3.36 \(\int \frac{\sin (c+d x)}{(a+b x)^3} \, dx\)
Optimal. Leaf size=104 \[ -\frac{d^2 \sin \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (\frac{a d}{b}+d x\right )}{2 b^3}-\frac{d^2 \cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (x d+\frac{a d}{b}\right )}{2 b^3}-\frac{d \cos (c+d x)}{2 b^2 (a+b x)}-\frac{\sin (c+d x)}{2 b (a+b x)^2} \]
[Out]
-(d*Cos[c + d*x])/(2*b^2*(a + b*x)) - (d^2*CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/(2*b^3) - Sin[c + d*x]
/(2*b*(a + b*x)^2) - (d^2*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/(2*b^3)
________________________________________________________________________________________
Rubi [A] time = 0.127049, antiderivative size = 104, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used =
{3297, 3303, 3299, 3302} \[ -\frac{d^2 \sin \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (\frac{a d}{b}+d x\right )}{2 b^3}-\frac{d^2 \cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (x d+\frac{a d}{b}\right )}{2 b^3}-\frac{d \cos (c+d x)}{2 b^2 (a+b x)}-\frac{\sin (c+d x)}{2 b (a+b x)^2} \]
Antiderivative was successfully verified.
[In]
Int[Sin[c + d*x]/(a + b*x)^3,x]
[Out]
-(d*Cos[c + d*x])/(2*b^2*(a + b*x)) - (d^2*CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/(2*b^3) - Sin[c + d*x]
/(2*b*(a + b*x)^2) - (d^2*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/(2*b^3)
Rule 3297
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[((c + d*x)^(m + 1)*Sin[e + f*x])/(d*(
m + 1)), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[
m, -1]
Rule 3303
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]
Rule 3299
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]
Rule 3302
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]
Rubi steps
\begin{align*} \int \frac{\sin (c+d x)}{(a+b x)^3} \, dx &=-\frac{\sin (c+d x)}{2 b (a+b x)^2}+\frac{d \int \frac{\cos (c+d x)}{(a+b x)^2} \, dx}{2 b}\\ &=-\frac{d \cos (c+d x)}{2 b^2 (a+b x)}-\frac{\sin (c+d x)}{2 b (a+b x)^2}-\frac{d^2 \int \frac{\sin (c+d x)}{a+b x} \, dx}{2 b^2}\\ &=-\frac{d \cos (c+d x)}{2 b^2 (a+b x)}-\frac{\sin (c+d x)}{2 b (a+b x)^2}-\frac{\left (d^2 \cos \left (c-\frac{a d}{b}\right )\right ) \int \frac{\sin \left (\frac{a d}{b}+d x\right )}{a+b x} \, dx}{2 b^2}-\frac{\left (d^2 \sin \left (c-\frac{a d}{b}\right )\right ) \int \frac{\cos \left (\frac{a d}{b}+d x\right )}{a+b x} \, dx}{2 b^2}\\ &=-\frac{d \cos (c+d x)}{2 b^2 (a+b x)}-\frac{d^2 \text{Ci}\left (\frac{a d}{b}+d x\right ) \sin \left (c-\frac{a d}{b}\right )}{2 b^3}-\frac{\sin (c+d x)}{2 b (a+b x)^2}-\frac{d^2 \cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (\frac{a d}{b}+d x\right )}{2 b^3}\\ \end{align*}
Mathematica [A] time = 0.648289, size = 87, normalized size = 0.84 \[ -\frac{d^2 \sin \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (d \left (\frac{a}{b}+x\right )\right )+d^2 \cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (d \left (\frac{a}{b}+x\right )\right )+\frac{b (d (a+b x) \cos (c+d x)+b \sin (c+d x))}{(a+b x)^2}}{2 b^3} \]
Antiderivative was successfully verified.
