3.711 \(\int \frac {1}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))} \, dx\)

Optimal. Leaf size=181 \[ \frac {2 b \left (-2 a^2 d+a b c+b^2 d\right ) \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (e+f x)\right )+b}{\sqrt {a^2-b^2}}\right )}{f \left (a^2-b^2\right )^{3/2} (b c-a d)^2}+\frac {b^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))}+\frac {2 d^2 \tan ^{-1}\left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{f \sqrt {c^2-d^2} (b c-a d)^2} \]

[Out]

2*b*(-2*a^2*d+a*b*c+b^2*d)*arctan((b+a*tan(1/2*f*x+1/2*e))/(a^2-b^2)^(1/2))/(a^2-b^2)^(3/2)/(-a*d+b*c)^2/f+b^2
*cos(f*x+e)/(a^2-b^2)/(-a*d+b*c)/f/(a+b*sin(f*x+e))+2*d^2*arctan((d+c*tan(1/2*f*x+1/2*e))/(c^2-d^2)^(1/2))/(-a
*d+b*c)^2/f/(c^2-d^2)^(1/2)

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Rubi [A]  time = 0.44, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2802, 3001, 2660, 618, 204} \[ \frac {2 b \left (-2 a^2 d+a b c+b^2 d\right ) \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (e+f x)\right )+b}{\sqrt {a^2-b^2}}\right )}{f \left (a^2-b^2\right )^{3/2} (b c-a d)^2}+\frac {b^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))}+\frac {2 d^2 \tan ^{-1}\left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{f \sqrt {c^2-d^2} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])),x]

[Out]

(2*b*(a*b*c - 2*a^2*d + b^2*d)*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(3/2)*(b*c - a*d
)^2*f) + (2*d^2*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((b*c - a*d)^2*Sqrt[c^2 - d^2]*f) + (b^2*Cos
[e + f*x])/((a^2 - b^2)*(b*c - a*d)*f*(a + b*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 2802

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -S
imp[(b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 -
b^2)), x] + Dist[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n
*Simp[a*(b*c - a*d)*(m + 1) + b^2*d*(m + n + 2) - (b^2*c + b*(b*c - a*d)*(m + 1))*Sin[e + f*x] - b^2*d*(m + n
+ 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] &
& NeQ[c^2 - d^2, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*n] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !
(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) || EqQ[a, 0])))

Rule 3001

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)])), x_Symbol] :> Dist[(A*b - a*B)/(b*c - a*d), Int[1/(a + b*Sin[e + f*x]), x], x] + Dist[(B*c - A
*d)/(b*c - a*d), Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0]
&& NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rubi steps

\begin {align*} \int \frac {1}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))} \, dx &=\frac {b^2 \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))}-\frac {\int \frac {-a b c+a^2 d-b^2 d-a b d \sin (e+f x)}{(a+b \sin (e+f x)) (c+d \sin (e+f x))} \, dx}{\left (a^2-b^2\right ) (b c-a d)}\\ &=\frac {b^2 \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))}+\frac {d^2 \int \frac {1}{c+d \sin (e+f x)} \, dx}{(b c-a d)^2}+\frac {\left (b \left (a b c-2 a^2 d+b^2 d\right )\right ) \int \frac {1}{a+b \sin (e+f x)} \, dx}{\left (a^2-b^2\right ) (b c-a d)^2}\\ &=\frac {b^2 \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))}+\frac {\left (2 d^2\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 )}{(b c-a d)^2 f}+\frac {\left (2 b \left (a b c-2 a^2 d+b^2 d\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (e+f x)\right )\right )}{\left (a^2-b^2\right ) (b c-a d)^2 f}\\ &=\frac {b^2 \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))}-\frac {\left (4 d^2\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 )}{(b c-a d)^2 f}-\frac {\left (4 b \left (a b c-2 a^2 d+b^2 d\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac {1}{2} (e+f x)\right )\right )}{\left (a^2-b^2\right ) (b c-a d)^2 f}\\ &=\frac {2 b \left (a b c-2 a^2 d+b^2 d\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} (b c-a d)^2 f}+\frac {2 d^2 \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{(b c-a d)^2 \sqrt {c^2-d^2} f}+\frac {b^2 \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))}\\ \end {align*}

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Mathematica [A]  time = 0.87, size = 178, normalized size = 0.98 \[ \frac {\frac {2 b \left (-2 a^2 d+a b c+b^2 d\right ) \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (e+f x)\right )+b}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} (b c-a d)^2}-\frac {b^2 \cos (e+f x)}{(a-b) (a+b) (a d-b c) (a+b \sin (e+f x))}+\frac {2 d^2 \tan ^{-1}\left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{\sqrt {c^2-d^2} (b c-a d)^2}}{f} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])),x]

[Out]

((2*b*(a*b*c - 2*a^2*d + b^2*d)*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(3/2)*(b*c - a*
d)^2) + (2*d^2*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((b*c - a*d)^2*Sqrt[c^2 - d^2]) - (b^2*Cos[e
+ f*x])/((a - b)*(a + b)*(-(b*c) + a*d)*(a + b*Sin[e + f*x])))/f

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fricas [B]  time = 140.16, size = 2871, normalized size = 15.86 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sin(f*x+e))^2/(c+d*sin(f*x+e)),x, algorithm="fricas")

[Out]

[1/2*((a^2*b^2*c^3 - a^2*b^2*c*d^2 - (2*a^3*b - a*b^3)*c^2*d + (2*a^3*b - a*b^3)*d^3 + (a*b^3*c^3 - a*b^3*c*d^
2 - (2*a^2*b^2 - b^4)*c^2*d + (2*a^2*b^2 - b^4)*d^3)*sin(f*x + e))*sqrt(-a^2 + b^2)*log(-((2*a^2 - b^2)*cos(f*
x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2 - 2*(a*cos(f*x + e)*sin(f*x + e) + b*cos(f*x + e))*sqrt(-a^2 + b^2))
/(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2)) - ((a^4*b - 2*a^2*b^3 + b^5)*d^2*sin(f*x + e) + (a^5 -
2*a^3*b^2 + a*b^4)*d^2)*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*((a^2*b^3 - b^5)*c^3 - (a^3*b^2 - a*b^4)*c^2*d - (a^2*b^3 - b^5)*c*d^2 + (a^3*b^2 - a*b^4)*d^
3)*cos(f*x + e))/(((a^4*b^3 - 2*a^2*b^5 + b^7)*c^4 - 2*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*c^3*d + (a^6*b - 3*a^4*b^
3 + 3*a^2*b^5 - b^7)*c^2*d^2 + 2*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*c*d^3 - (a^6*b - 2*a^4*b^3 + a^2*b^5)*d^4)*f*si
n(f*x + e) + ((a^5*b^2 - 2*a^3*b^4 + a*b^6)*c^4 - 2*(a^6*b - 2*a^4*b^3 + a^2*b^5)*c^3*d + (a^7 - 3*a^5*b^2 + 3
*a^3*b^4 - a*b^6)*c^2*d^2 + 2*(a^6*b - 2*a^4*b^3 + a^2*b^5)*c*d^3 - (a^7 - 2*a^5*b^2 + a^3*b^4)*d^4)*f), -1/2*
(2*(a^2*b^2*c^3 - a^2*b^2*c*d^2 - (2*a^3*b - a*b^3)*c^2*d + (2*a^3*b - a*b^3)*d^3 + (a*b^3*c^3 - a*b^3*c*d^2 -
(2*a^2*b^2 - b^4)*c^2*d + (2*a^2*b^2 - b^4)*d^3)*sin(f*x + e))*sqrt(a^2 - b^2)*arctan(-(a*sin(f*x + e) + b)/(
sqrt(a^2 - b^2)*cos(f*x + e))) + ((a^4*b - 2*a^2*b^3 + b^5)*d^2*sin(f*x + e) + (a^5 - 2*a^3*b^2 + a*b^4)*d^2)*
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*((a^2*b^
3 - b^5)*c^3 - (a^3*b^2 - a*b^4)*c^2*d - (a^2*b^3 - b^5)*c*d^2 + (a^3*b^2 - a*b^4)*d^3)*cos(f*x + e))/(((a^4*b
^3 - 2*a^2*b^5 + b^7)*c^4 - 2*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*c^3*d + (a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*c^2*
d^2 + 2*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*c*d^3 - (a^6*b - 2*a^4*b^3 + a^2*b^5)*d^4)*f*sin(f*x + e) + ((a^5*b^2 -
2*a^3*b^4 + a*b^6)*c^4 - 2*(a^6*b - 2*a^4*b^3 + a^2*b^5)*c^3*d + (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*c^2*d^2
+ 2*(a^6*b - 2*a^4*b^3 + a^2*b^5)*c*d^3 - (a^7 - 2*a^5*b^2 + a^3*b^4)*d^4)*f), -1/2*(2*((a^4*b - 2*a^2*b^3 +
b^5)*d^2*sin(f*x + e) + (a^5 - 2*a^3*b^2 + a*b^4)*d^2)*sqrt(c^2 - d^2)*arctan(-(c*sin(f*x + e) + d)/(sqrt(c^2
- d^2)*cos(f*x + e))) - (a^2*b^2*c^3 - a^2*b^2*c*d^2 - (2*a^3*b - a*b^3)*c^2*d + (2*a^3*b - a*b^3)*d^3 + (a*b^
3*c^3 - a*b^3*c*d^2 - (2*a^2*b^2 - b^4)*c^2*d + (2*a^2*b^2 - b^4)*d^3)*sin(f*x + e))*sqrt(-a^2 + b^2)*log(-((2
*a^2 - b^2)*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2 - 2*(a*cos(f*x + e)*sin(f*x + e) + b*cos(f*x + e))
*sqrt(-a^2 + b^2))/(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2)) - 2*((a^2*b^3 - b^5)*c^3 - (a^3*b^2
- a*b^4)*c^2*d - (a^2*b^3 - b^5)*c*d^2 + (a^3*b^2 - a*b^4)*d^3)*cos(f*x + e))/(((a^4*b^3 - 2*a^2*b^5 + b^7)*c^
4 - 2*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*c^3*d + (a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*c^2*d^2 + 2*(a^5*b^2 - 2*a^3
*b^4 + a*b^6)*c*d^3 - (a^6*b - 2*a^4*b^3 + a^2*b^5)*d^4)*f*sin(f*x + e) + ((a^5*b^2 - 2*a^3*b^4 + a*b^6)*c^4 -
2*(a^6*b - 2*a^4*b^3 + a^2*b^5)*c^3*d + (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*c^2*d^2 + 2*(a^6*b - 2*a^4*b^3
+ a^2*b^5)*c*d^3 - (a^7 - 2*a^5*b^2 + a^3*b^4)*d^4)*f), -((a^2*b^2*c^3 - a^2*b^2*c*d^2 - (2*a^3*b - a*b^3)*c^2
*d + (2*a^3*b - a*b^3)*d^3 + (a*b^3*c^3 - a*b^3*c*d^2 - (2*a^2*b^2 - b^4)*c^2*d + (2*a^2*b^2 - b^4)*d^3)*sin(f
*x + e))*sqrt(a^2 - b^2)*arctan(-(a*sin(f*x + e) + b)/(sqrt(a^2 - b^2)*cos(f*x + e))) + ((a^4*b - 2*a^2*b^3 +
b^5)*d^2*sin(f*x + e) + (a^5 - 2*a^3*b^2 + a*b^4)*d^2)*sqrt(c^2 - d^2)*arctan(-(c*sin(f*x + e) + d)/(sqrt(c^2
- d^2)*cos(f*x + e))) - ((a^2*b^3 - b^5)*c^3 - (a^3*b^2 - a*b^4)*c^2*d - (a^2*b^3 - b^5)*c*d^2 + (a^3*b^2 - a*
b^4)*d^3)*cos(f*x + e))/(((a^4*b^3 - 2*a^2*b^5 + b^7)*c^4 - 2*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*c^3*d + (a^6*b - 3
*a^4*b^3 + 3*a^2*b^5 - b^7)*c^2*d^2 + 2*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*c*d^3 - (a^6*b - 2*a^4*b^3 + a^2*b^5)*d^
4)*f*sin(f*x + e) + ((a^5*b^2 - 2*a^3*b^4 + a*b^6)*c^4 - 2*(a^6*b - 2*a^4*b^3 + a^2*b^5)*c^3*d + (a^7 - 3*a^5*
b^2 + 3*a^3*b^4 - a*b^6)*c^2*d^2 + 2*(a^6*b - 2*a^4*b^3 + a^2*b^5)*c*d^3 - (a^7 - 2*a^5*b^2 + a^3*b^4)*d^4)*f)
]

