3.362 \(\int \frac{1}{(\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x))^4} \, dx\)

Optimal. Leaf size=259 \[ -\frac{2 (c \cos (d+e x)-b \sin (d+e x))}{35 e \left (b^2+c^2\right )^{3/2} \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2}-\frac{3 (c \cos (d+e x)-b \sin (d+e x))}{35 e \left (b^2+c^2\right ) \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}-\frac{c \cos (d+e x)-b \sin (d+e x)}{7 e \sqrt{b^2+c^2} \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^4}-\frac{2 \left (c-\sqrt{b^2+c^2} \sin (d+e x)\right )}{35 c e \left (b^2+c^2\right )^{3/2} (c \cos (d+e x)-b \sin (d+e x))} \]

[Out]

-(c*Cos[d + e*x] - b*Sin[d + e*x])/(7*Sqrt[b^2 + c^2]*e*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^4)
- (3*(c*Cos[d + e*x] - b*Sin[d + e*x]))/(35*(b^2 + c^2)*e*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])
^3) - (2*(c*Cos[d + e*x] - b*Sin[d + e*x]))/(35*(b^2 + c^2)^(3/2)*e*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[
d + e*x])^2) - (2*(c - Sqrt[b^2 + c^2]*Sin[d + e*x]))/(35*c*(b^2 + c^2)^(3/2)*e*(c*Cos[d + e*x] - b*Sin[d + e*
x]))

________________________________________________________________________________________

Rubi [A]  time = 0.189269, antiderivative size = 259, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {3116, 3114} \[ -\frac{2 (c \cos (d+e x)-b \sin (d+e x))}{35 e \left (b^2+c^2\right )^{3/2} \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2}-\frac{3 (c \cos (d+e x)-b \sin (d+e x))}{35 e \left (b^2+c^2\right ) \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}-\frac{c \cos (d+e x)-b \sin (d+e x)}{7 e \sqrt{b^2+c^2} \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^4}-\frac{2 \left (c-\sqrt{b^2+c^2} \sin (d+e x)\right )}{35 c e \left (b^2+c^2\right )^{3/2} (c \cos (d+e x)-b \sin (d+e x))} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(-4),x]

[Out]

-(c*Cos[d + e*x] - b*Sin[d + e*x])/(7*Sqrt[b^2 + c^2]*e*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^4)
- (3*(c*Cos[d + e*x] - b*Sin[d + e*x]))/(35*(b^2 + c^2)*e*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])
^3) - (2*(c*Cos[d + e*x] - b*Sin[d + e*x]))/(35*(b^2 + c^2)^(3/2)*e*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[
d + e*x])^2) - (2*(c - Sqrt[b^2 + c^2]*Sin[d + e*x]))/(35*c*(b^2 + c^2)^(3/2)*e*(c*Cos[d + e*x] - b*Sin[d + e*
x]))

Rule 3116

Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^(n_), x_Symbol] :> Simp[((c*Cos[d +
e*x] - b*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^n)/(a*e*(2*n + 1)), x] + Dist[(n + 1)/(a*(2*n +
1)), Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[a^2 - b^2 -
c^2, 0] && LtQ[n, -1]

Rule 3114

Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^(-1), x_Symbol] :> -Simp[(c - a*Sin
[d + e*x])/(c*e*(c*Cos[d + e*x] - b*Sin[d + e*x])), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[a^2 - b^2 - c^2, 0]

