3.9 \(\int \frac{\cos (a+b x) \sin (a+b x)}{(c+d x)^4} \, dx\)
Optimal. Leaf size=144 \[ -\frac{2 b^3 \cos \left (2 a-\frac{2 b c}{d}\right ) \text{CosIntegral}\left (\frac{2 b c}{d}+2 b x\right )}{3 d^4}+\frac{2 b^3 \sin \left (2 a-\frac{2 b c}{d}\right ) \text{Si}\left (\frac{2 b c}{d}+2 b x\right )}{3 d^4}+\frac{b^2 \sin (2 a+2 b x)}{3 d^3 (c+d x)}-\frac{b \cos (2 a+2 b x)}{6 d^2 (c+d x)^2}-\frac{\sin (2 a+2 b x)}{6 d (c+d x)^3} \]
[Out]
-(b*Cos[2*a + 2*b*x])/(6*d^2*(c + d*x)^2) - (2*b^3*Cos[2*a - (2*b*c)/d]*CosIntegral[(2*b*c)/d + 2*b*x])/(3*d^4
) - Sin[2*a + 2*b*x]/(6*d*(c + d*x)^3) + (b^2*Sin[2*a + 2*b*x])/(3*d^3*(c + d*x)) + (2*b^3*Sin[2*a - (2*b*c)/d
]*SinIntegral[(2*b*c)/d + 2*b*x])/(3*d^4)
________________________________________________________________________________________
Rubi [A] time = 0.197655, antiderivative size = 144, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used =
{4406, 12, 3297, 3303, 3299, 3302} \[ -\frac{2 b^3 \cos \left (2 a-\frac{2 b c}{d}\right ) \text{CosIntegral}\left (\frac{2 b c}{d}+2 b x\right )}{3 d^4}+\frac{2 b^3 \sin \left (2 a-\frac{2 b c}{d}\right ) \text{Si}\left (\frac{2 b c}{d}+2 b x\right )}{3 d^4}+\frac{b^2 \sin (2 a+2 b x)}{3 d^3 (c+d x)}-\frac{b \cos (2 a+2 b x)}{6 d^2 (c+d x)^2}-\frac{\sin (2 a+2 b x)}{6 d (c+d x)^3} \]
Antiderivative was successfully verified.
[In]
Int[(Cos[a + b*x]*Sin[a + b*x])/(c + d*x)^4,x]
[Out]
-(b*Cos[2*a + 2*b*x])/(6*d^2*(c + d*x)^2) - (2*b^3*Cos[2*a - (2*b*c)/d]*CosIntegral[(2*b*c)/d + 2*b*x])/(3*d^4
) - Sin[2*a + 2*b*x]/(6*d*(c + d*x)^3) + (b^2*Sin[2*a + 2*b*x])/(3*d^3*(c + d*x)) + (2*b^3*Sin[2*a - (2*b*c)/d
]*SinIntegral[(2*b*c)/d + 2*b*x])/(3*d^4)
Rule 4406
Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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{\cos (a+b x) \sin (a+b x)}{(c+d x)^4} \, dx &=\int \frac{\sin (2 a+2 b x)}{2 (c+d x)^4} \, dx\\ &=\frac{1}{2} \int \frac{\sin (2 a+2 b x)}{(c+d x)^4} \, dx\\ &=-\frac{\sin (2 a+2 b x)}{6 d (c+d x)^3}+\frac{b \int \frac{\cos (2 a+2 b x)}{(c+d x)^3} \, dx}{3 d}\\ &=-\frac{b \cos (2 a+2 b x)}{6 d^2 (c+d x)^2}-\frac{\sin (2 a+2 b x)}{6 d (c+d x)^3}-\frac{b^2 \int \frac{\sin (2 a+2 b x)}{(c+d x)^2} \, dx}{3 d^2}\\ &=-\frac{b \cos (2 a+2 b x)}{6 d^2 (c+d x)^2}-\frac{\sin (2 a+2 b x)}{6 d (c+d x)^3}+\frac{b^2 \sin (2 a+2 b x)}{3 d^3 (c+d x)}-\frac{\left (2 b^3\right ) \int \frac{\cos (2 a+2 b x)}{c+d x} \, dx}{3 d^3}\\ &=-\frac{b \cos (2 a+2 b x)}{6 d^2 (c+d x)^2}-\frac{\sin (2 a+2 b x)}{6 d (c+d x)^3}+\frac{b^2 \sin (2 a+2 b x)}{3 d^3 (c+d x)}-\frac{\left (2 b^3 \cos \left (2 a-\frac{2 b c}{d}\right )\right ) \int \frac{\cos \left (\frac{2 b c}{d}+2 b x\right )}{c+d x} \, dx}{3 d^3}+\frac{\left (2 b^3 \sin \left (2 a-\frac{2 b c}{d}\right )\right ) \int \frac{\sin \left (\frac{2 b c}{d}+2 b x\right )}{c+d x} \, dx}{3 d^3}\\ &=-\frac{b \cos (2 a+2 b x)}{6 d^2 (c+d x)^2}-\frac{2 b^3 \cos \left (2 a-\frac{2 b c}{d}\right ) \text{Ci}\left (\frac{2 b c}{d}+2 b x\right )}{3 d^4}-\frac{\sin (2 a+2 b x)}{6 d (c+d x)^3}+\frac{b^2 \sin (2 a+2 b x)}{3 d^3 (c+d x)}+\frac{2 b^3 \sin \left (2 a-\frac{2 b c}{d}\right ) \text{Si}\left (\frac{2 b c}{d}+2 b x\right )}{3 d^4}\\ \end{align*}
