3.27 \(\int \frac{x^3 \sin (c+d x)}{(a+b x)^2} \, dx\)

Optimal. Leaf size=181 \[ \frac{3 a^2 \sin \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (\frac{a d}{b}+d x\right )}{b^4}-\frac{a^3 d \cos \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (\frac{a d}{b}+d x\right )}{b^5}+\frac{a^3 d \sin \left (c-\frac{a d}{b}\right ) \text{Si}\left (x d+\frac{a d}{b}\right )}{b^5}+\frac{3 a^2 \cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (x d+\frac{a d}{b}\right )}{b^4}+\frac{a^3 \sin (c+d x)}{b^4 (a+b x)}+\frac{2 a \cos (c+d x)}{b^3 d}+\frac{\sin (c+d x)}{b^2 d^2}-\frac{x \cos (c+d x)}{b^2 d} \]

[Out]

(2*a*Cos[c + d*x])/(b^3*d) - (x*Cos[c + d*x])/(b^2*d) - (a^3*d*Cos[c - (a*d)/b]*CosIntegral[(a*d)/b + d*x])/b^
5 + (3*a^2*CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/b^4 + Sin[c + d*x]/(b^2*d^2) + (a^3*Sin[c + d*x])/(b^4
*(a + b*x)) + (3*a^2*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/b^4 + (a^3*d*Sin[c - (a*d)/b]*SinIntegral[(a
*d)/b + d*x])/b^5

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Rubi [A]  time = 0.408469, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.471, Rules used = {6742, 2638, 3296, 2637, 3297, 3303, 3299, 3302} \[ \frac{3 a^2 \sin \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (\frac{a d}{b}+d x\right )}{b^4}-\frac{a^3 d \cos \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (\frac{a d}{b}+d x\right )}{b^5}+\frac{a^3 d \sin \left (c-\frac{a d}{b}\right ) \text{Si}\left (x d+\frac{a d}{b}\right )}{b^5}+\frac{3 a^2 \cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (x d+\frac{a d}{b}\right )}{b^4}+\frac{a^3 \sin (c+d x)}{b^4 (a+b x)}+\frac{2 a \cos (c+d x)}{b^3 d}+\frac{\sin (c+d x)}{b^2 d^2}-\frac{x \cos (c+d x)}{b^2 d} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*Sin[c + d*x])/(a + b*x)^2,x]

[Out]

(2*a*Cos[c + d*x])/(b^3*d) - (x*Cos[c + d*x])/(b^2*d) - (a^3*d*Cos[c - (a*d)/b]*CosIntegral[(a*d)/b + d*x])/b^
5 + (3*a^2*CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/b^4 + Sin[c + d*x]/(b^2*d^2) + (a^3*Sin[c + d*x])/(b^4
*(a + b*x)) + (3*a^2*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/b^4 + (a^3*d*Sin[c - (a*d)/b]*SinIntegral[(a
*d)/b + d*x])/b^5

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 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[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^3 \sin (c+d x)}{(a+b x)^2} \, dx &=\int \left (-\frac{2 a \sin (c+d x)}{b^3}+\frac{x \sin (c+d x)}{b^2}-\frac{a^3 \sin (c+d x)}{b^3 (a+b x)^2}+\frac{3 a^2 \sin (c+d x)}{b^3 (a+b x)}\right ) \, dx\\ &=-\frac{(2 a) \int \sin (c+d x) \, dx}{b^3}+\frac{\left (3 a^2\right ) \int \frac{\sin (c+d x)}{a+b x} \, dx}{b^3}-\frac{a^3 \int \frac{\sin (c+d x)}{(a+b x)^2} \, dx}{b^3}+\frac{\int x \sin (c+d x) \, dx}{b^2}\\ &=\frac{2 a \cos (c+d x)}{b^3 d}-\frac{x \cos (c+d x)}{b^2 d}+\frac{a^3 \sin (c+d x)}{b^4 (a+b x)}+\frac{\int \cos (c+d x) \, dx}{b^2 d}-\frac{\left (a^3 d\right ) \int \frac{\cos (c+d x)}{a+b x} \, dx}{b^4}+\frac{\left (3 a^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}{b^3}+\frac{\left (3 a^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}{b^3}\\ &=\frac{2 a \cos (c+d x)}{b^3 d}-\frac{x \cos (c+d x)}{b^2 d}+\frac{3 a^2 \text{Ci}\left (\frac{a d}{b}+d x\right ) \sin \left (c-\frac{a d}{b}\right )}{b^4}+\frac{\sin (c+d x)}{b^2 d^2}+\frac{a^3 \sin (c+d x)}{b^4 (a+b x)}+\frac{3 a^2 \cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (\frac{a d}{b}+d x\right )}{b^4}-\frac{\left (a^3 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^4}+\frac{\left (a^3 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^4}\\ &=\frac{2 a \cos (c+d x)}{b^3 d}-\frac{x \cos (c+d x)}{b^2 d}-\frac{a^3 d \cos \left (c-\frac{a d}{b}\right ) \text{Ci}\left (\frac{a d}{b}+d x\right )}{b^5}+\frac{3 a^2 \text{Ci}\left (\frac{a d}{b}+d x\right ) \sin \left (c-\frac{a d}{b}\right )}{b^4}+\frac{\sin (c+d x)}{b^2 d^2}+\frac{a^3 \sin (c+d x)}{b^4 (a+b x)}+\frac{3 a^2 \cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (\frac{a d}{b}+d x\right )}{b^4}+\frac{a^3 d \sin \left (c-\frac{a d}{b}\right ) \text{Si}\left (\frac{a d}{b}+d x\right )}{b^5}\\ \end{align*}

