d^6 + 72*A*B*a^6*c^6*d^5 - 48*A*B*a^6*c^7*d^4))/(2*c*d^9 + d^10 + c^2*d^8) + (8*tan(e/2 + (f*
x)/2)*(144*A^2*a^6*c^2*d^10 - 164*A^2*a^6*c^3*d^9 - 136*A^2*a^6*c^4*d^8 + 136*A^2*a^6*c^5*d^7 + 32*A^2*a^6*c^6
*d^6 - 32*A^2*a^6*c^7*d^5 - 100*B^2*a^6*c^2*d^10 - 299*B^2*a^6*c^3*d^9 + 494*B^2*a^6*c^4*d^8 + 91*B^2*a^6*c^5*
d^7 - 504*B^2*a^6*c^6*d^6 + 156*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 + 36*A^2*a^6*c*d^11
+ 94*B^2*a^6*c*d^11 + 88*A*B*a^6*c^2*d^10 - 628*A*B*a^6*c^3*d^9 + 208*A*B*a^6*c^4*d^8 + 572*A*B*a^6*c^5*d^7 -
320*A*B*a^6*c^6*d^6 - 144*A*B*a^6*c^7*d^5 + 96*A*B*a^6*c^8*d^4 + 144*A*B*a^6*c*d^11))/(2*c*d^10 + d^11 + c^2*
d^9) + (a^3*(-(c + d)^3*(c - d)^3)^(1/2)*((8*tan(e/2 + (f*x)/2)*(24*A*a^3*c*d^13 + 8*B*a^3*c*d^13 - 8*A*a^3*c^
2*d^12 - 40*A*a^3*c^3*d^11 + 8*A*a^3*c^4*d^10 + 16*A*a^3*c^5*d^9 - 32*B*a^3*c^2*d^12 - 8*B*a^3*c^3*d^11 + 56*B
*a^3*c^4*d^10 - 24*B*a^3*c^6*d^8))/(2*c*d^10 + d^11 + c^2*d^9) - (8*(12*A*a^3*c*d^12 + 14*B*a^3*c*d^12 + 4*A*a
^3*c^2*d^11 - 12*A*a^3*c^3*d^10 - 4*A*a^3*c^4*d^9 - 20*B*a^3*c^3*d^10 + 6*B*a^3*c^5*d^8))/(2*c*d^9 + d^10 + c^
2*d^8) + (a^3*((8*(4*c^2*d^13 + 8*c^3*d^12 + 4*c^4*d^11))/(2*c*d^9 + d^10 + c^2*d^8) + (8*tan(e/2 + (f*x)/2)*(
12*c*d^15 + 24*c^2*d^14 + 4*c^3*d^13 - 16*c^4*d^12 - 8*c^5*d^11))/(2*c*d^10 + d^11 + c^2*d^9))*(-(c + d)^3*(c
- d)^3)^(1/2)*(3*A*d^2 - 3*B*c^2 + B*d^2 + 2*A*c*d - 3*B*c*d))/(3*c*d^6 + d^7 + 3*c^2*d^5 + c^3*d^4))*(3*A*d^2
- 3*B*c^2 + B*d^2 + 2*A*c*d - 3*B*c*d))/(3*c*d^6 + d^7 + 3*c^2*d^5 + c^3*d^4))*(3*A*d^2 - 3*B*c^2 + B*d^2 + 2
*A*c*d - 3*B*c*d))/(3*c*d^6 + d^7 + 3*c^2*d^5 + c^3*d^4) - (a^3*(-(c + d)^3*(c - d)^3)^(1/2)*((8*(36*A^2*a^6*c
^2*d^9 + 24*A^2*a^6*c^3*d^8 - 44*A^2*a^6*c^4*d^7 - 16*A^2*a^6*c^5*d^6 + 16*A^2*a^6*c^6*d^5 + 49*B^2*a^6*c^2*d^