Mathematica [A] time = 0.660525, size = 164, normalized size = 1.14 \[ \frac{-4 b^3 (c+d x)^3 \left (\cos \left (2 a-\frac{2 b c}{d}\right ) \text{CosIntegral}\left (\frac{2 b (c+d x)}{d}\right )-\sin \left (2 a-\frac{2 b c}{d}\right ) \text{Si}\left (\frac{2 b (c+d x)}{d}\right )\right )-d \cos (2 b x) \left (\sin (2 a) \left (d^2-2 b^2 (c+d x)^2\right )+b d \cos (2 a) (c+d x)\right )+d \sin (2 b x) \left (\cos (2 a) \left (2 b^2 (c+d x)^2-d^2\right )+b d \sin (2 a) (c+d x)\right )}{6 d^4 (c+d x)^3} \]
Antiderivative was successfully verified.
[In]
Integrate[(Cos[a + b*x]*Sin[a + b*x])/(c + d*x)^4,x]
[Out]
(-(d*Cos[2*b*x]*(b*d*(c + d*x)*Cos[2*a] + (d^2 - 2*b^2*(c + d*x)^2)*Sin[2*a])) + d*((-d^2 + 2*b^2*(c + d*x)^2)
*Cos[2*a] + b*d*(c + d*x)*Sin[2*a])*Sin[2*b*x] - 4*b^3*(c + d*x)^3*(Cos[2*a - (2*b*c)/d]*CosIntegral[(2*b*(c +
d*x))/d] - Sin[2*a - (2*b*c)/d]*SinIntegral[(2*b*(c + d*x))/d]))/(6*d^4*(c + d*x)^3)
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Maple [A] time = 0.021, size = 200, normalized size = 1.4 \begin{align*}{\frac{{b}^{3}}{4} \left ( -{\frac{2\,\sin \left ( 2\,bx+2\,a \right ) }{3\, \left ( \left ( bx+a \right ) d-ad+bc \right ) ^{3}d}}+{\frac{2}{3\,d} \left ( -{\frac{\cos \left ( 2\,bx+2\,a \right ) }{ \left ( \left ( bx+a \right ) d-ad+bc \right ) ^{2}d}}-{\frac{1}{d} \left ( -2\,{\frac{\sin \left ( 2\,bx+2\,a \right ) }{ \left ( \left ( bx+a \right ) d-ad+bc \right ) d}}+2\,{\frac{1}{d} \left ( 2\,{\frac{1}{d}{\it Si} \left ( 2\,bx+2\,a+2\,{\frac{-ad+bc}{d}} \right ) \sin \left ( 2\,{\frac{-ad+bc}{d}} \right ) }+2\,{\frac{1}{d}{\it Ci} \left ( 2\,bx+2\,a+2\,{\frac{-ad+bc}{d}} \right ) \cos \left ( 2\,{\frac{-ad+bc}{d}} \right ) } \right ) } \right ) } \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(b*x+a)*sin(b*x+a)/(d*x+c)^4,x)
[Out]
1/4*b^3*(-2/3*sin(2*b*x+2*a)/((b*x+a)*d-a*d+b*c)^3/d+2/3*(-cos(2*b*x+2*a)/((b*x+a)*d-a*d+b*c)^2/d-(-2*sin(2*b*
x+2*a)/((b*x+a)*d-a*d+b*c)/d+2*(2*Si(2*b*x+2*a+2*(-a*d+b*c)/d)*sin(2*(-a*d+b*c)/d)/d+2*Ci(2*b*x+2*a+2*(-a*d+b*
c)/d)*cos(2*(-a*d+b*c)/d)/d)/d)/d)/d)
________________________________________________________________________________________
Maxima [C] time = 2.11298, size = 336, normalized size = 2.33 \begin{align*} -\frac{b^{4}{\left (i \, E_{4}\left (\frac{2 i \, b c + 2 i \,{\left (b x + a\right )} d - 2 i \, a d}{d}\right ) - i \, E_{4}\left (-\frac{2 i \, b c + 2 i \,{\left (b x + a\right )} d - 2 i \, a d}{d}\right )\right )} \cos \left (-\frac{2 \,{\left (b c - a d\right )}}{d}\right ) + b^{4}{\left (E_{4}\left (\frac{2 i \, b c + 2 i \,{\left (b x + a\right )} d - 2 i \, a d}{d}\right ) + E_{4}\left (-\frac{2 i \, b c + 2 i \,{\left (b x + a\right )} d - 2 i \, a d}{d}\right )\right )} \sin \left (-\frac{2 \,{\left (b c - a d\right )}}{d}\right )}{4 \,{\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} +{\left (b x + a\right )}^{3} d^{4} - a^{3} d^{4} + 3 \,{\left (b c d^{3} - a d^{4}\right )}{\left (b x + a\right )}^{2} + 3 \,{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )}{\left (b x + a\right )}\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(b*x+a)*sin(b*x+a)/(d*x+c)^4,x, algorithm="maxima")
[Out]