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giac [A]  time = 0.24, size = 304, normalized size = 1.68 \[ \frac {2 \, {\left (\frac {{\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (c) + \arctan \left (\frac {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + d}{\sqrt {c^{2} - d^{2}}}\right )\right )} d^{2}}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {c^{2} - d^{2}}} + \frac {{\left (a b^{2} c - 2 \, a^{2} b d + b^{3} d\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (a) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{2} b^{2} c^{2} - b^{4} c^{2} - 2 \, a^{3} b c d + 2 \, a b^{3} c d + a^{4} d^{2} - a^{2} b^{2} d^{2}\right )} \sqrt {a^{2} - b^{2}}} + \frac {b^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + a b^{2}}{{\left (a^{3} b c - a b^{3} c - a^{4} d + a^{2} b^{2} d\right )} {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + a\right )}}\right )}}{f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sin(f*x+e))^2/(c+d*sin(f*x+e)),x, algorithm="giac")

[Out]

2*((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)))*d^2/((b^2*
c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(c^2 - d^2)) + (a*b^2*c - 2*a^2*b*d + b^3*d)*(pi*floor(1/2*(f*x + e)/pi + 1/2)*
sgn(a) + arctan((a*tan(1/2*f*x + 1/2*e) + b)/sqrt(a^2 - b^2)))/((a^2*b^2*c^2 - b^4*c^2 - 2*a^3*b*c*d + 2*a*b^3
*c*d + a^4*d^2 - a^2*b^2*d^2)*sqrt(a^2 - b^2)) + (b^3*tan(1/2*f*x + 1/2*e) + a*b^2)/((a^3*b*c - a*b^3*c - a^4*
d + a^2*b^2*d)*(a*tan(1/2*f*x + 1/2*e)^2 + 2*b*tan(1/2*f*x + 1/2*e) + a)))/f

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maple [B]  time = 0.38, size = 514, normalized size = 2.84 \[ \frac {2 d^{2} \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{f \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {c^{2}-d^{2}}}-\frac {2 b^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) d}{f \left (d a -c b \right )^{2} \left (\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a +2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) b +a \right ) \left (a^{2}-b^{2}\right )}+\frac {2 b^{4} \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) c}{f \left (d a -c b \right )^{2} \left (\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a +2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) b +a \right ) a \left (a^{2}-b^{2}\right )}-\frac {2 b^{2} d a}{f \left (d a -c b \right )^{2} \left (\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a +2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) b +a \right ) \left (a^{2}-b^{2}\right )}+\frac {2 b^{3} c}{f \left (d a -c b \right )^{2} \left (\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a +2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) b +a \right ) \left (a^{2}-b^{2}\right )}-\frac {4 b \arctan \left (\frac {2 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right ) a^{2} d}{f \left (d a -c b \right )^{2} \left (a^{2}-b^{2}\right )^{\frac {3}{2}}}+\frac {2 b^{2} \arctan \left (\frac {2 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right ) a c}{f \left (d a -c b \right )^{2} \left (a^{2}-b^{2}\right )^{\frac {3}{2}}}+\frac {2 b^{3} \arctan \left (\frac {2 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right ) d}{f \left (d a -c b \right )^{2} \left (a^{2}-b^{2}\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*sin(f*x+e))^2/(c+d*sin(f*x+e)),x)

[Out]

2/f*d^2/(a^2*d^2-2*a*b*c*d+b^2*c^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))-2
/f*b^3/(a*d-b*c)^2/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)/(a^2-b^2)*tan(1/2*f*x+1/2*e)*d+2/f*b^4/(a
*d-b*c)^2/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)/a/(a^2-b^2)*tan(1/2*f*x+1/2*e)*c-2/f*b^2/(a*d-b*c)
^2/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)/(a^2-b^2)*d*a+2/f*b^3/(a*d-b*c)^2/(tan(1/2*f*x+1/2*e)^2*a
+2*tan(1/2*f*x+1/2*e)*b+a)/(a^2-b^2)*c-4/f*b/(a*d-b*c)^2/(a^2-b^2)^(3/2)*arctan(1/2*(2*a*tan(1/2*f*x+1/2*e)+2*
b)/(a^2-b^2)^(1/2))*a^2*d+2/f*b^2/(a*d-b*c)^2/(a^2-b^2)^(3/2)*arctan(1/2*(2*a*tan(1/2*f*x+1/2*e)+2*b)/(a^2-b^2
)^(1/2))*a*c+2/f*b^3/(a*d-b*c)^2/(a^2-b^2)^(3/2)*arctan(1/2*(2*a*tan(1/2*f*x+1/2*e)+2*b)/(a^2-b^2)^(1/2))*d

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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(1/(a+b*sin(f*x+e))^2/(c+d*sin(f*x+e)),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?

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mupad [B]  time = 22.80, size = 24123, normalized size = 133.28 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*sin(e + f*x))^2*(c + d*sin(e + f*x))),x)

[Out]