9 - 70*B^2*a^6*c^3*d^8 - 59*B^2*a^6*c^4*d^7 + 144*B^2*a^6*c^5*d^6 - 24*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 + 84*A*B*a^6*c^2*d^9 - 32*A*B*a^6*c^3*d^8 - 148*A*B*a^6*c^4*d^7 + 88*A*B*a^6*c^5*d^6 + 72*A
*B*a^6*c^6*d^5 - 48*A*B*a^6*c^7*d^4))/(2*c*d^9 + d^10 + c^2*d^8) + (8*tan(e/2 + (f*x)/2)*(144*A^2*a^6*c^2*d^10
- 164*A^2*a^6*c^3*d^9 - 136*A^2*a^6*c^4*d^8 + 136*A^2*a^6*c^5*d^7 + 32*A^2*a^6*c^6*d^6 - 32*A^2*a^6*c^7*d^5 -
100*B^2*a^6*c^2*d^10 - 299*B^2*a^6*c^3*d^9 + 494*B^2*a^6*c^4*d^8 + 91*B^2*a^6*c^5*d^7 - 504*B^2*a^6*c^6*d^6 +
156*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 + 36*A^2*a^6*c*d^11 + 94*B^2*a^6*c*d^11 + 88*A
*B*a^6*c^2*d^10 - 628*A*B*a^6*c^3*d^9 + 208*A*B*a^6*c^4*d^8 + 572*A*B*a^6*c^5*d^7 - 320*A*B*a^6*c^6*d^6 - 144*
A*B*a^6*c^7*d^5 + 96*A*B*a^6*c^8*d^4 + 144*A*B*a^6*c*d^11))/(2*c*d^10 + d^11 + c^2*d^9) + (a^3*(-(c + d)^3*(c
- d)^3)^(1/2)*((8*(12*A*a^3*c*d^12 + 14*B*a^3*c*d^12 + 4*A*a^3*c^2*d^11 - 12*A*a^3*c^3*d^10 - 4*A*a^3*c^4*d^9
- 20*B*a^3*c^3*d^10 + 6*B*a^3*c^5*d^8))/(2*c*d^9 + d^10 + c^2*d^8) - (8*tan(e/2 + (f*x)/2)*(24*A*a^3*c*d^13 +
8*B*a^3*c*d^13 - 8*A*a^3*c^2*d^12 - 40*A*a^3*c^3*d^11 + 8*A*a^3*c^4*d^10 + 16*A*a^3*c^5*d^9 - 32*B*a^3*c^2*d^1
2 - 8*B*a^3*c^3*d^11 + 56*B*a^3*c^4*d^10 - 24*B*a^3*c^6*d^8))/(2*c*d^10 + d^11 + c^2*d^9) + (a^3*((8*(4*c^2*d^
13 + 8*c^3*d^12 + 4*c^4*d^11))/(2*c*d^9 + d^10 + c^2*d^8) + (8*tan(e/2 + (f*x)/2)*(12*c*d^15 + 24*c^2*d^14 + 4
*c^3*d^13 - 16*c^4*d^12 - 8*c^5*d^11))/(2*c*d^10 + d^11 + c^2*d^9))*(-(c + d)^3*(c - d)^3)^(1/2)*(3*A*d^2 - 3*
B*c^2 + B*d^2 + 2*A*c*d - 3*B*c*d))/(3*c*d^6 + d^7 + 3*c^2*d^5 + c^3*d^4))*(3*A*d^2 - 3*B*c^2 + B*d^2 + 2*A*c*
d - 3*B*c*d))/(3*c*d^6 + d^7 + 3*c^2*d^5 + c^3*d^4))*(3*A*d^2 - 3*B*c^2 + B*d^2 + 2*A*c*d - 3*B*c*d))/(3*c*d^6
+ d^7 + 3*c^2*d^5 + c^3*d^4)))*(-(c + d)^3*(c - d)^3)^(1/2)*(3*A*d^2 - 3*B*c^2 + B*d^2 + 2*A*c*d - 3*B*c*d)*2
i)/(f*(3*c*d^6 + d^7 + 3*c^2*d^5 + c^3*d^4))