3.56 \(\int \frac{(a+b x^2)^2 \sin (c+d x)}{x^5} \, dx\)

Optimal. Leaf size=177 \[ \frac{1}{24} a^2 d^4 \sin (c) \text{CosIntegral}(d x)+\frac{1}{24} a^2 d^4 \cos (c) \text{Si}(d x)+\frac{a^2 d^2 \sin (c+d x)}{24 x^2}+\frac{a^2 d^3 \cos (c+d x)}{24 x}-\frac{a^2 \sin (c+d x)}{4 x^4}-\frac{a^2 d \cos (c+d x)}{12 x^3}-a b d^2 \sin (c) \text{CosIntegral}(d x)-a b d^2 \cos (c) \text{Si}(d x)-\frac{a b \sin (c+d x)}{x^2}-\frac{a b d \cos (c+d x)}{x}+b^2 \sin (c) \text{CosIntegral}(d x)+b^2 \cos (c) \text{Si}(d x) \]

[Out]

-(a^2*d*Cos[c + d*x])/(12*x^3) - (a*b*d*Cos[c + d*x])/x + (a^2*d^3*Cos[c + d*x])/(24*x) + b^2*CosIntegral[d*x]
*Sin[c] - a*b*d^2*CosIntegral[d*x]*Sin[c] + (a^2*d^4*CosIntegral[d*x]*Sin[c])/24 - (a^2*Sin[c + d*x])/(4*x^4)
- (a*b*Sin[c + d*x])/x^2 + (a^2*d^2*Sin[c + d*x])/(24*x^2) + b^2*Cos[c]*SinIntegral[d*x] - a*b*d^2*Cos[c]*SinI
ntegral[d*x] + (a^2*d^4*Cos[c]*SinIntegral[d*x])/24

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Rubi [A]  time = 0.332833, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {3339, 3297, 3303, 3299, 3302} \[ \frac{1}{24} a^2 d^4 \sin (c) \text{CosIntegral}(d x)+\frac{1}{24} a^2 d^4 \cos (c) \text{Si}(d x)+\frac{a^2 d^2 \sin (c+d x)}{24 x^2}+\frac{a^2 d^3 \cos (c+d x)}{24 x}-\frac{a^2 \sin (c+d x)}{4 x^4}-\frac{a^2 d \cos (c+d x)}{12 x^3}-a b d^2 \sin (c) \text{CosIntegral}(d x)-a b d^2 \cos (c) \text{Si}(d x)-\frac{a b \sin (c+d x)}{x^2}-\frac{a b d \cos (c+d x)}{x}+b^2 \sin (c) \text{CosIntegral}(d x)+b^2 \cos (c) \text{Si}(d x) \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^2*Sin[c + d*x])/x^5,x]

[Out]

-(a^2*d*Cos[c + d*x])/(12*x^3) - (a*b*d*Cos[c + d*x])/x + (a^2*d^3*Cos[c + d*x])/(24*x) + b^2*CosIntegral[d*x]
*Sin[c] - a*b*d^2*CosIntegral[d*x]*Sin[c] + (a^2*d^4*CosIntegral[d*x]*Sin[c])/24 - (a^2*Sin[c + d*x])/(4*x^4)
- (a*b*Sin[c + d*x])/x^2 + (a^2*d^2*Sin[c + d*x])/(24*x^2) + b^2*Cos[c]*SinIntegral[d*x] - a*b*d^2*Cos[c]*SinI
ntegral[d*x] + (a^2*d^4*Cos[c]*SinIntegral[d*x])/24

Rule 3339

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*Sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Int[ExpandIntegran
d[Sin[c + d*x], (e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]