[In]
Integrate[Sin[c + d*x]/(a + b*x)^3,x]
[Out]
-(d^2*CosIntegral[d*(a/b + x)]*Sin[c - (a*d)/b] + (b*(d*(a + b*x)*Cos[c + d*x] + b*Sin[c + d*x]))/(a + b*x)^2
+ d^2*Cos[c - (a*d)/b]*SinIntegral[d*(a/b + x)])/(2*b^3)
________________________________________________________________________________________
Maple [A] time = 0.01, size = 145, normalized size = 1.4 \begin{align*}{d}^{2} \left ( -{\frac{\sin \left ( dx+c \right ) }{2\, \left ( \left ( dx+c \right ) b+da-cb \right ) ^{2}b}}+{\frac{1}{2\,b} \left ( -{\frac{\cos \left ( dx+c \right ) }{ \left ( \left ( dx+c \right ) b+da-cb \right ) b}}-{\frac{1}{b} \left ({\frac{1}{b}{\it Si} \left ( dx+c+{\frac{da-cb}{b}} \right ) \cos \left ({\frac{da-cb}{b}} \right ) }-{\frac{1}{b}{\it Ci} \left ( dx+c+{\frac{da-cb}{b}} \right ) \sin \left ({\frac{da-cb}{b}} \right ) } \right ) } \right ) } \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sin(d*x+c)/(b*x+a)^3,x)
[Out]
d^2*(-1/2*sin(d*x+c)/((d*x+c)*b+d*a-c*b)^2/b+1/2*(-cos(d*x+c)/((d*x+c)*b+d*a-c*b)/b-(Si(d*x+c+(a*d-b*c)/b)*cos
((a*d-b*c)/b)/b-Ci(d*x+c+(a*d-b*c)/b)*sin((a*d-b*c)/b)/b)/b)/b)
________________________________________________________________________________________
Maxima [C] time = 1.49303, size = 269, normalized size = 2.59 \begin{align*} \frac{d^{3}{\left (-i \, E_{3}\left (\frac{i \,{\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right ) + i \, E_{3}\left (-\frac{i \,{\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right )\right )} \cos \left (-\frac{b c - a d}{b}\right ) + d^{3}{\left (E_{3}\left (\frac{i \,{\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right ) + E_{3}\left (-\frac{i \,{\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right )\right )} \sin \left (-\frac{b c - a d}{b}\right )}{2 \,{\left ({\left (d x + c\right )}^{2} b^{3} + b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2} - 2 \,{\left (b^{3} c - a b^{2} d\right )}{\left (d x + c\right )}\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sin(d*x+c)/(b*x+a)^3,x, algorithm="maxima")
[Out]
1/2*(d^3*(-I*exp_integral_e(3, (I*(d*x + c)*b - I*b*c + I*a*d)/b) + I*exp_integral_e(3, -(I*(d*x + c)*b - I*b*
c + I*a*d)/b))*cos(-(b*c - a*d)/b) + d^3*(exp_integral_e(3, (I*(d*x + c)*b - I*b*c + I*a*d)/b) + exp_integral_
e(3, -(I*(d*x + c)*b - I*b*c + I*a*d)/b))*sin(-(b*c - a*d)/b))/(((d*x + c)^2*b^3 + b^3*c^2 - 2*a*b^2*c*d + a^2
*b*d^2 - 2*(b^3*c - a*b^2*d)*(d*x + c))*d)
________________________________________________________________________________________
Fricas [B] time = 1.40107, size = 471, normalized size = 4.53 \begin{align*} -\frac{2 \, b^{2} \sin \left (d x + c\right ) + 2 \,{\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \cos \left (-\frac{b c - a d}{b}\right ) \operatorname{Si}\left (\frac{b d x + a d}{b}\right ) + 2 \,{\left (b^{2} d x + a b d\right )} \cos \left (d x + c\right ) -{\left ({\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \operatorname{Ci}\left (\frac{b d x + a d}{b}\right ) +{\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \operatorname{Ci}\left (-\frac{b d x + a d}{b}\right )\right )} \sin \left (-\frac{b c - a d}{b}\right )}{4 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sin(d*x+c)/(b*x+a)^3,x, algorithm="fricas")
[Out]
-1/4*(2*b^2*sin(d*x + c) + 2*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*cos(-(b*c - a*d)/b)*sin_integral((b*d*x + a
*d)/b) + 2*(b^2*d*x + a*b*d)*cos(d*x + c) - ((b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*cos_integral((b*d*x + a*d)/
b) + (b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*cos_integral(-(b*d*x + a*d)/b))*sin(-(b*c - a*d)/b))/(b^5*x^2 + 2*a
*b^4*x + a^2*b^3)
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin{\left (c + d x \right )}}{\left (a + b x\right )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sin(d*x+c)/(b*x+a)**3,x)
[Out]
Integral(sin(c + d*x)/(a + b*x)**3, x)
________________________________________________________________________________________
Giac [C] time = 1.47393, size = 7731, normalized size = 74.34 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sin(d*x+c)/(b*x+a)^3,x, algorithm="giac")
