3.15 \(\int \frac{\sin ^2(a+b x)}{(c+d x)^4} \, dx\)

Optimal. Leaf size=162 \[ -\frac{2 b^3 \sin \left (2 a-\frac{2 b c}{d}\right ) \text{CosIntegral}\left (\frac{2 b c}{d}+2 b x\right )}{3 d^4}-\frac{2 b^3 \cos \left (2 a-\frac{2 b c}{d}\right ) \text{Si}\left (\frac{2 b c}{d}+2 b x\right )}{3 d^4}+\frac{2 b^2 \sin ^2(a+b x)}{3 d^3 (c+d x)}-\frac{b \sin (a+b x) \cos (a+b x)}{3 d^2 (c+d x)^2}-\frac{\sin ^2(a+b x)}{3 d (c+d x)^3}-\frac{b^2}{3 d^3 (c+d x)} \]

[Out]

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

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Rubi [A]  time = 0.180895, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {3314, 32, 3313, 12, 3303, 3299, 3302} \[ -\frac{2 b^3 \sin \left (2 a-\frac{2 b c}{d}\right ) \text{CosIntegral}\left (\frac{2 b c}{d}+2 b x\right )}{3 d^4}-\frac{2 b^3 \cos \left (2 a-\frac{2 b c}{d}\right ) \text{Si}\left (\frac{2 b c}{d}+2 b x\right )}{3 d^4}+\frac{2 b^2 \sin ^2(a+b x)}{3 d^3 (c+d x)}-\frac{b \sin (a+b x) \cos (a+b x)}{3 d^2 (c+d x)^2}-\frac{\sin ^2(a+b x)}{3 d (c+d x)^3}-\frac{b^2}{3 d^3 (c+d x)} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*x]^2/(c + d*x)^4,x]

[Out]

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

Rule 3314

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((c + d*x)^(m + 1)*(b*Si
n[e + f*x])^n)/(d*(m + 1)), x] + (Dist[(b^2*f^2*n*(n - 1))/(d^2*(m + 1)*(m + 2)), Int[(c + d*x)^(m + 2)*(b*Sin
[e + f*x])^(n - 2), x], x] - Dist[(f^2*n^2)/(d^2*(m + 1)*(m + 2)), Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^n, x
], x] - Simp[(b*f*n*(c + d*x)^(m + 2)*Cos[e + f*x]*(b*Sin[e + f*x])^(n - 1))/(d^2*(m + 1)*(m + 2)), x]) /; Fre
eQ[{b, c, d, e, f}, x] && GtQ[n, 1] && LtQ[m, -2]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 3313

