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

Optimal. Leaf size=127 \[ \frac{b^2 \cos \left (4 a-\frac{4 b c}{d}\right ) \text{CosIntegral}\left (\frac{4 b c}{d}+4 b x\right )}{d^3}-\frac{b^2 \sin \left (4 a-\frac{4 b c}{d}\right ) \text{Si}\left (\frac{4 b c}{d}+4 b x\right )}{d^3}-\frac{b \sin (4 a+4 b x)}{4 d^2 (c+d x)}+\frac{\cos (4 a+4 b x)}{16 d (c+d x)^2}-\frac{1}{16 d (c+d x)^2} \]

[Out]

-1/(16*d*(c + d*x)^2) + Cos[4*a + 4*b*x]/(16*d*(c + d*x)^2) + (b^2*Cos[4*a - (4*b*c)/d]*CosIntegral[(4*b*c)/d
+ 4*b*x])/d^3 - (b*Sin[4*a + 4*b*x])/(4*d^2*(c + d*x)) - (b^2*Sin[4*a - (4*b*c)/d]*SinIntegral[(4*b*c)/d + 4*b
*x])/d^3

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Rubi [A]  time = 0.198285, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {4406, 3297, 3303, 3299, 3302} \[ \frac{b^2 \cos \left (4 a-\frac{4 b c}{d}\right ) \text{CosIntegral}\left (\frac{4 b c}{d}+4 b x\right )}{d^3}-\frac{b^2 \sin \left (4 a-\frac{4 b c}{d}\right ) \text{Si}\left (\frac{4 b c}{d}+4 b x\right )}{d^3}-\frac{b \sin (4 a+4 b x)}{4 d^2 (c+d x)}+\frac{\cos (4 a+4 b x)}{16 d (c+d x)^2}-\frac{1}{16 d (c+d x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(16*d*(c + d*x)^2) + Cos[4*a + 4*b*x]/(16*d*(c + d*x)^2) + (b^2*Cos[4*a - (4*b*c)/d]*CosIntegral[(4*b*c)/d
+ 4*b*x])/d^3 - (b*Sin[4*a + 4*b*x])/(4*d^2*(c + d*x)) - (b^2*Sin[4*a - (4*b*c)/d]*SinIntegral[(4*b*c)/d + 4*b
*x])/d^3

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 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 ^2(a+b x) \sin ^2(a+b x)}{(c+d x)^3} \, dx &=\int \left (\frac{1}{8 (c+d x)^3}-\frac{\cos (4 a+4 b x)}{8 (c+d x)^3}\right ) \, dx\\ &=-\frac{1}{16 d (c+d x)^2}-\frac{1}{8} \int \frac{\cos (4 a+4 b x)}{(c+d x)^3} \, dx\\ &=-\frac{1}{16 d (c+d x)^2}+\frac{\cos (4 a+4 b x)}{16 d (c+d x)^2}+\frac{b \int \frac{\sin (4 a+4 b x)}{(c+d x)^2} \, dx}{4 d}\\ &=-\frac{1}{16 d (c+d x)^2}+\frac{\cos (4 a+4 b x)}{16 d (c+d x)^2}-\frac{b \sin (4 a+4 b x)}{4 d^2 (c+d x)}+\frac{b^2 \int \frac{\cos (4 a+4 b x)}{c+d x} \, dx}{d^2}\\ &=-\frac{1}{16 d (c+d x)^2}+\frac{\cos (4 a+4 b x)}{16 d (c+d x)^2}-\frac{b \sin (4 a+4 b x)}{4 d^2 (c+d x)}+\frac{\left (b^2 \cos \left (4 a-\frac{4 b c}{d}\right )\right ) \int \frac{\cos \left (\frac{4 b c}{d}+4 b x\right )}{c+d x} \, dx}{d^2}-\frac{\left (b^2 \sin \left (4 a-\frac{4 b c}{d}\right )\right ) \int \frac{\sin \left (\frac{4 b c}{d}+4 b x\right )}{c+d x} \, dx}{d^2}\\ &=-\frac{1}{16 d (c+d x)^2}+\frac{\cos (4 a+4 b x)}{16 d (c+d x)^2}+\frac{b^2 \cos \left (4 a-\frac{4 b c}{d}\right ) \text{Ci}\left (\frac{4 b c}{d}+4 b x\right )}{d^3}-\frac{b \sin (4 a+4 b x)}{4 d^2 (c+d x)}-\frac{b^2 \sin \left (4 a-\frac{4 b c}{d}\right ) \text{Si}\left (\frac{4 b c}{d}+4 b x\right )}{d^3}\\ \end{align*}

