\(\int \frac {1}{a+b \cos (d+e x)+c \sin (d+e x)} \, dx\) [399]

Optimal result

Integrand size = 20, antiderivative size = 61 \[ \int \frac {1}{a+b \cos (d+e x)+c \sin (d+e x)} \, dx=\frac {2 \arctan \left (\frac {c+(a-b) \tan \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a^2-b^2-c^2}}\right )}{\sqrt {a^2-b^2-c^2} e} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3203, 632, 210} \[ \int \frac {1}{a+b \cos (d+e x)+c \sin (d+e x)} \, dx=\frac {2 \arctan \left (\frac {(a-b) \tan \left (\frac {1}{2} (d+e x)\right )+c}{\sqrt {a^2-b^2-c^2}}\right )}{e \sqrt {a^2-b^2-c^2}} \]

[In]

Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(-1),x]

[Out]

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

Rule 210

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

Rule 632

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

Rule 3203

Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^(-1), x_Symbol] :> Module[{f = Free
Factors[Tan[(d + e*x)/2], x]}, Dist[2*(f/e), Subst[Int[1/(a + b + 2*c*f*x + (a - b)*f^2*x^2), x], x, Tan[(d +
e*x)/2]/f], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {1}{a+b+2 c x+(a-b) x^2} \, dx,x,\tan \left (\frac {1}{2} (d+e x)\right )\right )}{e} \\ & = -\frac {4 \text {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2-c^2\right )-x^2} \, dx,x,2 c+2 (a-b) \tan \left (\frac {1}{2} (d+e x)\right )\right )}{e} \\ & = \frac {2 \arctan \left (\frac {c+(a-b) \tan \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a^2-b^2-c^2}}\right )}{\sqrt {a^2-b^2-c^2} e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.93 \[ \int \frac {1}{a+b \cos (d+e x)+c \sin (d+e x)} \, dx=-\frac {2 \text {arctanh}\left (\frac {c+(a-b) \tan \left (\frac {1}{2} (d+e x)\right )}{\sqrt {-a^2+b^2+c^2}}\right )}{\sqrt {-a^2+b^2+c^2} e} \]

[In]

Integrate[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(-1),x]

[Out]

(-2*ArcTanh[(c + (a - b)*Tan[(d + e*x)/2])/Sqrt[-a^2 + b^2 + c^2]])/(Sqrt[-a^2 + b^2 + c^2]*e)

Maple [A] (verified)

Time = 0.81 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00

method	result	size
derivativedivides	\(\frac {2 \arctan \left (\frac {2 \left (a -b \right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )+2 c}{2 \sqrt {a^{2}-b^{2}-c^{2}}}\right )}{e \sqrt {a^{2}-b^{2}-c^{2}}}\)	\(61\)
default	\(\frac {2 \arctan \left (\frac {2 \left (a -b \right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )+2 c}{2 \sqrt {a^{2}-b^{2}-c^{2}}}\right )}{e \sqrt {a^{2}-b^{2}-c^{2}}}\)	\(61\)
risch	\(-\frac {\ln \left ({\mathrm e}^{i \left (e x +d \right )}+\frac {i a c \sqrt {-a^{2}+b^{2}+c^{2}}+i b \,a^{2}-i b^{3}-i b \,c^{2}+a b \sqrt {-a^{2}+b^{2}+c^{2}}-a^{2} c +b^{2} c +c^{3}}{\left (b^{2}+c^{2}\right ) \sqrt {-a^{2}+b^{2}+c^{2}}}\right )}{\sqrt {-a^{2}+b^{2}+c^{2}}\, e}+\frac {\ln \left ({\mathrm e}^{i \left (e x +d \right )}+\frac {i a c \sqrt {-a^{2}+b^{2}+c^{2}}-i b \,a^{2}+i b^{3}+i b \,c^{2}+a b \sqrt {-a^{2}+b^{2}+c^{2}}+a^{2} c -b^{2} c -c^{3}}{\left (b^{2}+c^{2}\right ) \sqrt {-a^{2}+b^{2}+c^{2}}}\right )}{\sqrt {-a^{2}+b^{2}+c^{2}}\, e}\)	\(253\)

[In]

int(1/(a+b*cos(e*x+d)+c*sin(e*x+d)),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (56) = 112\).