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{\left (a+b x^2\right )^2 \sin (c+d x)}{x^5} \, dx &=\int \left (\frac{a^2 \sin (c+d x)}{x^5}+\frac{2 a b \sin (c+d x)}{x^3}+\frac{b^2 \sin (c+d x)}{x}\right ) \, dx\\ &=a^2 \int \frac{\sin (c+d x)}{x^5} \, dx+(2 a b) \int \frac{\sin (c+d x)}{x^3} \, dx+b^2 \int \frac{\sin (c+d x)}{x} \, dx\\ &=-\frac{a^2 \sin (c+d x)}{4 x^4}-\frac{a b \sin (c+d x)}{x^2}+\frac{1}{4} \left (a^2 d\right ) \int \frac{\cos (c+d x)}{x^4} \, dx+(a b d) \int \frac{\cos (c+d x)}{x^2} \, dx+\left (b^2 \cos (c)\right ) \int \frac{\sin (d x)}{x} \, dx+\left (b^2 \sin (c)\right ) \int \frac{\cos (d x)}{x} \, dx\\ &=-\frac{a^2 d \cos (c+d x)}{12 x^3}-\frac{a b d \cos (c+d x)}{x}+b^2 \text{Ci}(d x) \sin (c)-\frac{a^2 \sin (c+d x)}{4 x^4}-\frac{a b \sin (c+d x)}{x^2}+b^2 \cos (c) \text{Si}(d x)-\frac{1}{12} \left (a^2 d^2\right ) \int \frac{\sin (c+d x)}{x^3} \, dx-\left (a b d^2\right ) \int \frac{\sin (c+d x)}{x} \, dx\\ &=-\frac{a^2 d \cos (c+d x)}{12 x^3}-\frac{a b d \cos (c+d x)}{x}+b^2 \text{Ci}(d x) \sin (c)-\frac{a^2 \sin (c+d x)}{4 x^4}-\frac{a b \sin (c+d x)}{x^2}+\frac{a^2 d^2 \sin (c+d x)}{24 x^2}+b^2 \cos (c) \text{Si}(d x)-\frac{1}{24} \left (a^2 d^3\right ) \int \frac{\cos (c+d x)}{x^2} \, dx-\left (a b d^2 \cos (c)\right ) \int \frac{\sin (d x)}{x} \, dx-\left (a b d^2 \sin (c)\right ) \int \frac{\cos (d x)}{x} \, dx\\ &=-\frac{a^2 d \cos (c+d x)}{12 x^3}-\frac{a b d \cos (c+d x)}{x}+\frac{a^2 d^3 \cos (c+d x)}{24 x}+b^2 \text{Ci}(d x) \sin (c)-a b d^2 \text{Ci}(d x) \sin (c)-\frac{a^2 \sin (c+d x)}{4 x^4}-\frac{a b \sin (c+d x)}{x^2}+\frac{a^2 d^2 \sin (c+d x)}{24 x^2}+b^2 \cos (c) \text{Si}(d x)-a b d^2 \cos (c) \text{Si}(d x)+\frac{1}{24} \left (a^2 d^4\right ) \int \frac{\sin (c+d x)}{x} \, dx\\ &=-\frac{a^2 d \cos (c+d x)}{12 x^3}-\frac{a b d \cos (c+d x)}{x}+\frac{a^2 d^3 \cos (c+d x)}{24 x}+b^2 \text{Ci}(d x) \sin (c)-a b d^2 \text{Ci}(d x) \sin (c)-\frac{a^2 \sin (c+d x)}{4 x^4}-\frac{a b \sin (c+d x)}{x^2}+\frac{a^2 d^2 \sin (c+d x)}{24 x^2}+b^2 \cos (c) \text{Si}(d x)-a b d^2 \cos (c) \text{Si}(d x)+\frac{1}{24} \left (a^2 d^4 \cos (c)\right ) \int \frac{\sin (d x)}{x} \, dx+\frac{1}{24} \left (a^2 d^4 \sin (c)\right ) \int \frac{\cos (d x)}{x} \, dx\\ &=-\frac{a^2 d \cos (c+d x)}{12 x^3}-\frac{a b d \cos (c+d x)}{x}+\frac{a^2 d^3 \cos (c+d x)}{24 x}+b^2 \text{Ci}(d x) \sin (c)-a b d^2 \text{Ci}(d x) \sin (c)+\frac{1}{24} a^2 d^4 \text{Ci}(d x) \sin (c)-\frac{a^2 \sin (c+d x)}{4 x^4}-\frac{a b \sin (c+d x)}{x^2}+\frac{a^2 d^2 \sin (c+d x)}{24 x^2}+b^2 \cos (c) \text{Si}(d x)-a b d^2 \cos (c) \text{Si}(d x)+\frac{1}{24} a^2 d^4 \cos (c) \text{Si}(d x)\\ \end{align*}

Mathematica [A]  time = 0.458211, size = 122, normalized size = 0.69 \[ \frac{x^4 \sin (c) \left (a^2 d^4-24 a b d^2+24 b^2\right ) \text{CosIntegral}(d x)+x^4 \cos (c) \left (a^2 d^4-24 a b d^2+24 b^2\right ) \text{Si}(d x)+a \left (a \left (d^2 x^2-6\right )-24 b x^2\right ) \sin (c+d x)+a d x \left (a \left (d^2 x^2-2\right )-24 b x^2\right ) \cos (c+d x)}{24 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^2*Sin[c + d*x])/x^5,x]