Mathematica [A]  time = 0.88326, size = 105, normalized size = 0.83 \[ \frac{16 b^2 \cos \left (4 a-\frac{4 b c}{d}\right ) \text{CosIntegral}\left (\frac{4 b (c+d x)}{d}\right )-16 b^2 \sin \left (4 a-\frac{4 b c}{d}\right ) \text{Si}\left (\frac{4 b (c+d x)}{d}\right )+\frac{d (-4 b (c+d x) \sin (4 (a+b x))+d \cos (4 (a+b x))-d)}{(c+d x)^2}}{16 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(16*b^2*Cos[4*a - (4*b*c)/d]*CosIntegral[(4*b*(c + d*x))/d] + (d*(-d + d*Cos[4*(a + b*x)] - 4*b*(c + d*x)*Sin[
4*(a + b*x)]))/(c + d*x)^2 - 16*b^2*Sin[4*a - (4*b*c)/d]*SinIntegral[(4*b*(c + d*x))/d])/(16*d^3)

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Maple [A]  time = 0.024, size = 193, normalized size = 1.5 \begin{align*}{\frac{1}{b} \left ( -{\frac{{b}^{3}}{32} \left ( -2\,{\frac{\cos \left ( 4\,bx+4\,a \right ) }{ \left ( \left ( bx+a \right ) d-ad+bc \right ) ^{2}d}}-2\,{\frac{1}{d} \left ( -4\,{\frac{\sin \left ( 4\,bx+4\,a \right ) }{ \left ( \left ( bx+a \right ) d-ad+bc \right ) d}}+4\,{\frac{1}{d} \left ( 4\,{\frac{1}{d}{\it Si} \left ( 4\,bx+4\,a+4\,{\frac{-ad+bc}{d}} \right ) \sin \left ( 4\,{\frac{-ad+bc}{d}} \right ) }+4\,{\frac{1}{d}{\it Ci} \left ( 4\,bx+4\,a+4\,{\frac{-ad+bc}{d}} \right ) \cos \left ( 4\,{\frac{-ad+bc}{d}} \right ) } \right ) } \right ) } \right ) }-{\frac{{b}^{3}}{16\, \left ( \left ( bx+a \right ) d-ad+bc \right ) ^{2}d}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b*(-1/32*b^3*(-2*cos(4*b*x+4*a)/((b*x+a)*d-a*d+b*c)^2/d-2*(-4*sin(4*b*x+4*a)/((b*x+a)*d-a*d+b*c)/d+4*(4*Si(4
*b*x+4*a+4*(-a*d+b*c)/d)*sin(4*(-a*d+b*c)/d)/d+4*Ci(4*b*x+4*a+4*(-a*d+b*c)/d)*cos(4*(-a*d+b*c)/d)/d)/d)/d)-1/1
6*b^3/((b*x+a)*d-a*d+b*c)^2/d)

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Maxima [C]  time = 1.83377, size = 278, normalized size = 2.19 \begin{align*} \frac{64 \, b^{3}{\left (E_{3}\left (\frac{4 i \, b c + 4 i \,{\left (b x + a\right )} d - 4 i \, a d}{d}\right ) + E_{3}\left (-\frac{4 i \, b c + 4 i \,{\left (b x + a\right )} d - 4 i \, a d}{d}\right )\right )} \cos \left (-\frac{4 \,{\left (b c - a d\right )}}{d}\right ) - b^{3}{\left (64 i \, E_{3}\left (\frac{4 i \, b c + 4 i \,{\left (b x + a\right )} d - 4 i \, a d}{d}\right ) - 64 i \, E_{3}\left (-\frac{4 i \, b c + 4 i \,{\left (b x + a\right )} d - 4 i \, a d}{d}\right )\right )} \sin \left (-\frac{4 \,{\left (b c - a d\right )}}{d}\right ) - 64 \, b^{3}}{1024 \,{\left (b^{2} c^{2} d - 2 \, a b c d^{2} +{\left (b x + a\right )}^{2} d^{3} + a^{2} d^{3} + 2 \,{\left (b c d^{2} - a d^{3}\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)^2*sin(b*x+a)^2/(d*x+c)^3,x, algorithm="maxima")

