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

Optimal. Leaf size=149 \[ \frac{a^2 d \cos \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (\frac{a d}{b}+d x\right )}{b^4}-\frac{a^2 d \sin \left (c-\frac{a d}{b}\right ) \text{Si}\left (x d+\frac{a d}{b}\right )}{b^4}-\frac{a^2 \sin (c+d x)}{b^3 (a+b x)}-\frac{2 a \sin \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (\frac{a d}{b}+d x\right )}{b^3}-\frac{2 a \cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (x d+\frac{a d}{b}\right )}{b^3}-\frac{\cos (c+d x)}{b^2 d} \]

[Out]

-(Cos[c + d*x]/(b^2*d)) + (a^2*d*Cos[c - (a*d)/b]*CosIntegral[(a*d)/b + d*x])/b^4 - (2*a*CosIntegral[(a*d)/b +
d*x]*Sin[c - (a*d)/b])/b^3 - (a^2*Sin[c + d*x])/(b^3*(a + b*x)) - (2*a*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b +
d*x])/b^3 - (a^2*d*Sin[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/b^4

________________________________________________________________________________________

Rubi [A]  time = 0.363169, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353, Rules used = {6742, 2638, 3297, 3303, 3299, 3302} \[ \frac{a^2 d \cos \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (\frac{a d}{b}+d x\right )}{b^4}-\frac{a^2 d \sin \left (c-\frac{a d}{b}\right ) \text{Si}\left (x d+\frac{a d}{b}\right )}{b^4}-\frac{a^2 \sin (c+d x)}{b^3 (a+b x)}-\frac{2 a \sin \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (\frac{a d}{b}+d x\right )}{b^3}-\frac{2 a \cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (x d+\frac{a d}{b}\right )}{b^3}-\frac{\cos (c+d x)}{b^2 d} \]

Antiderivative was successfully verified.

[In]

Int[(x^2*Sin[c + d*x])/(a + b*x)^2,x]

[Out]

-(Cos[c + d*x]/(b^2*d)) + (a^2*d*Cos[c - (a*d)/b]*CosIntegral[(a*d)/b + d*x])/b^4 - (2*a*CosIntegral[(a*d)/b +
d*x]*Sin[c - (a*d)/b])/b^3 - (a^2*Sin[c + d*x])/(b^3*(a + b*x)) - (2*a*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b +
d*x])/b^3 - (a^2*d*Sin[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/b^4

Rule 6742

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

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

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^2 \sin (c+d x)}{(a+b x)^2} \, dx &=\int \left (\frac{\sin (c+d x)}{b^2}+\frac{a^2 \sin (c+d x)}{b^2 (a+b x)^2}-\frac{2 a \sin (c+d x)}{b^2 (a+b x)}\right ) \, dx\\ &=\frac{\int \sin (c+d x) \, dx}{b^2}-\frac{(2 a) \int \frac{\sin (c+d x)}{a+b x} \, dx}{b^2}+\frac{a^2 \int \frac{\sin (c+d x)}{(a+b x)^2} \, dx}{b^2}\\ &=-\frac{\cos (c+d x)}{b^2 d}-\frac{a^2 \sin (c+d x)}{b^3 (a+b x)}+\frac{\left (a^2 d\right ) \int \frac{\cos (c+d x)}{a+b x} \, dx}{b^3}-\frac{\left (2 a \cos \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{\left (2 a \sin \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{\cos (c+d x)}{b^2 d}-\frac{2 a \text{Ci}\left (\frac{a d}{b}+d x\right ) \sin \left (c-\frac{a d}{b}\right )}{b^3}-\frac{a^2 \sin (c+d x)}{b^3 (a+b x)}-\frac{2 a \cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (\frac{a d}{b}+d x\right )}{b^3}+\frac{\left (a^2 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^3}-\frac{\left (a^2 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^3}\\ &=-\frac{\cos (c+d x)}{b^2 d}+\frac{a^2 d \cos \left (c-\frac{a d}{b}\right ) \text{Ci}\left (\frac{a d}{b}+d x\right )}{b^4}-\frac{2 a \text{Ci}\left (\frac{a d}{b}+d x\right ) \sin \left (c-\frac{a d}{b}\right )}{b^3}-\frac{a^2 \sin (c+d x)}{b^3 (a+b x)}-\frac{2 a \cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (\frac{a d}{b}+d x\right )}{b^3}-\frac{a^2 d \sin \left (c-\frac{a d}{b}\right ) \text{Si}\left (\frac{a d}{b}+d x\right )}{b^4}\\ \end{align*}

