3.29 \(\int \frac{x \sin (c+d x)}{(a+b x)^2} \, dx\)

Optimal. Leaf size=124 \[ \frac{\sin \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (\frac{a d}{b}+d x\right )}{b^2}-\frac{a d \cos \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (\frac{a d}{b}+d x\right )}{b^3}+\frac{a d \sin \left (c-\frac{a d}{b}\right ) \text{Si}\left (x d+\frac{a d}{b}\right )}{b^3}+\frac{\cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (x d+\frac{a d}{b}\right )}{b^2}+\frac{a \sin (c+d x)}{b^2 (a+b x)} \]

[Out]

-((a*d*Cos[c - (a*d)/b]*CosIntegral[(a*d)/b + d*x])/b^3) + (CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/b^2 +
(a*Sin[c + d*x])/(b^2*(a + b*x)) + (Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/b^2 + (a*d*Sin[c - (a*d)/b]*
SinIntegral[(a*d)/b + d*x])/b^3

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Rubi [A]  time = 0.284939, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6742, 3297, 3303, 3299, 3302} \[ \frac{\sin \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (\frac{a d}{b}+d x\right )}{b^2}-\frac{a d \cos \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (\frac{a d}{b}+d x\right )}{b^3}+\frac{a d \sin \left (c-\frac{a d}{b}\right ) \text{Si}\left (x d+\frac{a d}{b}\right )}{b^3}+\frac{\cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (x d+\frac{a d}{b}\right )}{b^2}+\frac{a \sin (c+d x)}{b^2 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[(x*Sin[c + d*x])/(a + b*x)^2,x]

[Out]

-((a*d*Cos[c - (a*d)/b]*CosIntegral[(a*d)/b + d*x])/b^3) + (CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/b^2 +
(a*Sin[c + d*x])/(b^2*(a + b*x)) + (Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/b^2 + (a*d*Sin[c - (a*d)/b]*
SinIntegral[(a*d)/b + d*x])/b^3

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

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{x \sin (c+d x)}{(a+b x)^2} \, dx &=\int \left (-\frac{a \sin (c+d x)}{b (a+b x)^2}+\frac{\sin (c+d x)}{b (a+b x)}\right ) \, dx\\ &=\frac{\int \frac{\sin (c+d x)}{a+b x} \, dx}{b}-\frac{a \int \frac{\sin (c+d x)}{(a+b x)^2} \, dx}{b}\\ &=\frac{a \sin (c+d x)}{b^2 (a+b x)}-\frac{(a d) \int \frac{\cos (c+d x)}{a+b x} \, dx}{b^2}+\frac{\cos \left (c-\frac{a d}{b}\right ) \int \frac{\sin \left (\frac{a d}{b}+d x\right )}{a+b x} \, dx}{b}+\frac{\sin \left (c-\frac{a d}{b}\right ) \int \frac{\cos \left (\frac{a d}{b}+d x\right )}{a+b x} \, dx}{b}\\ &=\frac{\text{Ci}\left (\frac{a d}{b}+d x\right ) \sin \left (c-\frac{a d}{b}\right )}{b^2}+\frac{a \sin (c+d x)}{b^2 (a+b x)}+\frac{\cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (\frac{a d}{b}+d x\right )}{b^2}-\frac{\left (a d \cos \left (c-\frac{a d}{b}\right )\right ) \int \frac{\cos \left (\frac{a d}{b}+d x\right )}{a+b x} \, dx}{b^2}+\frac{\left (a d \sin \left (c-\frac{a d}{b}\right )\right ) \int \frac{\sin \left (\frac{a d}{b}+d x\right )}{a+b x} \, dx}{b^2}\\ &=-\frac{a d \cos \left (c-\frac{a d}{b}\right ) \text{Ci}\left (\frac{a d}{b}+d x\right )}{b^3}+\frac{\text{Ci}\left (\frac{a d}{b}+d x\right ) \sin \left (c-\frac{a d}{b}\right )}{b^2}+\frac{a \sin (c+d x)}{b^2 (a+b x)}+\frac{\cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (\frac{a d}{b}+d x\right )}{b^2}+\frac{a d \sin \left (c-\frac{a d}{b}\right ) \text{Si}\left (\frac{a d}{b}+d x\right )}{b^3}\\ \end{align*}

Mathematica [A]  time = 0.437234, size = 96, normalized size = 0.77 \[ \frac{\text{CosIntegral}\left (d \left (\frac{a}{b}+x\right )\right ) \left (b \sin \left (c-\frac{a d}{b}\right )-a d \cos \left (c-\frac{a d}{b}\right )\right )+\text{Si}\left (d \left (\frac{a}{b}+x\right )\right ) \left (a d \sin \left (c-\frac{a d}{b}\right )+b \cos \left (c-\frac{a d}{b}\right )\right )+\frac{a b \sin (c+d x)}{a+b x}}{b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(CosIntegral[d*(a/b + x)]*(-(a*d*Cos[c - (a*d)/b]) + b*Sin[c - (a*d)/b]) + (a*b*Sin[c + d*x])/(a + b*x) + (b*C
os[c - (a*d)/b] + a*d*Sin[c - (a*d)/b])*SinIntegral[d*(a/b + x)])/b^3

