3.26 \(\int \frac{x^4 \sin (c+d x)}{(a+b x)^2} \, dx\)
Optimal. Leaf size=233 \[ -\frac{4 a^3 \sin \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (\frac{a d}{b}+d x\right )}{b^5}+\frac{a^4 d \cos \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (\frac{a d}{b}+d x\right )}{b^6}-\frac{a^4 d \sin \left (c-\frac{a d}{b}\right ) \text{Si}\left (x d+\frac{a d}{b}\right )}{b^6}-\frac{4 a^3 \cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (x d+\frac{a d}{b}\right )}{b^5}-\frac{a^4 \sin (c+d x)}{b^5 (a+b x)}-\frac{3 a^2 \cos (c+d x)}{b^4 d}-\frac{2 a \sin (c+d x)}{b^3 d^2}+\frac{2 a x \cos (c+d x)}{b^3 d}+\frac{2 x \sin (c+d x)}{b^2 d^2}+\frac{2 \cos (c+d x)}{b^2 d^3}-\frac{x^2 \cos (c+d x)}{b^2 d} \]
[Out]
(2*Cos[c + d*x])/(b^2*d^3) - (3*a^2*Cos[c + d*x])/(b^4*d) + (2*a*x*Cos[c + d*x])/(b^3*d) - (x^2*Cos[c + d*x])/
(b^2*d) + (a^4*d*Cos[c - (a*d)/b]*CosIntegral[(a*d)/b + d*x])/b^6 - (4*a^3*CosIntegral[(a*d)/b + d*x]*Sin[c -
(a*d)/b])/b^5 - (2*a*Sin[c + d*x])/(b^3*d^2) + (2*x*Sin[c + d*x])/(b^2*d^2) - (a^4*Sin[c + d*x])/(b^5*(a + b*x
)) - (4*a^3*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/b^5 - (a^4*d*Sin[c - (a*d)/b]*SinIntegral[(a*d)/b + d
*x])/b^6
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Rubi [A] time = 0.508966, antiderivative size = 233, normalized size of antiderivative = 1.,
number of steps used = 15, 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} \[ -\frac{4 a^3 \sin \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (\frac{a d}{b}+d x\right )}{b^5}+\frac{a^4 d \cos \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (\frac{a d}{b}+d x\right )}{b^6}-\frac{a^4 d \sin \left (c-\frac{a d}{b}\right ) \text{Si}\left (x d+\frac{a d}{b}\right )}{b^6}-\frac{4 a^3 \cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (x d+\frac{a d}{b}\right )}{b^5}-\frac{a^4 \sin (c+d x)}{b^5 (a+b x)}-\frac{3 a^2 \cos (c+d x)}{b^4 d}-\frac{2 a \sin (c+d x)}{b^3 d^2}+\frac{2 a x \cos (c+d x)}{b^3 d}+\frac{2 x \sin (c+d x)}{b^2 d^2}+\frac{2 \cos (c+d x)}{b^2 d^3}-\frac{x^2 \cos (c+d x)}{b^2 d} \]
Antiderivative was successfully verified.