(d^2*atan(((d^2*(d^2 - c^2)^(1/2)*((32*tan(e/2 + (f*x)/2)*(a^3*b^5*c^6 - a^8*c*d^5 - 4*a*b^7*c^2*d^4 + a*b^7*c
^4*d^2 + 4*a^2*b^6*c*d^5 + 2*a^2*b^6*c^5*d - 13*a^4*b^4*c*d^5 - 5*a^4*b^4*c^5*d + 12*a^6*b^2*c*d^5 + a^7*b*c^2
*d^4 - 5*a^2*b^6*c^3*d^3 + 17*a^3*b^5*c^2*d^4 - 8*a^3*b^5*c^4*d^2 + 14*a^4*b^4*c^3*d^3 - 20*a^5*b^3*c^2*d^4 +
8*a^5*b^3*c^4*d^2 - 4*a^6*b^2*c^3*d^3))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5
*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d
^2) - (32*(a*b^7*c^3*d^3 - a^3*b^5*c*d^5 + a^3*b^5*c^5*d + 2*a^5*b^3*c*d^5 + 2*a^2*b^6*c^4*d^2 - 6*a^3*b^5*c^3
*d^3 + 2*a^4*b^4*c^2*d^4 - 5*a^4*b^4*c^4*d^2 + 8*a^5*b^3*c^3*d^3 - 3*a^6*b^2*c^2*d^4 - a^7*b*c*d^5))/(a^7*d^3
- b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*
a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) + (d^2*(d^2 - c^2)^(1/2)*((32*(a^3*b^7*c^7 -
a^5*b^5*c^7 + a^10*c^2*d^5 + a*b^9*c^5*d^2 + a^2*b^8*c^6*d - 6*a^4*b^6*c^6*d + a^5*b^5*c*d^6 + 5*a^6*b^4*c^6*d
- 3*a^7*b^3*c*d^6 - 5*a^9*b*c^3*d^4 - 4*a^2*b^8*c^4*d^3 + 6*a^3*b^7*c^3*d^4 - 7*a^3*b^7*c^5*d^2 - 4*a^4*b^6*c
^2*d^5 + 18*a^4*b^6*c^4*d^3 - 22*a^5*b^5*c^3*d^4 + 16*a^5*b^5*c^5*d^2 + 13*a^6*b^4*c^2*d^5 - 24*a^6*b^4*c^4*d^
3 + 21*a^7*b^3*c^3*d^4 - 10*a^7*b^3*c^5*d^2 - 10*a^8*b^2*c^2*d^5 + 10*a^8*b^2*c^4*d^3 + 2*a^9*b*c*d^6))/(a^7*d
^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d +
6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) + (32*tan(e/2 + (f*x)/2)*(2*a^10*c*d^6 + 2
*a^2*b^8*c^7 - 2*a^4*b^6*c^7 - 12*a^3*b^7*c^6*d + 10*a^5*b^5*c^6*d - 2*a^8*b^2*c*d^6 - 6*a^9*b*c^2*d^5 - 8*a^2
*b^8*c^5*d^2 + 12*a^3*b^7*c^4*d^3 - 8*a^4*b^6*c^3*d^4 + 26*a^4*b^6*c^5*d^2 + 2*a^5*b^5*c^2*d^5 - 24*a^5*b^5*c^
4*d^3 + 6*a^6*b^4*c^3*d^4 - 18*a^6*b^4*c^5*d^2 + 4*a^7*b^3*c^2*d^5 + 12*a^7*b^3*c^4*d^3 + 2*a^8*b^2*c^3*d^4 +
2*a*b^9*c^6*d))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d
^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) + (d^2*(d^2 - c^2)^(
1/2)*((32*(2*a^4*b^8*c^8 - a^2*b^10*c^8 - a^6*b^6*c^8 + a^12*c^2*d^6 + 2*a^3*b^9*c^7*d - 7*a^5*b^7*c^7*d - a^7
*b^5*c*d^7 + 4*a^7*b^5*c^7*d + 2*a^9*b^3*c*d^7 - 4*a^11*b*c^3*d^5 - 4*a^2*b^10*c^6*d^2 + 5*a^3*b^9*c^5*d^3 + 3
*a^4*b^8*c^6*d^2 - 5*a^5*b^7*c^3*d^5 - 10*a^5*b^7*c^5*d^3 + 4*a^6*b^6*c^2*d^6 + 5*a^6*b^6*c^4*d^4 + 6*a^6*b^6*
c^6*d^2 + 6*a^7*b^5*c^3*d^5 + 5*a^7*b^5*c^5*d^3 - 7*a^8*b^4*c^2*d^6 - 10*a^8*b^4*c^4*d^4 - 5*a^8*b^4*c^6*d^2 +
3*a^9*b^3*c^3*d^5 + 2*a^10*b^2*c^2*d^6 + 5*a^10*b^2*c^4*d^4 + a*b^11*c^7*d - a^11*b*c*d^7))/(a^7*d^3 - b^7*c^
3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*
c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) - (32*tan(e/2 + (f*x)/2)*(3*a*b^11*c^8 - 3*a^12*c*d^7
- 8*a^3*b^9*c^8 + 7*a^5*b^7*c^8 - 2*a^7*b^5*c^8 + 2*a^12*c^3*d^5 - 4*a*b^11*c^6*d^2 - 15*a^2*b^10*c^7*d + 40*
a^4*b^8*c^7*d + 4*a^6*b^6*c*d^7 - 35*a^6*b^6*c^7*d - 11*a^8*b^4*c*d^7 + 10*a^8*b^4*c^7*d + 10*a^10*b^2*c*d^7 +
15*a^11*b*c^2*d^6 - 10*a^11*b*c^4*d^4 + 20*a^2*b^10*c^5*d^3 - 40*a^3*b^9*c^4*d^4 + 41*a^3*b^9*c^6*d^2 + 40*a^
4*b^8*c^3*d^5 - 85*a^4*b^8*c^5*d^3 - 20*a^5*b^7*c^2*d^6 + 125*a^5*b^7*c^4*d^4 - 90*a^5*b^7*c^6*d^2 - 113*a^6*b
^6*c^3*d^5 + 130*a^6*b^6*c^5*d^3 + 55*a^7*b^5*c^2*d^6 - 140*a^7*b^5*c^4*d^4 + 73*a^7*b^5*c^6*d^2 + 108*a^8*b^4
*c^3*d^5 - 85*a^8*b^4*c^5*d^3 - 50*a^9*b^3*c^2*d^6 + 65*a^9*b^3*c^4*d^4 - 20*a^9*b^3*c^6*d^2 - 37*a^10*b^2*c^3
*d^5 + 20*a^10*b^2*c^5*d^3))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 -
3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2)))/(a^2*
d^4 - b^2*c^4 - a^2*c^2*d^2 + b^2*c^2*d^2 - 2*a*b*c*d^3 + 2*a*b*c^3*d)))/(a^2*d^4 - b^2*c^4 - a^2*c^2*d^2 + b^
2*c^2*d^2 - 2*a*b*c*d^3 + 2*a*b*c^3*d))*1i)/(a^2*d^4 - b^2*c^4 - a^2*c^2*d^2 + b^2*c^2*d^2 - 2*a*b*c*d^3 + 2*a
*b*c^3*d) - (d^2*(d^2 - c^2)^(1/2)*((32*(a*b^7*c^3*d^3 - a^3*b^5*c*d^5 + a^3*b^5*c^5*d + 2*a^5*b^3*c*d^5 + 2*a
^2*b^6*c^4*d^2 - 6*a^3*b^5*c^3*d^3 + 2*a^4*b^4*c^2*d^4 - 5*a^4*b^4*c^4*d^2 + 8*a^5*b^3*c^3*d^3 - 3*a^6*b^2*c^2
*d^4 - a^7*b*c*d^5))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^
5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) - (32*tan(e/2 +
(f*x)/2)*(a^3*b^5*c^6 - a^8*c*d^5 - 4*a*b^7*c^2*d^4 + a*b^7*c^4*d^2 + 4*a^2*b^6*c*d^5 + 2*a^2*b^6*c^5*d - 13*
a^4*b^4*c*d^5 - 5*a^4*b^4*c^5*d + 12*a^6*b^2*c*d^5 + a^7*b*c^2*d^4 - 5*a^2*b^6*c^3*d^3 + 17*a^3*b^5*c^2*d^4 -
8*a^3*b^5*c^4*d^2 + 14*a^4*b^4*c^3*d^3 - 20*a^5*b^3*c^2*d^4 + 8*a^5*b^3*c^4*d^2 - 4*a^6*b^2*c^3*d^3))/(a^7*d^3
- b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6
*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) + (d^2*(d^2 - c^2)^(1/2)*((32*(a^3*b^7*c^7 -
a^5*b^5*c^7 + a^10*c^2*d^5 + a*b^9*c^5*d^2 + a^2*b^8*c^6*d - 6*a^4*b^6*c^6*d + a^5*b^5*c*d^6 + 5*a^6*b^4*c^6*
d - 3*a^7*b^3*c*d^6 - 5*a^9*b*c^3*d^4 - 4*a^2*b^8*c^4*d^3 + 6*a^3*b^7*c^3*d^4 - 7*a^3*b^7*c^5*d^2 - 4*a^4*b^6*
c^2*d^5 + 18*a^4*b^6*c^4*d^3 - 22*a^5*b^5*c^3*d^4 + 16*a^5*b^5*c^5*d^2 + 13*a^6*b^4*c^2*d^5 - 24*a^6*b^4*c^4*d
^3 + 21*a^7*b^3*c^3*d^4 - 10*a^7*b^3*c^5*d^2 - 10*a^8*b^2*c^2*d^5 + 10*a^8*b^2*c^4*d^3 + 2*a^9*b*c*d^6))/(a^7*
d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d
+ 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) + (32*tan(e/2 + (f*x)/2)*(2*a^10*c*d^6 +
2*a^2*b^8*c^7 - 2*a^4*b^6*c^7 - 12*a^3*b^7*c^6*d + 10*a^5*b^5*c^6*d - 2*a^8*b^2*c*d^6 - 6*a^9*b*c^2*d^5 - 8*a^
2*b^8*c^5*d^2 + 12*a^3*b^7*c^4*d^3 - 8*a^4*b^6*c^3*d^4 + 26*a^4*b^6*c^5*d^2 + 2*a^5*b^5*c^2*d^5 - 24*a^5*b^5*c
^4*d^3 + 6*a^6*b^4*c^3*d^4 - 18*a^6*b^4*c^5*d^2 + 4*a^7*b^3*c^2*d^5 + 12*a^7*b^3*c^4*d^3 + 2*a^8*b^2*c^3*d^4 +