[Out]
-1/4*(b^2*d^2*x^2*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - b^2*d^2*
x^2*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*b^2*d^2*x^2*sin_int
egral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*b^2*d^2*x^2*real_part(cos_integral(d*x
+ a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b) + 2*b^2*d^2*x^2*real_part(cos_integral(-d*x - a*d/b))*ta
n(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b) - 2*b^2*d^2*x^2*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*t
an(1/2*c)*tan(1/2*a*d/b)^2 - 2*b^2*d^2*x^2*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan
(1/2*a*d/b)^2 + 2*a*b*d^2*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2
- 2*a*b*d^2*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 4*a*b*d^2*x
*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - b^2*d^2*x^2*imag_part(cos_integr
al(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 + b^2*d^2*x^2*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^
2*tan(1/2*c)^2 - 2*b^2*d^2*x^2*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)^2 + 4*b^2*d^2*x^2*imag_
part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b) - 4*b^2*d^2*x^2*imag_part(cos_integra
l(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b) + 8*b^2*d^2*x^2*sin_integral((b*d*x + a*d)/b)*tan(1/
2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b) + 4*a*b*d^2*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c
)^2*tan(1/2*a*d/b) + 4*a*b*d^2*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d
/b) - b^2*d^2*x^2*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + b^2*d^2*x^2*imag_part
(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 - 2*b^2*d^2*x^2*sin_integral((b*d*x + a*d)/b)*tan
(1/2*d*x)^2*tan(1/2*a*d/b)^2 - 4*a*b*d^2*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(
1/2*a*d/b)^2 - 4*a*b*d^2*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 +
b^2*d^2*x^2*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - b^2*d^2*x^2*imag_part(cos_int
egral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*b^2*d^2*x^2*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)^2*
tan(1/2*a*d/b)^2 + a^2*d^2*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 -
a^2*d^2*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a^2*d^2*sin_in
tegral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*b^2*d^2*x^2*real_part(cos_integral(d*
x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) + 2*b^2*d^2*x^2*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan
(1/2*c) - 2*a*b*d^2*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*a*b*d^2*x*imag_part
(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 - 4*a*b*d^2*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*d
*x)^2*tan(1/2*c)^2 - 2*b^2*d^2*x^2*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b) - 2*b^2*
d^2*x^2*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b) + 8*a*b*d^2*x*imag_part(cos_integr
al(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b) - 8*a*b*d^2*x*imag_part(cos_integral(-d*x - a*d/b))*
tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b) + 16*a*b*d^2*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c
)*tan(1/2*a*d/b) + 2*b^2*d^2*x^2*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b) + 2*b^2*d^2*
x^2*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b) + 2*a^2*d^2*real_part(cos_integral(d*x +
a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b) + 2*a^2*d^2*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*
d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b) - 2*a*b*d^2*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*a
*d/b)^2 + 2*a*b*d^2*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 - 4*a*b*d^2*x*sin_
integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 - 2*b^2*d^2*x^2*real_part(cos_integral(d*x + a*d/b))
*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*b^2*d^2*x^2*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2
- 2*a^2*d^2*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*a^2*d^2*real_
part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 + 2*a*b*d^2*x*imag_part(cos_integr