[Out]

(a*d*x*(-24*b*x^2 + a*(-2 + d^2*x^2))*Cos[c + d*x] + (24*b^2 - 24*a*b*d^2 + a^2*d^4)*x^4*CosIntegral[d*x]*Sin[
c] + a*(-24*b*x^2 + a*(-6 + d^2*x^2))*Sin[c + d*x] + (24*b^2 - 24*a*b*d^2 + a^2*d^4)*x^4*Cos[c]*SinIntegral[d*
x])/(24*x^4)

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Maple [A]  time = 0.022, size = 157, normalized size = 0.9 \begin{align*}{d}^{4} \left ({\frac{{b}^{2} \left ({\it Si} \left ( dx \right ) \cos \left ( c \right ) +{\it Ci} \left ( dx \right ) \sin \left ( c \right ) \right ) }{{d}^{4}}}+2\,{\frac{ab}{{d}^{2}} \left ( -1/2\,{\frac{\sin \left ( dx+c \right ) }{{d}^{2}{x}^{2}}}-1/2\,{\frac{\cos \left ( dx+c \right ) }{dx}}-1/2\,{\it Si} \left ( dx \right ) \cos \left ( c \right ) -1/2\,{\it Ci} \left ( dx \right ) \sin \left ( c \right ) \right ) }+{a}^{2} \left ( -{\frac{\sin \left ( dx+c \right ) }{4\,{x}^{4}{d}^{4}}}-{\frac{\cos \left ( dx+c \right ) }{12\,{d}^{3}{x}^{3}}}+{\frac{\sin \left ( dx+c \right ) }{24\,{d}^{2}{x}^{2}}}+{\frac{\cos \left ( dx+c \right ) }{24\,dx}}+{\frac{{\it Si} \left ( dx \right ) \cos \left ( c \right ) }{24}}+{\frac{{\it Ci} \left ( dx \right ) \sin \left ( c \right ) }{24}} \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2*sin(d*x+c)/x^5,x)

[Out]

d^4*(1/d^4*b^2*(Si(d*x)*cos(c)+Ci(d*x)*sin(c))+2/d^2*a*b*(-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))+a^2*(-1/4*sin(d*x+c)/x^4/d^4-1/12*cos(d*x+c)/x^3/d^3+1/24*sin(d*x+c)/x^2/d^2+1/2
4*cos(d*x+c)/x/d+1/24*Si(d*x)*cos(c)+1/24*Ci(d*x)*sin(c)))

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Maxima [C]  time = 69.4396, size = 298, normalized size = 1.68 \begin{align*} -\frac{{\left ({\left (a^{2}{\left (i \, \Gamma \left (-4, i \, d x\right ) - i \, \Gamma \left (-4, -i \, d x\right )\right )} \cos \left (c\right ) + a^{2}{\left (\Gamma \left (-4, i \, d x\right ) + \Gamma \left (-4, -i \, d x\right )\right )} \sin \left (c\right )\right )} d^{8} +{\left (a b{\left (-24 i \, \Gamma \left (-4, i \, d x\right ) + 24 i \, \Gamma \left (-4, -i \, d x\right )\right )} \cos \left (c\right ) - 24 \, a b{\left (\Gamma \left (-4, i \, d x\right ) + \Gamma \left (-4, -i \, d x\right )\right )} \sin \left (c\right )\right )} d^{6} +{\left (b^{2}{\left (24 i \, \Gamma \left (-4, i \, d x\right ) - 24 i \, \Gamma \left (-4, -i \, d x\right )\right )} \cos \left (c\right ) + 24 \, b^{2}{\left (\Gamma \left (-4, i \, d x\right ) + \Gamma \left (-4, -i \, d x\right )\right )} \sin \left (c\right )\right )} d^{4}\right )} x^{4} + 2 \,{\left (b^{2} d^{3} x^{3} + 2 \,{\left (a b d^{3} - b^{2} d\right )} x\right )} \cos \left (d x + c\right ) + 2 \,{\left (b^{2} d^{2} x^{2} + 6 \, a b d^{2} - 6 \, b^{2}\right )} \sin \left (d x + c\right )}{2 \, d^{4} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*sin(d*x+c)/x^5,x, algorithm="maxima")

[Out]