Rubi steps

\begin{align*} \int \frac{1}{\left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^4} \, dx &=-\frac{c \cos (d+e x)-b \sin (d+e x)}{7 \sqrt{b^2+c^2} e \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^4}+\frac{3 \int \frac{1}{\left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3} \, dx}{7 \sqrt{b^2+c^2}}\\ &=-\frac{c \cos (d+e x)-b \sin (d+e x)}{7 \sqrt{b^2+c^2} e \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^4}-\frac{3 (c \cos (d+e x)-b \sin (d+e x))}{35 \left (b^2+c^2\right ) e \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}+\frac{6 \int \frac{1}{\left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2} \, dx}{35 \left (b^2+c^2\right )}\\ &=-\frac{c \cos (d+e x)-b \sin (d+e x)}{7 \sqrt{b^2+c^2} e \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^4}-\frac{3 (c \cos (d+e x)-b \sin (d+e x))}{35 \left (b^2+c^2\right ) e \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}-\frac{2 (c \cos (d+e x)-b \sin (d+e x))}{35 \left (b^2+c^2\right )^{3/2} e \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2}+\frac{2 \int \frac{1}{\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)} \, dx}{35 \left (b^2+c^2\right )^{3/2}}\\ &=-\frac{c \cos (d+e x)-b \sin (d+e x)}{7 \sqrt{b^2+c^2} e \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^4}-\frac{3 (c \cos (d+e x)-b \sin (d+e x))}{35 \left (b^2+c^2\right ) e \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}-\frac{2 (c \cos (d+e x)-b \sin (d+e x))}{35 \left (b^2+c^2\right )^{3/2} e \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2}-\frac{2 \left (c-\sqrt{b^2+c^2} \sin (d+e x)\right )}{35 c \left (b^2+c^2\right )^{3/2} e (c \cos (d+e x)-b \sin (d+e x))}\\ \end{align*}

Mathematica [B]  time = 2.11432, size = 533, normalized size = 2.06 \[ \frac{-1295 b^4 c^2 \sin (d+e x)-189 b^4 c^2 \sin (3 (d+e x))+35 b^4 c^2 \sin (5 (d+e x))-15 b^4 c^2 \sin (7 (d+e x))+896 b^3 c^2 \sqrt{b^2+c^2} \sin (2 (d+e x))-2485 b^2 c^4 \sin (d+e x)-161 b^2 c^4 \sin (3 (d+e x))+35 b^2 c^4 \sin (5 (d+e x))+15 b^2 c^4 \sin (7 (d+e x))+896 b c^4 \sqrt{b^2+c^2} \sin (2 (d+e x))+56 b^3 c^3 \cos (3 (d+e x))+20 b^3 c^3 \cos (7 (d+e x))-1190 b c \left (b^2+c^2\right )^2 \cos (d+e x)+448 c \sqrt{b^2+c^2} \left (b^4-c^4\right ) \cos (2 (d+e x))+832 b^4 c \sqrt{b^2+c^2}+1664 b^2 c^3 \sqrt{b^2+c^2}+832 c^5 \sqrt{b^2+c^2}-112 b^5 c \cos (3 (d+e x))+28 b^5 c \cos (5 (d+e x))-6 b^5 c \cos (7 (d+e x))-35 b^6 \sin (d+e x)+21 b^6 \sin (3 (d+e x))-7 b^6 \sin (5 (d+e x))+b^6 \sin (7 (d+e x))+168 b c^5 \cos (3 (d+e x))-28 b c^5 \cos (5 (d+e x))-6 b c^5 \cos (7 (d+e x))-1225 c^6 \sin (d+e x)+49 c^6 \sin (3 (d+e x))-7 c^6 \sin (5 (d+e x))-c^6 \sin (7 (d+e x))}{1120 c e \left (b^2+c^2\right ) (b \sin (d+e x)-c \cos (d+e x))^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(-4),x]

[Out]

(832*b^4*c*Sqrt[b^2 + c^2] + 1664*b^2*c^3*Sqrt[b^2 + c^2] + 832*c^5*Sqrt[b^2 + c^2] - 1190*b*c*(b^2 + c^2)^2*C
os[d + e*x] + 448*c*Sqrt[b^2 + c^2]*(b^4 - c^4)*Cos[2*(d + e*x)] - 112*b^5*c*Cos[3*(d + e*x)] + 56*b^3*c^3*Cos
[3*(d + e*x)] + 168*b*c^5*Cos[3*(d + e*x)] + 28*b^5*c*Cos[5*(d + e*x)] - 28*b*c^5*Cos[5*(d + e*x)] - 6*b^5*c*C
os[7*(d + e*x)] + 20*b^3*c^3*Cos[7*(d + e*x)] - 6*b*c^5*Cos[7*(d + e*x)] - 35*b^6*Sin[d + e*x] - 1295*b^4*c^2*
Sin[d + e*x] - 2485*b^2*c^4*Sin[d + e*x] - 1225*c^6*Sin[d + e*x] + 896*b^3*c^2*Sqrt[b^2 + c^2]*Sin[2*(d + e*x)
] + 896*b*c^4*Sqrt[b^2 + c^2]*Sin[2*(d + e*x)] + 21*b^6*Sin[3*(d + e*x)] - 189*b^4*c^2*Sin[3*(d + e*x)] - 161*
b^2*c^4*Sin[3*(d + e*x)] + 49*c^6*Sin[3*(d + e*x)] - 7*b^6*Sin[5*(d + e*x)] + 35*b^4*c^2*Sin[5*(d + e*x)] + 35
*b^2*c^4*Sin[5*(d + e*x)] - 7*c^6*Sin[5*(d + e*x)] + b^6*Sin[7*(d + e*x)] - 15*b^4*c^2*Sin[7*(d + e*x)] + 15*b
^2*c^4*Sin[7*(d + e*x)] - c^6*Sin[7*(d + e*x)])/(1120*c*(b^2 + c^2)*e*(-(c*Cos[d + e*x]) + b*Sin[d + e*x])^7)

