\(\int \frac {\cos (a+b x) \sin (a+b x)}{(c+d x)^4} \, dx\) [9]

Optimal result

Integrand size = 20, antiderivative size = 144 \[ \int \frac {\cos (a+b x) \sin (a+b x)}{(c+d x)^4} \, dx=-\frac {b \cos (2 a+2 b x)}{6 d^2 (c+d x)^2}-\frac {2 b^3 \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right )}{3 d^4}-\frac {\sin (2 a+2 b x)}{6 d (c+d x)^3}+\frac {b^2 \sin (2 a+2 b x)}{3 d^3 (c+d x)}+\frac {2 b^3 \sin \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{3 d^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4491, 12, 3378, 3384, 3380, 3383} \[ \int \frac {\cos (a+b x) \sin (a+b x)}{(c+d x)^4} \, dx=-\frac {2 b^3 \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right )}{3 d^4}+\frac {2 b^3 \sin \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{3 d^4}+\frac {b^2 \sin (2 a+2 b x)}{3 d^3 (c+d x)}-\frac {b \cos (2 a+2 b x)}{6 d^2 (c+d x)^2}-\frac {\sin (2 a+2 b x)}{6 d (c+d x)^3} \]

[In]

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

[Out]

-1/6*(b*Cos[2*a + 2*b*x])/(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 12

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

Rule 3378

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 3380

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 3383

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]

Rule 3384

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 4491

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]

Rubi steps \begin{align*} \text {integral}& = \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 ) \operatorname {CosIntegral}\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] (verified)

Time = 0.62 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.14 \[ \int \frac {\cos (a+b x) \sin (a+b x)}{(c+d x)^4} \, dx=\frac {-d \cos (2 b x) \left (b d (c+d x) \cos (2 a)+\left (d^2-2 b^2 (c+d x)^2\right ) \sin (2 a)\right )+d \left (\left (-d^2+2 b^2 (c+d x)^2\right ) \cos (2 a)+b d (c+d x) \sin (2 a)\right ) \sin (2 b x)-4 b^3 (c+d x)^3 \left (\cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 b (c+d x)}{d}\right )-\sin \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b (c+d x)}{d}\right )\right )}{6 d^4 (c+d x)^3} \]

[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)

Maple [A] (verified)

Time = 0.77 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.39

method	result	size
derivativedivides	\(\frac {b^{3} \left (-\frac {2 \sin \left (2 x b +2 a \right )}{3 \left (-a d +c b +d \left (x b +a \right )\right )^{3} d}+\frac {-\frac {2 \cos \left (2 x b +2 a \right )}{3 \left (-a d +c b +d \left (x b +a \right )\right )^{2} d}-\frac {2 \left (-\frac {2 \sin \left (2 x b +2 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right ) d}+\frac {-\frac {4 \,\operatorname {Si}\left (-2 x b -2 a -\frac {2 \left (-a d +c b \right )}{d}\right ) \sin \left (\frac {-2 a d +2 c b}{d}\right )}{d}+\frac {4 \,\operatorname {Ci}\left (2 x b +2 a +\frac {-2 a d +2 c b}{d}\right ) \cos \left (\frac {-2 a d +2 c b}{d}\right )}{d}}{d}\right )}{3 d}}{d}\right )}{4}\)	\(200\)
default	\(\frac {b^{3} \left (-\frac {2 \sin \left (2 x b +2 a \right )}{3 \left (-a d +c b +d \left (x b +a \right )\right )^{3} d}+\frac {-\frac {2 \cos \left (2 x b +2 a \right )}{3 \left (-a d +c b +d \left (x b +a \right )\right )^{2} d}-\frac {2 \left (-\frac {2 \sin \left (2 x b +2 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right ) d}+\frac {-\frac {4 \,\operatorname {Si}\left (-2 x b -2 a -\frac {2 \left (-a d +c b \right )}{d}\right ) \sin \left (\frac {-2 a d +2 c b}{d}\right )}{d}+\frac {4 \,\operatorname {Ci}\left (2 x b +2 a +\frac {-2 a d +2 c b}{d}\right ) \cos \left (\frac {-2 a d +2 c b}{d}\right )}{d}}{d}\right )}{3 d}}{d}\right )}{4}\)	\(200\)
risch	\(\frac {b^{3} {\mathrm e}^{-\frac {2 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (2 i b x +2 i a -\frac {2 i \left (a d -c b \right )}{d}\right )}{3 d^{4}}+\frac {b^{3} {\mathrm e}^{\frac {2 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (-2 i b x -2 i a -\frac {2 \left (-i a d +i c b \right )}{d}\right )}{3 d^{4}}+\frac {i \left (2 i b^{4} d^{5} x^{4}+8 i b^{4} c \,d^{4} x^{3}+12 i b^{4} c^{2} d^{3} x^{2}+8 i b^{4} c^{3} d^{2} x +2 i b^{4} c^{4} d \right ) \cos \left (2 x b +2 a \right )}{12 d^{3} \left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +b^{3} c^{3}\right ) \left (d x +c \right )^{3}}-\frac {\left (-4 b^{5} d^{5} x^{5}-20 b^{5} c \,d^{4} x^{4}-40 b^{5} c^{2} d^{3} x^{3}-40 b^{5} c^{3} d^{2} x^{2}-20 b^{5} c^{4} d x +2 b^{3} d^{5} x^{3}-4 b^{5} c^{5}+6 b^{3} c \,d^{4} x^{2}+6 b^{3} c^{2} d^{3} x +2 b^{3} c^{3} d^{2}\right ) \sin \left (2 x b +2 a \right )}{12 d^{3} \left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +b^{3} c^{3}\right ) \left (d x +c \right )^{3}}\)	\(409\)