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Simp[((c + d*x)^(m + 1)*Sin[e + f*x]^
n)/(d*(m + 1)), x] - Dist[(f*n)/(d*(m + 1)), Int[ExpandTrigReduce[(c + d*x)^(m + 1), Cos[e + f*x]*Sin[e + f*x]
^(n - 1), x], x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && GeQ[m, -2] && LtQ[m, -1]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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{\sin ^2(a+b x)}{(c+d x)^4} \, dx &=-\frac{b \cos (a+b x) \sin (a+b x)}{3 d^2 (c+d x)^2}-\frac{\sin ^2(a+b x)}{3 d (c+d x)^3}+\frac{b^2 \int \frac{1}{(c+d x)^2} \, dx}{3 d^2}-\frac{\left (2 b^2\right ) \int \frac{\sin ^2(a+b x)}{(c+d x)^2} \, dx}{3 d^2}\\ &=-\frac{b^2}{3 d^3 (c+d x)}-\frac{b \cos (a+b x) \sin (a+b x)}{3 d^2 (c+d x)^2}-\frac{\sin ^2(a+b x)}{3 d (c+d x)^3}+\frac{2 b^2 \sin ^2(a+b x)}{3 d^3 (c+d x)}-\frac{\left (4 b^3\right ) \int \frac{\sin (2 a+2 b x)}{2 (c+d x)} \, dx}{3 d^3}\\ &=-\frac{b^2}{3 d^3 (c+d x)}-\frac{b \cos (a+b x) \sin (a+b x)}{3 d^2 (c+d x)^2}-\frac{\sin ^2(a+b x)}{3 d (c+d x)^3}+\frac{2 b^2 \sin ^2(a+b x)}{3 d^3 (c+d x)}-\frac{\left (2 b^3\right ) \int \frac{\sin (2 a+2 b x)}{c+d x} \, dx}{3 d^3}\\ &=-\frac{b^2}{3 d^3 (c+d x)}-\frac{b \cos (a+b x) \sin (a+b x)}{3 d^2 (c+d x)^2}-\frac{\sin ^2(a+b x)}{3 d (c+d x)^3}+\frac{2 b^2 \sin ^2(a+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{\sin \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{\cos \left (\frac{2 b c}{d}+2 b x\right )}{c+d x} \, dx}{3 d^3}\\ &=-\frac{b^2}{3 d^3 (c+d x)}-\frac{2 b^3 \text{Ci}\left (\frac{2 b c}{d}+2 b x\right ) \sin \left (2 a-\frac{2 b c}{d}\right )}{3 d^4}-\frac{b \cos (a+b x) \sin (a+b x)}{3 d^2 (c+d x)^2}-\frac{\sin ^2(a+b x)}{3 d (c+d x)^3}+\frac{2 b^2 \sin ^2(a+b x)}{3 d^3 (c+d x)}-\frac{2 b^3 \cos \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 = 1.21809, size = 122, normalized size = 0.75 \[ -\frac{4 b^3 \sin \left (2 a-\frac{2 b c}{d}\right ) \text{CosIntegral}\left (\frac{2 b (c+d x)}{d}\right )+\frac{d \left (\cos (2 (a+b x)) \left (2 b^2 (c+d x)^2-d^2\right )+d (b (c+d x) \sin (2 (a+b x))+d)\right )}{(c+d x)^3}+4 b^3 \cos \left (2 a-\frac{2 b c}{d}\right ) \text{Si}\left (\frac{2 b (c+d x)}{d}\right )}{6 d^4} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b*x]^2/(c + d*x)^4,x]

[Out]

-(4*b^3*CosIntegral[(2*b*(c + d*x))/d]*Sin[2*a - (2*b*c)/d] + (d*((-d^2 + 2*b^2*(c + d*x)^2)*Cos[2*(a + b*x)]
+ d*(d + b*(c + d*x)*Sin[2*(a + b*x)])))/(c + d*x)^3 + 4*b^3*Cos[2*a - (2*b*c)/d]*SinIntegral[(2*b*(c + d*x))/
d])/(6*d^4)

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Maple [A]  time = 0.008, size = 229, normalized size = 1.4 \begin{align*}{\frac{1}{b} \left ( -{\frac{{b}^{4}}{6\, \left ( \left ( bx+a \right ) d-da+cb \right ) ^{3}d}}-{\frac{{b}^{4}}{4} \left ( -{\frac{2\,\cos \left ( 2\,bx+2\,a \right ) }{3\, \left ( \left ( bx+a \right ) d-da+cb \right ) ^{3}d}}-{\frac{2}{3\,d} \left ( -{\frac{\sin \left ( 2\,bx+2\,a \right ) }{ \left ( \left ( bx+a \right ) d-da+cb \right ) ^{2}d}}+{\frac{1}{d} \left ( -2\,{\frac{\cos \left ( 2\,bx+2\,a \right ) }{ \left ( \left ( bx+a \right ) d-da+cb \right ) d}}-2\,{\frac{1}{d} \left ( 2\,{\frac{1}{d}{\it Si} \left ( 2\,bx+2\,a+2\,{\frac{-da+cb}{d}} \right ) \cos \left ( 2\,{\frac{-da+cb}{d}} \right ) }-2\,{\frac{1}{d}{\it Ci} \left ( 2\,bx+2\,a+2\,{\frac{-da+cb}{d}} \right ) \sin \left ( 2\,{\frac{-da+cb}{d}} \right ) } \right ) } \right ) } \right ) } \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(b*x+a)^2/(d*x+c)^4,x)

[Out]

