\(\int \frac {\cos (a+b x) \sin (a+b x)}{(c+d x)^3} \, dx\) [8]

Optimal result

Integrand size = 20, antiderivative size = 114 \[ \int \frac {\cos (a+b x) \sin (a+b x)}{(c+d x)^3} \, dx=-\frac {b \cos (2 a+2 b x)}{2 d^2 (c+d x)}-\frac {b^2 \operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{d^3}-\frac {\sin (2 a+2 b x)}{4 d (c+d x)^2}-\frac {b^2 \cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{d^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4491, 12, 3378, 3384, 3380, 3383} \[ \int \frac {\cos (a+b x) \sin (a+b x)}{(c+d x)^3} \, dx=-\frac {b^2 \sin \left (2 a-\frac {2 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right )}{d^3}-\frac {b^2 \cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{d^3}-\frac {b \cos (2 a+2 b x)}{2 d^2 (c+d x)}-\frac {\sin (2 a+2 b x)}{4 d (c+d x)^2} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 1.10 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.89 \[ \int \frac {\cos (a+b x) \sin (a+b x)}{(c+d x)^3} \, dx=-\frac {4 b^2 \operatorname {CosIntegral}\left (\frac {2 b (c+d x)}{d}\right ) \sin \left (2 a-\frac {2 b c}{d}\right )+\frac {d (2 b (c+d x) \cos (2 (a+b x))+d \sin (2 (a+b x)))}{(c+d x)^2}+4 b^2 \cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b (c+d x)}{d}\right )}{4 d^3} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.58 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.42

method	result	size
derivativedivides	\(\frac {b^{2} \left (-\frac {\sin \left (2 x b +2 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right )^{2} d}+\frac {-\frac {2 \cos \left (2 x b +2 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right ) d}-\frac {2 \left (-\frac {2 \,\operatorname {Si}\left (-2 x b -2 a -\frac {2 \left (-a d +c b \right )}{d}\right ) \cos \left (\frac {-2 a d +2 c b}{d}\right )}{d}-\frac {2 \,\operatorname {Ci}\left (2 x b +2 a +\frac {-2 a d +2 c b}{d}\right ) \sin \left (\frac {-2 a d +2 c b}{d}\right )}{d}\right )}{d}}{d}\right )}{4}\)	\(162\)
default	\(\frac {b^{2} \left (-\frac {\sin \left (2 x b +2 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right )^{2} d}+\frac {-\frac {2 \cos \left (2 x b +2 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right ) d}-\frac {2 \left (-\frac {2 \,\operatorname {Si}\left (-2 x b -2 a -\frac {2 \left (-a d +c b \right )}{d}\right ) \cos \left (\frac {-2 a d +2 c b}{d}\right )}{d}-\frac {2 \,\operatorname {Ci}\left (2 x b +2 a +\frac {-2 a d +2 c b}{d}\right ) \sin \left (\frac {-2 a d +2 c b}{d}\right )}{d}\right )}{d}}{d}\right )}{4}\)	\(162\)
risch	\(\frac {i b^{2} {\mathrm e}^{-\frac {2 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (2 i b x +2 i a -\frac {2 i \left (a d -c b \right )}{d}\right )}{2 d^{3}}-\frac {i b^{2} {\mathrm e}^{\frac {2 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (-2 i b x -2 i a -\frac {2 \left (-i a d +i c b \right )}{d}\right )}{2 d^{3}}+\frac {i \left (4 i b^{3} d^{3} x^{3}+12 i b^{3} c \,d^{2} x^{2}+12 i b^{3} c^{2} d x +4 i b^{3} c^{3}\right ) \cos \left (2 x b +2 a \right )}{8 d^{2} \left (x^{2} d^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}\right ) \left (d x +c \right )^{2}}-\frac {\left (2 x^{2} d^{2} b^{2}+4 b^{2} c d x +2 b^{2} c^{2}\right ) \sin \left (2 x b +2 a \right )}{8 d \left (x^{2} d^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}\right ) \left (d x +c \right )^{2}}\)	\(277\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.64 \[ \int \frac {\cos (a+b x) \sin (a+b x)}{(c+d x)^3} \, dx=\frac {b d^{2} x - d^{2} \cos \left (b x + a\right ) \sin \left (b x + a\right ) + b c d - 2 \, {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right )^{2} - 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \operatorname {Ci}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right )}{2 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\cos (a+b x) \sin (a+b x)}{(c+d x)^3} \, dx=\int \frac {\sin {\left (a + b x \right )} \cos {\left (a + b x \right )}}{\left (c + d x\right )^{3}}\, dx \]