[In]
Int[(x^4*Sin[c + d*x])/(a + b*x)^2,x]
[Out]
(2*Cos[c + d*x])/(b^2*d^3) - (3*a^2*Cos[c + d*x])/(b^4*d) + (2*a*x*Cos[c + d*x])/(b^3*d) - (x^2*Cos[c + d*x])/
(b^2*d) + (a^4*d*Cos[c - (a*d)/b]*CosIntegral[(a*d)/b + d*x])/b^6 - (4*a^3*CosIntegral[(a*d)/b + d*x]*Sin[c -
(a*d)/b])/b^5 - (2*a*Sin[c + d*x])/(b^3*d^2) + (2*x*Sin[c + d*x])/(b^2*d^2) - (a^4*Sin[c + d*x])/(b^5*(a + b*x
)) - (4*a^3*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/b^5 - (a^4*d*Sin[c - (a*d)/b]*SinIntegral[(a*d)/b + d
*x])/b^6
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^4 \sin (c+d x)}{(a+b x)^2} \, dx &=\int \left (\frac{3 a^2 \sin (c+d x)}{b^4}-\frac{2 a x \sin (c+d x)}{b^3}+\frac{x^2 \sin (c+d x)}{b^2}+\frac{a^4 \sin (c+d x)}{b^4 (a+b x)^2}-\frac{4 a^3 \sin (c+d x)}{b^4 (a+b x)}\right ) \, dx\\ &=\frac{\left (3 a^2\right ) \int \sin (c+d x) \, dx}{b^4}-\frac{\left (4 a^3\right ) \int \frac{\sin (c+d x)}{a+b x} \, dx}{b^4}+\frac{a^4 \int \frac{\sin (c+d x)}{(a+b x)^2} \, dx}{b^4}-\frac{(2 a) \int x \sin (c+d x) \, dx}{b^3}+\frac{\int x^2 \sin (c+d x) \, dx}{b^2}\\ &=-\frac{3 a^2 \cos (c+d x)}{b^4 d}+\frac{2 a x \cos (c+d x)}{b^3 d}-\frac{x^2 \cos (c+d x)}{b^2 d}-\frac{a^4 \sin (c+d x)}{b^5 (a+b x)}-\frac{(2 a) \int \cos (c+d x) \, dx}{b^3 d}+\frac{2 \int x \cos (c+d x) \, dx}{b^2 d}+\frac{\left (a^4 d\right ) \int \frac{\cos (c+d x)}{a+b x} \, dx}{b^5}-\frac{\left (4 a^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}{b^4}-\frac{\left (4 a^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}{b^4}\\ &=-\frac{3 a^2 \cos (c+d x)}{b^4 d}+\frac{2 a x \cos (c+d x)}{b^3 d}-\frac{x^2 \cos (c+d x)}{b^2 d}-\frac{4 a^3 \text{Ci}\left (\frac{a d}{b}+d x\right ) \sin \left (c-\frac{a d}{b}\right )}{b^5}-\frac{2 a \sin (c+d x)}{b^3 d^2}+\frac{2 x \sin (c+d x)}{b^2 d^2}-\frac{a^4 \sin (c+d x)}{b^5 (a+b x)}-\frac{4 a^3 \cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (\frac{a d}{b}+d x\right )}{b^5}-\frac{2 \int \sin (c+d x) \, dx}{b^2 d^2}+\frac{\left (a^4 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^5}-\frac{\left (a^4 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^5}\\ &=\frac{2 \cos (c+d x)}{b^2 d^3}-\frac{3 a^2 \cos (c+d x)}{b^4 d}+\frac{2 a x \cos (c+d x)}{b^3 d}-\frac{x^2 \cos (c+d x)}{b^2 d}+\frac{a^4 d \cos \left (c-\frac{a d}{b}\right ) \text{Ci}\left (\frac{a d}{b}+d x\right )}{b^6}-\frac{4 a^3 \text{Ci}\left (\frac{a d}{b}+d x\right ) \sin \left (c-\frac{a d}{b}\right )}{b^5}-\frac{2 a \sin (c+d x)}{b^3 d^2}+\frac{2 x \sin (c+d x)}{b^2 d^2}-\frac{a^4 \sin (c+d x)}{b^5 (a+b x)}-\frac{4 a^3 \cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (\frac{a d}{b}+d x\right )}{b^5}-\frac{a^4 d \sin \left (c-\frac{a d}{b}\right ) \text{Si}\left (\frac{a d}{b}+d x\right )}{b^6}\\ \end{align*}
Mathematica [A] time = 1.01506, size = 177, normalized size = 0.76 \[ \frac{-\frac{b \left (d \left (2 a^2 b^2+a^4 d^2-2 b^4 x^2\right ) \sin (c+d x)+b (a+b x) \left (3 a^2 d^2-2 a b d^2 x+b^2 \left (d^2 x^2-2\right )\right ) \cos (c+d x)\right )}{d^3 (a+b x)}+a^3 \text{CosIntegral}\left (d \left (\frac{a}{b}+x\right )\right ) \left (a d \cos \left (c-\frac{a d}{b}\right )-4 b \sin \left (c-\frac{a d}{b}\right )\right )-a^3 \text{Si}\left (d \left (\frac{a}{b}+x\right )\right ) \left (a d \sin \left (c-\frac{a d}{b}\right )+4 b \cos \left (c-\frac{a d}{b}\right )\right )}{b^6} \]
Antiderivative was successfully verified.