Mathematica [A]  time = 0.879869, size = 153, normalized size = 0.85 \[ \frac{\frac{b \left (\left (a^3 d^2+a b^2+b^3 x\right ) \sin (c+d x)+b d \left (2 a^2+a b x-b^2 x^2\right ) \cos (c+d x)\right )}{d^2 (a+b x)}+a^2 \left (-\text{CosIntegral}\left (d \left (\frac{a}{b}+x\right )\right )\right ) \left (a d \cos \left (c-\frac{a d}{b}\right )-3 b \sin \left (c-\frac{a d}{b}\right )\right )+a^2 \text{Si}\left (d \left (\frac{a}{b}+x\right )\right ) \left (a d \sin \left (c-\frac{a d}{b}\right )+3 b \cos \left (c-\frac{a d}{b}\right )\right )}{b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^3*Sin[c + d*x])/(a + b*x)^2,x]

[Out]

(-(a^2*CosIntegral[d*(a/b + x)]*(a*d*Cos[c - (a*d)/b] - 3*b*Sin[c - (a*d)/b])) + (b*(b*d*(2*a^2 + a*b*x - b^2*
x^2)*Cos[c + d*x] + (a*b^2 + a^3*d^2 + b^3*x)*Sin[c + d*x]))/(d^2*(a + b*x)) + a^2*(3*b*Cos[c - (a*d)/b] + a*d
*Sin[c - (a*d)/b])*SinIntegral[d*(a/b + x)])/b^5

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Maple [B]  time = 0.016, size = 848, normalized size = 4.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*sin(d*x+c)/(b*x+a)^2,x)

[Out]

1/d^4*((-2*a*d+2*b*c+b)*d^2/b^3*(sin(d*x+c)-(d*x+c)*cos(d*x+c))-(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*
d^2/b^3*(-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)+3/b^3*(a^2*d^2-2*a*b*c*d+b^2*c^2)*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)+3*d^2*c/b^2*cos(d*x+c)-3*(a^2*d^2-2*a*b*c*d+b^2*c^2)*d^2*c/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)+6/b
^2*(a*d-b*c)*d^2*c*(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)-3*d^2*(
a*d-b*c)/b*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)+3*d^2*c^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^3*(-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^3*sin(d*x+c)/(b*x+a)^2,x, algorithm="maxima")

[Out]