[In]

int(cos(b*x+a)*sin(b*x+a)/(d*x+c)^4,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.83 \[ \int \frac {\cos (a+b x) \sin (a+b x)}{(c+d x)^4} \, dx=\frac {b d^{3} x + b c d^{2} - 2 \, {\left (b d^{3} x + b c d^{2}\right )} \cos \left (b x + a\right )^{2} - 4 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Ci}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) + 2 \, {\left (2 \, b^{2} d^{3} x^{2} + 4 \, b^{2} c d^{2} x + 2 \, b^{2} c^{2} d - d^{3}\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right ) + 4 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right )}{6 \, {\left (d^{7} x^{3} + 3 \, c d^{6} x^{2} + 3 \, c^{2} d^{5} x + c^{3} d^{4}\right )}} \]

[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 - 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)*cos(-2*(b*c - a*d)/d)*cos_integral(2*(b*d*x + b*c)/d) + 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))/(d^7*x^3 + 3*c*d^6*x^2 + 3*c^2*d^5*x + c^3*d^4)

Sympy [F]

\[ \int \frac {\cos (a+b x) \sin (a+b x)}{(c+d x)^4} \, dx=\int \frac {\sin {\left (a + b x \right )} \cos {\left (a + b x \right )}}{\left (c + d x\right )^{4}}\, dx \]

[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)

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.40 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.74 \[ \int \frac {\cos (a+b x) \sin (a+b x)}{(c+d x)^4} \, dx=-\frac {b^{4} {\left (-i \, E_{4}\left (\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + i \, E_{4}\left (-\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + b^{4} {\left (E_{4}\left (\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + E_{4}\left (-\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )}{4 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} + {\left (b x + a\right )}^{3} d^{4} - a^{3} d^{4} + 3 \, {\left (b c d^{3} - a d^{4}\right )} {\left (b x + a\right )}^{2} + 3 \, {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} {\left (b x + a\right )}\right )} b} \]

[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 - I*(b*x + a)*d + I*a*d)/d) + I*exp_integral_e(4, -2*(-I*b*c - I*(b*
x + a)*d + I*a*d)/d))*cos(-2*(b*c - a*d)/d) + b^4*(exp_integral_e(4, 2*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) + e
xp_integral_e(4, -2*(-I*b*c - I*(b*x + a)*d + 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)

Giac [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.58 (sec) , antiderivative size = 7592, normalized size of antiderivative = 52.72 \[ \int \frac {\cos (a+b x) \sin (a+b x)}{(c+d x)^4} \, dx=\text {Too large to display} \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos (a+b x) \sin (a+b x)}{(c+d x)^4} \, dx=\int \frac {\cos \left (a+b\,x\right )\,\sin \left (a+b\,x\right )}{{\left (c+d\,x\right )}^4} \,d x \]

[In]

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

[Out]

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