[In]
Integrate[(x^4*Sin[c + d*x])/(a + b*x)^2,x]
[Out]
(a^3*CosIntegral[d*(a/b + x)]*(a*d*Cos[c - (a*d)/b] - 4*b*Sin[c - (a*d)/b]) - (b*(b*(a + b*x)*(3*a^2*d^2 - 2*a
*b*d^2*x + b^2*(-2 + d^2*x^2))*Cos[c + d*x] + d*(2*a^2*b^2 + a^4*d^2 - 2*b^4*x^2)*Sin[c + d*x]))/(d^3*(a + b*x
)) - a^3*(4*b*Cos[c - (a*d)/b] + a*d*Sin[c - (a*d)/b])*SinIntegral[d*(a/b + x)])/b^6
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Maple [B] time = 0.02, size = 1214, normalized size = 5.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^4*sin(d*x+c)/(b*x+a)^2,x)
[Out]
1/d^5*((3*a^2*d^2-6*a*b*c*d+3*b^2*c^2-2*a*b*d+2*b^2*c+b^2)*d^2/b^4*(-(d*x+c)^2*cos(d*x+c)+2*cos(d*x+c)+2*(d*x+
c)*sin(d*x+c))+(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)*d^2/b^4*(-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)-4/b^4*(a^3*
d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*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)*s
in((a*d-b*c)/b)/b)-4*d^2*c*(-2*a*d+2*b*c+b)/b^3*(sin(d*x+c)-(d*x+c)*cos(d*x+c))+4*(a^3*d^3-3*a^2*b*c*d^2+3*a*b
^2*c^2*d-b^3*c^3)*d^2*c/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)-12/b^3*(a^2*d^2-2*a*b*c*d+b^2*c^2)*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)-6*d^2*c^2/b^2*cos(d*x+c)+6*(a^2*d^2-2*a*b*c*d+b^2*c^2)*
d^2*c^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)-12/b^2*(a*d-b*c)*d^2*c^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)+4*d^2*(a*d-b*c)/b*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)-4*d^2*c^3/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^4*(-sin(d*x+c)/((d*x+c)*b+d*a-c*b)/b+(Si(d*x+c+(a*d-b*c)/b)*s
in((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^4*sin(d*x+c)/(b*x+a)^2,x, algorithm="maxima")
[Out]
Timed out
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Fricas [A] time = 1.85265, size = 794, normalized size = 3.41 \begin{align*} -\frac{2 \,{\left (b^{5} d^{2} x^{3} - a b^{4} d^{2} x^{2} + 3 \, a^{3} b^{2} d^{2} - 2 \, a b^{4} +{\left (a^{2} b^{3} d^{2} - 2 \, b^{5}\right )} x\right )} \cos \left (d x + c\right ) -{\left ({\left (a^{4} b d^{4} x + a^{5} d^{4}\right )} \operatorname{Ci}\left (\frac{b d x + a d}{b}\right ) +{\left (a^{4} b d^{4} x + a^{5} d^{4}\right )} \operatorname{Ci}\left (-\frac{b d x + a d}{b}\right ) - 8 \,{\left (a^{3} b^{2} d^{3} x + a^{4} b d^{3}\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^{4} b d^{3} - 2 \, b^{5} d x^{2} + 2 \, a^{2} b^{3} d\right )} \sin \left (d x + c\right ) - 2 \,{\left (2 \,{\left (a^{3} b^{2} d^{3} x + a^{4} b d^{3}\right )} \operatorname{Ci}\left (\frac{b d x + a d}{b}\right ) + 2 \,{\left (a^{3} b^{2} d^{3} x + a^{4} b d^{3}\right )} \operatorname{Ci}\left (-\frac{b d x + a d}{b}\right ) +{\left (a^{4} b d^{4} x + a^{5} d^{4}\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^{7} d^{3} x + a b^{6} d^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^4*sin(d*x+c)/(b*x+a)^2,x, algorithm="fricas")
[Out]
-1/2*(2*(b^5*d^2*x^3 - a*b^4*d^2*x^2 + 3*a^3*b^2*d^2 - 2*a*b^4 + (a^2*b^3*d^2 - 2*b^5)*x)*cos(d*x + c) - ((a^4
*b*d^4*x + a^5*d^4)*cos_integral((b*d*x + a*d)/b) + (a^4*b*d^4*x + a^5*d^4)*cos_integral(-(b*d*x + a*d)/b) - 8
*(a^3*b^2*d^3*x + a^4*b*d^3)*sin_integral((b*d*x + a*d)/b))*cos(-(b*c - a*d)/b) + 2*(a^4*b*d^3 - 2*b^5*d*x^2 +
2*a^2*b^3*d)*sin(d*x + c) - 2*(2*(a^3*b^2*d^3*x + a^4*b*d^3)*cos_integral((b*d*x + a*d)/b) + 2*(a^3*b^2*d^3*x
+ a^4*b*d^3)*cos_integral(-(b*d*x + a*d)/b) + (a^4*b*d^4*x + a^5*d^4)*sin_integral((b*d*x + a*d)/b))*sin(-(b*
c - a*d)/b))/(b^7*d^3*x + a*b^6*d^3)
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4} \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**4*sin(d*x+c)/(b*x+a)**2,x)
[Out]
Integral(x**4*sin(c + d*x)/(a + b*x)**2, x)
________________________________________________________________________________________
Giac [C] time = 1.43496, size = 8586, normalized size = 36.85 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^4*sin(d*x+c)/(b*x+a)^2,x, algorithm="giac")
[Out]