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

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Maxima [C]  time = 1.88609, size = 346, normalized size = 2.14 \begin{align*} \frac{3 \, 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 )} \cos \left (-\frac{2 \,{\left (b c - a d\right )}}{d}\right ) - b^{4}{\left (3 i \, E_{4}\left (\frac{2 i \, b c + 2 i \,{\left (b x + a\right )} d - 2 i \, a d}{d}\right ) - 3 i \, 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 ) - 2 \, b^{4}}{12 \,{\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(sin(b*x+a)^2/(d*x+c)^4,x, algorithm="maxima")

[Out]

1/12*(3*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))*cos(-2*(b*c - a*d)/d) - b^4*(3*I*exp_integral_e(4, (2*I*b*c + 2*I*(b*x + a)*d - 2*I*a
*d)/d) - 3*I*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) - 2*b^4)/((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 = 1.92985, size = 733, normalized size = 4.52 \begin{align*} \frac{b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d - d^{3} -{\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 )^{2} -{\left (b d^{3} x + b c d^{2}\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right ) - 2 \,{\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 )} \cos \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 ) -{\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 )} \sin \left (-\frac{2 \,{\left (b c - a d\right )}}{d}\right )}{3 \,{\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(sin(b*x+a)^2/(d*x+c)^4,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin ^{2}{\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(sin(b*x+a)**2/(d*x+c)**4,x)

[Out]

Integral(sin(a + b*x)**2/(c + d*x)**4, x)

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Giac [C]  time = 1.77156, size = 10573, normalized size = 65.27 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x+a)^2/(d*x+c)^4,x, algorithm="giac")

[Out]