al(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 2*a*b*d^2*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^
2*tan(1/2*a*d/b)^2 + 4*a*b*d^2*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*b^2*d*x*tan(1
/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + b^2*d^2*x^2*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2 - b^
2*d^2*x^2*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2 + 2*b^2*d^2*x^2*sin_integral((b*d*x + a*d)/b)*t
an(1/2*d*x)^2 + 4*a*b*d^2*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) + 4*a*b*d^2*x*real_
part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) - b^2*d^2*x^2*imag_part(cos_integral(d*x + a*d/b))*
tan(1/2*c)^2 + b^2*d^2*x^2*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2 - 2*b^2*d^2*x^2*sin_integral((b*
d*x + a*d)/b)*tan(1/2*c)^2 - a^2*d^2*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 + a^2*d^
2*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*a^2*d^2*sin_integral((b*d*x + a*d)/b)*
tan(1/2*d*x)^2*tan(1/2*c)^2 - 4*a*b*d^2*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b) -
4*a*b*d^2*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b) + 4*b^2*d^2*x^2*imag_part(cos
_integral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) - 4*b^2*d^2*x^2*imag_part(cos_integral(-d*x - a*d/b))*tan(1/
2*c)*tan(1/2*a*d/b) + 8*b^2*d^2*x^2*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)*tan(1/2*a*d/b) + 4*a^2*d^2*imag_p
art(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b) - 4*a^2*d^2*imag_part(cos_integral(-d*
x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b) + 8*a^2*d^2*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*
tan(1/2*c)*tan(1/2*a*d/b) + 4*a*b*d^2*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b) + 4*a
*b*d^2*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b) - b^2*d^2*x^2*imag_part(cos_integra
l(d*x + a*d/b))*tan(1/2*a*d/b)^2 + b^2*d^2*x^2*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b)^2 - 2*b^2*
d^2*x^2*sin_integral((b*d*x + a*d)/b)*tan(1/2*a*d/b)^2 - a^2*d^2*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*
d*x)^2*tan(1/2*a*d/b)^2 + a^2*d^2*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 - 2*a^
2*d^2*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 - 4*a*b*d^2*x*real_part(cos_integral(d*x +
a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2 - 4*a*b*d^2*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*
d/b)^2 + a^2*d^2*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - a^2*d^2*imag_part(cos_in
tegral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a^2*d^2*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)^2*tan
(1/2*a*d/b)^2 + 2*a*b*d*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a*b*d^2*x*imag_part(cos_integral(d*x
+ a*d/b))*tan(1/2*d*x)^2 - 2*a*b*d^2*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2 + 4*a*b*d^2*x*sin_
integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2 + 2*b^2*d^2*x^2*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c) + 2*b
^2*d^2*x^2*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c) + 2*a^2*d^2*real_part(cos_integral(d*x + a*d/b))*t
an(1/2*d*x)^2*tan(1/2*c) + 2*a^2*d^2*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) - 2*a*b*d
^2*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2 + 2*a*b*d^2*x*imag_part(cos_integral(-d*x - a*d/b))*tan
(1/2*c)^2 - 4*a*b*d^2*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)^2 + 2*b^2*d*x*tan(1/2*d*x)^2*tan(1/2*c)^2 - 2
*b^2*d^2*x^2*real_part(cos_integral(d*x + a*d/b))*tan(1/2*a*d/b) - 2*b^2*d^2*x^2*real_part(cos_integral(-d*x -
a*d/b))*tan(1/2*a*d/b) - 2*a^2*d^2*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b) - 2*a^2
*d^2*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b) + 8*a*b*d^2*x*imag_part(cos_integral(
d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) - 8*a*b*d^2*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2
*a*d/b) + 16*a*b*d^2*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)*tan(1/2*a*d/b) + 2*a^2*d^2*real_part(cos_integ
