3.25 \(\int \frac{\sin (c+d x)}{x^3 (a+b x)} \, dx\)

Optimal. Leaf size=189 \[ \frac{b^2 \sin (c) \text{CosIntegral}(d x)}{a^3}-\frac{b^2 \sin \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (\frac{a d}{b}+d x\right )}{a^3}+\frac{b^2 \cos (c) \text{Si}(d x)}{a^3}-\frac{b^2 \cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (x d+\frac{a d}{b}\right )}{a^3}-\frac{b d \cos (c) \text{CosIntegral}(d x)}{a^2}+\frac{b d \sin (c) \text{Si}(d x)}{a^2}+\frac{b \sin (c+d x)}{a^2 x}-\frac{d^2 \sin (c) \text{CosIntegral}(d x)}{2 a}-\frac{d^2 \cos (c) \text{Si}(d x)}{2 a}-\frac{\sin (c+d x)}{2 a x^2}-\frac{d \cos (c+d x)}{2 a x} \]

[Out]

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

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Rubi [A]  time = 0.490763, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {6742, 3297, 3303, 3299, 3302} \[ \frac{b^2 \sin (c) \text{CosIntegral}(d x)}{a^3}-\frac{b^2 \sin \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (\frac{a d}{b}+d x\right )}{a^3}+\frac{b^2 \cos (c) \text{Si}(d x)}{a^3}-\frac{b^2 \cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (x d+\frac{a d}{b}\right )}{a^3}-\frac{b d \cos (c) \text{CosIntegral}(d x)}{a^2}+\frac{b d \sin (c) \text{Si}(d x)}{a^2}+\frac{b \sin (c+d x)}{a^2 x}-\frac{d^2 \sin (c) \text{CosIntegral}(d x)}{2 a}-\frac{d^2 \cos (c) \text{Si}(d x)}{2 a}-\frac{\sin (c+d x)}{2 a x^2}-\frac{d \cos (c+d x)}{2 a x} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]/(x^3*(a + b*x)),x]

[Out]