Mathematica [A]  time = 0.773836, size = 117, normalized size = 0.79 \[ \frac{b \left (-\frac{a^2 \sin (c+d x)}{a+b x}-\frac{b \cos (c+d x)}{d}\right )+a \text{CosIntegral}\left (d \left (\frac{a}{b}+x\right )\right ) \left (a d \cos \left (c-\frac{a d}{b}\right )-2 b \sin \left (c-\frac{a d}{b}\right )\right )-a \text{Si}\left (d \left (\frac{a}{b}+x\right )\right ) \left (a d \sin \left (c-\frac{a d}{b}\right )+2 b \cos \left (c-\frac{a d}{b}\right )\right )}{b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*CosIntegral[d*(a/b + x)]*(a*d*Cos[c - (a*d)/b] - 2*b*Sin[c - (a*d)/b]) + b*(-((b*Cos[c + d*x])/d) - (a^2*Si
n[c + d*x])/(a + b*x)) - a*(2*b*Cos[c - (a*d)/b] + a*d*Sin[c - (a*d)/b])*SinIntegral[d*(a/b + x)])/b^4

________________________________________________________________________________________

Maple [B]  time = 0.013, size = 553, normalized size = 3.7 \begin{align*}{\frac{1}{{d}^{3}} \left ( -{\frac{{d}^{2}\cos \left ( dx+c \right ) }{{b}^{2}}}+{\frac{ \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ){d}^{2}}{{b}^{2}} \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 ) }-2\,{\frac{ \left ( da-cb \right ){d}^{2}}{{b}^{2}} \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 ) }+2\,{\frac{ \left ( da-cb \right ){d}^{2}c}{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 ) }-2\,{\frac{{d}^{2}c}{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}^{2} \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^2*sin(d*x+c)/(b*x+a)^2,x)

[Out]

1/d^3*(-d^2/b^2*cos(d*x+c)+(a^2*d^2-2*a*b*c*d+b^2*c^2)*d^2/b^2*(-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)-2/b^2*(a*d-b*c)*d^2*(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)+2/b*(a*d-b*c)*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)-2*d^2*c/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^2*(-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))

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Fricas [A]  time = 1.81944, size = 626, normalized size = 4.2 \begin{align*} -\frac{2 \, a^{2} b d \sin \left (d x + c\right ) + 2 \,{\left (b^{3} x + a b^{2}\right )} \cos \left (d x + c\right ) -{\left ({\left (a^{2} b d^{2} x + a^{3} d^{2}\right )} \operatorname{Ci}\left (\frac{b d x + a d}{b}\right ) +{\left (a^{2} b d^{2} x + a^{3} d^{2}\right )} \operatorname{Ci}\left (-\frac{b d x + a d}{b}\right ) - 4 \,{\left (a b^{2} d x + a^{2} b d\right )} \operatorname{Si}\left (\frac{b d x + a d}{b}\right )\right )} \cos \left (-\frac{b c - a d}{b}\right ) - 2 \,{\left ({\left (a b^{2} d x + a^{2} b d\right )} \operatorname{Ci}\left (\frac{b d x + a d}{b}\right ) +{\left (a b^{2} d x + a^{2} b d\right )} \operatorname{Ci}\left (-\frac{b d x + a d}{b}\right ) +{\left (a^{2} b d^{2} x + a^{3} d^{2}\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^{5} d x + a b^{4} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*a^2*b*d*sin(d*x + c) + 2*(b^3*x + a*b^2)*cos(d*x + c) - ((a^2*b*d^2*x + a^3*d^2)*cos_integral((b*d*x +
a*d)/b) + (a^2*b*d^2*x + a^3*d^2)*cos_integral(-(b*d*x + a*d)/b) - 4*(a*b^2*d*x + a^2*b*d)*sin_integral((b*d*
x + a*d)/b))*cos(-(b*c - a*d)/b) - 2*((a*b^2*d*x + a^2*b*d)*cos_integral((b*d*x + a*d)/b) + (a*b^2*d*x + a^2*b
*d)*cos_integral(-(b*d*x + a*d)/b) + (a^2*b*d^2*x + a^3*d^2)*sin_integral((b*d*x + a*d)/b))*sin(-(b*c - a*d)/b
))/(b^5*d*x + a*b^4*d)