2*a*b^9*c^6*d))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*
d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) - (d^2*(d^2 - c^2)^
(1/2)*((32*(2*a^4*b^8*c^8 - a^2*b^10*c^8 - a^6*b^6*c^8 + a^12*c^2*d^6 + 2*a^3*b^9*c^7*d - 7*a^5*b^7*c^7*d - a^
7*b^5*c*d^7 + 4*a^7*b^5*c^7*d + 2*a^9*b^3*c*d^7 - 4*a^11*b*c^3*d^5 - 4*a^2*b^10*c^6*d^2 + 5*a^3*b^9*c^5*d^3 +
3*a^4*b^8*c^6*d^2 - 5*a^5*b^7*c^3*d^5 - 10*a^5*b^7*c^5*d^3 + 4*a^6*b^6*c^2*d^6 + 5*a^6*b^6*c^4*d^4 + 6*a^6*b^6
*c^6*d^2 + 6*a^7*b^5*c^3*d^5 + 5*a^7*b^5*c^5*d^3 - 7*a^8*b^4*c^2*d^6 - 10*a^8*b^4*c^4*d^4 - 5*a^8*b^4*c^6*d^2
+ 3*a^9*b^3*c^3*d^5 + 2*a^10*b^2*c^2*d^6 + 5*a^10*b^2*c^4*d^4 + a*b^11*c^7*d - a^11*b*c*d^7))/(a^7*d^3 - b^7*c
^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3
*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) - (32*tan(e/2 + (f*x)/2)*(3*a*b^11*c^8 - 3*a^12*c*d^
7 - 8*a^3*b^9*c^8 + 7*a^5*b^7*c^8 - 2*a^7*b^5*c^8 + 2*a^12*c^3*d^5 - 4*a*b^11*c^6*d^2 - 15*a^2*b^10*c^7*d + 40
*a^4*b^8*c^7*d + 4*a^6*b^6*c*d^7 - 35*a^6*b^6*c^7*d - 11*a^8*b^4*c*d^7 + 10*a^8*b^4*c^7*d + 10*a^10*b^2*c*d^7
+ 15*a^11*b*c^2*d^6 - 10*a^11*b*c^4*d^4 + 20*a^2*b^10*c^5*d^3 - 40*a^3*b^9*c^4*d^4 + 41*a^3*b^9*c^6*d^2 + 40*a
^4*b^8*c^3*d^5 - 85*a^4*b^8*c^5*d^3 - 20*a^5*b^7*c^2*d^6 + 125*a^5*b^7*c^4*d^4 - 90*a^5*b^7*c^6*d^2 - 113*a^6*
b^6*c^3*d^5 + 130*a^6*b^6*c^5*d^3 + 55*a^7*b^5*c^2*d^6 - 140*a^7*b^5*c^4*d^4 + 73*a^7*b^5*c^6*d^2 + 108*a^8*b^
4*c^3*d^5 - 85*a^8*b^4*c^5*d^3 - 50*a^9*b^3*c^2*d^6 + 65*a^9*b^3*c^4*d^4 - 20*a^9*b^3*c^6*d^2 - 37*a^10*b^2*c^
3*d^5 + 20*a^10*b^2*c^5*d^3))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 -
3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2)))/(a^2
*d^4 - b^2*c^4 - a^2*c^2*d^2 + b^2*c^2*d^2 - 2*a*b*c*d^3 + 2*a*b*c^3*d)))/(a^2*d^4 - b^2*c^4 - a^2*c^2*d^2 + b
^2*c^2*d^2 - 2*a*b*c*d^3 + 2*a*b*c^3*d))*1i)/(a^2*d^4 - b^2*c^4 - a^2*c^2*d^2 + b^2*c^2*d^2 - 2*a*b*c*d^3 + 2*
a*b*c^3*d))/((64*(2*a^2*b^4*c^2*d^3 - 3*a^3*b^3*c*d^4 + a^3*b^3*c^3*d^2 - 3*a^4*b^2*c^2*d^3 + a*b^5*c*d^4 + 2*
a^5*b*c*d^4))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2
- 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) + (64*tan(e/2 + (f*x)/
2)*(2*a^2*b^4*c*d^4 - 4*a^4*b^2*c*d^4 + 2*a^3*b^3*c^2*d^3))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 +
a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6
*c^2*d - 3*a^6*b*c*d^2) - (d^2*(d^2 - c^2)^(1/2)*((32*tan(e/2 + (f*x)/2)*(a^3*b^5*c^6 - a^8*c*d^5 - 4*a*b^7*c^
2*d^4 + a*b^7*c^4*d^2 + 4*a^2*b^6*c*d^5 + 2*a^2*b^6*c^5*d - 13*a^4*b^4*c*d^5 - 5*a^4*b^4*c^5*d + 12*a^6*b^2*c*
d^5 + a^7*b*c^2*d^4 - 5*a^2*b^6*c^3*d^3 + 17*a^3*b^5*c^2*d^4 - 8*a^3*b^5*c^4*d^2 + 14*a^4*b^4*c^3*d^3 - 20*a^5
*b^3*c^2*d^4 + 8*a^5*b^3*c^4*d^2 - 4*a^6*b^2*c^3*d^3))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*
b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*
d - 3*a^6*b*c*d^2) - (32*(a*b^7*c^3*d^3 - a^3*b^5*c*d^5 + a^3*b^5*c^5*d + 2*a^5*b^3*c*d^5 + 2*a^2*b^6*c^4*d^2
- 6*a^3*b^5*c^3*d^3 + 2*a^4*b^4*c^2*d^4 - 5*a^4*b^4*c^4*d^2 + 8*a^5*b^3*c^3*d^3 - 3*a^6*b^2*c^2*d^4 - a^7*b*c*
d^5))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3
*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) + (d^2*(d^2 - c^2)^(1/2)*((32*
(a^3*b^7*c^7 - a^5*b^5*c^7 + a^10*c^2*d^5 + a*b^9*c^5*d^2 + a^2*b^8*c^6*d - 6*a^4*b^6*c^6*d + a^5*b^5*c*d^6 +
5*a^6*b^4*c^6*d - 3*a^7*b^3*c*d^6 - 5*a^9*b*c^3*d^4 - 4*a^2*b^8*c^4*d^3 + 6*a^3*b^7*c^3*d^4 - 7*a^3*b^7*c^5*d^
2 - 4*a^4*b^6*c^2*d^5 + 18*a^4*b^6*c^4*d^3 - 22*a^5*b^5*c^3*d^4 + 16*a^5*b^5*c^5*d^2 + 13*a^6*b^4*c^2*d^5 - 24
*a^6*b^4*c^4*d^3 + 21*a^7*b^3*c^3*d^4 - 10*a^7*b^3*c^5*d^2 - 10*a^8*b^2*c^2*d^5 + 10*a^8*b^2*c^4*d^3 + 2*a^9*b
*c*d^6))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*
a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) + (32*tan(e/2 + (f*x)/2)*(2
*a^10*c*d^6 + 2*a^2*b^8*c^7 - 2*a^4*b^6*c^7 - 12*a^3*b^7*c^6*d + 10*a^5*b^5*c^6*d - 2*a^8*b^2*c*d^6 - 6*a^9*b*
c^2*d^5 - 8*a^2*b^8*c^5*d^2 + 12*a^3*b^7*c^4*d^3 - 8*a^4*b^6*c^3*d^4 + 26*a^4*b^6*c^5*d^2 + 2*a^5*b^5*c^2*d^5
- 24*a^5*b^5*c^4*d^3 + 6*a^6*b^4*c^3*d^4 - 18*a^6*b^4*c^5*d^2 + 4*a^7*b^3*c^2*d^5 + 12*a^7*b^3*c^4*d^3 + 2*a^8
*b^2*c^3*d^4 + 2*a*b^9*c^6*d))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3
- 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) + (d^
2*(d^2 - c^2)^(1/2)*((32*(2*a^4*b^8*c^8 - a^2*b^10*c^8 - a^6*b^6*c^8 + a^12*c^2*d^6 + 2*a^3*b^9*c^7*d - 7*a^5*
b^7*c^7*d - a^7*b^5*c*d^7 + 4*a^7*b^5*c^7*d + 2*a^9*b^3*c*d^7 - 4*a^11*b*c^3*d^5 - 4*a^2*b^10*c^6*d^2 + 5*a^3*
b^9*c^5*d^3 + 3*a^4*b^8*c^6*d^2 - 5*a^5*b^7*c^3*d^5 - 10*a^5*b^7*c^5*d^3 + 4*a^6*b^6*c^2*d^6 + 5*a^6*b^6*c^4*d
^4 + 6*a^6*b^6*c^6*d^2 + 6*a^7*b^5*c^3*d^5 + 5*a^7*b^5*c^5*d^3 - 7*a^8*b^4*c^2*d^6 - 10*a^8*b^4*c^4*d^4 - 5*a^
8*b^4*c^6*d^2 + 3*a^9*b^3*c^3*d^5 + 2*a^10*b^2*c^2*d^6 + 5*a^10*b^2*c^4*d^4 + a*b^11*c^7*d - a^11*b*c*d^7))/(a
^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2
*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) - (32*tan(e/2 + (f*x)/2)*(3*a*b^11*c^8
- 3*a^12*c*d^7 - 8*a^3*b^9*c^8 + 7*a^5*b^7*c^8 - 2*a^7*b^5*c^8 + 2*a^12*c^3*d^5 - 4*a*b^11*c^6*d^2 - 15*a^2*b
^10*c^7*d + 40*a^4*b^8*c^7*d + 4*a^6*b^6*c*d^7 - 35*a^6*b^6*c^7*d - 11*a^8*b^4*c*d^7 + 10*a^8*b^4*c^7*d + 10*a
^10*b^2*c*d^7 + 15*a^11*b*c^2*d^6 - 10*a^11*b*c^4*d^4 + 20*a^2*b^10*c^5*d^3 - 40*a^3*b^9*c^4*d^4 + 41*a^3*b^9*
c^6*d^2 + 40*a^4*b^8*c^3*d^5 - 85*a^4*b^8*c^5*d^3 - 20*a^5*b^7*c^2*d^6 + 125*a^5*b^7*c^4*d^4 - 90*a^5*b^7*c^6*
d^2 - 113*a^6*b^6*c^3*d^5 + 130*a^6*b^6*c^5*d^3 + 55*a^7*b^5*c^2*d^6 - 140*a^7*b^5*c^4*d^4 + 73*a^7*b^5*c^6*d^