Time = 0.30 (sec) , antiderivative size = 434, normalized size of antiderivative = 7.11 \[ \int \frac {1}{a+b \cos (d+e x)+c \sin (d+e x)} \, dx=\left [-\frac {\sqrt {-a^{2} + b^{2} + c^{2}} \log \left (-\frac {a^{2} b^{2} - 2 \, b^{4} - c^{4} - {\left (a^{2} + 3 \, b^{2}\right )} c^{2} - {\left (2 \, a^{2} b^{2} - b^{4} - 2 \, a^{2} c^{2} + c^{4}\right )} \cos \left (e x + d\right )^{2} - 2 \, {\left (a b^{3} + a b c^{2}\right )} \cos \left (e x + d\right ) - 2 \, {\left (a b^{2} c + a c^{3} - {\left (b c^{3} - {\left (2 \, a^{2} b - b^{3}\right )} c\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right ) + 2 \, {\left (2 \, a b c \cos \left (e x + d\right )^{2} - a b c + {\left (b^{2} c + c^{3}\right )} \cos \left (e x + d\right ) - {\left (b^{3} + b c^{2} + {\left (a b^{2} - a c^{2}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )\right )} \sqrt {-a^{2} + b^{2} + c^{2}}}{2 \, a b \cos \left (e x + d\right ) + {\left (b^{2} - c^{2}\right )} \cos \left (e x + d\right )^{2} + a^{2} + c^{2} + 2 \, {\left (b c \cos \left (e x + d\right ) + a c\right )} \sin \left (e x + d\right )}\right )}{2 \, {\left (a^{2} - b^{2} - c^{2}\right )} e}, \frac {\arctan \left (-\frac {{\left (a b \cos \left (e x + d\right ) + a c \sin \left (e x + d\right ) + b^{2} + c^{2}\right )} \sqrt {a^{2} - b^{2} - c^{2}}}{{\left (c^{3} - {\left (a^{2} - b^{2}\right )} c\right )} \cos \left (e x + d\right ) + {\left (a^{2} b - b^{3} - b c^{2}\right )} \sin \left (e x + d\right )}\right )}{\sqrt {a^{2} - b^{2} - c^{2}} e}\right ] \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3179 vs. \(2 (46) = 92\).

Time = 113.18 (sec) , antiderivative size = 3179, normalized size of antiderivative = 52.11 \[ \int \frac {1}{a+b \cos (d+e x)+c \sin (d+e x)} \, dx=\text {Too large to display} \]

[In]

integrate(1/(a+b*cos(e*x+d)+c*sin(e*x+d)),x)

[Out]