-1/4*(b^4*(I*exp_integral_e(4, (2*I*b*c + 2*I*(b*x + a)*d - 2*I*a*d)/d) - I*exp_integral_e(4, -(2*I*b*c + 2*I*
(b*x + a)*d - 2*I*a*d)/d))*cos(-2*(b*c - a*d)/d) + b^4*(exp_integral_e(4, (2*I*b*c + 2*I*(b*x + a)*d - 2*I*a*d
)/d) + exp_integral_e(4, -(2*I*b*c + 2*I*(b*x + a)*d - 2*I*a*d)/d))*sin(-2*(b*c - a*d)/d))/((b^3*c^3*d - 3*a*b
^2*c^2*d^2 + 3*a^2*b*c*d^3 + (b*x + a)^3*d^4 - a^3*d^4 + 3*(b*c*d^3 - a*d^4)*(b*x + a)^2 + 3*(b^2*c^2*d^2 - 2*
a*b*c*d^3 + a^2*d^4)*(b*x + a))*b)
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Fricas [B] time = 0.53332, size = 703, normalized size = 4.88 \begin{align*} \frac{b d^{3} x + b c d^{2} - 2 \,{\left (b d^{3} x + b c d^{2}\right )} \cos \left (b x + a\right )^{2} + 2 \,{\left (2 \, b^{2} d^{3} x^{2} + 4 \, b^{2} c d^{2} x + 2 \, b^{2} c^{2} d - d^{3}\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right ) + 4 \,{\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \sin \left (-\frac{2 \,{\left (b c - a d\right )}}{d}\right ) \operatorname{Si}\left (\frac{2 \,{\left (b d x + b c\right )}}{d}\right ) - 2 \,{\left ({\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \operatorname{Ci}\left (\frac{2 \,{\left (b d x + b c\right )}}{d}\right ) +{\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \operatorname{Ci}\left (-\frac{2 \,{\left (b d x + b c\right )}}{d}\right )\right )} \cos \left (-\frac{2 \,{\left (b c - a d\right )}}{d}\right )}{6 \,{\left (d^{7} x^{3} + 3 \, c d^{6} x^{2} + 3 \, c^{2} d^{5} x + c^{3} d^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(b*x+a)*sin(b*x+a)/(d*x+c)^4,x, algorithm="fricas")
[Out]
1/6*(b*d^3*x + b*c*d^2 - 2*(b*d^3*x + b*c*d^2)*cos(b*x + a)^2 + 2*(2*b^2*d^3*x^2 + 4*b^2*c*d^2*x + 2*b^2*c^2*d
- d^3)*cos(b*x + a)*sin(b*x + a) + 4*(b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + b^3*c^3)*sin(-2*(b*c -
a*d)/d)*sin_integral(2*(b*d*x + b*c)/d) - 2*((b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + b^3*c^3)*cos_int
egral(2*(b*d*x + b*c)/d) + (b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + b^3*c^3)*cos_integral(-2*(b*d*x +
b*c)/d))*cos(-2*(b*c - a*d)/d))/(d^7*x^3 + 3*c*d^6*x^2 + 3*c^2*d^5*x + c^3*d^4)
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin{\left (a + b x \right )} \cos{\left (a + b x \right )}}{\left (c + d x\right )^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(b*x+a)*sin(b*x+a)/(d*x+c)**4,x)
[Out]
Integral(sin(a + b*x)*cos(a + b*x)/(c + d*x)**4, x)
________________________________________________________________________________________
Giac [C] time = 1.80927, size = 10249, normalized size = 71.17 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(b*x+a)*sin(b*x+a)/(d*x+c)^4,x, algorithm="giac")
[Out]
-1/6*(2*b^3*d^3*x^3*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(a)^2*tan(b*c/d)^2 + 2*b^3*d^3*x^3*
real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a)^2*tan(b*c/d)^2 - 4*b^3*d^3*x^3*imag_part(cos_integ
ral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(a)^2*tan(b*c/d) + 4*b^3*d^3*x^3*imag_part(cos_integral(-2*b*x - 2*b*c/d))
*tan(b*x)^2*tan(a)^2*tan(b*c/d) - 8*b^3*d^3*x^3*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(a)^2*tan(b*c/d)