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Giac [C]  time = 1.42059, size = 8462, normalized size = 56.79 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(a^2*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^2*b*d*x*r
eal_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*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) + 2*a^2*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^2*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^2*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^2*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^2*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 - 2*a*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 + 2*a*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^3*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^3*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 - 4*a*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^2*b*d
*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 - a^2*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^2*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^2*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)
- 2*a^3*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^3*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) - 4*a*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*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^3*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^2*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 - a^2*b*d*x*real_pa
rt(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + 2*a^3*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^3*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 + 4*a*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*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^3*d*s
in_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*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^2*b*d*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/
2*a*d/b)^2 - 2*a^2*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 + 2*a^2
*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 - 4*a^2*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^2*b*d*x*imag_part(cos_integral(d*x + a*d/b)
)*tan(1/2*d*x)^2*tan(1/2*c) + 2*a^2*b*d*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) - 4*
a^2*b*d*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c) + 2*a*b^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^2*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2
- a^3*d*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 - a^3*d*real_part(cos_integral(-d*x
- a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 + 4*a*b^2*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)^2 +
2*a^2*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^2*b*d*x*imag_part(cos_int
egral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b) + 4*a^2*b*d*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*
tan(1/2*a*d/b) - 8*a*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) + 8*a
*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^3*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^3*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) - 16*a*b^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*a^2*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^2*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^2*b*d*x*sin_integral((b*d*x + a*d)/b)*
tan(1/2*c)^2*tan(1/2*a*d/b) - 4*a^2*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) - 4*a^2*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^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^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^3*d*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*
a*d/b)^2 - a^3*d*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + 4*a*b^2*x*sin_integra
l((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + 2*a^2*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^2*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^2*b*
d*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)*tan(1/2*a*d/b)^2 + 4*a^2*b*real_part(cos_integral(d*x + a*d/b))*t
an(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 + 4*a^2*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^2*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a*b^