1/2*(a^4*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^4*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^4*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^4*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^4*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^4*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^4*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^4*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 - 4*a^3*b^2*x*imag_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 + 4*a^3*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^5*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^5*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 - 8*a^3*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
^4*b*d*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 - a^4*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^4*b*d*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*ta
n(1/2*c)*tan(1/2*a*d/b) + 4*a^4*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^5*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^5*d*ima
g_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) - 8*a^3*b^2*x*real_part(cos_inte
gral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b) - 8*a^3*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^5*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^4*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^4*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^5*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^5*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 + 8*a^3*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 + 8*a^3*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^5*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^4*b*d*x*real_part(co
s_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + a^4*b*d*x*real_part(cos_integral(-d*x - a*d/b))*tan(1
/2*c)^2*tan(1/2*a*d/b)^2 - 4*a^4*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^4*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 - 8*a^4*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^4*b*d*x*imag_part(cos_integra
l(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) + 2*a^4*b*d*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*t
an(1/2*c) - 4*a^4*b*d*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c) + 4*a^3*b^2*x*imag_part(cos_in
tegral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 - 4*a^3*b^2*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d
*x)^2*tan(1/2*c)^2 - a^5*d*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 - a^5*d*real_part(
cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 + 8*a^3*b^2*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*
x)^2*tan(1/2*c)^2 + 2*a^4*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^4*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^4*b*d*x*sin_integral((b*d*x + a*d
)/b)*tan(1/2*d*x)^2*tan(1/2*a*d/b) - 16*a^3*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) + 16*a^3*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^5*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^5*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) - 32*a^3*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^4*b*d*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*t
an(1/2*a*d/b) + 2*a^4*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^4*b*d*x*si
n_integral((b*d*x + a*d)/b)*tan(1/2*c)^2*tan(1/2*a*d/b) - 8*a^4*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) - 8*a^4*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^3*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 - 4*a^3*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^5*d*real_part(cos_integral(d*x +
a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 - a^5*d*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*
a*d/b)^2 + 8*a^3*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^4*b*d*x*imag_part(c
os_integral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*a^4*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^4*b*d*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)*tan(1/2*a*d/b)^2 + 8*a^4*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 + 8*a^4*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 - 4*a^3*b^2*x*imag_part(cos_integral(d*x + a*d/b))*ta
n(1/2*c)^2*tan(1/2*a*d/b)^2 + 4*a^3*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^5*d*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + a^5*d*real_part(cos_integral(-d*x