+ 4*b^3*d^3*x^3*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(a)*tan(b*c/d)^2 - 4*b^3*d^3*x^3*imag_
part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a)*tan(b*c/d)^2 + 8*b^3*d^3*x^3*sin_integral(2*(b*d*x + b*
c)/d)*tan(b*x)^2*tan(a)*tan(b*c/d)^2 + 6*b^3*c*d^2*x^2*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan
(a)^2*tan(b*c/d)^2 + 6*b^3*c*d^2*x^2*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a)^2*tan(b*c/d)^
2 - 2*b^3*d^3*x^3*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(a)^2 - 2*b^3*d^3*x^3*real_part(cos_i
ntegral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a)^2 + 8*b^3*d^3*x^3*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*
x)^2*tan(a)*tan(b*c/d) + 8*b^3*d^3*x^3*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a)*tan(b*c/d)
- 12*b^3*c*d^2*x^2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(a)^2*tan(b*c/d) + 12*b^3*c*d^2*x^2*
imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a)^2*tan(b*c/d) - 24*b^3*c*d^2*x^2*sin_integral(2*(b*
d*x + b*c)/d)*tan(b*x)^2*tan(a)^2*tan(b*c/d) - 2*b^3*d^3*x^3*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)
^2*tan(b*c/d)^2 - 2*b^3*d^3*x^3*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(b*c/d)^2 + 12*b^3*c*d
^2*x^2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(a)*tan(b*c/d)^2 - 12*b^3*c*d^2*x^2*imag_part(co
s_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a)*tan(b*c/d)^2 + 24*b^3*c*d^2*x^2*sin_integral(2*(b*d*x + b*c)/d
)*tan(b*x)^2*tan(a)*tan(b*c/d)^2 + 2*b^3*d^3*x^3*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)^2*tan(b*c/d)^
2 + 2*b^3*d^3*x^3*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2*tan(b*c/d)^2 + 6*b^3*c^2*d*x*real_part(co
s_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(a)^2*tan(b*c/d)^2 + 6*b^3*c^2*d*x*real_part(cos_integral(-2*b*x -
2*b*c/d))*tan(b*x)^2*tan(a)^2*tan(b*c/d)^2 - 4*b^3*d^3*x^3*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2
*tan(a) + 4*b^3*d^3*x^3*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a) - 8*b^3*d^3*x^3*sin_integr
al(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(a) - 6*b^3*c*d^2*x^2*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*
tan(a)^2 - 6*b^3*c*d^2*x^2*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a)^2 + 4*b^3*d^3*x^3*imag_
part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(b*c/d) - 4*b^3*d^3*x^3*imag_part(cos_integral(-2*b*x - 2*b*
c/d))*tan(b*x)^2*tan(b*c/d) + 8*b^3*d^3*x^3*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(b*c/d) + 24*b^3*c*d
^2*x^2*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(a)*tan(b*c/d) + 24*b^3*c*d^2*x^2*real_part(cos_
integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a)*tan(b*c/d) - 4*b^3*d^3*x^3*imag_part(cos_integral(2*b*x + 2*b*c/
d))*tan(a)^2*tan(b*c/d) + 4*b^3*d^3*x^3*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2*tan(b*c/d) - 8*b^3*
d^3*x^3*sin_integral(2*(b*d*x + b*c)/d)*tan(a)^2*tan(b*c/d) - 12*b^3*c^2*d*x*imag_part(cos_integral(2*b*x + 2*
b*c/d))*tan(b*x)^2*tan(a)^2*tan(b*c/d) + 12*b^3*c^2*d*x*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*t
an(a)^2*tan(b*c/d) - 24*b^3*c^2*d*x*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(a)^2*tan(b*c/d) - 6*b^3*c*d