-1/3*(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)^2 - 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)^2 + 2*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)^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*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*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)^2 - 2*b^3*d^3*x^3*real_pa
rt(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a)*tan(b*c/d)^2 + 3*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)^2 - 3*b^3*c*d^2*x^2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*ta
n(b*x)^2*tan(a)^2*tan(b*c/d)^2 + 6*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*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(a)^2 + b^3*d^3*x^3*imag_part(cos_int
egral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a)^2 - 2*b^3*d^3*x^3*sin_integral(2*(b*d*x + 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(a)*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) + 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) + 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) + 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) - 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)^2 + 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)^2 - 2*b^3*d^3*x^3*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^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)*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)*tan(b*c/d)^2 + b^3*d^3*x^3*imag_part(cos_integr
al(2*b*x + 2*b*c/d))*tan(a)^2*tan(b*c/d)^2 - b^3*d^3*x^3*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2*ta
n(b*c/d)^2 + 2*b^3*d^3*x^3*sin_integral(2*(b*d*x + b*c)/d)*tan(a)^2*tan(b*c/d)^2 + 3*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)^2 - 3*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)^2 + 6*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)^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*b^3*d^3*x^3*real_p
art(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a) - 3*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 + 3*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 - 6*b^3
*c*d^2*x^2*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(a)^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*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
) + 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) - 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) + 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*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*b^3*d^3*x^3*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2*tan(b*c/d) + 6*b^3*c^2*d*x*real_par
t(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*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*x)^2*tan(a)^2*tan(b*c/d) - 3*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)^2 + 3*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)^2 - 6*b^3*c*
d^2*x^2*sin_integral(2*(b*d*x + 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(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)*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)*tan(b*c/d)^2 - 6*b^3*c^2*d*x*real_pa
rt(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a)*tan(b*c/d)^2 + 3*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)^2 - 3*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)^2 + 6*b^3*c*d^2*x^2*sin_integral(2*(b*d*x + b*c)/d)*tan(a)^2*tan(b*c/d)^2 + b^2*d^3*x^2*tan(b*x)^2*tan(
a)^2*tan(b*c/d)^2 + 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)^2 - 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)^2 + 2*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*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2
- b^3*d^3*x^3*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2 + 2*b^3*d^3*x^3*sin_integral(2*(b*d*x + 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*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) - b^3*d^3*x^3*imag_part(cos_integral(2*b*x + 2*b*
c/d))*tan(a)^2 + b^3*d^3*x^3*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2 - 2*b^3*d^3*x^3*sin_integral(2
*(b*d*x + b*c)/d)*tan(a)^2 - 3*b^3*c^2*d*x*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(a)^2 + 3*b^
3*c^2*d*x*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a)^2 - 6*b^3*c^2*d*x*sin_integral(2*(b*d*x
+ 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(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) + 4*b^3*d^3*x^3*imag_part(c