________________________________________________________________________________________

Maple [B]  time = 0.318, size = 823, normalized size = 3.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*cos(e*x+d)+c*sin(e*x+d)+(b^2+c^2)^(1/2))^4,x)

[Out]

-2/e/c^6*((8*b^4+8*b^2*c^2+c^4+8*(b^2+c^2)^(1/2)*b^3+4*(b^2+c^2)^(1/2)*b*c^2)/c^2*tan(1/2*d+1/2*e*x)^6+3*(16*(
b^2+c^2)^(1/2)*b^4+12*(b^2+c^2)^(1/2)*b^2*c^2+(b^2+c^2)^(1/2)*c^4+16*b^5+20*b^3*c^2+5*b*c^4)/c^3*tan(1/2*d+1/2
*e*x)^5+2*(80*(b^2+c^2)^(1/2)*b^5+84*(b^2+c^2)^(1/2)*b^3*c^2+17*(b^2+c^2)^(1/2)*b*c^4+80*b^6+124*b^4*c^2+49*b^
2*c^4+3*c^6)/c^4*tan(1/2*d+1/2*e*x)^4+2*(160*b^7+288*b^5*c^2+150*b^3*c^4+20*b*c^6+160*(b^2+c^2)^(1/2)*b^6+208*
(b^2+c^2)^(1/2)*b^4*c^2+66*(b^2+c^2)^(1/2)*b^2*c^4+3*(b^2+c^2)^(1/2)*c^6)/c^5*tan(1/2*d+1/2*e*x)^3+3/5*(640*b^
7*(b^2+c^2)^(1/2)+992*(b^2+c^2)^(1/2)*b^5*c^2+440*(b^2+c^2)^(1/2)*b^3*c^4+50*(b^2+c^2)^(1/2)*b*c^6+640*b^8+131
2*b^6*c^2+856*b^4*c^4+186*b^2*c^6+7*c^8)/c^6*tan(1/2*d+1/2*e*x)^2+1/5*(1280*b^9+2944*b^7*c^2+2288*b^5*c^4+676*
b^3*c^6+57*b*c^8+1280*(b^2+c^2)^(1/2)*b^8+2304*(b^2+c^2)^(1/2)*b^6*c^2+1296*(b^2+c^2)^(1/2)*b^4*c^4+236*(b^2+c
^2)^(1/2)*b^2*c^6+7*(b^2+c^2)^(1/2)*c^8)/c^7*tan(1/2*d+1/2*e*x)+4/35*(640*(b^2+c^2)^(1/2)*b^9+1312*(b^2+c^2)^(
1/2)*b^7*c^2+896*(b^2+c^2)^(1/2)*b^5*c^4+238*(b^2+c^2)^(1/2)*b^3*c^6+21*(b^2+c^2)^(1/2)*b*c^8+640*b^10+1632*b^
8*c^2+1472*b^6*c^4+562*b^4*c^6+85*b^2*c^8+3*c^10)/c^8)/(tan(1/2*d+1/2*e*x)^2+2/c*(b^2+c^2)^(1/2)*tan(1/2*d+1/2
*e*x)+2*b/c*tan(1/2*d+1/2*e*x)+2/c^2*(b^2+c^2)^(1/2)*b+2/c^2*b^2+1)^3/(tan(1/2*d+1/2*e*x)+1/c*(b^2+c^2)^(1/2)+
b/c)

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*cos(e*x+d)+c*sin(e*x+d)+(b^2+c^2)^(1/2))^4,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