[In]

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

[Out]

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

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.34 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.76 \[ \int \frac {\cos (a+b x) \sin (a+b x)}{(c+d x)^3} \, dx=-\frac {b^{3} {\left (-i \, E_{3}\left (\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + i \, E_{3}\left (-\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + b^{3} {\left (E_{3}\left (\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + E_{3}\left (-\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )}{4 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + {\left (b x + a\right )}^{2} d^{3} + a^{2} d^{3} + 2 \, {\left (b c d^{2} - a d^{3}\right )} {\left (b x + a\right )}\right )} b} \]

[In]

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

[Out]

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

Giac [C] (verification not implemented)

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

Time = 0.46 (sec) , antiderivative size = 5398, normalized size of antiderivative = 47.35 \[ \int \frac {\cos (a+b x) \sin (a+b x)}{(c+d x)^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/2*(b^2*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 - b^2*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 + 2*b^2*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 + 2*b^2*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*b^2*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*b^2*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 - 2*b^2*d^2*x^2*real_pa
rt(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a)*tan(b*c/d)^2 + 2*b^2*c*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 - 2*b^2*c*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 + 4*b^2*c*d*x*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(a)^2*tan(b*c/d)^2 - b^2*d^
2*x^2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(a)^2 + b^2*d^2*x^2*imag_part(cos_integral(-2*b*x
- 2*b*c/d))*tan(b*x)^2*tan(a)^2 - 2*b^2*d^2*x^2*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(a)^2 + 4*b^2*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) - 4*b^2*d^2*x^2*imag_part(cos_int
egral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a)*tan(b*c/d) + 8*b^2*d^2*x^2*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)
^2*tan(a)*tan(b*c/d) + 4*b^2*c*d*x*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(a)^2*tan(b*c/d) + 4
*b^2*c*d*x*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a)^2*tan(b*c/d) - b^2*d^2*x^2*imag_part(co
s_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(b*c/d)^2 + b^2*d^2*x^2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*t
an(b*x)^2*tan(b*c/d)^2 - 2*b^2*d^2*x^2*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(b*c/d)^2 - 4*b^2*c*d*x*r
eal_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(a)*tan(b*c/d)^2 - 4*b^2*c*d*x*real_part(cos_integral(-2
*b*x - 2*b*c/d))*tan(b*x)^2*tan(a)*tan(b*c/d)^2 + b^2*d^2*x^2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)^
2*tan(b*c/d)^2 - b^2*d^2*x^2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2*tan(b*c/d)^2 + 2*b^2*d^2*x^2*s
in_integral(2*(b*d*x + b*c)/d)*tan(a)^2*tan(b*c/d)^2 + b^2*c^2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*
x)^2*tan(a)^2*tan(b*c/d)^2 - b^2*c^2*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^2*c^2*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(a)^2*tan(b*c/d)^2 + 2*b^2*d^2*x^2*real_part(cos_i
ntegral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(a) + 2*b^2*d^2*x^2*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)