os_integral(2*b*x + 2*b*c/d))*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)*tan(b*c/d) + 8*b^3*d^3*x^3*sin_integral(2*(b*d*x + b*c)/d)*tan(a)*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)*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)*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)*tan(b*c/
d) + 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) + 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*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*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)
- b^3*d^3*x^3*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*c/d)^2 + b^3*d^3*x^3*imag_part(cos_integral(-2*b
*x - 2*b*c/d))*tan(b*c/d)^2 - 2*b^3*d^3*x^3*sin_integral(2*(b*d*x + b*c)/d)*tan(b*c/d)^2 - 3*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)^2 + 3*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)^2 - 6*b^3*c^2*d*x*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^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)*tan(b*c/d)^2 - 6*b^3*c*d^2*x^2*real_part(cos_integ
ral(-2*b*x - 2*b*c/d))*tan(a)*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)*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)*tan(b*c/d)^2 + 3*b^3*
c^2*d*x*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)^2*tan(b*c/d)^2 - 3*b^3*c^2*d*x*imag_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*sin_integral(2*(b*d*x + b*c)/d)*tan(a)^2*tan(b*c/d)^2
+ 2*b^2*c*d^2*x*tan(b*x)^2*tan(a)^2*tan(b*c/d)^2 + 3*b^3*c*d^2*x^2*imag_part(cos_integral(2*b*x + 2*b*c/d))*t
an(b*x)^2 - 3*b^3*c*d^2*x^2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2 + 6*b^3*c*d^2*x^2*sin_integra
l(2*(b*d*x + 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(a) + 2*b^3*d^3*x^
3*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a) + 6*b^3*c^2*d*x*real_part(cos_integral(2*b*x + 2*b*c/d))*ta
n(b*x)^2*tan(a) + 6*b^3*c^2*d*x*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a) - 3*b^3*c*d^2*x^2*
imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)^2 + 3*b^3*c*d^2*x^2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*
tan(a)^2 - 6*b^3*c*d^2*x^2*sin_integral(2*(b*d*x + b*c)/d)*tan(a)^2 + b^2*d^3*x^2*tan(b*x)^2*tan(a)^2 - b^3*c^
3*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(a)^2 + b^3*c^3*imag_part(cos_integral(-2*b*x - 2*b*c
/d))*tan(b*x)^2*tan(a)^2 - 2*b^3*c^3*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(a)^2 - 2*b^3*d^3*x^3*real_
part(cos_integral(2*b*x + 2*b*c/d))*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) - 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) - 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) + 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) - 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) + 24*b^3
*c*d^2*x^2*sin_integral(2*(b*d*x + 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(b*x)^2*tan(a)*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)*tan(b
*c/d) + 8*b^3*c^3*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(a)*tan(b*c/d) + 6*b^3*c^2*d*x*real_part(cos_i
ntegral(2*b*x + 2*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(a)
^2*tan(b*c/d) - 3*b^3*c*d^2*x^2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*c/d)^2 + 3*b^3*c*d^2*x^2*imag_p
art(cos_integral(-2*b*x - 2*b*c/d))*tan(b*c/d)^2 - 6*b^3*c*d^2*x^2*sin_integral(2*(b*d*x + b*c)/d)*tan(b*c/d)^
2 - b^2*d^3*x^2*tan(b*x)^2*tan(b*c/d)^2 - b^3*c^3*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(b*c/
d)^2 + b^3*c^3*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(b*c/d)^2 - 2*b^3*c^3*sin_integral(2*(b
*d*x + 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(a)*tan(b*c
/d)^2 - 6*b^3*c^2*d*x*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)*tan(b*c/d)^2 - 4*b^2*d^3*x^2*tan(b*x)*t
an(a)*tan(b*c/d)^2 - b^2*d^3*x^2*tan(a)^2*tan(b*c/d)^2 + b^3*c^3*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(
a)^2*tan(b*c/d)^2 - b^3*c^3*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2*tan(b*c/d)^2 + 2*b^3*c^3*sin_in
tegral(2*(b*d*x + b*c)/d)*tan(a)^2*tan(b*c/d)^2 + b^2*c^2*d*tan(b*x)^2*tan(a)^2*tan(b*c/d)^2 + b^3*d^3*x^3*ima
g_part(cos_integral(2*b*x + 2*b*c/d)) - b^3*d^3*x^3*imag_part(cos_integral(-2*b*x - 2*b*c/d)) + 2*b^3*d^3*x^3*
sin_integral(2*(b*d*x + b*c)/d) + 3*b^3*c^2*d*x*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2 - 3*b^3*c^
2*d*x*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2 + 6*b^3*c^2*d*x*sin_integral(2*(b*d*x + 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(a) + 6*b^3*c*d^2*x^2*real_part(cos_inte
gral(-2*b*x - 2*b*c/d))*tan(a) + 2*b^3*c^3*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(a) + 2*b^3*