[Out]

1/1024*(64*b^3*(exp_integral_e(3, (4*I*b*c + 4*I*(b*x + a)*d - 4*I*a*d)/d) + exp_integral_e(3, -(4*I*b*c + 4*I
*(b*x + a)*d - 4*I*a*d)/d))*cos(-4*(b*c - a*d)/d) - b^3*(64*I*exp_integral_e(3, (4*I*b*c + 4*I*(b*x + a)*d - 4
*I*a*d)/d) - 64*I*exp_integral_e(3, -(4*I*b*c + 4*I*(b*x + a)*d - 4*I*a*d)/d))*sin(-4*(b*c - a*d)/d) - 64*b^3)
/((b^2*c^2*d - 2*a*b*c*d^2 + (b*x + a)^2*d^3 + a^2*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x + a))*b)

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Fricas [B]  time = 0.564002, size = 582, normalized size = 4.58 \begin{align*} \frac{d^{2} \cos \left (b x + a\right )^{4} - d^{2} \cos \left (b x + a\right )^{2} - 2 \,{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \sin \left (-\frac{4 \,{\left (b c - a d\right )}}{d}\right ) \operatorname{Si}\left (\frac{4 \,{\left (b d x + b c\right )}}{d}\right ) +{\left ({\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \operatorname{Ci}\left (\frac{4 \,{\left (b d x + b c\right )}}{d}\right ) +{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \operatorname{Ci}\left (-\frac{4 \,{\left (b d x + b c\right )}}{d}\right )\right )} \cos \left (-\frac{4 \,{\left (b c - a d\right )}}{d}\right ) - 2 \,{\left (2 \,{\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right )^{3} -{\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{2 \,{\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [C]  time = 1.64119, size = 7560, normalized size = 59.53 \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)^2*sin(b*x+a)^2/(d*x+c)^3,x, algorithm="giac")

[Out]