2 + 108*a^8*b^4*c^3*d^5 - 85*a^8*b^4*c^5*d^3 - 50*a^9*b^3*c^2*d^6 + 65*a^9*b^3*c^4*d^4 - 20*a^9*b^3*c^6*d^2 -
37*a^10*b^2*c^3*d^5 + 20*a^10*b^2*c^5*d^3))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2
*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b
*c*d^2)))/(a^2*d^4 - b^2*c^4 - a^2*c^2*d^2 + b^2*c^2*d^2 - 2*a*b*c*d^3 + 2*a*b*c^3*d)))/(a^2*d^4 - b^2*c^4 - a
^2*c^2*d^2 + b^2*c^2*d^2 - 2*a*b*c*d^3 + 2*a*b*c^3*d)))/(a^2*d^4 - b^2*c^4 - a^2*c^2*d^2 + b^2*c^2*d^2 - 2*a*b
*c*d^3 + 2*a*b*c^3*d) - (d^2*(d^2 - c^2)^(1/2)*((32*(a*b^7*c^3*d^3 - a^3*b^5*c*d^5 + a^3*b^5*c^5*d + 2*a^5*b^3
*c*d^5 + 2*a^2*b^6*c^4*d^2 - 6*a^3*b^5*c^3*d^3 + 2*a^4*b^4*c^2*d^4 - 5*a^4*b^4*c^4*d^2 + 8*a^5*b^3*c^3*d^3 - 3
*a^6*b^2*c^2*d^4 - a^7*b*c*d^5))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^
3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) - (
32*tan(e/2 + (f*x)/2)*(a^3*b^5*c^6 - a^8*c*d^5 - 4*a*b^7*c^2*d^4 + a*b^7*c^4*d^2 + 4*a^2*b^6*c*d^5 + 2*a^2*b^6
*c^5*d - 13*a^4*b^4*c*d^5 - 5*a^4*b^4*c^5*d + 12*a^6*b^2*c*d^5 + a^7*b*c^2*d^4 - 5*a^2*b^6*c^3*d^3 + 17*a^3*b^
5*c^2*d^4 - 8*a^3*b^5*c^4*d^2 + 14*a^4*b^4*c^3*d^3 - 20*a^5*b^3*c^2*d^4 + 8*a^5*b^3*c^4*d^2 - 4*a^6*b^2*c^3*d^
3))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b
^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) + (d^2*(d^2 - c^2)^(1/2)*((32*(a
^3*b^7*c^7 - a^5*b^5*c^7 + a^10*c^2*d^5 + a*b^9*c^5*d^2 + a^2*b^8*c^6*d - 6*a^4*b^6*c^6*d + a^5*b^5*c*d^6 + 5*
a^6*b^4*c^6*d - 3*a^7*b^3*c*d^6 - 5*a^9*b*c^3*d^4 - 4*a^2*b^8*c^4*d^3 + 6*a^3*b^7*c^3*d^4 - 7*a^3*b^7*c^5*d^2
- 4*a^4*b^6*c^2*d^5 + 18*a^4*b^6*c^4*d^3 - 22*a^5*b^5*c^3*d^4 + 16*a^5*b^5*c^5*d^2 + 13*a^6*b^4*c^2*d^5 - 24*a
^6*b^4*c^4*d^3 + 21*a^7*b^3*c^3*d^4 - 10*a^7*b^3*c^5*d^2 - 10*a^8*b^2*c^2*d^5 + 10*a^8*b^2*c^4*d^3 + 2*a^9*b*c
*d^6))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^
3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) + (32*tan(e/2 + (f*x)/2)*(2*a
^10*c*d^6 + 2*a^2*b^8*c^7 - 2*a^4*b^6*c^7 - 12*a^3*b^7*c^6*d + 10*a^5*b^5*c^6*d - 2*a^8*b^2*c*d^6 - 6*a^9*b*c^
2*d^5 - 8*a^2*b^8*c^5*d^2 + 12*a^3*b^7*c^4*d^3 - 8*a^4*b^6*c^3*d^4 + 26*a^4*b^6*c^5*d^2 + 2*a^5*b^5*c^2*d^5 -
24*a^5*b^5*c^4*d^3 + 6*a^6*b^4*c^3*d^4 - 18*a^6*b^4*c^5*d^2 + 4*a^7*b^3*c^2*d^5 + 12*a^7*b^3*c^4*d^3 + 2*a^8*b
^2*c^3*d^4 + 2*a*b^9*c^6*d))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 -
3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) - (d^2*
(d^2 - c^2)^(1/2)*((32*(2*a^4*b^8*c^8 - a^2*b^10*c^8 - a^6*b^6*c^8 + a^12*c^2*d^6 + 2*a^3*b^9*c^7*d - 7*a^5*b^
7*c^7*d - a^7*b^5*c*d^7 + 4*a^7*b^5*c^7*d + 2*a^9*b^3*c*d^7 - 4*a^11*b*c^3*d^5 - 4*a^2*b^10*c^6*d^2 + 5*a^3*b^
9*c^5*d^3 + 3*a^4*b^8*c^6*d^2 - 5*a^5*b^7*c^3*d^5 - 10*a^5*b^7*c^5*d^3 + 4*a^6*b^6*c^2*d^6 + 5*a^6*b^6*c^4*d^4
+ 6*a^6*b^6*c^6*d^2 + 6*a^7*b^5*c^3*d^5 + 5*a^7*b^5*c^5*d^3 - 7*a^8*b^4*c^2*d^6 - 10*a^8*b^4*c^4*d^4 - 5*a^8*
b^4*c^6*d^2 + 3*a^9*b^3*c^3*d^5 + 2*a^10*b^2*c^2*d^6 + 5*a^10*b^2*c^4*d^4 + a*b^11*c^7*d - a^11*b*c*d^7))/(a^7
*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d
+ 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) - (32*tan(e/2 + (f*x)/2)*(3*a*b^11*c^8 -
3*a^12*c*d^7 - 8*a^3*b^9*c^8 + 7*a^5*b^7*c^8 - 2*a^7*b^5*c^8 + 2*a^12*c^3*d^5 - 4*a*b^11*c^6*d^2 - 15*a^2*b^1
0*c^7*d + 40*a^4*b^8*c^7*d + 4*a^6*b^6*c*d^7 - 35*a^6*b^6*c^7*d - 11*a^8*b^4*c*d^7 + 10*a^8*b^4*c^7*d + 10*a^1
0*b^2*c*d^7 + 15*a^11*b*c^2*d^6 - 10*a^11*b*c^4*d^4 + 20*a^2*b^10*c^5*d^3 - 40*a^3*b^9*c^4*d^4 + 41*a^3*b^9*c^
6*d^2 + 40*a^4*b^8*c^3*d^5 - 85*a^4*b^8*c^5*d^3 - 20*a^5*b^7*c^2*d^6 + 125*a^5*b^7*c^4*d^4 - 90*a^5*b^7*c^6*d^
2 - 113*a^6*b^6*c^3*d^5 + 130*a^6*b^6*c^5*d^3 + 55*a^7*b^5*c^2*d^6 - 140*a^7*b^5*c^4*d^4 + 73*a^7*b^5*c^6*d^2
+ 108*a^8*b^4*c^3*d^5 - 85*a^8*b^4*c^5*d^3 - 50*a^9*b^3*c^2*d^6 + 65*a^9*b^3*c^4*d^4 - 20*a^9*b^3*c^6*d^2 - 37
*a^10*b^2*c^3*d^5 + 20*a^10*b^2*c^5*d^3))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a
^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c
*d^2)))/(a^2*d^4 - b^2*c^4 - a^2*c^2*d^2 + b^2*c^2*d^2 - 2*a*b*c*d^3 + 2*a*b*c^3*d)))/(a^2*d^4 - b^2*c^4 - a^2
*c^2*d^2 + b^2*c^2*d^2 - 2*a*b*c*d^3 + 2*a*b*c^3*d)))/(a^2*d^4 - b^2*c^4 - a^2*c^2*d^2 + b^2*c^2*d^2 - 2*a*b*c
*d^3 + 2*a*b*c^3*d)))*(d^2 - c^2)^(1/2)*2i)/(f*(a^2*d^4 - b^2*c^4 - a^2*c^2*d^2 + b^2*c^2*d^2 - 2*a*b*c*d^3 +
2*a*b*c^3*d)) - ((2*b^2)/((a^2 - b^2)*(a*d - b*c)) + (2*b^3*tan(e/2 + (f*x)/2))/(a*(a^2 - b^2)*(a*d - b*c)))/(
f*(a + 2*b*tan(e/2 + (f*x)/2) + a*tan(e/2 + (f*x)/2)^2)) + (b*atan(((b*(-(a + b)^3*(a - b)^3)^(1/2)*((32*tan(e
/2 + (f*x)/2)*(a^3*b^5*c^6 - a^8*c*d^5 - 4*a*b^7*c^2*d^4 + a*b^7*c^4*d^2 + 4*a^2*b^6*c*d^5 + 2*a^2*b^6*c^5*d -
13*a^4*b^4*c*d^5 - 5*a^4*b^4*c^5*d + 12*a^6*b^2*c*d^5 + a^7*b*c^2*d^4 - 5*a^2*b^6*c^3*d^3 + 17*a^3*b^5*c^2*d^
4 - 8*a^3*b^5*c^4*d^2 + 14*a^4*b^4*c^3*d^3 - 20*a^5*b^3*c^2*d^4 + 8*a^5*b^3*c^4*d^2 - 4*a^6*b^2*c^3*d^3))/(a^7
*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d
+ 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) - (32*(a*b^7*c^3*d^3 - a^3*b^5*c*d^5 + a
^3*b^5*c^5*d + 2*a^5*b^3*c*d^5 + 2*a^2*b^6*c^4*d^2 - 6*a^3*b^5*c^3*d^3 + 2*a^4*b^4*c^2*d^4 - 5*a^4*b^4*c^4*d^2
+ 8*a^5*b^3*c^3*d^3 - 3*a^6*b^2*c^2*d^4 - a^7*b*c*d^5))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^
3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^
2*d - 3*a^6*b*c*d^2) + (b*(-(a + b)^3*(a - b)^3)^(1/2)*((32*(a^3*b^7*c^7 - a^5*b^5*c^7 + a^10*c^2*d^5 + a*b^9*
c^5*d^2 + a^2*b^8*c^6*d - 6*a^4*b^6*c^6*d + a^5*b^5*c*d^6 + 5*a^6*b^4*c^6*d - 3*a^7*b^3*c*d^6 - 5*a^9*b*c^3*d^