-(d*Cos[c + d*x])/(2*a*x) - (b*d*Cos[c]*CosIntegral[d*x])/a^2 + (b^2*CosIntegral[d*x]*Sin[c])/a^3 - (d^2*CosIn
tegral[d*x]*Sin[c])/(2*a) - (b^2*CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/a^3 - Sin[c + d*x]/(2*a*x^2) + (
b*Sin[c + d*x])/(a^2*x) + (b^2*Cos[c]*SinIntegral[d*x])/a^3 - (d^2*Cos[c]*SinIntegral[d*x])/(2*a) + (b*d*Sin[c
]*SinIntegral[d*x])/a^2 - (b^2*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/a^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{\sin (c+d x)}{x^3 (a+b x)} \, dx &=\int \left (\frac{\sin (c+d x)}{a x^3}-\frac{b \sin (c+d x)}{a^2 x^2}+\frac{b^2 \sin (c+d x)}{a^3 x}-\frac{b^3 \sin (c+d x)}{a^3 (a+b x)}\right ) \, dx\\ &=\frac{\int \frac{\sin (c+d x)}{x^3} \, dx}{a}-\frac{b \int \frac{\sin (c+d x)}{x^2} \, dx}{a^2}+\frac{b^2 \int \frac{\sin (c+d x)}{x} \, dx}{a^3}-\frac{b^3 \int \frac{\sin (c+d x)}{a+b x} \, dx}{a^3}\\ &=-\frac{\sin (c+d x)}{2 a x^2}+\frac{b \sin (c+d x)}{a^2 x}+\frac{d \int \frac{\cos (c+d x)}{x^2} \, dx}{2 a}-\frac{(b d) \int \frac{\cos (c+d x)}{x} \, dx}{a^2}+\frac{\left (b^2 \cos (c)\right ) \int \frac{\sin (d x)}{x} \, dx}{a^3}-\frac{\left (b^3 \cos \left (c-\frac{a d}{b}\right )\right ) \int \frac{\sin \left (\frac{a d}{b}+d x\right )}{a+b x} \, dx}{a^3}+\frac{\left (b^2 \sin (c)\right ) \int \frac{\cos (d x)}{x} \, dx}{a^3}-\frac{\left (b^3 \sin \left (c-\frac{a d}{b}\right )\right ) \int \frac{\cos \left (\frac{a d}{b}+d x\right )}{a+b x} \, dx}{a^3}\\ &=-\frac{d \cos (c+d x)}{2 a x}+\frac{b^2 \text{Ci}(d x) \sin (c)}{a^3}-\frac{b^2 \text{Ci}\left (\frac{a d}{b}+d x\right ) \sin \left (c-\frac{a d}{b}\right )}{a^3}-\frac{\sin (c+d x)}{2 a x^2}+\frac{b \sin (c+d x)}{a^2 x}+\frac{b^2 \cos (c) \text{Si}(d x)}{a^3}-\frac{b^2 \cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (\frac{a d}{b}+d x\right )}{a^3}-\frac{d^2 \int \frac{\sin (c+d x)}{x} \, dx}{2 a}-\frac{(b d \cos (c)) \int \frac{\cos (d x)}{x} \, dx}{a^2}+\frac{(b d \sin (c)) \int \frac{\sin (d x)}{x} \, dx}{a^2}\\ &=-\frac{d \cos (c+d x)}{2 a x}-\frac{b d \cos (c) \text{Ci}(d x)}{a^2}+\frac{b^2 \text{Ci}(d x) \sin (c)}{a^3}-\frac{b^2 \text{Ci}\left (\frac{a d}{b}+d x\right ) \sin \left (c-\frac{a d}{b}\right )}{a^3}-\frac{\sin (c+d x)}{2 a x^2}+\frac{b \sin (c+d x)}{a^2 x}+\frac{b^2 \cos (c) \text{Si}(d x)}{a^3}+\frac{b d \sin (c) \text{Si}(d x)}{a^2}-\frac{b^2 \cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (\frac{a d}{b}+d x\right )}{a^3}-\frac{\left (d^2 \cos (c)\right ) \int \frac{\sin (d x)}{x} \, dx}{2 a}-\frac{\left (d^2 \sin (c)\right ) \int \frac{\cos (d x)}{x} \, dx}{2 a}\\ &=-\frac{d \cos (c+d x)}{2 a x}-\frac{b d \cos (c) \text{Ci}(d x)}{a^2}+\frac{b^2 \text{Ci}(d x) \sin (c)}{a^3}-\frac{d^2 \text{Ci}(d x) \sin (c)}{2 a}-\frac{b^2 \text{Ci}\left (\frac{a d}{b}+d x\right ) \sin \left (c-\frac{a d}{b}\right )}{a^3}-\frac{\sin (c+d x)}{2 a x^2}+\frac{b \sin (c+d x)}{a^2 x}+\frac{b^2 \cos (c) \text{Si}(d x)}{a^3}-\frac{d^2 \cos (c) \text{Si}(d x)}{2 a}+\frac{b d \sin (c) \text{Si}(d x)}{a^2}-\frac{b^2 \cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (\frac{a d}{b}+d x\right )}{a^3}\\ \end{align*}

Mathematica [A]  time = 0.648618, size = 176, normalized size = 0.93 \[ -\frac{x^2 \text{CosIntegral}(d x) \left (\sin (c) \left (a^2 d^2-2 b^2\right )+2 a b d \cos (c)\right )+a^2 d^2 x^2 \cos (c) \text{Si}(d x)+a^2 \sin (c+d x)+a^2 d x \cos (c+d x)+2 b^2 x^2 \sin \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (d \left (\frac{a}{b}+x\right )\right )+2 b^2 x^2 \cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (d \left (\frac{a}{b}+x\right )\right )-2 a b d x^2 \sin (c) \text{Si}(d x)-2 a b x \sin (c+d x)-2 b^2 x^2 \cos (c) \text{Si}(d x)}{2 a^3 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[c + d*x]/(x^3*(a + b*x)),x]