________________________________________________________________________________________

Fricas [B]  time = 5.61137, size = 1643, normalized size = 6.34 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*cos(e*x+d)+c*sin(e*x+d)+(b^2+c^2)^(1/2))^4,x, algorithm="fricas")

[Out]

1/35*(2*(b^7 - 21*b^5*c^2 + 35*b^3*c^4 - 7*b*c^6)*cos(e*x + d)^7 - 7*(b^7 - 15*b^5*c^2 + 15*b^3*c^4 - b*c^6)*c
os(e*x + d)^5 - 14*(5*b^5*c^2 - 5*b^3*c^4 - 2*b*c^6)*cos(e*x + d)^3 - 7*(5*b^7 + 15*b^5*c^2 + 20*b^3*c^4 + 8*b
*c^6)*cos(e*x + d) - (35*b^6*c + 105*b^4*c^3 + 112*b^2*c^5 + 40*c^7 - 2*(7*b^6*c - 35*b^4*c^3 + 21*b^2*c^5 - c
^7)*cos(e*x + d)^6 + (35*b^6*c - 105*b^4*c^3 + 21*b^2*c^5 + c^7)*cos(e*x + d)^4 + 2*(35*b^4*c^3 + 7*b^2*c^5 -
4*c^7)*cos(e*x + d)^2)*sin(e*x + d) + 4*(3*b^6 + 16*b^4*c^2 + 23*b^2*c^4 + 10*c^6 + 7*(b^6 + b^4*c^2 - b^2*c^4
- c^6)*cos(e*x + d)^2 + 14*(b^5*c + 2*b^3*c^3 + b*c^5)*cos(e*x + d)*sin(e*x + d))*sqrt(b^2 + c^2))/((7*b^10*c
- 21*b^8*c^3 - 42*b^6*c^5 + 6*b^4*c^7 + 19*b^2*c^9 - c^11)*e*cos(e*x + d)^7 - 7*(3*b^10*c - 4*b^8*c^3 - 14*b^
6*c^5 - 4*b^4*c^7 + 3*b^2*c^9)*e*cos(e*x + d)^5 + 7*(3*b^10*c + b^8*c^3 - 7*b^6*c^5 - 5*b^4*c^7)*e*cos(e*x + d
)^3 - 7*(b^10*c + 2*b^8*c^3 + b^6*c^5)*e*cos(e*x + d) - ((b^11 - 19*b^9*c^2 - 6*b^7*c^4 + 42*b^5*c^6 + 21*b^3*
c^8 - 7*b*c^10)*e*cos(e*x + d)^6 - (3*b^11 - 36*b^9*c^2 - 46*b^7*c^4 + 28*b^5*c^6 + 35*b^3*c^8)*e*cos(e*x + d)
^4 + 3*(b^11 - 5*b^9*c^2 - 13*b^7*c^4 - 7*b^5*c^6)*e*cos(e*x + d)^2 - (b^11 + 2*b^9*c^2 + b^7*c^4)*e)*sin(e*x
+ d))