1/8*(4*b^2*d^2*x^2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d)^2 + 4*b^2*d^2
*x^2*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d)^2 - 8*b^2*d^2*x^2*imag_par
t(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d) + 8*b^2*d^2*x^2*imag_part(cos_integral(-
4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d) - 16*b^2*d^2*x^2*sin_integral(4*(b*d*x + b*c)/d)*tan(2*
b*x)^2*tan(2*a)^2*tan(2*b*c/d) + 8*b^2*d^2*x^2*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a)*
tan(2*b*c/d)^2 - 8*b^2*d^2*x^2*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d)^2
+ 16*b^2*d^2*x^2*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d)^2 + 8*b^2*c*d*x*real_part(
cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d)^2 + 8*b^2*c*d*x*real_part(cos_integral(-4*
b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d)^2 - 4*b^2*d^2*x^2*real_part(cos_integral(4*b*x + 4*b*c/d)
)*tan(2*b*x)^2*tan(2*a)^2 - 4*b^2*d^2*x^2*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^2 +
16*b^2*d^2*x^2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d) + 16*b^2*d^2*x^2*re
al_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d) - 16*b^2*c*d*x*imag_part(cos_integr
al(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d) + 16*b^2*c*d*x*imag_part(cos_integral(-4*b*x - 4*b*c
/d))*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d) - 32*b^2*c*d*x*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2*tan(2*a)
^2*tan(2*b*c/d) - 4*b^2*d^2*x^2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*b*c/d)^2 - 4*b^2*d
^2*x^2*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*b*c/d)^2 + 16*b^2*c*d*x*imag_part(cos_inte
gral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d)^2 - 16*b^2*c*d*x*imag_part(cos_integral(-4*b*x - 4*b
*c/d))*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d)^2 + 32*b^2*c*d*x*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2*tan(2*
a)*tan(2*b*c/d)^2 + 4*b^2*d^2*x^2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d)^2 + 4*b^2*d
^2*x^2*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d)^2 + 4*b^2*c^2*real_part(cos_integral(
4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d)^2 + 4*b^2*c^2*real_part(cos_integral(-4*b*x - 4*b*c/d))
*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d)^2 - 8*b^2*d^2*x^2*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*
tan(2*a) + 8*b^2*d^2*x^2*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a) - 16*b^2*d^2*x^2*sin_
integral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2*tan(2*a) - 8*b^2*c*d*x*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2
*b*x)^2*tan(2*a)^2 - 8*b^2*c*d*x*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^2 + 8*b^2*d^2
*x^2*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*b*c/d) - 8*b^2*d^2*x^2*imag_part(cos_integral
(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*b*c/d) + 16*b^2*d^2*x^2*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2*ta
n(2*b*c/d) + 32*b^2*c*d*x*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d) + 32*b^2
*c*d*x*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d) - 8*b^2*d^2*x^2*imag_part(
cos_integral(4*b*x + 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d) + 8*b^2*d^2*x^2*imag_part(cos_integral(-4*b*x - 4*b*c/d
))*tan(2*a)^2*tan(2*b*c/d) - 16*b^2*d^2*x^2*sin_integral(4*(b*d*x + b*c)/d)*tan(2*a)^2*tan(2*b*c/d) - 8*b^2*c^
2*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d) + 8*b^2*c^2*imag_part(cos_inte
gral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d) - 16*b^2*c^2*sin_integral(4*(b*d*x + b*c)/d)*tan(
2*b*x)^2*tan(2*a)^2*tan(2*b*c/d) - 8*b^2*c*d*x*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*b*c
/d)^2 - 8*b^2*c*d*x*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*b*c/d)^2 + 8*b^2*d^2*x^2*imag
_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)*tan(2*b*c/d)^2 - 8*b^2*d^2*x^2*imag_part(cos_integral(-4*b*x - 4
*b*c/d))*tan(2*a)*tan(2*b*c/d)^2 + 16*b^2*d^2*x^2*sin_integral(4*(b*d*x + b*c)/d)*tan(2*a)*tan(2*b*c/d)^2 + 8*