- a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 8*a^3*b^2*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)^2*tan(1/2*a*d/
b)^2 + a^4*b*d*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2 + a^4*b*d*x*real_part(cos_integral(-d*x -
a*d/b))*tan(1/2*d*x)^2 - 2*a^5*d*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) + 2*a^5*d*ima
g_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) - 8*a^3*b^2*x*real_part(cos_integral(d*x + a*d/b)
)*tan(1/2*d*x)^2*tan(1/2*c) - 8*a^3*b^2*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) - 4*
a^5*d*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c) - a^4*b*d*x*real_part(cos_integral(d*x + a*d/b))
*tan(1/2*c)^2 - a^4*b*d*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2 + 4*a^4*b*imag_part(cos_integral(
d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 - 4*a^4*b*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1
/2*c)^2 + 8*a^4*b*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*a^5*d*imag_part(cos_integral(d
*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b) - 2*a^5*d*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(
1/2*a*d/b) + 8*a^3*b^2*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b) + 8*a^3*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^5*d*sin_integral((b*d*x + a*d)/b)*tan(1/
2*d*x)^2*tan(1/2*a*d/b) + 4*a^4*b*d*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) + 4*a^4*b
*d*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) - 16*a^4*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^4*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) - 32*a^4*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^5*d*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b) + 2*a^5*d*imag_part(cos_integral(-d
*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b) - 8*a^3*b^2*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1
/2*a*d/b) - 8*a^3*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^5*d*sin_integr
al((b*d*x + a*d)/b)*tan(1/2*c)^2*tan(1/2*a*d/b) - a^4*b*d*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*a*d/b
)^2 - a^4*b*d*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b)^2 + 4*a^4*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^4*b*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2
*a*d/b)^2 + 8*a^4*b*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + 2*a^5*d*imag_part(cos_inte
gral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*a^5*d*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(
1/2*a*d/b)^2 + 8*a^3*b^2*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2 + 8*a^3*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^5*d*sin_integral((b*d*x + a*d)/b)*tan(1/2*
c)*tan(1/2*a*d/b)^2 + 4*a^4*b*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 - 4*a^4*b*imag_part(cos_integral(d*x
+ a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 4*a^4*b*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a
*d/b)^2 - 8*a^4*b*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 4*a^4*b*tan(1/2*d*x)*tan(1/2*c
)^2*tan(1/2*a*d/b)^2 - 4*a^3*b^2*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2 + 4*a^3*b^2*x*imag_part
(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2 + a^5*d*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2 + a^5*
d*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2 - 8*a^3*b^2*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x
)^2 - 2*a^4*b*d*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c) + 2*a^4*b*d*x*imag_part(cos_integral(-d*x -
a*d/b))*tan(1/2*c) - 4*a^4*b*d*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*c) - 8*a^4*b*real_part(cos_integral(d*x
+ a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) - 8*a^4*b*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)
+ 4*a^3*b^2*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2 - 4*a^3*b^2*x*imag_part(cos_integral(-d*x - a
*d/b))*tan(1/2*c)^2 - a^5*d*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2 - a^5*d*real_part(cos_integral(-
d*x - a*d/b))*tan(1/2*c)^2 + 8*a^3*b^2*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)^2 + 2*a^4*b*d*x*imag_part(co
s_integral(d*x + a*d/b))*tan(1/2*a*d/b) - 2*a^4*b*d*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b) + 4
