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

Optimal. Leaf size=185 \[ \frac {2 b^2 \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (e+f x)\right )+b}{\sqrt {a^2-b^2}}\right )}{f \sqrt {a^2-b^2} (b c-a d)^2}+\frac {2 d \left (a c d-b \left (2 c^2-d^2\right )\right ) \tan ^{-1}\left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{f \left (c^2-d^2\right )^{3/2} (b c-a d)^2}-\frac {d^2 \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))} \]

[Out]

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

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Rubi [A]  time = 0.47, antiderivative size = 185, 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} \[ \frac {2 b^2 \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (e+f x)\right )+b}{\sqrt {a^2-b^2}}\right )}{f \sqrt {a^2-b^2} (b c-a d)^2}+\frac {2 d \left (a c d-b \left (2 c^2-d^2\right )\right ) \tan ^{-1}\left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{f \left (c^2-d^2\right )^{3/2} (b c-a d)^2}-\frac {d^2 \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2660

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[(2*e)/d, Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]

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

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Mathematica [A]  time = 0.93, size = 165, normalized size = 0.89 \[ \frac {\frac {2 b^2 \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (e+f x)\right )+b}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+\frac {2 d \left (a c d+b \left (d^2-2 c^2\right )\right ) \tan ^{-1}\left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{\left (c^2-d^2\right )^{3/2}}+\frac {d^2 (a d-b c) \cos (e+f x)}{(c-d) (c+d) (c+d \sin (e+f x))}}{f (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 174.52, size = 2882, normalized size = 15.58 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.22, size = 308, normalized size = 1.66 \[ \frac {2 \, {\left (\frac {{\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (a) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )} b^{2}}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {a^{2} - b^{2}}} - \frac {{\left (2 \, b c^{2} d - a c d^{2} - b d^{3}\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (c) + \arctan \left (\frac {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + d}{\sqrt {c^{2} - d^{2}}}\right )\right )}}{{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} - b^{2} c^{2} d^{2} + 2 \, a b c d^{3} - a^{2} d^{4}\right )} \sqrt {c^{2} - d^{2}}} - \frac {d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + c d^{2}}{{\left (b c^{4} - a c^{3} d - b c^{2} d^{2} + a c d^{3}\right )} {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + c\right )}}\right )}}{f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.34, size = 514, normalized size = 2.78 \[ \frac {2 d^{4} \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) a}{f \left (d a -c b \right )^{2} \left (\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c +2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) d +c \right ) c \left (c^{2}-d^{2}\right )}-\frac {2 d^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) b}{f \left (d a -c b \right )^{2} \left (\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c +2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) d +c \right ) \left (c^{2}-d^{2}\right )}+\frac {2 d^{3} a}{f \left (d a -c b \right )^{2} \left (\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c +2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) d +c \right ) \left (c^{2}-d^{2}\right )}-\frac {2 d^{2} c b}{f \left (d a -c b \right )^{2} \left (\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c +2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) d +c \right ) \left (c^{2}-d^{2}\right )}+\frac {2 d^{2} \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right ) a c}{f \left (d a -c b \right )^{2} \left (c^{2}-d^{2}\right )^{\frac {3}{2}}}-\frac {4 d \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right ) b \,c^{2}}{f \left (d a -c b \right )^{2} \left (c^{2}-d^{2}\right )^{\frac {3}{2}}}+\frac {2 d^{3} \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right ) b}{f \left (d a -c b \right )^{2} \left (c^{2}-d^{2}\right )^{\frac {3}{2}}}+\frac {2 b^{2} \arctan \left (\frac {2 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{f \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {a^{2}-b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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))/(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*b^2-4*a^2>0)', see `assume?`
for more details)Is 4*b^2-4*a^2 positive or negative?

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mupad [B]  time = 22.74, size = 24122, normalized size = 130.39 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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