[Out]

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

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Maple [A]  time = 0.013, size = 202, normalized size = 1.1 \begin{align*}{d}^{2} \left ( -{\frac{b}{d{a}^{2}} \left ( -{\frac{\sin \left ( dx+c \right ) }{dx}}-{\it Si} \left ( dx \right ) \sin \left ( c \right ) +{\it Ci} \left ( dx \right ) \cos \left ( c \right ) \right ) }+{\frac{1}{a} \left ( -{\frac{\sin \left ( dx+c \right ) }{2\,{d}^{2}{x}^{2}}}-{\frac{\cos \left ( dx+c \right ) }{2\,dx}}-{\frac{{\it Si} \left ( dx \right ) \cos \left ( c \right ) }{2}}-{\frac{{\it Ci} \left ( dx \right ) \sin \left ( c \right ) }{2}} \right ) }-{\frac{{b}^{3}}{{d}^{2}{a}^{3}} \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 ) }+{\frac{{b}^{2} \left ({\it Si} \left ( dx \right ) \cos \left ( c \right ) +{\it Ci} \left ( dx \right ) \sin \left ( c \right ) \right ) }{{d}^{2}{a}^{3}}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)/x^3/(b*x+a),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (d x + c\right )}{{\left (b x + a\right )} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)/x^3/(b*x+a),x, algorithm="maxima")

[Out]

integrate(sin(d*x + c)/((b*x + a)*x^3), x)

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Fricas [A]  time = 1.90702, size = 649, normalized size = 3.43 \begin{align*} -\frac{4 \, b^{2} x^{2} \cos \left (-\frac{b c - a d}{b}\right ) \operatorname{Si}\left (\frac{b d x + a d}{b}\right ) + 2 \, a^{2} d x \cos \left (d x + c\right ) + 2 \,{\left (a b d x^{2} \operatorname{Ci}\left (d x\right ) + a b d x^{2} \operatorname{Ci}\left (-d x\right ) +{\left (a^{2} d^{2} - 2 \, b^{2}\right )} x^{2} \operatorname{Si}\left (d x\right )\right )} \cos \left (c\right ) - 2 \,{\left (2 \, a b x - a^{2}\right )} \sin \left (d x + c\right ) -{\left (4 \, a b d x^{2} \operatorname{Si}\left (d x\right ) -{\left (a^{2} d^{2} - 2 \, b^{2}\right )} x^{2} \operatorname{Ci}\left (d x\right ) -{\left (a^{2} d^{2} - 2 \, b^{2}\right )} x^{2} \operatorname{Ci}\left (-d x\right )\right )} \sin \left (c\right ) - 2 \,{\left (b^{2} x^{2} \operatorname{Ci}\left (\frac{b d x + a d}{b}\right ) + b^{2} x^{2} \operatorname{Ci}\left (-\frac{b d x + a d}{b}\right )\right )} \sin \left (-\frac{b c - a d}{b}\right )}{4 \, a^{3} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)/x^3/(b*x+a),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin{\left (c + d x \right )}}{x^{3} \left (a + b x\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)/x**3/(b*x+a),x)

[Out]

Integral(sin(c + d*x)/(x**3*(a + b*x)), x)

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Giac [C]  time = 1.33616, size = 6163, normalized size = 32.61 \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)/x^3/(b*x+a),x, algorithm="giac")

[Out]