*a^4*b*d*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*a*d/b) + 8*a^4*b*real_part(cos_integral(d*x + a*d/b))*tan(1/2
*d*x)^2*tan(1/2*a*d/b) + 8*a^4*b*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b) - 16*a^3*
b^2*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) + 16*a^3*b^2*x*imag_part(cos_integral(-d*
x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) + 4*a^5*d*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)
+ 4*a^5*d*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) - 32*a^3*b^2*x*sin_integral((b*d*x
+ a*d)/b)*tan(1/2*c)*tan(1/2*a*d/b) - 8*a^4*b*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)
- 8*a^4*b*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b) + 4*a^3*b^2*x*imag_part(cos_integ
ral(d*x + a*d/b))*tan(1/2*a*d/b)^2 - 4*a^3*b^2*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b)^2 - a^5*
d*real_part(cos_integral(d*x + a*d/b))*tan(1/2*a*d/b)^2 - a^5*d*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*
a*d/b)^2 + 8*a^3*b^2*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*a*d/b)^2 + 8*a^4*b*real_part(cos_integral(d*x + a
*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2 + 8*a^4*b*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2
+ a^4*b*d*x*real_part(cos_integral(d*x + a*d/b)) + a^4*b*d*x*real_part(cos_integral(-d*x - a*d/b)) - 4*a^4*b*
imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2 + 4*a^4*b*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*
x)^2 - 8*a^4*b*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2 - 2*a^5*d*imag_part(cos_integral(d*x + a*d/b))*tan
(1/2*c) + 2*a^5*d*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c) - 8*a^3*b^2*x*real_part(cos_integral(d*x +
a*d/b))*tan(1/2*c) - 8*a^3*b^2*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c) - 4*a^5*d*sin_integral((b*d*
x + a*d)/b)*tan(1/2*c) + 4*a^4*b*tan(1/2*d*x)^2*tan(1/2*c) + 4*a^4*b*imag_part(cos_integral(d*x + a*d/b))*tan(
1/2*c)^2 - 4*a^4*b*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2 + 8*a^4*b*sin_integral((b*d*x + a*d)/b)*
tan(1/2*c)^2 + 4*a^4*b*tan(1/2*d*x)*tan(1/2*c)^2 + 2*a^5*d*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*a*d/b)
- 2*a^5*d*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b) + 8*a^3*b^2*x*real_part(cos_integral(d*x + a*d
/b))*tan(1/2*a*d/b) + 8*a^3*b^2*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b) + 4*a^5*d*sin_integral(
(b*d*x + a*d)/b)*tan(1/2*a*d/b) - 16*a^4*b*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) + 16
*a^4*b*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) - 32*a^4*b*sin_integral((b*d*x + a*d)/b
)*tan(1/2*c)*tan(1/2*a*d/b) + 4*a^4*b*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*a*d/b)^2 - 4*a^4*b*imag_par
t(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b)^2 + 8*a^4*b*sin_integral((b*d*x + a*d)/b)*tan(1/2*a*d/b)^2 - 4*a^
4*b*tan(1/2*d*x)*tan(1/2*a*d/b)^2 - 4*a^4*b*tan(1/2*c)*tan(1/2*a*d/b)^2 - 4*a^3*b^2*x*imag_part(cos_integral(d
*x + a*d/b)) + 4*a^3*b^2*x*imag_part(cos_integral(-d*x - a*d/b)) + a^5*d*real_part(cos_integral(d*x + a*d/b))
+ a^5*d*real_part(cos_integral(-d*x - a*d/b)) - 8*a^3*b^2*x*sin_integral((b*d*x + a*d)/b) - 8*a^4*b*real_part(
cos_integral(d*x + a*d/b))*tan(1/2*c) - 8*a^4*b*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c) + 8*a^4*b*rea
l_part(cos_integral(d*x + a*d/b))*tan(1/2*a*d/b) + 8*a^4*b*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b
) - 4*a^4*b*imag_part(cos_integral(d*x + a*d/b)) + 4*a^4*b*imag_part(cos_integral(-d*x - a*d/b)) - 8*a^4*b*sin
_integral((b*d*x + a*d)/b) - 4*a^4*b*tan(1/2*d*x) - 4*a^4*b*tan(1/2*c))/(b^7*x*tan(1/2*d*x)^2*tan(1/2*c)^2*tan
(1/2*a*d/b)^2 + a*b^6*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + b^7*x*tan(1/2*d*x)^2*tan(1/2*c)^2 + b^7*x
*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + b^7*x*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + a*b^6*tan(1/2*d*x)^2*tan(1/2*c)^2 + a
*b^6*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + a*b^6*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + b^7*x*tan(1/2*d*x)^2 + b^7*x*tan(
1/2*c)^2 + b^7*x*tan(1/2*a*d/b)^2 + a*b^6*tan(1/2*d*x)^2 + a*b^6*tan(1/2*c)^2 + a*b^6*tan(1/2*a*d/b)^2 + b^7*x
+ a*b^6)