c^3*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a) - 3*b^3*c^2*d*x*imag_part(cos_integral(2*b*x +
2*b*c/d))*tan(a)^2 + 3*b^3*c^2*d*x*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2 - 6*b^3*c^2*d*x*sin_int
egral(2*(b*d*x + b*c)/d)*tan(a)^2 + 2*b^2*c*d^2*x*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*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*b^3*c
^3*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(b*c/d) - 2*b^3*c^3*real_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_integral(2*b*x + 2*b*c/d))*tan(a)*tan(b*c/d) -
12*b^3*c^2*d*x*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)*tan(b*c/d) + 24*b^3*c^2*d*x*sin_integral(2*(b*
d*x + 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(a)^2*tan(b*c/d) + 2*b
^3*c^3*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2*tan(b*c/d) - 3*b^3*c^2*d*x*imag_part(cos_integral(2*
b*x + 2*b*c/d))*tan(b*c/d)^2 + 3*b^3*c^2*d*x*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*c/d)^2 - 6*b^3*c^
2*d*x*sin_integral(2*(b*d*x + b*c)/d)*tan(b*c/d)^2 - 2*b^2*c*d^2*x*tan(b*x)^2*tan(b*c/d)^2 - 2*b^3*c^3*real_pa
rt(cos_integral(2*b*x + 2*b*c/d))*tan(a)*tan(b*c/d)^2 - 2*b^3*c^3*real_part(cos_integral(-2*b*x - 2*b*c/d))*ta
n(a)*tan(b*c/d)^2 - 8*b^2*c*d^2*x*tan(b*x)*tan(a)*tan(b*c/d)^2 - b*d^3*x*tan(b*x)^2*tan(a)*tan(b*c/d)^2 - 2*b^
2*c*d^2*x*tan(a)^2*tan(b*c/d)^2 - b*d^3*x*tan(b*x)*tan(a)^2*tan(b*c/d)^2 + 3*b^3*c*d^2*x^2*imag_part(cos_integ
ral(2*b*x + 2*b*c/d)) - 3*b^3*c*d^2*x^2*imag_part(cos_integral(-2*b*x - 2*b*c/d)) + 6*b^3*c*d^2*x^2*sin_integr
al(2*(b*d*x + b*c)/d) - b^2*d^3*x^2*tan(b*x)^2 + b^3*c^3*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2 -
b^3*c^3*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2 + 2*b^3*c^3*sin_integral(2*(b*d*x + b*c)/d)*tan(
b*x)^2 + 6*b^3*c^2*d*x*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(a) + 6*b^3*c^2*d*x*real_part(cos_integral(
-2*b*x - 2*b*c/d))*tan(a) - 4*b^2*d^3*x^2*tan(b*x)*tan(a) - b^2*d^3*x^2*tan(a)^2 - b^3*c^3*imag_part(cos_integ
ral(2*b*x + 2*b*c/d))*tan(a)^2 + b^3*c^3*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2 - 2*b^3*c^3*sin_in
tegral(2*(b*d*x + b*c)/d)*tan(a)^2 + b^2*c^2*d*tan(b*x)^2*tan(a)^2 - 6*b^3*c^2*d*x*real_part(cos_integral(2*b*
x + 2*b*c/d))*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) + 4*b^3*c^3*imag
_part(cos_integral(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))*t
an(a)*tan(b*c/d) + 8*b^3*c^3*sin_integral(2*(b*d*x + b*c)/d)*tan(a)*tan(b*c/d) + b^2*d^3*x^2*tan(b*c/d)^2 - b^
3*c^3*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*c/d)^2 + b^3*c^3*imag_part(cos_integral(-2*b*x - 2*b*c/d)
)*tan(b*c/d)^2 - 2*b^3*c^3*sin_integral(2*(b*d*x + b*c)/d)*tan(b*c/d)^2 - b^2*c^2*d*tan(b*x)^2*tan(b*c/d)^2 -
4*b^2*c^2*d*tan(b*x)*tan(a)*tan(b*c/d)^2 - b*c*d^2*tan(b*x)^2*tan(a)*tan(b*c/d)^2 - b^2*c^2*d*tan(a)^2*tan(b*c
/d)^2 - b*c*d^2*tan(b*x)*tan(a)^2*tan(b*c/d)^2 + 3*b^3*c^2*d*x*imag_part(cos_integral(2*b*x + 2*b*c/d)) - 3*b^
3*c^2*d*x*imag_part(cos_integral(-2*b*x - 2*b*c/d)) + 6*b^3*c^2*d*x*sin_integral(2*(b*d*x + b*c)/d) - 2*b^2*c*
d^2*x*tan(b*x)^2 + 2*b^3*c^3*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(a) + 2*b^3*c^3*real_part(cos_integra
l(-2*b*x - 2*b*c/d))*tan(a) - 8*b^2*c*d^2*x*tan(b*x)*tan(a) - b*d^3*x*tan(b*x)^2*tan(a) - 2*b^2*c*d^2*x*tan(a)
^2 - b*d^3*x*tan(b*x)*tan(a)^2 - 2*b^3*c^3*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*c/d) - 2*b^3*c^3*rea
l_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*c/d) + 2*b^2*c*d^2*x*tan(b*c/d)^2 + b*d^3*x*tan(b*x)*tan(b*c/d)^2
+ b*d^3*x*tan(a)*tan(b*c/d)^2 + b^2*d^3*x^2 + b^3*c^3*imag_part(cos_integral(2*b*x + 2*b*c/d)) - b^3*c^3*imag
_part(cos_integral(-2*b*x - 2*b*c/d)) + 2*b^3*c^3*sin_integral(2*(b*d*x + b*c)/d) - b^2*c^2*d*tan(b*x)^2 - 4*b
^2*c^2*d*tan(b*x)*tan(a) - b*c*d^2*tan(b*x)^2*tan(a) - b^2*c^2*d*tan(a)^2 - b*c*d^2*tan(b*x)*tan(a)^2 + b^2*c^
2*d*tan(b*c/d)^2 + b*c*d^2*tan(b*x)*tan(b*c/d)^2 + d^3*tan(b*x)^2*tan(b*c/d)^2 + b*c*d^2*tan(a)*tan(b*c/d)^2 +
2*d^3*tan(b*x)*tan(a)*tan(b*c/d)^2 + d^3*tan(a)^2*tan(b*c/d)^2 + 2*b^2*c*d^2*x + b*d^3*x*tan(b*x) + b*d^3*x*t
an(a) + b^2*c^2*d + b*c*d^2*tan(b*x) + d^3*tan(b*x)^2 + b*c*d^2*tan(a) + 2*d^3*tan(b*x)*tan(a) + d^3*tan(a)^2)
/(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*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 + 3*c*d^6*x^2*tan(b*x)^2*tan(b*c/d)^2 + 3*c*d^6*x^2*tan(a)^2*ta
n(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*ta
n(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*tan(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)