Piecewise((x/(a + b*cos(d) + c*sin(d)), Eq(e, 0)), (log(b/c + tan(d/2 + e*x/2))/(c*e), Eq(a, b)), (32*b**5/(32
*b**6*e*tan(d/2 + e*x/2) - 16*b**5*c*e + 32*b**5*e*sqrt(b**2 + c**2)*tan(d/2 + e*x/2) + 48*b**4*c**2*e*tan(d/2
+ e*x/2) - 16*b**4*c*e*sqrt(b**2 + c**2) - 20*b**3*c**3*e + 32*b**3*c**2*e*sqrt(b**2 + c**2)*tan(d/2 + e*x/2)
+ 18*b**2*c**4*e*tan(d/2 + e*x/2) - 12*b**2*c**3*e*sqrt(b**2 + c**2) - 5*b*c**5*e + 6*b*c**4*e*sqrt(b**2 + c*
*2)*tan(d/2 + e*x/2) + c**6*e*tan(d/2 + e*x/2) - c**5*e*sqrt(b**2 + c**2)) + 32*b**4*sqrt(b**2 + c**2)/(32*b**
6*e*tan(d/2 + e*x/2) - 16*b**5*c*e + 32*b**5*e*sqrt(b**2 + c**2)*tan(d/2 + e*x/2) + 48*b**4*c**2*e*tan(d/2 + e
*x/2) - 16*b**4*c*e*sqrt(b**2 + c**2) - 20*b**3*c**3*e + 32*b**3*c**2*e*sqrt(b**2 + c**2)*tan(d/2 + e*x/2) + 1
8*b**2*c**4*e*tan(d/2 + e*x/2) - 12*b**2*c**3*e*sqrt(b**2 + c**2) - 5*b*c**5*e + 6*b*c**4*e*sqrt(b**2 + c**2)*
tan(d/2 + e*x/2) + c**6*e*tan(d/2 + e*x/2) - c**5*e*sqrt(b**2 + c**2)) + 40*b**3*c**2/(32*b**6*e*tan(d/2 + e*x
/2) - 16*b**5*c*e + 32*b**5*e*sqrt(b**2 + c**2)*tan(d/2 + e*x/2) + 48*b**4*c**2*e*tan(d/2 + e*x/2) - 16*b**4*c
*e*sqrt(b**2 + c**2) - 20*b**3*c**3*e + 32*b**3*c**2*e*sqrt(b**2 + c**2)*tan(d/2 + e*x/2) + 18*b**2*c**4*e*tan
(d/2 + e*x/2) - 12*b**2*c**3*e*sqrt(b**2 + c**2) - 5*b*c**5*e + 6*b*c**4*e*sqrt(b**2 + c**2)*tan(d/2 + e*x/2)
+ c**6*e*tan(d/2 + e*x/2) - c**5*e*sqrt(b**2 + c**2)) + 24*b**2*c**2*sqrt(b**2 + c**2)/(32*b**6*e*tan(d/2 + e*
x/2) - 16*b**5*c*e + 32*b**5*e*sqrt(b**2 + c**2)*tan(d/2 + e*x/2) + 48*b**4*c**2*e*tan(d/2 + e*x/2) - 16*b**4*
c*e*sqrt(b**2 + c**2) - 20*b**3*c**3*e + 32*b**3*c**2*e*sqrt(b**2 + c**2)*tan(d/2 + e*x/2) + 18*b**2*c**4*e*ta
n(d/2 + e*x/2) - 12*b**2*c**3*e*sqrt(b**2 + c**2) - 5*b*c**5*e + 6*b*c**4*e*sqrt(b**2 + c**2)*tan(d/2 + e*x/2)
+ c**6*e*tan(d/2 + e*x/2) - c**5*e*sqrt(b**2 + c**2)) + 10*b*c**4/(32*b**6*e*tan(d/2 + e*x/2) - 16*b**5*c*e +
32*b**5*e*sqrt(b**2 + c**2)*tan(d/2 + e*x/2) + 48*b**4*c**2*e*tan(d/2 + e*x/2) - 16*b**4*c*e*sqrt(b**2 + c**2
) - 20*b**3*c**3*e + 32*b**3*c**2*e*sqrt(b**2 + c**2)*tan(d/2 + e*x/2) + 18*b**2*c**4*e*tan(d/2 + e*x/2) - 12*
b**2*c**3*e*sqrt(b**2 + c**2) - 5*b*c**5*e + 6*b*c**4*e*sqrt(b**2 + c**2)*tan(d/2 + e*x/2) + c**6*e*tan(d/2 +
e*x/2) - c**5*e*sqrt(b**2 + c**2)) + 2*c**4*sqrt(b**2 + c**2)/(32*b**6*e*tan(d/2 + e*x/2) - 16*b**5*c*e + 32*b
**5*e*sqrt(b**2 + c**2)*tan(d/2 + e*x/2) + 48*b**4*c**2*e*tan(d/2 + e*x/2) - 16*b**4*c*e*sqrt(b**2 + c**2) - 2
0*b**3*c**3*e + 32*b**3*c**2*e*sqrt(b**2 + c**2)*tan(d/2 + e*x/2) + 18*b**2*c**4*e*tan(d/2 + e*x/2) - 12*b**2*
c**3*e*sqrt(b**2 + c**2) - 5*b*c**5*e + 6*b*c**4*e*sqrt(b**2 + c**2)*tan(d/2 + e*x/2) + c**6*e*tan(d/2 + e*x/2
) - c**5*e*sqrt(b**2 + c**2)), Eq(a, -sqrt(b**2 + c**2))), (32*b**5/(32*b**6*e*tan(d/2 + e*x/2) - 16*b**5*c*e