Timed out

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Fricas [A]  time = 1.7347, size = 718, normalized size = 3.97 \begin{align*} -\frac{2 \,{\left (b^{4} d x^{2} - a b^{3} d x - 2 \, a^{2} b^{2} d\right )} \cos \left (d x + c\right ) +{\left ({\left (a^{3} b d^{3} x + a^{4} d^{3}\right )} \operatorname{Ci}\left (\frac{b d x + a d}{b}\right ) +{\left (a^{3} b d^{3} x + a^{4} d^{3}\right )} \operatorname{Ci}\left (-\frac{b d x + a d}{b}\right ) - 6 \,{\left (a^{2} b^{2} d^{2} x + a^{3} b d^{2}\right )} \operatorname{Si}\left (\frac{b d x + a d}{b}\right )\right )} \cos \left (-\frac{b c - a d}{b}\right ) - 2 \,{\left (a^{3} b d^{2} + b^{4} x + a b^{3}\right )} \sin \left (d x + c\right ) +{\left (3 \,{\left (a^{2} b^{2} d^{2} x + a^{3} b d^{2}\right )} \operatorname{Ci}\left (\frac{b d x + a d}{b}\right ) + 3 \,{\left (a^{2} b^{2} d^{2} x + a^{3} b d^{2}\right )} \operatorname{Ci}\left (-\frac{b d x + a d}{b}\right ) + 2 \,{\left (a^{3} b d^{3} x + a^{4} d^{3}\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^{6} d^{2} x + a b^{5} d^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*sin(d*x+c)/(b*x+a)^2,x, algorithm="fricas")

[Out]

-1/2*(2*(b^4*d*x^2 - a*b^3*d*x - 2*a^2*b^2*d)*cos(d*x + c) + ((a^3*b*d^3*x + a^4*d^3)*cos_integral((b*d*x + a*
d)/b) + (a^3*b*d^3*x + a^4*d^3)*cos_integral(-(b*d*x + a*d)/b) - 6*(a^2*b^2*d^2*x + a^3*b*d^2)*sin_integral((b
*d*x + a*d)/b))*cos(-(b*c - a*d)/b) - 2*(a^3*b*d^2 + b^4*x + a*b^3)*sin(d*x + c) + (3*(a^2*b^2*d^2*x + a^3*b*d
^2)*cos_integral((b*d*x + a*d)/b) + 3*(a^2*b^2*d^2*x + a^3*b*d^2)*cos_integral(-(b*d*x + a*d)/b) + 2*(a^3*b*d^
3*x + a^4*d^3)*sin_integral((b*d*x + a*d)/b))*sin(-(b*c - a*d)/b))/(b^6*d^2*x + a*b^5*d^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \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**3*sin(d*x+c)/(b*x+a)**2,x)

[Out]

Integral(x**3*sin(c + d*x)/(a + b*x)**2, x)

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Giac [C]  time = 1.42808, size = 8586, normalized size = 47.44 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*sin(d*x+c)/(b*x+a)^2,x, algorithm="giac")

[Out]