b^2*c^2*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d)^2 - 8*b^2*c^2*imag_part(co
s_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d)^2 + 16*b^2*c^2*sin_integral(4*(b*d*x + b*c)/d
)*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d)^2 + 8*b^2*c*d*x*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)^2*tan(2
*b*c/d)^2 + 8*b^2*c*d*x*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d)^2 + 4*b^2*d^2*x^2*re
al_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2 + 4*b^2*d^2*x^2*real_part(cos_integral(-4*b*x - 4*b*c/d))*
tan(2*b*x)^2 - 16*b^2*c*d*x*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a) + 16*b^2*c*d*x*imag
_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a) - 32*b^2*c*d*x*sin_integral(4*(b*d*x + b*c)/d)*tan
(2*b*x)^2*tan(2*a) - 4*b^2*d^2*x^2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)^2 - 4*b^2*d^2*x^2*real_pa
rt(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2 - 4*b^2*c^2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^
2*tan(2*a)^2 - 4*b^2*c^2*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^2 + 16*b^2*c*d*x*imag
_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*b*c/d) - 16*b^2*c*d*x*imag_part(cos_integral(-4*b*x -
4*b*c/d))*tan(2*b*x)^2*tan(2*b*c/d) + 32*b^2*c*d*x*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2*tan(2*b*c/d) +
16*b^2*d^2*x^2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)*tan(2*b*c/d) + 16*b^2*d^2*x^2*real_part(cos_
integral(-4*b*x - 4*b*c/d))*tan(2*a)*tan(2*b*c/d) + 16*b^2*c^2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*
b*x)^2*tan(2*a)*tan(2*b*c/d) + 16*b^2*c^2*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a)*tan(
2*b*c/d) - 16*b^2*c*d*x*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d) + 16*b^2*c*d*x*imag_p
art(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d) - 32*b^2*c*d*x*sin_integral(4*(b*d*x + b*c)/d)*tan
(2*a)^2*tan(2*b*c/d) - 4*b^2*d^2*x^2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*c/d)^2 - 4*b^2*d^2*x^2*r
eal_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*c/d)^2 - 4*b^2*c^2*real_part(cos_integral(4*b*x + 4*b*c/d))*t
an(2*b*x)^2*tan(2*b*c/d)^2 - 4*b^2*c^2*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*b*c/d)^2 +
16*b^2*c*d*x*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)*tan(2*b*c/d)^2 - 16*b^2*c*d*x*imag_part(cos_in
tegral(-4*b*x - 4*b*c/d))*tan(2*a)*tan(2*b*c/d)^2 + 32*b^2*c*d*x*sin_integral(4*(b*d*x + b*c)/d)*tan(2*a)*tan(
2*b*c/d)^2 + 4*b*d^2*x*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d)^2 + 4*b^2*c^2*real_part(cos_integral(4*b*x + 4*b*c/d
))*tan(2*a)^2*tan(2*b*c/d)^2 + 4*b^2*c^2*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d)^2 +
4*b*d^2*x*tan(2*b*x)*tan(2*a)^2*tan(2*b*c/d)^2 + 8*b^2*c*d*x*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b
*x)^2 + 8*b^2*c*d*x*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2 - 8*b^2*d^2*x^2*imag_part(cos_integ
ral(4*b*x + 4*b*c/d))*tan(2*a) + 8*b^2*d^2*x^2*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a) - 16*b^2*d^2
*x^2*sin_integral(4*(b*d*x + b*c)/d)*tan(2*a) - 8*b^2*c^2*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^
2*tan(2*a) + 8*b^2*c^2*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a) - 16*b^2*c^2*sin_integr
al(4*(b*d*x + b*c)/d)*tan(2*b*x)^2*tan(2*a) - 8*b^2*c*d*x*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)^2
- 8*b^2*c*d*x*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2 + 8*b^2*d^2*x^2*imag_part(cos_integral(4*b*
x + 4*b*c/d))*tan(2*b*c/d) - 8*b^2*d^2*x^2*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*c/d) + 16*b^2*d^2
*x^2*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*c/d) + 8*b^2*c^2*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b
*x)^2*tan(2*b*c/d) - 8*b^2*c^2*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*b*c/d) + 16*b^2*c^
2*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2*tan(2*b*c/d) + 32*b^2*c*d*x*real_part(cos_integral(4*b*x + 4*b*