- 32*b**5*e*sqrt(b**2 + c**2)*tan(d/2 + e*x/2) + 48*b**4*c**2*e*tan(d/2 + e*x/2) + 16*b**4*c*e*sqrt(b**2 + c**
2) - 20*b**3*c**3*e - 32*b**3*c**2*e*sqrt(b**2 + c**2)*tan(d/2 + e*x/2) + 18*b**2*c**4*e*tan(d/2 + e*x/2) + 12
*b**2*c**3*e*sqrt(b**2 + c**2) - 5*b*c**5*e - 6*b*c**4*e*sqrt(b**2 + c**2)*tan(d/2 + e*x/2) + c**6*e*tan(d/2 +
e*x/2) + c**5*e*sqrt(b**2 + c**2)) - 32*b**4*sqrt(b**2 + c**2)/(32*b**6*e*tan(d/2 + e*x/2) - 16*b**5*c*e - 32
*b**5*e*sqrt(b**2 + c**2)*tan(d/2 + e*x/2) + 48*b**4*c**2*e*tan(d/2 + e*x/2) + 16*b**4*c*e*sqrt(b**2 + c**2) -
20*b**3*c**3*e - 32*b**3*c**2*e*sqrt(b**2 + c**2)*tan(d/2 + e*x/2) + 18*b**2*c**4*e*tan(d/2 + e*x/2) + 12*b**
2*c**3*e*sqrt(b**2 + c**2) - 5*b*c**5*e - 6*b*c**4*e*sqrt(b**2 + c**2)*tan(d/2 + e*x/2) + c**6*e*tan(d/2 + e*x
/2) + c**5*e*sqrt(b**2 + c**2)) + 40*b**3*c**2/(32*b**6*e*tan(d/2 + e*x/2) - 16*b**5*c*e - 32*b**5*e*sqrt(b**2
+ c**2)*tan(d/2 + e*x/2) + 48*b**4*c**2*e*tan(d/2 + e*x/2) + 16*b**4*c*e*sqrt(b**2 + c**2) - 20*b**3*c**3*e -
32*b**3*c**2*e*sqrt(b**2 + c**2)*tan(d/2 + e*x/2) + 18*b**2*c**4*e*tan(d/2 + e*x/2) + 12*b**2*c**3*e*sqrt(b**
2 + c**2) - 5*b*c**5*e - 6*b*c**4*e*sqrt(b**2 + c**2)*tan(d/2 + e*x/2) + c**6*e*tan(d/2 + e*x/2) + c**5*e*sqrt
(b**2 + c**2)) - 24*b**2*c**2*sqrt(b**2 + c**2)/(32*b**6*e*tan(d/2 + e*x/2) - 16*b**5*c*e - 32*b**5*e*sqrt(b**
2 + c**2)*tan(d/2 + e*x/2) + 48*b**4*c**2*e*tan(d/2 + e*x/2) + 16*b**4*c*e*sqrt(b**2 + c**2) - 20*b**3*c**3*e
- 32*b**3*c**2*e*sqrt(b**2 + c**2)*tan(d/2 + e*x/2) + 18*b**2*c**4*e*tan(d/2 + e*x/2) + 12*b**2*c**3*e*sqrt(b*
*2 + c**2) - 5*b*c**5*e - 6*b*c**4*e*sqrt(b**2 + c**2)*tan(d/2 + e*x/2) + c**6*e*tan(d/2 + e*x/2) + c**5*e*sqr
t(b**2 + c**2)) + 10*b*c**4/(32*b**6*e*tan(d/2 + e*x/2) - 16*b**5*c*e - 32*b**5*e*sqrt(b**2 + c**2)*tan(d/2 +
e*x/2) + 48*b**4*c**2*e*tan(d/2 + e*x/2) + 16*b**4*c*e*sqrt(b**2 + c**2) - 20*b**3*c**3*e - 32*b**3*c**2*e*sqr
t(b**2 + c**2)*tan(d/2 + e*x/2) + 18*b**2*c**4*e*tan(d/2 + e*x/2) + 12*b**2*c**3*e*sqrt(b**2 + c**2) - 5*b*c**
5*e - 6*b*c**4*e*sqrt(b**2 + c**2)*tan(d/2 + e*x/2) + c**6*e*tan(d/2 + e*x/2) + c**5*e*sqrt(b**2 + c**2)) - 2*
c**4*sqrt(b**2 + c**2)/(32*b**6*e*tan(d/2 + e*x/2) - 16*b**5*c*e - 32*b**5*e*sqrt(b**2 + c**2)*tan(d/2 + e*x/2
) + 48*b**4*c**2*e*tan(d/2 + e*x/2) + 16*b**4*c*e*sqrt(b**2 + c**2) - 20*b**3*c**3*e - 32*b**3*c**2*e*sqrt(b**
2 + c**2)*tan(d/2 + e*x/2) + 18*b**2*c**4*e*tan(d/2 + e*x/2) + 12*b**2*c**3*e*sqrt(b**2 + c**2) - 5*b*c**5*e -
6*b*c**4*e*sqrt(b**2 + c**2)*tan(d/2 + e*x/2) + c**6*e*tan(d/2 + e*x/2) + c**5*e*sqrt(b**2 + c**2)), Eq(a, sq
rt(b**2 + c**2))), (log(c/(a - b) + tan(d/2 + e*x/2) - sqrt(-a**2 + b**2 + c**2)/(a - b))/(e*sqrt(-a**2 + b**2
+ c**2)) - log(c/(a - b) + tan(d/2 + e*x/2) + sqrt(-a**2 + b**2 + c**2)/(a - b))/(e*sqrt(-a**2 + b**2 + c**2)
), True))