-1/2*(a^3*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^3*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^3*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^3*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^3*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^3*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^3*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^3*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 - 3*a^2*b^2*x*imag_part(c
os_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 3*a^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^4*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^4*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 - 6*a^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^3*b*d*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 - a^3*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^3*b*d*x*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) + 4*a^3*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^4*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^4*d*im
ag_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) - 6*a^2*b^2*x*real_part(cos_int
egral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b) - 6*a^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^4*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^3*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^3*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^4*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^4*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 + 6*a^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 + 6*a^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^4*d*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^3*b*d*x*real_part(c
os_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + a^3*b*d*x*real_part(cos_integral(-d*x - a*d/b))*tan(
1/2*c)^2*tan(1/2*a*d/b)^2 - 3*a^3*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 + 3*a^3*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 - 6*a^3*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^3*b*d*x*imag_part(cos_integr
al(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) + 2*a^3*b*d*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*
tan(1/2*c) - 4*a^3*b*d*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c) + 3*a^2*b^2*x*imag_part(cos_i
ntegral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 - 3*a^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^4*d*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 - a^4*d*real_part
(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 + 6*a^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^3*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^3*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^3*b*d*x*sin_integral((b*d*x + a*
d)/b)*tan(1/2*d*x)^2*tan(1/2*a*d/b) - 12*a^2*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) + 12*a^2*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^4*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^4*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) - 24*a^2*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^3*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^3*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^3*b*d*x*s
in_integral((b*d*x + a*d)/b)*tan(1/2*c)^2*tan(1/2*a*d/b) - 6*a^3*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) - 6*a^3*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) + 3*a^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 - 3*a^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^4*d*real_part(cos_integral(d*x
+ a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 - a^4*d*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2
*a*d/b)^2 + 6*a^2*b^2*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + 2*a^3*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^3*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^3*b*d*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)*tan(1/2*a*d/b)^2 + 6*a^3*b*rea
l_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 + 6*a^3*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 - 3*a^2*b^2*x*imag_part(cos_integral(d*x + a*d/b))*t
an(1/2*c)^2*tan(1/2*a*d/b)^2 + 3*a^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
+ a^4*d*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + a^4*d*real_part(cos_integral(-d*
x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 6*a^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^3*b*d*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2 + a^3*b*d*x*real_part(cos_integral(-d*x
- a*d/b))*tan(1/2*d*x)^2 - 2*a^4*d*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) + 2*a^4*d*im
ag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) - 6*a^2*b^2*x*real_part(cos_integral(d*x + a*d/b
))*tan(1/2*d*x)^2*tan(1/2*c) - 6*a^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^4*d*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c) - a^3*b*d*x*real_part(cos_integral(d*x + a*d/b)
)*tan(1/2*c)^2 - a^3*b*d*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2 + 3*a^3*b*imag_part(cos_integral
(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 - 3*a^3*b*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(
1/2*c)^2 + 6*a^3*b*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*a^4*d*imag_part(cos_integral(
d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b) - 2*a^4*d*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan
(1/2*a*d/b) + 6*a^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) + 6*a^2*b^2*x*rea
l_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b) + 4*a^4*d*sin_integral((b*d*x + a*d)/b)*tan(1
/2*d*x)^2*tan(1/2*a*d/b) + 4*a^3*b*d*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) + 4*a^3*
b*d*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) - 12*a^3*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) + 12*a^3*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) - 24*a^3*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^4*d*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b) + 2*a^4*d*imag_part(cos_integral(-
d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b) - 6*a^2*b^2*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(
1/2*a*d/b) - 6*a^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^4*d*sin_integ
ral((b*d*x + a*d)/b)*tan(1/2*c)^2*tan(1/2*a*d/b) - a^3*b*d*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*a*d/