4 - 4*a^2*b^8*c^4*d^3 + 6*a^3*b^7*c^3*d^4 - 7*a^3*b^7*c^5*d^2 - 4*a^4*b^6*c^2*d^5 + 18*a^4*b^6*c^4*d^3 - 22*a^
5*b^5*c^3*d^4 + 16*a^5*b^5*c^5*d^2 + 13*a^6*b^4*c^2*d^5 - 24*a^6*b^4*c^4*d^3 + 21*a^7*b^3*c^3*d^4 - 10*a^7*b^3
*c^5*d^2 - 10*a^8*b^2*c^2*d^5 + 10*a^8*b^2*c^4*d^3 + 2*a^9*b*c*d^6))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*
b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d
+ 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) + (32*tan(e/2 + (f*x)/2)*(2*a^10*c*d^6 + 2*a^2*b^8*c^7 - 2*a^4*b^6*c^7 - 12*a
^3*b^7*c^6*d + 10*a^5*b^5*c^6*d - 2*a^8*b^2*c*d^6 - 6*a^9*b*c^2*d^5 - 8*a^2*b^8*c^5*d^2 + 12*a^3*b^7*c^4*d^3 -
8*a^4*b^6*c^3*d^4 + 26*a^4*b^6*c^5*d^2 + 2*a^5*b^5*c^2*d^5 - 24*a^5*b^5*c^4*d^3 + 6*a^6*b^4*c^3*d^4 - 18*a^6*
b^4*c^5*d^2 + 4*a^7*b^3*c^2*d^5 + 12*a^7*b^3*c^4*d^3 + 2*a^8*b^2*c^3*d^4 + 2*a*b^9*c^6*d))/(a^7*d^3 - b^7*c^3
+ 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*
d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) + (b*((32*(2*a^4*b^8*c^8 - a^2*b^10*c^8 - a^6*b^6*c^8 +
a^12*c^2*d^6 + 2*a^3*b^9*c^7*d - 7*a^5*b^7*c^7*d - a^7*b^5*c*d^7 + 4*a^7*b^5*c^7*d + 2*a^9*b^3*c*d^7 - 4*a^11
*b*c^3*d^5 - 4*a^2*b^10*c^6*d^2 + 5*a^3*b^9*c^5*d^3 + 3*a^4*b^8*c^6*d^2 - 5*a^5*b^7*c^3*d^5 - 10*a^5*b^7*c^5*d
^3 + 4*a^6*b^6*c^2*d^6 + 5*a^6*b^6*c^4*d^4 + 6*a^6*b^6*c^6*d^2 + 6*a^7*b^5*c^3*d^5 + 5*a^7*b^5*c^5*d^3 - 7*a^8
*b^4*c^2*d^6 - 10*a^8*b^4*c^4*d^4 - 5*a^8*b^4*c^6*d^2 + 3*a^9*b^3*c^3*d^5 + 2*a^10*b^2*c^2*d^6 + 5*a^10*b^2*c^
4*d^4 + a*b^11*c^7*d - a^11*b*c*d^7))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b
^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2
) - (32*tan(e/2 + (f*x)/2)*(3*a*b^11*c^8 - 3*a^12*c*d^7 - 8*a^3*b^9*c^8 + 7*a^5*b^7*c^8 - 2*a^7*b^5*c^8 + 2*a^
12*c^3*d^5 - 4*a*b^11*c^6*d^2 - 15*a^2*b^10*c^7*d + 40*a^4*b^8*c^7*d + 4*a^6*b^6*c*d^7 - 35*a^6*b^6*c^7*d - 11
*a^8*b^4*c*d^7 + 10*a^8*b^4*c^7*d + 10*a^10*b^2*c*d^7 + 15*a^11*b*c^2*d^6 - 10*a^11*b*c^4*d^4 + 20*a^2*b^10*c^
5*d^3 - 40*a^3*b^9*c^4*d^4 + 41*a^3*b^9*c^6*d^2 + 40*a^4*b^8*c^3*d^5 - 85*a^4*b^8*c^5*d^3 - 20*a^5*b^7*c^2*d^6
+ 125*a^5*b^7*c^4*d^4 - 90*a^5*b^7*c^6*d^2 - 113*a^6*b^6*c^3*d^5 + 130*a^6*b^6*c^5*d^3 + 55*a^7*b^5*c^2*d^6 -
140*a^7*b^5*c^4*d^4 + 73*a^7*b^5*c^6*d^2 + 108*a^8*b^4*c^3*d^5 - 85*a^8*b^4*c^5*d^3 - 50*a^9*b^3*c^2*d^6 + 65
*a^9*b^3*c^4*d^4 - 20*a^9*b^3*c^6*d^2 - 37*a^10*b^2*c^3*d^5 + 20*a^10*b^2*c^5*d^3))/(a^7*d^3 - b^7*c^3 + 2*a^2
*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3
*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2))*(-(a + b)^3*(a - b)^3)^(1/2)*(b^2*d - 2*a^2*d + a*b*c))/(a^8*
d^2 - b^8*c^2 + 3*a^2*b^6*c^2 - 3*a^4*b^4*c^2 + a^6*b^2*c^2 - a^2*b^6*d^2 + 3*a^4*b^4*d^2 - 3*a^6*b^2*d^2 + 2*
a*b^7*c*d - 2*a^7*b*c*d - 6*a^3*b^5*c*d + 6*a^5*b^3*c*d))*(b^2*d - 2*a^2*d + a*b*c))/(a^8*d^2 - b^8*c^2 + 3*a^
2*b^6*c^2 - 3*a^4*b^4*c^2 + a^6*b^2*c^2 - a^2*b^6*d^2 + 3*a^4*b^4*d^2 - 3*a^6*b^2*d^2 + 2*a*b^7*c*d - 2*a^7*b*
c*d - 6*a^3*b^5*c*d + 6*a^5*b^3*c*d))*(b^2*d - 2*a^2*d + a*b*c)*1i)/(a^8*d^2 - b^8*c^2 + 3*a^2*b^6*c^2 - 3*a^4
*b^4*c^2 + a^6*b^2*c^2 - a^2*b^6*d^2 + 3*a^4*b^4*d^2 - 3*a^6*b^2*d^2 + 2*a*b^7*c*d - 2*a^7*b*c*d - 6*a^3*b^5*c
*d + 6*a^5*b^3*c*d) - (b*(-(a + b)^3*(a - b)^3)^(1/2)*((32*(a*b^7*c^3*d^3 - a^3*b^5*c*d^5 + a^3*b^5*c^5*d + 2*
a^5*b^3*c*d^5 + 2*a^2*b^6*c^4*d^2 - 6*a^3*b^5*c^3*d^3 + 2*a^4*b^4*c^2*d^4 - 5*a^4*b^4*c^4*d^2 + 8*a^5*b^3*c^3*
d^3 - 3*a^6*b^2*c^2*d^4 - a^7*b*c*d^5))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5
*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d
^2) - (32*tan(e/2 + (f*x)/2)*(a^3*b^5*c^6 - a^8*c*d^5 - 4*a*b^7*c^2*d^4 + a*b^7*c^4*d^2 + 4*a^2*b^6*c*d^5 + 2*
a^2*b^6*c^5*d - 13*a^4*b^4*c*d^5 - 5*a^4*b^4*c^5*d + 12*a^6*b^2*c*d^5 + a^7*b*c^2*d^4 - 5*a^2*b^6*c^3*d^3 + 17
*a^3*b^5*c^2*d^4 - 8*a^3*b^5*c^4*d^2 + 14*a^4*b^4*c^3*d^3 - 20*a^5*b^3*c^2*d^4 + 8*a^5*b^3*c^4*d^2 - 4*a^6*b^2
*c^3*d^3))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 -
6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) + (b*(-(a + b)^3*(a - b)^
3)^(1/2)*((32*(a^3*b^7*c^7 - a^5*b^5*c^7 + a^10*c^2*d^5 + a*b^9*c^5*d^2 + a^2*b^8*c^6*d - 6*a^4*b^6*c^6*d + a^
5*b^5*c*d^6 + 5*a^6*b^4*c^6*d - 3*a^7*b^3*c*d^6 - 5*a^9*b*c^3*d^4 - 4*a^2*b^8*c^4*d^3 + 6*a^3*b^7*c^3*d^4 - 7*
a^3*b^7*c^5*d^2 - 4*a^4*b^6*c^2*d^5 + 18*a^4*b^6*c^4*d^3 - 22*a^5*b^5*c^3*d^4 + 16*a^5*b^5*c^5*d^2 + 13*a^6*b^
4*c^2*d^5 - 24*a^6*b^4*c^4*d^3 + 21*a^7*b^3*c^3*d^4 - 10*a^7*b^3*c^5*d^2 - 10*a^8*b^2*c^2*d^5 + 10*a^8*b^2*c^4
*d^3 + 2*a^9*b*c*d^6))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*
b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) + (32*tan(e/2
+ (f*x)/2)*(2*a^10*c*d^6 + 2*a^2*b^8*c^7 - 2*a^4*b^6*c^7 - 12*a^3*b^7*c^6*d + 10*a^5*b^5*c^6*d - 2*a^8*b^2*c*
d^6 - 6*a^9*b*c^2*d^5 - 8*a^2*b^8*c^5*d^2 + 12*a^3*b^7*c^4*d^3 - 8*a^4*b^6*c^3*d^4 + 26*a^4*b^6*c^5*d^2 + 2*a^
5*b^5*c^2*d^5 - 24*a^5*b^5*c^4*d^3 + 6*a^6*b^4*c^3*d^4 - 18*a^6*b^4*c^5*d^2 + 4*a^7*b^3*c^2*d^5 + 12*a^7*b^3*c
^4*d^3 + 2*a^8*b^2*c^3*d^4 + 2*a*b^9*c^6*d))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 -
2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*
b*c*d^2) - (b*((32*(2*a^4*b^8*c^8 - a^2*b^10*c^8 - a^6*b^6*c^8 + a^12*c^2*d^6 + 2*a^3*b^9*c^7*d - 7*a^5*b^7*c^
7*d - a^7*b^5*c*d^7 + 4*a^7*b^5*c^7*d + 2*a^9*b^3*c*d^7 - 4*a^11*b*c^3*d^5 - 4*a^2*b^10*c^6*d^2 + 5*a^3*b^9*c^
5*d^3 + 3*a^4*b^8*c^6*d^2 - 5*a^5*b^7*c^3*d^5 - 10*a^5*b^7*c^5*d^3 + 4*a^6*b^6*c^2*d^6 + 5*a^6*b^6*c^4*d^4 + 6
*a^6*b^6*c^6*d^2 + 6*a^7*b^5*c^3*d^5 + 5*a^7*b^5*c^5*d^3 - 7*a^8*b^4*c^2*d^6 - 10*a^8*b^4*c^4*d^4 - 5*a^8*b^4*