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Maple [B]  time = 0.012, size = 315, normalized size = 2.5 \begin{align*}{\frac{1}{{d}^{2}} \left ( -{\frac{{d}^{2} \left ( da-cb \right ) }{b} \left ( -{\frac{\sin \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 ) \sin \left ({\frac{da-cb}{b}} \right ) }+{\frac{1}{b}{\it Ci} \left ( dx+c+{\frac{da-cb}{b}} \right ) \cos \left ({\frac{da-cb}{b}} \right ) } \right ) } \right ) }+{\frac{{d}^{2}}{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 ) }-{d}^{2}c \left ( -{\frac{\sin \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 ) \sin \left ({\frac{da-cb}{b}} \right ) }+{\frac{1}{b}{\it Ci} \left ( dx+c+{\frac{da-cb}{b}} \right ) \cos \left ({\frac{da-cb}{b}} \right ) } \right ) } \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d^2*(-d^2*(a*d-b*c)/b*(-sin(d*x+c)/((d*x+c)*b+d*a-c*b)/b+(Si(d*x+c+(a*d-b*c)/b)*sin((a*d-b*c)/b)/b+Ci(d*x+c+
(a*d-b*c)/b)*cos((a*d-b*c)/b)/b)/b)+d^2/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)-d^2*c*(-sin(d*x+c)/((d*x+c)*b+d*a-c*b)/b+(Si(d*x+c+(a*d-b*c)/b)*sin((a*d-b*c)/b)/b+Ci(d*x+c+(a
*d-b*c)/b)*cos((a*d-b*c)/b)/b)/b))

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Maxima [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(x*sin(d*x+c)/(b*x+a)^2,x, algorithm="maxima")

[Out]

Timed out

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Fricas [A]  time = 1.7182, size = 513, normalized size = 4.14 \begin{align*} \frac{2 \, a b \sin \left (d x + c\right ) -{\left ({\left (a b d x + a^{2} d\right )} \operatorname{Ci}\left (\frac{b d x + a d}{b}\right ) +{\left (a b d x + a^{2} d\right )} \operatorname{Ci}\left (-\frac{b d x + a d}{b}\right ) - 2 \,{\left (b^{2} x + a b\right )} \operatorname{Si}\left (\frac{b d x + a d}{b}\right )\right )} \cos \left (-\frac{b c - a d}{b}\right ) -{\left ({\left (b^{2} x + a b\right )} \operatorname{Ci}\left (\frac{b d x + a d}{b}\right ) +{\left (b^{2} x + a b\right )} \operatorname{Ci}\left (-\frac{b d x + a d}{b}\right ) + 2 \,{\left (a b d x + a^{2} d\right )} \operatorname{Si}\left (\frac{b d x + a d}{b}\right )\right )} \sin \left (-\frac{b c - a d}{b}\right )}{2 \,{\left (b^{4} x + a b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*a*b*sin(d*x + c) - ((a*b*d*x + a^2*d)*cos_integral((b*d*x + a*d)/b) + (a*b*d*x + a^2*d)*cos_integral(-(
b*d*x + a*d)/b) - 2*(b^2*x + a*b)*sin_integral((b*d*x + a*d)/b))*cos(-(b*c - a*d)/b) - ((b^2*x + a*b)*cos_inte
gral((b*d*x + a*d)/b) + (b^2*x + a*b)*cos_integral(-(b*d*x + a*d)/b) + 2*(a*b*d*x + a^2*d)*sin_integral((b*d*x
+ a*d)/b))*sin(-(b*c - a*d)/b))/(b^4*x + a*b^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \sin{\left (c + d x \right )}}{\left (a + b x\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sin(d*x+c)/(b*x+a)**2,x)

[Out]

Integral(x*sin(c + d*x)/(a + b*x)**2, x)