b)^2 - a^3*b*d*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b)^2 + 3*a^3*b*imag_part(cos_integral(d*x +
a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 - 3*a^3*b*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/
2*a*d/b)^2 + 6*a^3*b*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + 2*a^4*d*imag_part(cos_int
egral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*a^4*d*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan
(1/2*a*d/b)^2 + 6*a^2*b^2*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2 + 6*a^2*b^2*x*rea
l_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2 + 4*a^4*d*sin_integral((b*d*x + a*d)/b)*tan(1/2
*c)*tan(1/2*a*d/b)^2 + 4*a^3*b*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 - 3*a^3*b*imag_part(cos_integral(d*x
+ a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 3*a^3*b*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*
a*d/b)^2 - 6*a^3*b*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 4*a^3*b*tan(1/2*d*x)*tan(1/2*
c)^2*tan(1/2*a*d/b)^2 - 3*a^2*b^2*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2 + 3*a^2*b^2*x*imag_par
t(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2 + a^4*d*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2 + a^4
*d*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2 - 6*a^2*b^2*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*
x)^2 - 2*a^3*b*d*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c) + 2*a^3*b*d*x*imag_part(cos_integral(-d*x -
a*d/b))*tan(1/2*c) - 4*a^3*b*d*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*c) - 6*a^3*b*real_part(cos_integral(d*
x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) - 6*a^3*b*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c
) + 3*a^2*b^2*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2 - 3*a^2*b^2*x*imag_part(cos_integral(-d*x -
a*d/b))*tan(1/2*c)^2 - a^4*d*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2 - a^4*d*real_part(cos_integral(
-d*x - a*d/b))*tan(1/2*c)^2 + 6*a^2*b^2*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)^2 + 2*a^3*b*d*x*imag_part(c
os_integral(d*x + a*d/b))*tan(1/2*a*d/b) - 2*a^3*b*d*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b) +
4*a^3*b*d*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*a*d/b) + 6*a^3*b*real_part(cos_integral(d*x + a*d/b))*tan(1/
2*d*x)^2*tan(1/2*a*d/b) + 6*a^3*b*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b) - 12*a^2
*b^2*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) + 12*a^2*b^2*x*imag_part(cos_integral(-d
*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) + 4*a^4*d*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b
) + 4*a^4*d*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) - 24*a^2*b^2*x*sin_integral((b*d*x
+ a*d)/b)*tan(1/2*c)*tan(1/2*a*d/b) - 6*a^3*b*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b
) - 6*a^3*b*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b) + 3*a^2*b^2*x*imag_part(cos_inte
gral(d*x + a*d/b))*tan(1/2*a*d/b)^2 - 3*a^2*b^2*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b)^2 - a^4
*d*real_part(cos_integral(d*x + a*d/b))*tan(1/2*a*d/b)^2 - a^4*d*real_part(cos_integral(-d*x - a*d/b))*tan(1/2
*a*d/b)^2 + 6*a^2*b^2*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*a*d/b)^2 + 6*a^3*b*real_part(cos_integral(d*x +
a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2 + 6*a^3*b*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^
2 + a^3*b*d*x*real_part(cos_integral(d*x + a*d/b)) + a^3*b*d*x*real_part(cos_integral(-d*x - a*d/b)) - 3*a^3*b
*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2 + 3*a^3*b*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d
*x)^2 - 6*a^3*b*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2 - 2*a^4*d*imag_part(cos_integral(d*x + a*d/b))*ta
n(1/2*c) + 2*a^4*d*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c) - 6*a^2*b^2*x*real_part(cos_integral(d*x +
a*d/b))*tan(1/2*c) - 6*a^2*b^2*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c) - 4*a^4*d*sin_integral((b*d
*x + a*d)/b)*tan(1/2*c) + 4*a^3*b*tan(1/2*d*x)^2*tan(1/2*c) + 3*a^3*b*imag_part(cos_integral(d*x + a*d/b))*tan
(1/2*c)^2 - 3*a^3*b*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2 + 6*a^3*b*sin_integral((b*d*x + a*d)/b)
*tan(1/2*c)^2 + 4*a^3*b*tan(1/2*d*x)*tan(1/2*c)^2 + 2*a^4*d*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*a*d/b
) - 2*a^4*d*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b) + 6*a^2*b^2*x*real_part(cos_integral(d*x + a*
d/b))*tan(1/2*a*d/b) + 6*a^2*b^2*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b) + 4*a^4*d*sin_integral
((b*d*x + a*d)/b)*tan(1/2*a*d/b) - 12*a^3*b*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) + 1
2*a^3*b*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) - 24*a^3*b*sin_integral((b*d*x + a*d)/
b)*tan(1/2*c)*tan(1/2*a*d/b) + 3*a^3*b*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*a*d/b)^2 - 3*a^3*b*imag_pa
rt(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b)^2 + 6*a^3*b*sin_integral((b*d*x + a*d)/b)*tan(1/2*a*d/b)^2 - 4*a
^3*b*tan(1/2*d*x)*tan(1/2*a*d/b)^2 - 4*a^3*b*tan(1/2*c)*tan(1/2*a*d/b)^2 - 3*a^2*b^2*x*imag_part(cos_integral(
d*x + a*d/b)) + 3*a^2*b^2*x*imag_part(cos_integral(-d*x - a*d/b)) + a^4*d*real_part(cos_integral(d*x + a*d/b))
+ a^4*d*real_part(cos_integral(-d*x - a*d/b)) - 6*a^2*b^2*x*sin_integral((b*d*x + a*d)/b) - 6*a^3*b*real_part
(cos_integral(d*x + a*d/b))*tan(1/2*c) - 6*a^3*b*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c) + 6*a^3*b*re
al_part(cos_integral(d*x + a*d/b))*tan(1/2*a*d/b) + 6*a^3*b*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/
b) - 3*a^3*b*imag_part(cos_integral(d*x + a*d/b)) + 3*a^3*b*imag_part(cos_integral(-d*x - a*d/b)) - 6*a^3*b*si
n_integral((b*d*x + a*d)/b) - 4*a^3*b*tan(1/2*d*x) - 4*a^3*b*tan(1/2*c))/(b^6*x*tan(1/2*d*x)^2*tan(1/2*c)^2*ta
n(1/2*a*d/b)^2 + a*b^5*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + b^6*x*tan(1/2*d*x)^2*tan(1/2*c)^2 + b^6*
x*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + b^6*x*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + a*b^5*tan(1/2*d*x)^2*tan(1/2*c)^2 +
a*b^5*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + a*b^5*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + b^6*x*tan(1/2*d*x)^2 + b^6*x*tan
(1/2*c)^2 + b^6*x*tan(1/2*a*d/b)^2 + a*b^5*tan(1/2*d*x)^2 + a*b^5*tan(1/2*c)^2 + a*b^5*tan(1/2*a*d/b)^2 + b^6*
x + a*b^5)