-1/2*(((a^2*(I*gamma(-4, I*d*x) - I*gamma(-4, -I*d*x))*cos(c) + a^2*(gamma(-4, I*d*x) + gamma(-4, -I*d*x))*sin
(c))*d^8 + (a*b*(-24*I*gamma(-4, I*d*x) + 24*I*gamma(-4, -I*d*x))*cos(c) - 24*a*b*(gamma(-4, I*d*x) + gamma(-4
, -I*d*x))*sin(c))*d^6 + (b^2*(24*I*gamma(-4, I*d*x) - 24*I*gamma(-4, -I*d*x))*cos(c) + 24*b^2*(gamma(-4, I*d*
x) + gamma(-4, -I*d*x))*sin(c))*d^4)*x^4 + 2*(b^2*d^3*x^3 + 2*(a*b*d^3 - b^2*d)*x)*cos(d*x + c) + 2*(b^2*d^2*x
^2 + 6*a*b*d^2 - 6*b^2)*sin(d*x + c))/(d^4*x^4)

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Fricas [A]  time = 1.73441, size = 409, normalized size = 2.31 \begin{align*} \frac{2 \,{\left (a^{2} d^{4} - 24 \, a b d^{2} + 24 \, b^{2}\right )} x^{4} \cos \left (c\right ) \operatorname{Si}\left (d x\right ) - 2 \,{\left (2 \, a^{2} d x -{\left (a^{2} d^{3} - 24 \, a b d\right )} x^{3}\right )} \cos \left (d x + c\right ) + 2 \,{\left ({\left (a^{2} d^{2} - 24 \, a b\right )} x^{2} - 6 \, a^{2}\right )} \sin \left (d x + c\right ) +{\left ({\left (a^{2} d^{4} - 24 \, a b d^{2} + 24 \, b^{2}\right )} x^{4} \operatorname{Ci}\left (d x\right ) +{\left (a^{2} d^{4} - 24 \, a b d^{2} + 24 \, b^{2}\right )} x^{4} \operatorname{Ci}\left (-d x\right )\right )} \sin \left (c\right )}{48 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*sin(d*x+c)/x^5,x, algorithm="fricas")

[Out]

1/48*(2*(a^2*d^4 - 24*a*b*d^2 + 24*b^2)*x^4*cos(c)*sin_integral(d*x) - 2*(2*a^2*d*x - (a^2*d^3 - 24*a*b*d)*x^3
)*cos(d*x + c) + 2*((a^2*d^2 - 24*a*b)*x^2 - 6*a^2)*sin(d*x + c) + ((a^2*d^4 - 24*a*b*d^2 + 24*b^2)*x^4*cos_in
tegral(d*x) + (a^2*d^4 - 24*a*b*d^2 + 24*b^2)*x^4*cos_integral(-d*x))*sin(c))/x^4

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x^{2}\right )^{2} \sin{\left (c + d x \right )}}{x^{5}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2*sin(d*x+c)/x**5,x)

[Out]

Integral((a + b*x**2)**2*sin(c + d*x)/x**5, x)

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Giac [C]  time = 1.19146, size = 2021, normalized size = 11.42 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*sin(d*x+c)/x^5,x, algorithm="giac")

[Out]