^2*x^2*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(b*c/d)^2 - 6*b^3*c*d^2*x^2*real_part(cos_integr
al(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(b*c/d)^2 + 4*b^3*d^3*x^3*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)*
tan(b*c/d)^2 - 4*b^3*d^3*x^3*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)*tan(b*c/d)^2 + 8*b^3*d^3*x^3*sin
_integral(2*(b*d*x + b*c)/d)*tan(a)*tan(b*c/d)^2 + 12*b^3*c^2*d*x*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan
(b*x)^2*tan(a)*tan(b*c/d)^2 - 12*b^3*c^2*d*x*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a)*tan(b
*c/d)^2 + 24*b^3*c^2*d*x*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(a)*tan(b*c/d)^2 + 6*b^3*c*d^2*x^2*real
_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)^2*tan(b*c/d)^2 + 6*b^3*c*d^2*x^2*real_part(cos_integral(-2*b*x - 2
*b*c/d))*tan(a)^2*tan(b*c/d)^2 + 2*b^3*c^3*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(a)^2*tan(b*
c/d)^2 + 2*b^3*c^3*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a)^2*tan(b*c/d)^2 + 2*b^3*d^3*x^3*
real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2 + 2*b^3*d^3*x^3*real_part(cos_integral(-2*b*x - 2*b*c/d))*
tan(b*x)^2 - 12*b^3*c*d^2*x^2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(a) + 12*b^3*c*d^2*x^2*im
ag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a) - 24*b^3*c*d^2*x^2*sin_integral(2*(b*d*x + b*c)/d)*t
an(b*x)^2*tan(a) - 2*b^3*d^3*x^3*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)^2 - 2*b^3*d^3*x^3*real_part(c
os_integral(-2*b*x - 2*b*c/d))*tan(a)^2 - 6*b^3*c^2*d*x*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*ta
n(a)^2 - 6*b^3*c^2*d*x*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a)^2 + 12*b^3*c*d^2*x^2*imag_p
art(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(b*c/d) - 12*b^3*c*d^2*x^2*imag_part(cos_integral(-2*b*x - 2*
b*c/d))*tan(b*x)^2*tan(b*c/d) + 24*b^3*c*d^2*x^2*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(b*c/d) + 8*b^3
*d^3*x^3*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)*tan(b*c/d) + 8*b^3*d^3*x^3*real_part(cos_integral(-2*
b*x - 2*b*c/d))*tan(a)*tan(b*c/d) + 24*b^3*c^2*d*x*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(a)*
tan(b*c/d) + 24*b^3*c^2*d*x*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a)*tan(b*c/d) - 12*b^3*c*
d^2*x^2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)^2*tan(b*c/d) + 12*b^3*c*d^2*x^2*imag_part(cos_integral
(-2*b*x - 2*b*c/d))*tan(a)^2*tan(b*c/d) - 24*b^3*c*d^2*x^2*sin_integral(2*(b*d*x + b*c)/d)*tan(a)^2*tan(b*c/d)
- 4*b^3*c^3*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(a)^2*tan(b*c/d) + 4*b^3*c^3*imag_part(cos
_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a)^2*tan(b*c/d) - 8*b^3*c^3*sin_integral(2*(b*d*x + b*c)/d)*tan(b*
x)^2*tan(a)^2*tan(b*c/d) - 2*b^3*d^3*x^3*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*c/d)^2 - 2*b^3*d^3*x^3
*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*c/d)^2 - 6*b^3*c^2*d*x*real_part(cos_integral(2*b*x + 2*b*c/d
))*tan(b*x)^2*tan(b*c/d)^2 - 6*b^3*c^2*d*x*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(b*c/d)^2 +