2*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + a^3*d*real_part(cos_integral(d*x + a
*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + a^3*d*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)
^2 - 4*a*b^2*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + a^2*b*d*x*real_part(cos_integral(
d*x + a*d/b))*tan(1/2*d*x)^2 + a^2*b*d*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2 - 2*a^3*d*imag_p
art(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) + 2*a^3*d*imag_part(cos_integral(-d*x - a*d/b))*tan(1
/2*d*x)^2*tan(1/2*c) - 4*a*b^2*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) - 4*a*b^2*x*re
al_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) - 4*a^3*d*sin_integral((b*d*x + a*d)/b)*tan(1/2*
d*x)^2*tan(1/2*c) - a^2*b*d*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2 - a^2*b*d*x*real_part(cos_inte
gral(-d*x - a*d/b))*tan(1/2*c)^2 + 2*a^2*b*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 -
2*a^2*b*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 + 4*a^2*b*sin_integral((b*d*x + a*d)
/b)*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*a^3*d*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b) -
2*a^3*d*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b) + 4*a*b^2*x*real_part(cos_integra
l(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b) + 4*a*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^3*d*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*a*d/b) + 4*a^2*b*d*x*real_part(
cos_integral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) + 4*a^2*b*d*x*real_part(cos_integral(-d*x - a*d/b))*tan(1
/2*c)*tan(1/2*a*d/b) - 8*a^2*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^2*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) - 16*a^2*b*sin_integra
l((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b) - 2*a^3*d*imag_part(cos_integral(d*x + a*d/b))*tan
(1/2*c)^2*tan(1/2*a*d/b) + 2*a^3*d*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b) - 4*a*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*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^3*d*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)^2*tan(1/2*a*d/b) - a^2*
b*d*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*a*d/b)^2 - a^2*b*d*x*real_part(cos_integral(-d*x - a*d/b))*
tan(1/2*a*d/b)^2 + 2*a^2*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^2*b*imag
_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + 4*a^2*b*sin_integral((b*d*x + a*d)/b)*tan(
1/2*d*x)^2*tan(1/2*a*d/b)^2 + 2*a^3*d*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*a^3
*d*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2 + 4*a*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*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^3*d*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)*tan(1/2*a*d/b)^2 + 4*a^2*b*tan(1/2*d*x)^2*tan(1/2*c)*ta
n(1/2*a*d/b)^2 - 2*a^2*b*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a^2*b*imag_par
t(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 4*a^2*b*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)
^2*tan(1/2*a*d/b)^2 + 4*a^2*b*tan(1/2*d*x)*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 2*a*b^2*x*imag_part(cos_integral(d*
x + a*d/b))*tan(1/2*d*x)^2 + 2*a*b^2*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2 + a^3*d*real_part(
cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2 + a^3*d*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2 - 4*a*b
^2*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2 - 2*a^2*b*d*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c
) + 2*a^2*b*d*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c) - 4*a^2*b*d*x*sin_integral((b*d*x + a*d)/b)*t
an(1/2*c) - 4*a^2*b*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) - 4*a^2*b*real_part(cos_int
egral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) + 2*a*b^2*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2 -
2*a*b^2*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2 - a^3*d*real_part(cos_integral(d*x + a*d/b))*tan
(1/2*c)^2 - a^3*d*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2 + 4*a*b^2*x*sin_integral((b*d*x + a*d)/b)
*tan(1/2*c)^2 + 2*a^2*b*d*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*a*d/b) - 2*a^2*b*d*x*imag_part(cos_in