-1/48*(a^2*d^4*x^4*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 - a^2*d^4*x^4*imag_part(cos_integr
al(-d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*a^2*d^4*x^4*sin_integral(d*x)*tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*a^2*d^
4*x^4*real_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c) - 2*a^2*d^4*x^4*real_part(cos_integral(-d*x))*tan
(1/2*d*x)^2*tan(1/2*c) - a^2*d^4*x^4*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2 + a^2*d^4*x^4*imag_part(cos_i
ntegral(-d*x))*tan(1/2*d*x)^2 - 2*a^2*d^4*x^4*sin_integral(d*x)*tan(1/2*d*x)^2 + a^2*d^4*x^4*imag_part(cos_int
egral(d*x))*tan(1/2*c)^2 - a^2*d^4*x^4*imag_part(cos_integral(-d*x))*tan(1/2*c)^2 + 2*a^2*d^4*x^4*sin_integral
(d*x)*tan(1/2*c)^2 - 24*a*b*d^2*x^4*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 + 24*a*b*d^2*x^4*
imag_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 - 48*a*b*d^2*x^4*sin_integral(d*x)*tan(1/2*d*x)^2*ta
n(1/2*c)^2 - 2*a^2*d^4*x^4*real_part(cos_integral(d*x))*tan(1/2*c) - 2*a^2*d^4*x^4*real_part(cos_integral(-d*x
))*tan(1/2*c) + 48*a*b*d^2*x^4*real_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c) + 48*a*b*d^2*x^4*real_pa
rt(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c) - 2*a^2*d^3*x^3*tan(1/2*d*x)^2*tan(1/2*c)^2 - a^2*d^4*x^4*ima
g_part(cos_integral(d*x)) + a^2*d^4*x^4*imag_part(cos_integral(-d*x)) - 2*a^2*d^4*x^4*sin_integral(d*x) + 24*a
*b*d^2*x^4*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2 - 24*a*b*d^2*x^4*imag_part(cos_integral(-d*x))*tan(1/2*
d*x)^2 + 48*a*b*d^2*x^4*sin_integral(d*x)*tan(1/2*d*x)^2 - 24*a*b*d^2*x^4*imag_part(cos_integral(d*x))*tan(1/2
*c)^2 + 24*a*b*d^2*x^4*imag_part(cos_integral(-d*x))*tan(1/2*c)^2 - 48*a*b*d^2*x^4*sin_integral(d*x)*tan(1/2*c
)^2 + 24*b^2*x^4*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 - 24*b^2*x^4*imag_part(cos_integral(
-d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 + 48*b^2*x^4*sin_integral(d*x)*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*a^2*d^3*x^3*
tan(1/2*d*x)^2 + 48*a*b*d^2*x^4*real_part(cos_integral(d*x))*tan(1/2*c) + 48*a*b*d^2*x^4*real_part(cos_integra
l(-d*x))*tan(1/2*c) + 8*a^2*d^3*x^3*tan(1/2*d*x)*tan(1/2*c) - 48*b^2*x^4*real_part(cos_integral(d*x))*tan(1/2*
d*x)^2*tan(1/2*c) - 48*b^2*x^4*real_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c) + 2*a^2*d^3*x^3*tan(1/2
*c)^2 + 48*a*b*d*x^3*tan(1/2*d*x)^2*tan(1/2*c)^2 + 24*a*b*d^2*x^4*imag_part(cos_integral(d*x)) - 24*a*b*d^2*x^
4*imag_part(cos_integral(-d*x)) + 48*a*b*d^2*x^4*sin_integral(d*x) - 24*b^2*x^4*imag_part(cos_integral(d*x))*t
an(1/2*d*x)^2 + 24*b^2*x^4*imag_part(cos_integral(-d*x))*tan(1/2*d*x)^2 - 48*b^2*x^4*sin_integral(d*x)*tan(1/2
*d*x)^2 + 4*a^2*d^2*x^2*tan(1/2*d*x)^2*tan(1/2*c) + 24*b^2*x^4*imag_part(cos_integral(d*x))*tan(1/2*c)^2 - 24*
b^2*x^4*imag_part(cos_integral(-d*x))*tan(1/2*c)^2 + 48*b^2*x^4*sin_integral(d*x)*tan(1/2*c)^2 + 4*a^2*d^2*x^2
*tan(1/2*d*x)*tan(1/2*c)^2 - 2*a^2*d^3*x^3 - 48*a*b*d*x^3*tan(1/2*d*x)^2 - 48*b^2*x^4*real_part(cos_integral(d
*x))*tan(1/2*c) - 48*b^2*x^4*real_part(cos_integral(-d*x))*tan(1/2*c) - 192*a*b*d*x^3*tan(1/2*d*x)*tan(1/2*c)
- 48*a*b*d*x^3*tan(1/2*c)^2 + 4*a^2*d*x*tan(1/2*d*x)^2*tan(1/2*c)^2 - 24*b^2*x^4*imag_part(cos_integral(d*x))
+ 24*b^2*x^4*imag_part(cos_integral(-d*x)) - 48*b^2*x^4*sin_integral(d*x) - 4*a^2*d^2*x^2*tan(1/2*d*x) - 4*a^2
*d^2*x^2*tan(1/2*c) - 96*a*b*x^2*tan(1/2*d*x)^2*tan(1/2*c) - 96*a*b*x^2*tan(1/2*d*x)*tan(1/2*c)^2 + 48*a*b*d*x
^3 - 4*a^2*d*x*tan(1/2*d*x)^2 - 16*a^2*d*x*tan(1/2*d*x)*tan(1/2*c) - 4*a^2*d*x*tan(1/2*c)^2 + 96*a*b*x^2*tan(1
/2*d*x) + 96*a*b*x^2*tan(1/2*c) - 24*a^2*tan(1/2*d*x)^2*tan(1/2*c) - 24*a^2*tan(1/2*d*x)*tan(1/2*c)^2 + 4*a^2*
d*x + 24*a^2*tan(1/2*d*x) + 24*a^2*tan(1/2*c))/(x^4*tan(1/2*d*x)^2*tan(1/2*c)^2 + x^4*tan(1/2*d*x)^2 + x^4*tan
(1/2*c)^2 + x^4)