1/4*(a^2*d^2*x^2*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - a^2*d^2*x^2*imag_
part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a^2*d^2*x^2*sin_integral(d*x)*tan(1/
2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 2*a^2*d^2*x^2*real_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c)*
tan(1/2*a*d/b)^2 - 2*a^2*d^2*x^2*real_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 + 2*
a*b*d*x^2*real_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a*b*d*x^2*real_part(co
s_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + a^2*d^2*x^2*imag_part(cos_integral(d*x))*tan(
1/2*d*x)^2*tan(1/2*c)^2 - a^2*d^2*x^2*imag_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*a^2*d^2*x^
2*sin_integral(d*x)*tan(1/2*d*x)^2*tan(1/2*c)^2 - a^2*d^2*x^2*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(
1/2*a*d/b)^2 + a^2*d^2*x^2*imag_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 - 2*a^2*d^2*x^2*sin_i
ntegral(d*x)*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + 4*a*b*d*x^2*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2
*c)*tan(1/2*a*d/b)^2 - 4*a*b*d*x^2*imag_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 +
8*a*b*d*x^2*sin_integral(d*x)*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 + a^2*d^2*x^2*imag_part(cos_integral(
d*x))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - a^2*d^2*x^2*imag_part(cos_integral(-d*x))*tan(1/2*c)^2*tan(1/2*a*d/b)^2
+ 2*a^2*d^2*x^2*sin_integral(d*x)*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 2*b^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*x^2*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/
2*c)^2*tan(1/2*a*d/b)^2 + 2*b^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*x^2*imag_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 4*b^2*x^2*si
n_integral(d*x)*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 4*b^2*x^2*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*d^2*x^2*real_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c) -
2*a^2*d^2*x^2*real_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c) + 2*a*b*d*x^2*real_part(cos_integral(d*
x))*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*a*b*d*x^2*real_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 - 4*b^
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) - 4*b^2*x^2*real_part(co
s_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^2*real_part(cos_integral(d*x)
)*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 - 2*a*b*d*x^2*real_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2
- 2*a^2*d^2*x^2*real_part(cos_integral(d*x))*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*a^2*d^2*x^2*real_part(cos_integra
l(-d*x))*tan(1/2*c)*tan(1/2*a*d/b)^2 + 4*b^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 + 4*b^2*x^2*real_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 + 4*b^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 + 4*b^2*x^2*real_part(co
s_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 + 2*a*b*d*x^2*real_part(cos_integral(d*x))*tan(1/
2*c)^2*tan(1/2*a*d/b)^2 + 2*a*b*d*x^2*real_part(cos_integral(-d*x))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 2*a^2*d*x*
tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - a^2*d^2*x^2*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2 + a^2*d
^2*x^2*imag_part(cos_integral(-d*x))*tan(1/2*d*x)^2 - 2*a^2*d^2*x^2*sin_integral(d*x)*tan(1/2*d*x)^2 + 4*a*b*d
*x^2*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c) - 4*a*b*d*x^2*imag_part(cos_integral(-d*x))*tan(1/