c^6*d^2 + 3*a^9*b^3*c^3*d^5 + 2*a^10*b^2*c^2*d^6 + 5*a^10*b^2*c^4*d^4 + a*b^11*c^7*d - a^11*b*c*d^7))/(a^7*d^3
- b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6
*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) - (32*tan(e/2 + (f*x)/2)*(3*a*b^11*c^8 - 3*a
^12*c*d^7 - 8*a^3*b^9*c^8 + 7*a^5*b^7*c^8 - 2*a^7*b^5*c^8 + 2*a^12*c^3*d^5 - 4*a*b^11*c^6*d^2 - 15*a^2*b^10*c^
7*d + 40*a^4*b^8*c^7*d + 4*a^6*b^6*c*d^7 - 35*a^6*b^6*c^7*d - 11*a^8*b^4*c*d^7 + 10*a^8*b^4*c^7*d + 10*a^10*b^
2*c*d^7 + 15*a^11*b*c^2*d^6 - 10*a^11*b*c^4*d^4 + 20*a^2*b^10*c^5*d^3 - 40*a^3*b^9*c^4*d^4 + 41*a^3*b^9*c^6*d^
2 + 40*a^4*b^8*c^3*d^5 - 85*a^4*b^8*c^5*d^3 - 20*a^5*b^7*c^2*d^6 + 125*a^5*b^7*c^4*d^4 - 90*a^5*b^7*c^6*d^2 -
113*a^6*b^6*c^3*d^5 + 130*a^6*b^6*c^5*d^3 + 55*a^7*b^5*c^2*d^6 - 140*a^7*b^5*c^4*d^4 + 73*a^7*b^5*c^6*d^2 + 10
8*a^8*b^4*c^3*d^5 - 85*a^8*b^4*c^5*d^3 - 50*a^9*b^3*c^2*d^6 + 65*a^9*b^3*c^4*d^4 - 20*a^9*b^3*c^6*d^2 - 37*a^1
0*b^2*c^3*d^5 + 20*a^10*b^2*c^5*d^3))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b
^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2
))*(-(a + b)^3*(a - b)^3)^(1/2)*(b^2*d - 2*a^2*d + a*b*c))/(a^8*d^2 - b^8*c^2 + 3*a^2*b^6*c^2 - 3*a^4*b^4*c^2
+ a^6*b^2*c^2 - a^2*b^6*d^2 + 3*a^4*b^4*d^2 - 3*a^6*b^2*d^2 + 2*a*b^7*c*d - 2*a^7*b*c*d - 6*a^3*b^5*c*d + 6*a^
5*b^3*c*d))*(b^2*d - 2*a^2*d + a*b*c))/(a^8*d^2 - b^8*c^2 + 3*a^2*b^6*c^2 - 3*a^4*b^4*c^2 + a^6*b^2*c^2 - a^2*
b^6*d^2 + 3*a^4*b^4*d^2 - 3*a^6*b^2*d^2 + 2*a*b^7*c*d - 2*a^7*b*c*d - 6*a^3*b^5*c*d + 6*a^5*b^3*c*d))*(b^2*d -
2*a^2*d + a*b*c)*1i)/(a^8*d^2 - b^8*c^2 + 3*a^2*b^6*c^2 - 3*a^4*b^4*c^2 + a^6*b^2*c^2 - a^2*b^6*d^2 + 3*a^4*b
^4*d^2 - 3*a^6*b^2*d^2 + 2*a*b^7*c*d - 2*a^7*b*c*d - 6*a^3*b^5*c*d + 6*a^5*b^3*c*d))/((64*(2*a^2*b^4*c^2*d^3 -
3*a^3*b^3*c*d^4 + a^3*b^3*c^3*d^2 - 3*a^4*b^2*c^2*d^3 + a*b^5*c*d^4 + 2*a^5*b*c*d^4))/(a^7*d^3 - b^7*c^3 + 2*
a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2
+ 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) + (64*tan(e/2 + (f*x)/2)*(2*a^2*b^4*c*d^4 - 4*a^4*b^2*c*d^4
+ 2*a^3*b^3*c^2*d^3))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*
b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) - (b*(-(a + b
)^3*(a - b)^3)^(1/2)*((32*tan(e/2 + (f*x)/2)*(a^3*b^5*c^6 - a^8*c*d^5 - 4*a*b^7*c^2*d^4 + a*b^7*c^4*d^2 + 4*a^
2*b^6*c*d^5 + 2*a^2*b^6*c^5*d - 13*a^4*b^4*c*d^5 - 5*a^4*b^4*c^5*d + 12*a^6*b^2*c*d^5 + a^7*b*c^2*d^4 - 5*a^2*
b^6*c^3*d^3 + 17*a^3*b^5*c^2*d^4 - 8*a^3*b^5*c^4*d^2 + 14*a^4*b^4*c^3*d^3 - 20*a^5*b^3*c^2*d^4 + 8*a^5*b^3*c^4
*d^2 - 4*a^6*b^2*c^3*d^3))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*
a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) - (32*(a*
b^7*c^3*d^3 - a^3*b^5*c*d^5 + a^3*b^5*c^5*d + 2*a^5*b^3*c*d^5 + 2*a^2*b^6*c^4*d^2 - 6*a^3*b^5*c^3*d^3 + 2*a^4*
b^4*c^2*d^4 - 5*a^4*b^4*c^4*d^2 + 8*a^5*b^3*c^3*d^3 - 3*a^6*b^2*c^2*d^4 - a^7*b*c*d^5))/(a^7*d^3 - b^7*c^3 + 2
*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2
+ 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) + (b*(-(a + b)^3*(a - b)^3)^(1/2)*((32*(a^3*b^7*c^7 - a^5*
b^5*c^7 + a^10*c^2*d^5 + a*b^9*c^5*d^2 + a^2*b^8*c^6*d - 6*a^4*b^6*c^6*d + a^5*b^5*c*d^6 + 5*a^6*b^4*c^6*d - 3
*a^7*b^3*c*d^6 - 5*a^9*b*c^3*d^4 - 4*a^2*b^8*c^4*d^3 + 6*a^3*b^7*c^3*d^4 - 7*a^3*b^7*c^5*d^2 - 4*a^4*b^6*c^2*d
^5 + 18*a^4*b^6*c^4*d^3 - 22*a^5*b^5*c^3*d^4 + 16*a^5*b^5*c^5*d^2 + 13*a^6*b^4*c^2*d^5 - 24*a^6*b^4*c^4*d^3 +
21*a^7*b^3*c^3*d^4 - 10*a^7*b^3*c^5*d^2 - 10*a^8*b^2*c^2*d^5 + 10*a^8*b^2*c^4*d^3 + 2*a^9*b*c*d^6))/(a^7*d^3 -
b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a
^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) + (32*tan(e/2 + (f*x)/2)*(2*a^10*c*d^6 + 2*a^2
*b^8*c^7 - 2*a^4*b^6*c^7 - 12*a^3*b^7*c^6*d + 10*a^5*b^5*c^6*d - 2*a^8*b^2*c*d^6 - 6*a^9*b*c^2*d^5 - 8*a^2*b^8
*c^5*d^2 + 12*a^3*b^7*c^4*d^3 - 8*a^4*b^6*c^3*d^4 + 26*a^4*b^6*c^5*d^2 + 2*a^5*b^5*c^2*d^5 - 24*a^5*b^5*c^4*d^
3 + 6*a^6*b^4*c^3*d^4 - 18*a^6*b^4*c^5*d^2 + 4*a^7*b^3*c^2*d^5 + 12*a^7*b^3*c^4*d^3 + 2*a^8*b^2*c^3*d^4 + 2*a*
b^9*c^6*d))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 -
6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) + (b*((32*(2*a^4*b^8*c^8
- a^2*b^10*c^8 - a^6*b^6*c^8 + a^12*c^2*d^6 + 2*a^3*b^9*c^7*d - 7*a^5*b^7*c^7*d - a^7*b^5*c*d^7 + 4*a^7*b^5*c
^7*d + 2*a^9*b^3*c*d^7 - 4*a^11*b*c^3*d^5 - 4*a^2*b^10*c^6*d^2 + 5*a^3*b^9*c^5*d^3 + 3*a^4*b^8*c^6*d^2 - 5*a^5
*b^7*c^3*d^5 - 10*a^5*b^7*c^5*d^3 + 4*a^6*b^6*c^2*d^6 + 5*a^6*b^6*c^4*d^4 + 6*a^6*b^6*c^6*d^2 + 6*a^7*b^5*c^3*
d^5 + 5*a^7*b^5*c^5*d^3 - 7*a^8*b^4*c^2*d^6 - 10*a^8*b^4*c^4*d^4 - 5*a^8*b^4*c^6*d^2 + 3*a^9*b^3*c^3*d^5 + 2*a
^10*b^2*c^2*d^6 + 5*a^10*b^2*c^4*d^4 + a*b^11*c^7*d - a^11*b*c*d^7))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*
b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d
+ 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) - (32*tan(e/2 + (f*x)/2)*(3*a*b^11*c^8 - 3*a^12*c*d^7 - 8*a^3*b^9*c^8 + 7*a^5
*b^7*c^8 - 2*a^7*b^5*c^8 + 2*a^12*c^3*d^5 - 4*a*b^11*c^6*d^2 - 15*a^2*b^10*c^7*d + 40*a^4*b^8*c^7*d + 4*a^6*b^
6*c*d^7 - 35*a^6*b^6*c^7*d - 11*a^8*b^4*c*d^7 + 10*a^8*b^4*c^7*d + 10*a^10*b^2*c*d^7 + 15*a^11*b*c^2*d^6 - 10*
a^11*b*c^4*d^4 + 20*a^2*b^10*c^5*d^3 - 40*a^3*b^9*c^4*d^4 + 41*a^3*b^9*c^6*d^2 + 40*a^4*b^8*c^3*d^5 - 85*a^4*b
^8*c^5*d^3 - 20*a^5*b^7*c^2*d^6 + 125*a^5*b^7*c^4*d^4 - 90*a^5*b^7*c^6*d^2 - 113*a^6*b^6*c^3*d^5 + 130*a^6*b^6
*c^5*d^3 + 55*a^7*b^5*c^2*d^6 - 140*a^7*b^5*c^4*d^4 + 73*a^7*b^5*c^6*d^2 + 108*a^8*b^4*c^3*d^5 - 85*a^8*b^4*c^