12*b^3*c*d^2*x^2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)*tan(b*c/d)^2 - 12*b^3*c*d^2*x^2*imag_part(co
s_integral(-2*b*x - 2*b*c/d))*tan(a)*tan(b*c/d)^2 + 24*b^3*c*d^2*x^2*sin_integral(2*(b*d*x + b*c)/d)*tan(a)*ta
n(b*c/d)^2 + 4*b^2*d^3*x^2*tan(b*x)^2*tan(a)*tan(b*c/d)^2 + 4*b^3*c^3*imag_part(cos_integral(2*b*x + 2*b*c/d))
*tan(b*x)^2*tan(a)*tan(b*c/d)^2 - 4*b^3*c^3*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a)*tan(b*
c/d)^2 + 8*b^3*c^3*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(a)*tan(b*c/d)^2 + 6*b^3*c^2*d*x*real_part(co
s_integral(2*b*x + 2*b*c/d))*tan(a)^2*tan(b*c/d)^2 + 6*b^3*c^2*d*x*real_part(cos_integral(-2*b*x - 2*b*c/d))*t
an(a)^2*tan(b*c/d)^2 + 4*b^2*d^3*x^2*tan(b*x)*tan(a)^2*tan(b*c/d)^2 + 6*b^3*c*d^2*x^2*real_part(cos_integral(2
*b*x + 2*b*c/d))*tan(b*x)^2 + 6*b^3*c*d^2*x^2*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2 - 4*b^3*d^3
*x^3*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(a) + 4*b^3*d^3*x^3*imag_part(cos_integral(-2*b*x - 2*b*c/d))
*tan(a) - 8*b^3*d^3*x^3*sin_integral(2*(b*d*x + b*c)/d)*tan(a) - 12*b^3*c^2*d*x*imag_part(cos_integral(2*b*x +
2*b*c/d))*tan(b*x)^2*tan(a) + 12*b^3*c^2*d*x*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a) - 24
*b^3*c^2*d*x*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(a) - 6*b^3*c*d^2*x^2*real_part(cos_integral(2*b*x
+ 2*b*c/d))*tan(a)^2 - 6*b^3*c*d^2*x^2*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2 - 2*b^3*c^3*real_par
t(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(a)^2 - 2*b^3*c^3*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan
(b*x)^2*tan(a)^2 + 4*b^3*d^3*x^3*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*c/d) - 4*b^3*d^3*x^3*imag_part
(cos_integral(-2*b*x - 2*b*c/d))*tan(b*c/d) + 8*b^3*d^3*x^3*sin_integral(2*(b*d*x + b*c)/d)*tan(b*c/d) + 12*b^
3*c^2*d*x*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(b*c/d) - 12*b^3*c^2*d*x*imag_part(cos_integr
al(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(b*c/d) + 24*b^3*c^2*d*x*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(b*
c/d) + 24*b^3*c*d^2*x^2*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)*tan(b*c/d) + 24*b^3*c*d^2*x^2*real_par
t(cos_integral(-2*b*x - 2*b*c/d))*tan(a)*tan(b*c/d) + 8*b^3*c^3*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b
*x)^2*tan(a)*tan(b*c/d) + 8*b^3*c^3*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a)*tan(b*c/d) - 1
2*b^3*c^2*d*x*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)^2*tan(b*c/d) + 12*b^3*c^2*d*x*imag_part(cos_inte
gral(-2*b*x - 2*b*c/d))*tan(a)^2*tan(b*c/d) - 24*b^3*c^2*d*x*sin_integral(2*(b*d*x + b*c)/d)*tan(a)^2*tan(b*c/
d) - 6*b^3*c*d^2*x^2*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*c/d)^2 - 6*b^3*c*d^2*x^2*real_part(cos_int
egral(-2*b*x - 2*b*c/d))*tan(b*c/d)^2 - 2*b^3*c^3*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(b*c/
d)^2 - 2*b^3*c^3*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(b*c/d)^2 + 12*b^3*c^2*d*x*imag_part(
cos_integral(2*b*x + 2*b*c/d))*tan(a)*tan(b*c/d)^2 - 12*b^3*c^2*d*x*imag_part(cos_integral(-2*b*x - 2*b*c/d))*
tan(a)*tan(b*c/d)^2 + 24*b^3*c^2*d*x*sin_integral(2*(b*d*x + b*c)/d)*tan(a)*tan(b*c/d)^2 + 8*b^2*c*d^2*x*tan(b