2*d*x)^2*tan(1/2*c) + 8*a*b*d*x^2*sin_integral(d*x)*tan(1/2*d*x)^2*tan(1/2*c) + a^2*d^2*x^2*imag_part(cos_inte
gral(d*x))*tan(1/2*c)^2 - a^2*d^2*x^2*imag_part(cos_integral(-d*x))*tan(1/2*c)^2 + 2*a^2*d^2*x^2*sin_integral(
d*x)*tan(1/2*c)^2 + 2*b^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*x^2*ima
g_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*b^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*x^2*imag_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 - 4*b^2*x^2*sin_in
tegral(d*x)*tan(1/2*d*x)^2*tan(1/2*c)^2 + 4*b^2*x^2*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)^2
- 8*b^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) + 8*b^2*x^2*imag_par
t(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b) - 16*b^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) - a^2*d^2*x^2*imag_part(cos_integral(d*x))*tan(1/2*a*d/b)^2 + a^2
*d^2*x^2*imag_part(cos_integral(-d*x))*tan(1/2*a*d/b)^2 - 2*a^2*d^2*x^2*sin_integral(d*x)*tan(1/2*a*d/b)^2 + 2
*b^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*x^2*imag_part(cos_integr
al(d*x))*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 - 2*b^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*x^2*imag_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + 4*b^2*x^2*sin_integra
l(d*x)*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + 4*b^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*x^2*imag_part(cos_integral(d*x))*tan(1/2*c)*tan(1/2*a*d/b)^2 - 4*a*b*d*x^2*imag_part(cos_integral
(-d*x))*tan(1/2*c)*tan(1/2*a*d/b)^2 + 8*a*b*d*x^2*sin_integral(d*x)*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*b^2*x^2*im
ag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 2*b^2*x^2*imag_part(cos_integral(d*x))*tan(
1/2*c)^2*tan(1/2*a*d/b)^2 + 2*b^2*x^2*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*
b^2*x^2*imag_part(cos_integral(-d*x))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 4*b^2*x^2*sin_integral(d*x)*tan(1/2*c)^2
*tan(1/2*a*d/b)^2 - 4*b^2*x^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*x^2*real_p
art(cos_integral(d*x))*tan(1/2*d*x)^2 - 2*a*b*d*x^2*real_part(cos_integral(-d*x))*tan(1/2*d*x)^2 - 2*a^2*d^2*x
^2*real_part(cos_integral(d*x))*tan(1/2*c) - 2*a^2*d^2*x^2*real_part(cos_integral(-d*x))*tan(1/2*c) - 4*b^2*x^
2*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) + 4*b^2*x^2*real_part(cos_integral(d*x))*tan(
1/2*d*x)^2*tan(1/2*c) - 4*b^2*x^2*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) + 4*b^2*x^2*
real_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c) + 2*a*b*d*x^2*real_part(cos_integral(d*x))*tan(1/2*c)^
2 + 2*a*b*d*x^2*real_part(cos_integral(-d*x))*tan(1/2*c)^2 - 2*a^2*d*x*tan(1/2*d*x)^2*tan(1/2*c)^2 + 4*b^2*x^2
*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^2*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^2*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*
d/b) - 4*b^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*b*d*x^2*real_part(cos
_integral(d*x))*tan(1/2*a*d/b)^2 - 2*a*b*d*x^2*real_part(cos_integral(-d*x))*tan(1/2*a*d/b)^2 + 2*a^2*d*x*tan(
1/2*d*x)^2*tan(1/2*a*d/b)^2 + 4*b^2*x^2*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2 + 4*b
^2*x^2*real_part(cos_integral(d*x))*tan(1/2*c)*tan(1/2*a*d/b)^2 + 4*b^2*x^2*real_part(cos_integral(-d*x - a*d/
b))*tan(1/2*c)*tan(1/2*a*d/b)^2 + 4*b^2*x^2*real_part(cos_integral(-d*x))*tan(1/2*c)*tan(1/2*a*d/b)^2 + 8*a^2*