5*d^3 - 50*a^9*b^3*c^2*d^6 + 65*a^9*b^3*c^4*d^4 - 20*a^9*b^3*c^6*d^2 - 37*a^10*b^2*c^3*d^5 + 20*a^10*b^2*c^5*d
^3))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*
b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2))*(-(a + b)^3*(a - b)^3)^(1/2)*(
b^2*d - 2*a^2*d + a*b*c))/(a^8*d^2 - b^8*c^2 + 3*a^2*b^6*c^2 - 3*a^4*b^4*c^2 + a^6*b^2*c^2 - a^2*b^6*d^2 + 3*a
^4*b^4*d^2 - 3*a^6*b^2*d^2 + 2*a*b^7*c*d - 2*a^7*b*c*d - 6*a^3*b^5*c*d + 6*a^5*b^3*c*d))*(b^2*d - 2*a^2*d + a*
b*c))/(a^8*d^2 - b^8*c^2 + 3*a^2*b^6*c^2 - 3*a^4*b^4*c^2 + a^6*b^2*c^2 - a^2*b^6*d^2 + 3*a^4*b^4*d^2 - 3*a^6*b
^2*d^2 + 2*a*b^7*c*d - 2*a^7*b*c*d - 6*a^3*b^5*c*d + 6*a^5*b^3*c*d))*(b^2*d - 2*a^2*d + a*b*c))/(a^8*d^2 - b^8
*c^2 + 3*a^2*b^6*c^2 - 3*a^4*b^4*c^2 + a^6*b^2*c^2 - a^2*b^6*d^2 + 3*a^4*b^4*d^2 - 3*a^6*b^2*d^2 + 2*a*b^7*c*d
- 2*a^7*b*c*d - 6*a^3*b^5*c*d + 6*a^5*b^3*c*d) - (b*(-(a + b)^3*(a - b)^3)^(1/2)*((32*(a*b^7*c^3*d^3 - a^3*b^
5*c*d^5 + a^3*b^5*c^5*d + 2*a^5*b^3*c*d^5 + 2*a^2*b^6*c^4*d^2 - 6*a^3*b^5*c^3*d^3 + 2*a^4*b^4*c^2*d^4 - 5*a^4*
b^4*c^4*d^2 + 8*a^5*b^3*c^3*d^3 - 3*a^6*b^2*c^2*d^4 - a^7*b*c*d^5))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b
^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d +
3*a*b^6*c^2*d - 3*a^6*b*c*d^2) - (32*tan(e/2 + (f*x)/2)*(a^3*b^5*c^6 - a^8*c*d^5 - 4*a*b^7*c^2*d^4 + a*b^7*c^
4*d^2 + 4*a^2*b^6*c*d^5 + 2*a^2*b^6*c^5*d - 13*a^4*b^4*c*d^5 - 5*a^4*b^4*c^5*d + 12*a^6*b^2*c*d^5 + a^7*b*c^2*
d^4 - 5*a^2*b^6*c^3*d^3 + 17*a^3*b^5*c^2*d^4 - 8*a^3*b^5*c^4*d^2 + 14*a^4*b^4*c^3*d^3 - 20*a^5*b^3*c^2*d^4 + 8
*a^5*b^3*c^4*d^2 - 4*a^6*b^2*c^3*d^3))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*
b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^
2) + (b*(-(a + b)^3*(a - b)^3)^(1/2)*((32*(a^3*b^7*c^7 - a^5*b^5*c^7 + a^10*c^2*d^5 + a*b^9*c^5*d^2 + a^2*b^8*
c^6*d - 6*a^4*b^6*c^6*d + a^5*b^5*c*d^6 + 5*a^6*b^4*c^6*d - 3*a^7*b^3*c*d^6 - 5*a^9*b*c^3*d^4 - 4*a^2*b^8*c^4*
d^3 + 6*a^3*b^7*c^3*d^4 - 7*a^3*b^7*c^5*d^2 - 4*a^4*b^6*c^2*d^5 + 18*a^4*b^6*c^4*d^3 - 22*a^5*b^5*c^3*d^4 + 16
*a^5*b^5*c^5*d^2 + 13*a^6*b^4*c^2*d^5 - 24*a^6*b^4*c^4*d^3 + 21*a^7*b^3*c^3*d^4 - 10*a^7*b^3*c^5*d^2 - 10*a^8*
b^2*c^2*d^5 + 10*a^8*b^2*c^4*d^3 + 2*a^9*b*c*d^6))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*
d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d -
3*a^6*b*c*d^2) + (32*tan(e/2 + (f*x)/2)*(2*a^10*c*d^6 + 2*a^2*b^8*c^7 - 2*a^4*b^6*c^7 - 12*a^3*b^7*c^6*d + 10*
a^5*b^5*c^6*d - 2*a^8*b^2*c*d^6 - 6*a^9*b*c^2*d^5 - 8*a^2*b^8*c^5*d^2 + 12*a^3*b^7*c^4*d^3 - 8*a^4*b^6*c^3*d^4
+ 26*a^4*b^6*c^5*d^2 + 2*a^5*b^5*c^2*d^5 - 24*a^5*b^5*c^4*d^3 + 6*a^6*b^4*c^3*d^4 - 18*a^6*b^4*c^5*d^2 + 4*a^
7*b^3*c^2*d^5 + 12*a^7*b^3*c^4*d^3 + 2*a^8*b^2*c^3*d^4 + 2*a*b^9*c^6*d))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 -
a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^
2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) - (b*((32*(2*a^4*b^8*c^8 - a^2*b^10*c^8 - a^6*b^6*c^8 + a^12*c^2*d^6 + 2*
a^3*b^9*c^7*d - 7*a^5*b^7*c^7*d - a^7*b^5*c*d^7 + 4*a^7*b^5*c^7*d + 2*a^9*b^3*c*d^7 - 4*a^11*b*c^3*d^5 - 4*a^2
*b^10*c^6*d^2 + 5*a^3*b^9*c^5*d^3 + 3*a^4*b^8*c^6*d^2 - 5*a^5*b^7*c^3*d^5 - 10*a^5*b^7*c^5*d^3 + 4*a^6*b^6*c^2
*d^6 + 5*a^6*b^6*c^4*d^4 + 6*a^6*b^6*c^6*d^2 + 6*a^7*b^5*c^3*d^5 + 5*a^7*b^5*c^5*d^3 - 7*a^8*b^4*c^2*d^6 - 10*
a^8*b^4*c^4*d^4 - 5*a^8*b^4*c^6*d^2 + 3*a^9*b^3*c^3*d^5 + 2*a^10*b^2*c^2*d^6 + 5*a^10*b^2*c^4*d^4 + a*b^11*c^7
*d - a^11*b*c*d^7))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5
*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) - (32*tan(e/2 +
(f*x)/2)*(3*a*b^11*c^8 - 3*a^12*c*d^7 - 8*a^3*b^9*c^8 + 7*a^5*b^7*c^8 - 2*a^7*b^5*c^8 + 2*a^12*c^3*d^5 - 4*a*b
^11*c^6*d^2 - 15*a^2*b^10*c^7*d + 40*a^4*b^8*c^7*d + 4*a^6*b^6*c*d^7 - 35*a^6*b^6*c^7*d - 11*a^8*b^4*c*d^7 + 1
0*a^8*b^4*c^7*d + 10*a^10*b^2*c*d^7 + 15*a^11*b*c^2*d^6 - 10*a^11*b*c^4*d^4 + 20*a^2*b^10*c^5*d^3 - 40*a^3*b^9
*c^4*d^4 + 41*a^3*b^9*c^6*d^2 + 40*a^4*b^8*c^3*d^5 - 85*a^4*b^8*c^5*d^3 - 20*a^5*b^7*c^2*d^6 + 125*a^5*b^7*c^4
*d^4 - 90*a^5*b^7*c^6*d^2 - 113*a^6*b^6*c^3*d^5 + 130*a^6*b^6*c^5*d^3 + 55*a^7*b^5*c^2*d^6 - 140*a^7*b^5*c^4*d
^4 + 73*a^7*b^5*c^6*d^2 + 108*a^8*b^4*c^3*d^5 - 85*a^8*b^4*c^5*d^3 - 50*a^9*b^3*c^2*d^6 + 65*a^9*b^3*c^4*d^4 -
20*a^9*b^3*c^6*d^2 - 37*a^10*b^2*c^3*d^5 + 20*a^10*b^2*c^5*d^3))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3
*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3
*a*b^6*c^2*d - 3*a^6*b*c*d^2))*(-(a + b)^3*(a - b)^3)^(1/2)*(b^2*d - 2*a^2*d + a*b*c))/(a^8*d^2 - b^8*c^2 + 3*
a^2*b^6*c^2 - 3*a^4*b^4*c^2 + a^6*b^2*c^2 - a^2*b^6*d^2 + 3*a^4*b^4*d^2 - 3*a^6*b^2*d^2 + 2*a*b^7*c*d - 2*a^7*
b*c*d - 6*a^3*b^5*c*d + 6*a^5*b^3*c*d))*(b^2*d - 2*a^2*d + a*b*c))/(a^8*d^2 - b^8*c^2 + 3*a^2*b^6*c^2 - 3*a^4*
b^4*c^2 + a^6*b^2*c^2 - a^2*b^6*d^2 + 3*a^4*b^4*d^2 - 3*a^6*b^2*d^2 + 2*a*b^7*c*d - 2*a^7*b*c*d - 6*a^3*b^5*c*
d + 6*a^5*b^3*c*d))*(b^2*d - 2*a^2*d + a*b*c))/(a^8*d^2 - b^8*c^2 + 3*a^2*b^6*c^2 - 3*a^4*b^4*c^2 + a^6*b^2*c^
2 - a^2*b^6*d^2 + 3*a^4*b^4*d^2 - 3*a^6*b^2*d^2 + 2*a*b^7*c*d - 2*a^7*b*c*d - 6*a^3*b^5*c*d + 6*a^5*b^3*c*d)))
*(-(a + b)^3*(a - b)^3)^(1/2)*(b^2*d - 2*a^2*d + a*b*c)*2i)/(f*(a^8*d^2 - b^8*c^2 + 3*a^2*b^6*c^2 - 3*a^4*b^4*
c^2 + a^6*b^2*c^2 - a^2*b^6*d^2 + 3*a^4*b^4*d^2 - 3*a^6*b^2*d^2 + 2*a*b^7*c*d - 2*a^7*b*c*d - 6*a^3*b^5*c*d +
6*a^5*b^3*c*d))

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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(1/(a+b*sin(f*x+e))**2/(c+d*sin(f*x+e)),x)

[Out]

Timed out

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