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{a+b \cos (d+e x)+c \sin (d+e x)} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(1/(a+b*cos(e*x+d)+c*sin(e*x+d)),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(c^2+b^2-a^2>0)', see `assume?`
for more de

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.46 \[ \int \frac {1}{a+b \cos (d+e x)+c \sin (d+e x)} \, dx=-\frac {2 \, {\left (\pi \left \lfloor \frac {e x + d}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) - b \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) + c}{\sqrt {a^{2} - b^{2} - c^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2} - c^{2}} e} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 28.84 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.23 \[ \int \frac {1}{a+b \cos (d+e x)+c \sin (d+e x)} \, dx=\frac {2\,\mathrm {atan}\left (\frac {c}{\sqrt {a^2-b^2-c^2}}+\frac {\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\left (2\,a-2\,b\right )}{2\,\sqrt {a^2-b^2-c^2}}\right )}{e\,\sqrt {a^2-b^2-c^2}} \]

[In]

int(1/(a + b*cos(d + e*x) + c*sin(d + e*x)),x)

[Out]

(2*atan(c/(a^2 - b^2 - c^2)^(1/2) + (tan(d/2 + (e*x)/2)*(2*a - 2*b))/(2*(a^2 - b^2 - c^2)^(1/2))))/(e*(a^2 - b
^2 - c^2)^(1/2))