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*cos(e*x+d)+c*sin(e*x+d)+(b**2+c**2)**(1/2))**4,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.17106, size = 809, normalized size = 3.12 \begin{align*} -\frac{2 \,{\left (2560 \, b^{10} + 6528 \, b^{8} c^{2} + 5888 \, b^{6} c^{4} + 2248 \, b^{4} c^{6} + 340 \, b^{2} c^{8} + 12 \, c^{10} + 35 \,{\left (8 \, b^{4} c^{6} + 8 \, b^{2} c^{8} + c^{10} + 4 \,{\left (2 \, b^{3} c^{6} + b c^{8}\right )} \sqrt{b^{2} + c^{2}}\right )} \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right )^{6} + 105 \,{\left (16 \, b^{5} c^{5} + 20 \, b^{3} c^{7} + 5 \, b c^{9} +{\left (16 \, b^{4} c^{5} + 12 \, b^{2} c^{7} + c^{9}\right )} \sqrt{b^{2} + c^{2}}\right )} \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right )^{5} + 70 \,{\left (80 \, b^{6} c^{4} + 124 \, b^{4} c^{6} + 49 \, b^{2} c^{8} + 3 \, c^{10} +{\left (80 \, b^{5} c^{4} + 84 \, b^{3} c^{6} + 17 \, b c^{8}\right )} \sqrt{b^{2} + c^{2}}\right )} \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right )^{4} + 70 \,{\left (160 \, b^{7} c^{3} + 288 \, b^{5} c^{5} + 150 \, b^{3} c^{7} + 20 \, b c^{9} +{\left (160 \, b^{6} c^{3} + 208 \, b^{4} c^{5} + 66 \, b^{2} c^{7} + 3 \, c^{9}\right )} \sqrt{b^{2} + c^{2}}\right )} \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right )^{3} + 21 \,{\left (640 \, b^{8} c^{2} + 1312 \, b^{6} c^{4} + 856 \, b^{4} c^{6} + 186 \, b^{2} c^{8} + 7 \, c^{10} + 2 \,{\left (320 \, b^{7} c^{2} + 496 \, b^{5} c^{4} + 220 \, b^{3} c^{6} + 25 \, b c^{8}\right )} \sqrt{b^{2} + c^{2}}\right )} \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right )^{2} + 7 \,{\left (1280 \, b^{9} c + 2944 \, b^{7} c^{3} + 2288 \, b^{5} c^{5} + 676 \, b^{3} c^{7} + 57 \, b c^{9} +{\left (1280 \, b^{8} c + 2304 \, b^{6} c^{3} + 1296 \, b^{4} c^{5} + 236 \, b^{2} c^{7} + 7 \, c^{9}\right )} \sqrt{b^{2} + c^{2}}\right )} \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right ) + 4 \,{\left (640 \, b^{9} + 1312 \, b^{7} c^{2} + 896 \, b^{5} c^{4} + 238 \, b^{3} c^{6} + 21 \, b c^{8}\right )} \sqrt{b^{2} + c^{2}}\right )} e^{\left (-1\right )}}{35 \,{\left (c \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right ) + b + \sqrt{b^{2} + c^{2}}\right )}^{7} c^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*cos(e*x+d)+c*sin(e*x+d)+(b^2+c^2)^(1/2))^4,x, algorithm="giac")

[Out]

-2/35*(2560*b^10 + 6528*b^8*c^2 + 5888*b^6*c^4 + 2248*b^4*c^6 + 340*b^2*c^8 + 12*c^10 + 35*(8*b^4*c^6 + 8*b^2*
c^8 + c^10 + 4*(2*b^3*c^6 + b*c^8)*sqrt(b^2 + c^2))*tan(1/2*x*e + 1/2*d)^6 + 105*(16*b^5*c^5 + 20*b^3*c^7 + 5*
b*c^9 + (16*b^4*c^5 + 12*b^2*c^7 + c^9)*sqrt(b^2 + c^2))*tan(1/2*x*e + 1/2*d)^5 + 70*(80*b^6*c^4 + 124*b^4*c^6
+ 49*b^2*c^8 + 3*c^10 + (80*b^5*c^4 + 84*b^3*c^6 + 17*b*c^8)*sqrt(b^2 + c^2))*tan(1/2*x*e + 1/2*d)^4 + 70*(16
0*b^7*c^3 + 288*b^5*c^5 + 150*b^3*c^7 + 20*b*c^9 + (160*b^6*c^3 + 208*b^4*c^5 + 66*b^2*c^7 + 3*c^9)*sqrt(b^2 +
c^2))*tan(1/2*x*e + 1/2*d)^3 + 21*(640*b^8*c^2 + 1312*b^6*c^4 + 856*b^4*c^6 + 186*b^2*c^8 + 7*c^10 + 2*(320*b
^7*c^2 + 496*b^5*c^4 + 220*b^3*c^6 + 25*b*c^8)*sqrt(b^2 + c^2))*tan(1/2*x*e + 1/2*d)^2 + 7*(1280*b^9*c + 2944*
b^7*c^3 + 2288*b^5*c^5 + 676*b^3*c^7 + 57*b*c^9 + (1280*b^8*c + 2304*b^6*c^3 + 1296*b^4*c^5 + 236*b^2*c^7 + 7*
c^9)*sqrt(b^2 + c^2))*tan(1/2*x*e + 1/2*d) + 4*(640*b^9 + 1312*b^7*c^2 + 896*b^5*c^4 + 238*b^3*c^6 + 21*b*c^8)
*sqrt(b^2 + c^2))*e^(-1)/((c*tan(1/2*x*e + 1/2*d) + b + sqrt(b^2 + c^2))^7*c^7)