*x)^2*tan(a)*tan(b*c/d)^2 + 2*b^3*c^3*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)^2*tan(b*c/d)^2 + 2*b^3*c
^3*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2*tan(b*c/d)^2 + 8*b^2*c*d^2*x*tan(b*x)*tan(a)^2*tan(b*c/d
)^2 + b*d^3*x*tan(b*x)^2*tan(a)^2*tan(b*c/d)^2 + 2*b^3*d^3*x^3*real_part(cos_integral(2*b*x + 2*b*c/d)) + 2*b^
3*d^3*x^3*real_part(cos_integral(-2*b*x - 2*b*c/d)) + 6*b^3*c^2*d*x*real_part(cos_integral(2*b*x + 2*b*c/d))*t
an(b*x)^2 + 6*b^3*c^2*d*x*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2 - 12*b^3*c*d^2*x^2*imag_part(co
s_integral(2*b*x + 2*b*c/d))*tan(a) + 12*b^3*c*d^2*x^2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a) - 24*b
^3*c*d^2*x^2*sin_integral(2*(b*d*x + b*c)/d)*tan(a) + 4*b^2*d^3*x^2*tan(b*x)^2*tan(a) - 4*b^3*c^3*imag_part(co
s_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(a) + 4*b^3*c^3*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^
2*tan(a) - 8*b^3*c^3*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(a) - 6*b^3*c^2*d*x*real_part(cos_integral(
2*b*x + 2*b*c/d))*tan(a)^2 - 6*b^3*c^2*d*x*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2 + 4*b^2*d^3*x^2*
tan(b*x)*tan(a)^2 + 12*b^3*c*d^2*x^2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*c/d) - 12*b^3*c*d^2*x^2*im
ag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*c/d) + 24*b^3*c*d^2*x^2*sin_integral(2*(b*d*x + b*c)/d)*tan(b*c/
d) + 4*b^3*c^3*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(b*c/d) - 4*b^3*c^3*imag_part(cos_integr
al(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(b*c/d) + 8*b^3*c^3*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(b*c/d)
+ 24*b^3*c^2*d*x*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)*tan(b*c/d) + 24*b^3*c^2*d*x*real_part(cos_int
egral(-2*b*x - 2*b*c/d))*tan(a)*tan(b*c/d) - 4*b^3*c^3*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)^2*tan(b
*c/d) + 4*b^3*c^3*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2*tan(b*c/d) - 8*b^3*c^3*sin_integral(2*(b*
d*x + b*c)/d)*tan(a)^2*tan(b*c/d) - 6*b^3*c^2*d*x*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*c/d)^2 - 6*b^
3*c^2*d*x*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*c/d)^2 - 4*b^2*d^3*x^2*tan(b*x)*tan(b*c/d)^2 - 4*b^2
*d^3*x^2*tan(a)*tan(b*c/d)^2 + 4*b^3*c^3*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)*tan(b*c/d)^2 - 4*b^3*
c^3*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)*tan(b*c/d)^2 + 8*b^3*c^3*sin_integral(2*(b*d*x + b*c)/d)*
tan(a)*tan(b*c/d)^2 + 4*b^2*c^2*d*tan(b*x)^2*tan(a)*tan(b*c/d)^2 + 4*b^2*c^2*d*tan(b*x)*tan(a)^2*tan(b*c/d)^2
+ b*c*d^2*tan(b*x)^2*tan(a)^2*tan(b*c/d)^2 + 6*b^3*c*d^2*x^2*real_part(cos_integral(2*b*x + 2*b*c/d)) + 6*b^3*
c*d^2*x^2*real_part(cos_integral(-2*b*x - 2*b*c/d)) + 2*b^3*c^3*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b
*x)^2 + 2*b^3*c^3*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2 - 12*b^3*c^2*d*x*imag_part(cos_integral
(2*b*x + 2*b*c/d))*tan(a) + 12*b^3*c^2*d*x*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a) - 24*b^3*c^2*d*x*s
in_integral(2*(b*d*x + b*c)/d)*tan(a) + 8*b^2*c*d^2*x*tan(b*x)^2*tan(a) - 2*b^3*c^3*real_part(cos_integral(2*b