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Giac [C]  time = 1.39868, size = 8118, normalized size = 65.47 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(a*b*d*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)^2 + a*b*d*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)^2 - 2*a*b*d*x*imag_part(cos_integ
ral(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*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) - 4*a*b*d*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*a*b*d*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)^2
- 2*a*b*d*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)^2 + 4*a*b*d*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*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 + b^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 + a^2*d*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*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 - 2*b^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 - a*b*d*x*real_part(cos_integral(d
*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 - a*b*d*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/
2*c)^2 + 4*a*b*d*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) + 4*a*b*d*x*r
eal_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*a^2*d*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*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*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) - 2*b^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^2*d*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) - a*b*d*x*real_part(cos_int
egral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 - a*b*d*x*real_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*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)^
2 - 2*a^2*d*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)^2 + 2*b^2*x*real_pa
rt(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*b^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^2*d*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*t
an(1/2*c)*tan(1/2*a*d/b)^2 + a*b*d*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + a*b*
d*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - a*b*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*b*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*t
an(1/2*c)^2*tan(1/2*a*d/b)^2 - 2*a*b*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 - 2*a*b*d*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) + 2*a*b*d*x*imag_part(cos_integra
l(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) - 4*a*b*d*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)
+ b^2*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 - b^2*x*imag_part(cos_integral(-d*x
- a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 - a^2*d*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2
- a^2*d*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*b^2*x*sin_integral((b*d*x + a*d
)/b)*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*a*b*d*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b
) - 2*a*b*d*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b) + 4*a*b*d*x*sin_integral((b*
d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*a*d/b) - 4*b^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) + 4*b^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)
+ 4*a^2*d*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) + 4*a^2*d*real_part(c
os_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b) - 8*b^2*x*sin_integral((b*d*x + a*d)/b)*ta
n(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b) - 2*a*b*d*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a
*d/b) + 2*a*b*d*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b) - 4*a*b*d*x*sin_integral((
b*d*x + a*d)/b)*tan(1/2*c)^2*tan(1/2*a*d/b) - 2*a*b*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*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*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 - b^2*x*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*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*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + 2*b^2*x*sin_integ
ral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + 2*a*b*d*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*
c)*tan(1/2*a*d/b)^2 - 2*a*b*d*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2 + 4*a*b*d*x*
sin_integral((b*d*x + a*d)/b)*tan(1/2*c)*tan(1/2*a*d/b)^2 + 2*a*b*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*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*ta
n(1/2*a*d/b)^2 - b^2*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + b^2*x*imag_part(co
s_integral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + a^2*d*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c
)^2*tan(1/2*a*d/b)^2 + a^2*d*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 2*b^2*x*sin
_integral((b*d*x + a*d)/b)*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + a*b*d*x*real_part(cos_integral(d*x + a*d/b))*tan(1/
2*d*x)^2 + a*b*d*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2 - 2*a^2*d*imag_part(cos_integral(d*x +
a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) + 2*a^2*d*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) -
2*b^2*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) - 2*b^2*x*real_part(cos_integral(-d*x
- a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) - 4*a^2*d*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c) - a*b*d*
x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2 - a*b*d*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)
^2 + a*b*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 - a*b*imag_part(cos_integral(-d*x -
a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*a*b*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*a^2*
d*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*imag_part(cos_integral(-d*x - a
*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b) + 2*b^2*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d
/b) + 2*b^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^2*d*sin_integral((b*d*
x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*a*d/b) + 4*a*b*d*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*
a*d/b) + 4*a*b*d*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) - 4*a*b*imag_part(cos_integ
ral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b) + 4*a*b*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*b*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*a^2*d*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b) + 2*a^2*d*imag_part(cos_integr