tegral(-d*x - a*d/b))*tan(1/2*a*d/b) + 4*a^2*b*d*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*a*d/b) + 4*a^2*b*real
_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b) + 4*a^2*b*real_part(cos_integral(-d*x - a*d/b))
*tan(1/2*d*x)^2*tan(1/2*a*d/b) - 8*a*b^2*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) + 8*
a*b^2*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) + 4*a^3*d*real_part(cos_integral(d*x +
a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) + 4*a^3*d*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) -
16*a*b^2*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)*tan(1/2*a*d/b) - 4*a^2*b*real_part(cos_integral(d*x + a*d
/b))*tan(1/2*c)^2*tan(1/2*a*d/b) - 4*a^2*b*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b) +
2*a*b^2*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*a*d/b)^2 - 2*a*b^2*x*imag_part(cos_integral(-d*x - a*d
/b))*tan(1/2*a*d/b)^2 - a^3*d*real_part(cos_integral(d*x + a*d/b))*tan(1/2*a*d/b)^2 - a^3*d*real_part(cos_inte
gral(-d*x - a*d/b))*tan(1/2*a*d/b)^2 + 4*a*b^2*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*a*d/b)^2 + 4*a^2*b*real
_part(cos_integral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2 + 4*a^2*b*real_part(cos_integral(-d*x - a*d/b))*t
an(1/2*c)*tan(1/2*a*d/b)^2 + a^2*b*d*x*real_part(cos_integral(d*x + a*d/b)) + a^2*b*d*x*real_part(cos_integral
(-d*x - a*d/b)) - 2*a^2*b*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2 + 2*a^2*b*imag_part(cos_integral
(-d*x - a*d/b))*tan(1/2*d*x)^2 - 4*a^2*b*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2 - 2*a^3*d*imag_part(cos_
integral(d*x + a*d/b))*tan(1/2*c) + 2*a^3*d*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c) - 4*a*b^2*x*real_
part(cos_integral(d*x + a*d/b))*tan(1/2*c) - 4*a*b^2*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c) - 4*a^
3*d*sin_integral((b*d*x + a*d)/b)*tan(1/2*c) + 4*a^2*b*tan(1/2*d*x)^2*tan(1/2*c) + 2*a^2*b*imag_part(cos_integ
ral(d*x + a*d/b))*tan(1/2*c)^2 - 2*a^2*b*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2 + 4*a^2*b*sin_inte
gral((b*d*x + a*d)/b)*tan(1/2*c)^2 + 4*a^2*b*tan(1/2*d*x)*tan(1/2*c)^2 + 2*a^3*d*imag_part(cos_integral(d*x +
a*d/b))*tan(1/2*a*d/b) - 2*a^3*d*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b) + 4*a*b^2*x*real_part(co
s_integral(d*x + a*d/b))*tan(1/2*a*d/b) + 4*a*b^2*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b) + 4*a
^3*d*sin_integral((b*d*x + a*d)/b)*tan(1/2*a*d/b) - 8*a^2*b*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)*ta
n(1/2*a*d/b) + 8*a^2*b*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) - 16*a^2*b*sin_integral
((b*d*x + a*d)/b)*tan(1/2*c)*tan(1/2*a*d/b) + 2*a^2*b*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*a*d/b)^2 -
2*a^2*b*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b)^2 + 4*a^2*b*sin_integral((b*d*x + a*d)/b)*tan(1/2
*a*d/b)^2 - 4*a^2*b*tan(1/2*d*x)*tan(1/2*a*d/b)^2 - 4*a^2*b*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*a*b^2*x*imag_part(
cos_integral(d*x + a*d/b)) + 2*a*b^2*x*imag_part(cos_integral(-d*x - a*d/b)) + a^3*d*real_part(cos_integral(d*
x + a*d/b)) + a^3*d*real_part(cos_integral(-d*x - a*d/b)) - 4*a*b^2*x*sin_integral((b*d*x + a*d)/b) - 4*a^2*b*
real_part(cos_integral(d*x + a*d/b))*tan(1/2*c) - 4*a^2*b*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c) + 4
*a^2*b*real_part(cos_integral(d*x + a*d/b))*tan(1/2*a*d/b) + 4*a^2*b*real_part(cos_integral(-d*x - a*d/b))*tan
(1/2*a*d/b) - 2*a^2*b*imag_part(cos_integral(d*x + a*d/b)) + 2*a^2*b*imag_part(cos_integral(-d*x - a*d/b)) - 4
*a^2*b*sin_integral((b*d*x + a*d)/b) - 4*a^2*b*tan(1/2*d*x) - 4*a^2*b*tan(1/2*c))/(b^5*x*tan(1/2*d*x)^2*tan(1/
2*c)^2*tan(1/2*a*d/b)^2 + a*b^4*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + b^5*x*tan(1/2*d*x)^2*tan(1/2*c)
^2 + b^5*x*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + b^5*x*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + a*b^4*tan(1/2*d*x)^2*tan(1/
2*c)^2 + a*b^4*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + a*b^4*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + b^5*x*tan(1/2*d*x)^2 +
b^5*x*tan(1/2*c)^2 + b^5*x*tan(1/2*a*d/b)^2 + a*b^4*tan(1/2*d*x)^2 + a*b^4*tan(1/2*c)^2 + a*b^4*tan(1/2*a*d/b)
^2 + b^5*x + a*b^4)