*x + 2*b*c/d))*tan(a)^2 - 2*b^3*c^3*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2 + 8*b^2*c*d^2*x*tan(b*x
)*tan(a)^2 + b*d^3*x*tan(b*x)^2*tan(a)^2 + 12*b^3*c^2*d*x*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*c/d)
- 12*b^3*c^2*d*x*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*c/d) + 24*b^3*c^2*d*x*sin_integral(2*(b*d*x +
b*c)/d)*tan(b*c/d) + 8*b^3*c^3*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)*tan(b*c/d) + 8*b^3*c^3*real_pa
rt(cos_integral(-2*b*x - 2*b*c/d))*tan(a)*tan(b*c/d) - 2*b^3*c^3*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(
b*c/d)^2 - 2*b^3*c^3*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*c/d)^2 - 8*b^2*c*d^2*x*tan(b*x)*tan(b*c/d
)^2 - b*d^3*x*tan(b*x)^2*tan(b*c/d)^2 - 8*b^2*c*d^2*x*tan(a)*tan(b*c/d)^2 - 4*b*d^3*x*tan(b*x)*tan(a)*tan(b*c/
d)^2 - b*d^3*x*tan(a)^2*tan(b*c/d)^2 + 6*b^3*c^2*d*x*real_part(cos_integral(2*b*x + 2*b*c/d)) + 6*b^3*c^2*d*x*
real_part(cos_integral(-2*b*x - 2*b*c/d)) - 4*b^2*d^3*x^2*tan(b*x) - 4*b^2*d^3*x^2*tan(a) - 4*b^3*c^3*imag_par
t(cos_integral(2*b*x + 2*b*c/d))*tan(a) + 4*b^3*c^3*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a) - 8*b^3*c
^3*sin_integral(2*(b*d*x + b*c)/d)*tan(a) + 4*b^2*c^2*d*tan(b*x)^2*tan(a) + 4*b^2*c^2*d*tan(b*x)*tan(a)^2 + b*
c*d^2*tan(b*x)^2*tan(a)^2 + 4*b^3*c^3*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*c/d) - 4*b^3*c^3*imag_par
t(cos_integral(-2*b*x - 2*b*c/d))*tan(b*c/d) + 8*b^3*c^3*sin_integral(2*(b*d*x + b*c)/d)*tan(b*c/d) - 4*b^2*c^
2*d*tan(b*x)*tan(b*c/d)^2 - b*c*d^2*tan(b*x)^2*tan(b*c/d)^2 - 4*b^2*c^2*d*tan(a)*tan(b*c/d)^2 - 4*b*c*d^2*tan(
b*x)*tan(a)*tan(b*c/d)^2 - 2*d^3*tan(b*x)^2*tan(a)*tan(b*c/d)^2 - b*c*d^2*tan(a)^2*tan(b*c/d)^2 - 2*d^3*tan(b*
x)*tan(a)^2*tan(b*c/d)^2 + 2*b^3*c^3*real_part(cos_integral(2*b*x + 2*b*c/d)) + 2*b^3*c^3*real_part(cos_integr
al(-2*b*x - 2*b*c/d)) - 8*b^2*c*d^2*x*tan(b*x) - b*d^3*x*tan(b*x)^2 - 8*b^2*c*d^2*x*tan(a) - 4*b*d^3*x*tan(b*x
)*tan(a) - b*d^3*x*tan(a)^2 + b*d^3*x*tan(b*c/d)^2 - 4*b^2*c^2*d*tan(b*x) - b*c*d^2*tan(b*x)^2 - 4*b^2*c^2*d*t
an(a) - 4*b*c*d^2*tan(b*x)*tan(a) - 2*d^3*tan(b*x)^2*tan(a) - b*c*d^2*tan(a)^2 - 2*d^3*tan(b*x)*tan(a)^2 + b*c
*d^2*tan(b*c/d)^2 + 2*d^3*tan(b*x)*tan(b*c/d)^2 + 2*d^3*tan(a)*tan(b*c/d)^2 + b*d^3*x + b*c*d^2 + 2*d^3*tan(b*
x) + 2*d^3*tan(a))/(d^7*x^3*tan(b*x)^2*tan(a)^2*tan(b*c/d)^2 + 3*c*d^6*x^2*tan(b*x)^2*tan(a)^2*tan(b*c/d)^2 +
d^7*x^3*tan(b*x)^2*tan(a)^2 + d^7*x^3*tan(b*x)^2*tan(b*c/d)^2 + d^7*x^3*tan(a)^2*tan(b*c/d)^2 + 3*c^2*d^5*x*ta
n(b*x)^2*tan(a)^2*tan(b*c/d)^2 + 3*c*d^6*x^2*tan(b*x)^2*tan(a)^2 + 3*c*d^6*x^2*tan(b*x)^2*tan(b*c/d)^2 + 3*c*d
^6*x^2*tan(a)^2*tan(b*c/d)^2 + c^3*d^4*tan(b*x)^2*tan(a)^2*tan(b*c/d)^2 + d^7*x^3*tan(b*x)^2 + d^7*x^3*tan(a)^
2 + 3*c^2*d^5*x*tan(b*x)^2*tan(a)^2 + d^7*x^3*tan(b*c/d)^2 + 3*c^2*d^5*x*tan(b*x)^2*tan(b*c/d)^2 + 3*c^2*d^5*x
*tan(a)^2*tan(b*c/d)^2 + 3*c*d^6*x^2*tan(b*x)^2 + 3*c*d^6*x^2*tan(a)^2 + c^3*d^4*tan(b*x)^2*tan(a)^2 + 3*c*d^6
*x^2*tan(b*c/d)^2 + c^3*d^4*tan(b*x)^2*tan(b*c/d)^2 + c^3*d^4*tan(a)^2*tan(b*c/d)^2 + d^7*x^3 + 3*c^2*d^5*x*ta
n(b*x)^2 + 3*c^2*d^5*x*tan(a)^2 + 3*c^2*d^5*x*tan(b*c/d)^2 + 3*c*d^6*x^2 + c^3*d^4*tan(b*x)^2 + c^3*d^4*tan(a)
^2 + c^3*d^4*tan(b*c/d)^2 + 3*c^2*d^5*x + c^3*d^4)