al(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b) - 2*b^2*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(
1/2*a*d/b) - 2*b^2*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b) - 4*a^2*d*sin_integral(
(b*d*x + a*d)/b)*tan(1/2*c)^2*tan(1/2*a*d/b) - a*b*d*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*a*d/b)^2 -
a*b*d*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b)^2 + a*b*imag_part(cos_integral(d*x + a*d/b))*tan
(1/2*d*x)^2*tan(1/2*a*d/b)^2 - a*b*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*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + 2*a^2*d*imag_part(cos_integral(d*x + a*d/b)
)*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*a^2*d*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2 + 2*
b^2*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2 + 2*b^2*x*real_part(cos_integral(-d*x -
a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2 + 4*a^2*d*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)*tan(1/2*a*d/b)^2 + 4*a
*b*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 - a*b*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*
a*d/b)^2 + a*b*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 2*a*b*sin_integral((b*d*x
+ a*d)/b)*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 4*a*b*tan(1/2*d*x)*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - b^2*x*imag_part(
cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2 + b^2*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2 + a^2*d
*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2 + a^2*d*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x
)^2 - 2*b^2*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2 - 2*a*b*d*x*imag_part(cos_integral(d*x + a*d/b))*ta
n(1/2*c) + 2*a*b*d*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c) - 4*a*b*d*x*sin_integral((b*d*x + a*d)/b
)*tan(1/2*c) - 2*a*b*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) - 2*a*b*real_part(cos_inte
gral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) + b^2*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2 - b^2*
x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2 - a^2*d*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2
- a^2*d*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2 + 2*b^2*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)
^2 + 2*a*b*d*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*a*d/b) - 2*a*b*d*x*imag_part(cos_integral(-d*x - a
*d/b))*tan(1/2*a*d/b) + 4*a*b*d*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*a*d/b) + 2*a*b*real_part(cos_integral(
d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b) + 2*a*b*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1
/2*a*d/b) - 4*b^2*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) + 4*b^2*x*imag_part(cos_int
egral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) + 4*a^2*d*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)*tan(1
/2*a*d/b) + 4*a^2*d*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) - 8*b^2*x*sin_integral((b*
d*x + a*d)/b)*tan(1/2*c)*tan(1/2*a*d/b) - 2*a*b*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/
b) - 2*a*b*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b) + b^2*x*imag_part(cos_integral(d*
x + a*d/b))*tan(1/2*a*d/b)^2 - b^2*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b)^2 - a^2*d*real_part(
cos_integral(d*x + a*d/b))*tan(1/2*a*d/b)^2 - a^2*d*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b)^2 + 2
*b^2*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*a*d/b)^2 + 2*a*b*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)*
tan(1/2*a*d/b)^2 + 2*a*b*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2 + a*b*d*x*real_part
(cos_integral(d*x + a*d/b)) + a*b*d*x*real_part(cos_integral(-d*x - a*d/b)) - a*b*imag_part(cos_integral(d*x +
a*d/b))*tan(1/2*d*x)^2 + a*b*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2 - 2*a*b*sin_integral((b*d*x
+ a*d)/b)*tan(1/2*d*x)^2 - 2*a^2*d*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c) + 2*a^2*d*imag_part(cos_in
tegral(-d*x - a*d/b))*tan(1/2*c) - 2*b^2*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c) - 2*b^2*x*real_part
(cos_integral(-d*x - a*d/b))*tan(1/2*c) - 4*a^2*d*sin_integral((b*d*x + a*d)/b)*tan(1/2*c) + 4*a*b*tan(1/2*d*x
)^2*tan(1/2*c) + a*b*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2 - a*b*imag_part(cos_integral(-d*x - a*d
/b))*tan(1/2*c)^2 + 2*a*b*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)^2 + 4*a*b*tan(1/2*d*x)*tan(1/2*c)^2 + 2*a^2
*d*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*a*d/b) - 2*a^2*d*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2
*a*d/b) + 2*b^2*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*a*d/b) + 2*b^2*x*real_part(cos_integral(-d*x -
a*d/b))*tan(1/2*a*d/b) + 4*a^2*d*sin_integral((b*d*x + a*d)/b)*tan(1/2*a*d/b) - 4*a*b*imag_part(cos_integral(d
*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) + 4*a*b*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)
- 8*a*b*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)*tan(1/2*a*d/b) + a*b*imag_part(cos_integral(d*x + a*d/b))*ta
n(1/2*a*d/b)^2 - a*b*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b)^2 + 2*a*b*sin_integral((b*d*x + a*d)
/b)*tan(1/2*a*d/b)^2 - 4*a*b*tan(1/2*d*x)*tan(1/2*a*d/b)^2 - 4*a*b*tan(1/2*c)*tan(1/2*a*d/b)^2 - b^2*x*imag_pa
rt(cos_integral(d*x + a*d/b)) + b^2*x*imag_part(cos_integral(-d*x - a*d/b)) + a^2*d*real_part(cos_integral(d*x
+ a*d/b)) + a^2*d*real_part(cos_integral(-d*x - a*d/b)) - 2*b^2*x*sin_integral((b*d*x + a*d)/b) - 2*a*b*real_
part(cos_integral(d*x + a*d/b))*tan(1/2*c) - 2*a*b*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c) + 2*a*b*re
al_part(cos_integral(d*x + a*d/b))*tan(1/2*a*d/b) + 2*a*b*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b)
- a*b*imag_part(cos_integral(d*x + a*d/b)) + a*b*imag_part(cos_integral(-d*x - a*d/b)) - 2*a*b*sin_integral((
b*d*x + a*d)/b) - 4*a*b*tan(1/2*d*x) - 4*a*b*tan(1/2*c))/((b^3*x*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2
+ a*b^2*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + b^3*x*tan(1/2*d*x)^2*tan(1/2*c)^2 + b^3*x*tan(1/2*d*x)^
2*tan(1/2*a*d/b)^2 + b^3*x*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + a*b^2*tan(1/2*d*x)^2*tan(1/2*c)^2 + a*b^2*tan(1/2*d
*x)^2*tan(1/2*a*d/b)^2 + a*b^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + b^3*x*tan(1/2*d*x)^2 + b^3*x*tan(1/2*c)^2 + b^3
*x*tan(1/2*a*d/b)^2 + a*b^2*tan(1/2*d*x)^2 + a*b^2*tan(1/2*c)^2 + a*b^2*tan(1/2*a*d/b)^2 + b^3*x + a*b^2)*b)