d*x*tan(1/2*d*x)*tan(1/2*c)*tan(1/2*a*d/b)^2 - 8*a*b*x*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 + 2*a^2*d*x*
tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 8*a*b*x*tan(1/2*d*x)*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - a^2*d^2*x^2*imag_part(cos
_integral(d*x)) + a^2*d^2*x^2*imag_part(cos_integral(-d*x)) - 2*a^2*d^2*x^2*sin_integral(d*x) - 2*b^2*x^2*imag
_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2 + 2*b^2*x^2*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2 + 2*b^
2*x^2*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2 - 2*b^2*x^2*imag_part(cos_integral(-d*x))*tan(1/2*d
*x)^2 + 4*b^2*x^2*sin_integral(d*x)*tan(1/2*d*x)^2 - 4*b^2*x^2*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2 +
4*a*b*d*x^2*imag_part(cos_integral(d*x))*tan(1/2*c) - 4*a*b*d*x^2*imag_part(cos_integral(-d*x))*tan(1/2*c) + 8
*a*b*d*x^2*sin_integral(d*x)*tan(1/2*c) + 2*b^2*x^2*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2 - 2*b^2*
x^2*imag_part(cos_integral(d*x))*tan(1/2*c)^2 - 2*b^2*x^2*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2 +
2*b^2*x^2*imag_part(cos_integral(-d*x))*tan(1/2*c)^2 - 4*b^2*x^2*sin_integral(d*x)*tan(1/2*c)^2 + 4*b^2*x^2*s
in_integral((b*d*x + a*d)/b)*tan(1/2*c)^2 - 8*b^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*x^2*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) - 16*b^2*x^2*sin_integral((
b*d*x + a*d)/b)*tan(1/2*c)*tan(1/2*a*d/b) + 2*b^2*x^2*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*a*d/b)^2 +
2*b^2*x^2*imag_part(cos_integral(d*x))*tan(1/2*a*d/b)^2 - 2*b^2*x^2*imag_part(cos_integral(-d*x - a*d/b))*tan(
1/2*a*d/b)^2 - 2*b^2*x^2*imag_part(cos_integral(-d*x))*tan(1/2*a*d/b)^2 + 4*b^2*x^2*sin_integral(d*x)*tan(1/2*
a*d/b)^2 + 4*b^2*x^2*sin_integral((b*d*x + a*d)/b)*tan(1/2*a*d/b)^2 + 4*a^2*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*
a*d/b)^2 + 4*a^2*tan(1/2*d*x)*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 2*a*b*d*x^2*real_part(cos_integral(d*x)) - 2*a*b
*d*x^2*real_part(cos_integral(-d*x)) + 2*a^2*d*x*tan(1/2*d*x)^2 - 4*b^2*x^2*real_part(cos_integral(d*x + a*d/b
))*tan(1/2*c) + 4*b^2*x^2*real_part(cos_integral(d*x))*tan(1/2*c) - 4*b^2*x^2*real_part(cos_integral(-d*x - a*
d/b))*tan(1/2*c) + 4*b^2*x^2*real_part(cos_integral(-d*x))*tan(1/2*c) + 8*a^2*d*x*tan(1/2*d*x)*tan(1/2*c) - 8*
a*b*x*tan(1/2*d*x)^2*tan(1/2*c) + 2*a^2*d*x*tan(1/2*c)^2 - 8*a*b*x*tan(1/2*d*x)*tan(1/2*c)^2 + 4*b^2*x^2*real_
part(cos_integral(d*x + a*d/b))*tan(1/2*a*d/b) + 4*b^2*x^2*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b
) - 2*a^2*d*x*tan(1/2*a*d/b)^2 + 8*a*b*x*tan(1/2*d*x)*tan(1/2*a*d/b)^2 + 8*a*b*x*tan(1/2*c)*tan(1/2*a*d/b)^2 -
2*b^2*x^2*imag_part(cos_integral(d*x + a*d/b)) + 2*b^2*x^2*imag_part(cos_integral(d*x)) + 2*b^2*x^2*imag_part
(cos_integral(-d*x - a*d/b)) - 2*b^2*x^2*imag_part(cos_integral(-d*x)) + 4*b^2*x^2*sin_integral(d*x) - 4*b^2*x
^2*sin_integral((b*d*x + a*d)/b) + 4*a^2*tan(1/2*d*x)^2*tan(1/2*c) + 4*a^2*tan(1/2*d*x)*tan(1/2*c)^2 - 4*a^2*t
an(1/2*d*x)*tan(1/2*a*d/b)^2 - 4*a^2*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*a^2*d*x + 8*a*b*x*tan(1/2*d*x) + 8*a*b*x*
tan(1/2*c) - 4*a^2*tan(1/2*d*x) - 4*a^2*tan(1/2*c))/(a^3*x^2*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + a^
3*x^2*tan(1/2*d*x)^2*tan(1/2*c)^2 + a^3*x^2*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + a^3*x^2*tan(1/2*c)^2*tan(1/2*a*d
/b)^2 + a^3*x^2*tan(1/2*d*x)^2 + a^3*x^2*tan(1/2*c)^2 + a^3*x^2*tan(1/2*a*d/b)^2 + a^3*x^2)