c/d))*tan(2*a)*tan(2*b*c/d) + 32*b^2*c*d*x*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)*tan(2*b*c/d) - 8
*b^2*c^2*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d) + 8*b^2*c^2*imag_part(cos_integral(-
4*b*x - 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d) - 16*b^2*c^2*sin_integral(4*(b*d*x + b*c)/d)*tan(2*a)^2*tan(2*b*c/d)
- 8*b^2*c*d*x*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*c/d)^2 - 8*b^2*c*d*x*real_part(cos_integral(-4
*b*x - 4*b*c/d))*tan(2*b*c/d)^2 + 8*b^2*c^2*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)*tan(2*b*c/d)^2 -
8*b^2*c^2*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)*tan(2*b*c/d)^2 + 16*b^2*c^2*sin_integral(4*(b*d*
x + b*c)/d)*tan(2*a)*tan(2*b*c/d)^2 + 4*b*c*d*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d)^2 + 4*b*c*d*tan(2*b*x)*tan(2*
a)^2*tan(2*b*c/d)^2 + 4*b^2*d^2*x^2*real_part(cos_integral(4*b*x + 4*b*c/d)) + 4*b^2*d^2*x^2*real_part(cos_int
egral(-4*b*x - 4*b*c/d)) + 4*b^2*c^2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2 + 4*b^2*c^2*real_pa
rt(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2 - 16*b^2*c*d*x*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*
a) + 16*b^2*c*d*x*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a) - 32*b^2*c*d*x*sin_integral(4*(b*d*x + b*
c)/d)*tan(2*a) + 4*b*d^2*x*tan(2*b*x)^2*tan(2*a) - 4*b^2*c^2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)
^2 - 4*b^2*c^2*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2 + 4*b*d^2*x*tan(2*b*x)*tan(2*a)^2 + 16*b^2
*c*d*x*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*c/d) - 16*b^2*c*d*x*imag_part(cos_integral(-4*b*x - 4*
b*c/d))*tan(2*b*c/d) + 32*b^2*c*d*x*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*c/d) + 16*b^2*c^2*real_part(cos_in
tegral(4*b*x + 4*b*c/d))*tan(2*a)*tan(2*b*c/d) + 16*b^2*c^2*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)
*tan(2*b*c/d) - 4*b^2*c^2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*c/d)^2 - 4*b^2*c^2*real_part(cos_in
tegral(-4*b*x - 4*b*c/d))*tan(2*b*c/d)^2 - 4*b*d^2*x*tan(2*b*x)*tan(2*b*c/d)^2 - 4*b*d^2*x*tan(2*a)*tan(2*b*c/
d)^2 + 8*b^2*c*d*x*real_part(cos_integral(4*b*x + 4*b*c/d)) + 8*b^2*c*d*x*real_part(cos_integral(-4*b*x - 4*b*
c/d)) - 8*b^2*c^2*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a) + 8*b^2*c^2*imag_part(cos_integral(-4*b*x
- 4*b*c/d))*tan(2*a) - 16*b^2*c^2*sin_integral(4*(b*d*x + b*c)/d)*tan(2*a) + 4*b*c*d*tan(2*b*x)^2*tan(2*a) + 4
*b*c*d*tan(2*b*x)*tan(2*a)^2 + 8*b^2*c^2*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*c/d) - 8*b^2*c^2*ima
g_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*c/d) + 16*b^2*c^2*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*c/d)
- 4*b*c*d*tan(2*b*x)*tan(2*b*c/d)^2 - d^2*tan(2*b*x)^2*tan(2*b*c/d)^2 - 4*b*c*d*tan(2*a)*tan(2*b*c/d)^2 - 2*d^
2*tan(2*b*x)*tan(2*a)*tan(2*b*c/d)^2 - d^2*tan(2*a)^2*tan(2*b*c/d)^2 + 4*b^2*c^2*real_part(cos_integral(4*b*x
+ 4*b*c/d)) + 4*b^2*c^2*real_part(cos_integral(-4*b*x - 4*b*c/d)) - 4*b*d^2*x*tan(2*b*x) - 4*b*d^2*x*tan(2*a)
- 4*b*c*d*tan(2*b*x) - d^2*tan(2*b*x)^2 - 4*b*c*d*tan(2*a) - 2*d^2*tan(2*b*x)*tan(2*a) - d^2*tan(2*a)^2)/(d^5*
x^2*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d)^2 + 2*c*d^4*x*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d)^2 + d^5*x^2*tan(2*
b*x)^2*tan(2*a)^2 + d^5*x^2*tan(2*b*x)^2*tan(2*b*c/d)^2 + d^5*x^2*tan(2*a)^2*tan(2*b*c/d)^2 + c^2*d^3*tan(2*b*
x)^2*tan(2*a)^2*tan(2*b*c/d)^2 + 2*c*d^4*x*tan(2*b*x)^2*tan(2*a)^2 + 2*c*d^4*x*tan(2*b*x)^2*tan(2*b*c/d)^2 + 2
*c*d^4*x*tan(2*a)^2*tan(2*b*c/d)^2 + d^5*x^2*tan(2*b*x)^2 + d^5*x^2*tan(2*a)^2 + c^2*d^3*tan(2*b*x)^2*tan(2*a)
^2 + d^5*x^2*tan(2*b*c/d)^2 + c^2*d^3*tan(2*b*x)^2*tan(2*b*c/d)^2 + c^2*d^3*tan(2*a)^2*tan(2*b*c/d)^2 + 2*c*d^
4*x*tan(2*b*x)^2 + 2*c*d^4*x*tan(2*a)^2 + 2*c*d^4*x*tan(2*b*c/d)^2 + d^5*x^2 + c^2*d^3*tan(2*b*x)^2 + c^2*d^3*
tan(2*a)^2 + c^2*d^3*tan(2*b*c/d)^2 + 2*c*d^4*x + c^2*d^3)