ral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b) + 2*a^2*d^2*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2*t
an(1/2*a*d/b) - 2*a*b*d^2*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*a*d/b)^2 + 2*a*b*d^2*x*imag_part(cos_
integral(-d*x - a*d/b))*tan(1/2*a*d/b)^2 - 4*a*b*d^2*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*a*d/b)^2 - 2*b^2*
d*x*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 - 2*a^2*d^2*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)
^2 - 2*a^2*d^2*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2 - 8*b^2*d*x*tan(1/2*d*x)*tan(
1/2*c)*tan(1/2*a*d/b)^2 - 2*b^2*d*x*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + b^2*d^2*x^2*imag_part(cos_integral(d*x + a
*d/b)) - b^2*d^2*x^2*imag_part(cos_integral(-d*x - a*d/b)) + 2*b^2*d^2*x^2*sin_integral((b*d*x + a*d)/b) + a^2
*d^2*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2 - a^2*d^2*imag_part(cos_integral(-d*x - a*d/b))*tan(1
/2*d*x)^2 + 2*a^2*d^2*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2 + 4*a*b*d^2*x*real_part(cos_integral(d*x +
a*d/b))*tan(1/2*c) + 4*a*b*d^2*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c) - a^2*d^2*imag_part(cos_inte
gral(d*x + a*d/b))*tan(1/2*c)^2 + a^2*d^2*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2 - 2*a^2*d^2*sin_i
ntegral((b*d*x + a*d)/b)*tan(1/2*c)^2 + 2*a*b*d*tan(1/2*d*x)^2*tan(1/2*c)^2 - 4*a*b*d^2*x*real_part(cos_integr
al(d*x + a*d/b))*tan(1/2*a*d/b) - 4*a*b*d^2*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b) + 4*a^2*d^2
*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) - 4*a^2*d^2*imag_part(cos_integral(-d*x - a*d/
b))*tan(1/2*c)*tan(1/2*a*d/b) + 8*a^2*d^2*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)*tan(1/2*a*d/b) - a^2*d^2*im
ag_part(cos_integral(d*x + a*d/b))*tan(1/2*a*d/b)^2 + a^2*d^2*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*
d/b)^2 - 2*a^2*d^2*sin_integral((b*d*x + a*d)/b)*tan(1/2*a*d/b)^2 - 2*a*b*d*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 -
8*a*b*d*tan(1/2*d*x)*tan(1/2*c)*tan(1/2*a*d/b)^2 - 4*b^2*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*a*b*d*
tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 4*b^2*tan(1/2*d*x)*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a*b*d^2*x*imag_part(cos_i
ntegral(d*x + a*d/b)) - 2*a*b*d^2*x*imag_part(cos_integral(-d*x - a*d/b)) + 4*a*b*d^2*x*sin_integral((b*d*x +
a*d)/b) - 2*b^2*d*x*tan(1/2*d*x)^2 + 2*a^2*d^2*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c) + 2*a^2*d^2*rea
l_part(cos_integral(-d*x - a*d/b))*tan(1/2*c) - 8*b^2*d*x*tan(1/2*d*x)*tan(1/2*c) - 2*b^2*d*x*tan(1/2*c)^2 - 2
*a^2*d^2*real_part(cos_integral(d*x + a*d/b))*tan(1/2*a*d/b) - 2*a^2*d^2*real_part(cos_integral(-d*x - a*d/b))
*tan(1/2*a*d/b) + 2*b^2*d*x*tan(1/2*a*d/b)^2 + a^2*d^2*imag_part(cos_integral(d*x + a*d/b)) - a^2*d^2*imag_par
t(cos_integral(-d*x - a*d/b)) + 2*a^2*d^2*sin_integral((b*d*x + a*d)/b) - 2*a*b*d*tan(1/2*d*x)^2 - 8*a*b*d*tan
(1/2*d*x)*tan(1/2*c) - 4*b^2*tan(1/2*d*x)^2*tan(1/2*c) - 2*a*b*d*tan(1/2*c)^2 - 4*b^2*tan(1/2*d*x)*tan(1/2*c)^
2 + 2*a*b*d*tan(1/2*a*d/b)^2 + 4*b^2*tan(1/2*d*x)*tan(1/2*a*d/b)^2 + 4*b^2*tan(1/2*c)*tan(1/2*a*d/b)^2 + 2*b^2
*d*x + 2*a*b*d + 4*b^2*tan(1/2*d*x) + 4*b^2*tan(1/2*c))/(b^5*x^2*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2
+ 2*a*b^4*x*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + b^5*x^2*tan(1/2*d*x)^2*tan(1/2*c)^2 + b^5*x^2*tan(1
/2*d*x)^2*tan(1/2*a*d/b)^2 + b^5*x^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + a^2*b^3*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1
/2*a*d/b)^2 + 2*a*b^4*x*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*a*b^4*x*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + 2*a*b^4*x*ta
n(1/2*c)^2*tan(1/2*a*d/b)^2 + b^5*x^2*tan(1/2*d*x)^2 + b^5*x^2*tan(1/2*c)^2 + a^2*b^3*tan(1/2*d*x)^2*tan(1/2*c
)^2 + b^5*x^2*tan(1/2*a*d/b)^2 + a^2*b^3*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + a^2*b^3*tan(1/2*c)^2*tan(1/2*a*d/b)
^2 + 2*a*b^4*x*tan(1/2*d*x)^2 + 2*a*b^4*x*tan(1/2*c)^2 + 2*a*b^4*x*tan(1/2*a*d/b)^2 + b^5*x^2 + a^2*b^3*tan(1/
2*d*x)^2 + a^2*b^3*tan(1/2*c)^2 + a^2*b^3*tan(1/2*a*d/b)^2 + 2*a*b^4*x + a^2*b^3)