^2*tan(a) - 2*b^2*c*d*x*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(a)^2 + 2*b^2*c*d*x*imag_part(c
os_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a)^2 - 4*b^2*c*d*x*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*ta
n(a)^2 - 2*b^2*d^2*x^2*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(b*c/d) - 2*b^2*d^2*x^2*real_par
t(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(b*c/d) + 8*b^2*c*d*x*imag_part(cos_integral(2*b*x + 2*b*c/d))
*tan(b*x)^2*tan(a)*tan(b*c/d) - 8*b^2*c*d*x*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a)*tan(b*
c/d) + 16*b^2*c*d*x*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(a)*tan(b*c/d) + 2*b^2*d^2*x^2*real_part(cos
_integral(2*b*x + 2*b*c/d))*tan(a)^2*tan(b*c/d) + 2*b^2*d^2*x^2*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(
a)^2*tan(b*c/d) + 2*b^2*c^2*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(a)^2*tan(b*c/d) + 2*b^2*c^
2*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a)^2*tan(b*c/d) - 2*b^2*c*d*x*imag_part(cos_integra
l(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(b*c/d)^2 + 2*b^2*c*d*x*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2
*tan(b*c/d)^2 - 4*b^2*c*d*x*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(b*c/d)^2 - 2*b^2*d^2*x^2*real_part(
cos_integral(2*b*x + 2*b*c/d))*tan(a)*tan(b*c/d)^2 - 2*b^2*d^2*x^2*real_part(cos_integral(-2*b*x - 2*b*c/d))*t
an(a)*tan(b*c/d)^2 - 2*b^2*c^2*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(a)*tan(b*c/d)^2 - 2*b^2
*c^2*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a)*tan(b*c/d)^2 + 2*b^2*c*d*x*imag_part(cos_inte
gral(2*b*x + 2*b*c/d))*tan(a)^2*tan(b*c/d)^2 - 2*b^2*c*d*x*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2*
tan(b*c/d)^2 + 4*b^2*c*d*x*sin_integral(2*(b*d*x + b*c)/d)*tan(a)^2*tan(b*c/d)^2 + b*d^2*x*tan(b*x)^2*tan(a)^2
*tan(b*c/d)^2 + b^2*d^2*x^2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2 - b^2*d^2*x^2*imag_part(cos_in
tegral(-2*b*x - 2*b*c/d))*tan(b*x)^2 + 2*b^2*d^2*x^2*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2 + 4*b^2*c*d*x*
real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(a) + 4*b^2*c*d*x*real_part(cos_integral(-2*b*x - 2*b*c
/d))*tan(b*x)^2*tan(a) - b^2*d^2*x^2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)^2 + b^2*d^2*x^2*imag_part
(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2 - 2*b^2*d^2*x^2*sin_integral(2*(b*d*x + b*c)/d)*tan(a)^2 - b^2*c^2*i
mag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(a)^2 + b^2*c^2*imag_part(cos_integral(-2*b*x - 2*b*c/d)
)*tan(b*x)^2*tan(a)^2 - 2*b^2*c^2*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(a)^2 - 4*b^2*c*d*x*real_part(
cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(b*c/d) - 4*b^2*c*d*x*real_part(cos_integral(-2*b*x - 2*b*c/d))*t
an(b*x)^2*tan(b*c/d) + 4*b^2*d^2*x^2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)*tan(b*c/d) - 4*b^2*d^2*x^
2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)*tan(b*c/d) + 8*b^2*d^2*x^2*sin_integral(2*(b*d*x + b*c)/d)*
tan(a)*tan(b*c/d) + 4*b^2*c^2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(a)*tan(b*c/d) - 4*b^2*c^
2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a)*tan(b*c/d) + 8*b^2*c^2*sin_integral(2*(b*d*x + b
*c)/d)*tan(b*x)^2*tan(a)*tan(b*c/d) + 4*b^2*c*d*x*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)^2*tan(b*c/d)
+ 4*b^2*c*d*x*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2*tan(b*c/d) - b^2*d^2*x^2*imag_part(cos_integ
ral(2*b*x + 2*b*c/d))*tan(b*c/d)^2 + b^2*d^2*x^2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*c/d)^2 - 2*b^
2*d^2*x^2*sin_integral(2*(b*d*x + b*c)/d)*tan(b*c/d)^2 - b^2*c^2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(
b*x)^2*tan(b*c/d)^2 + b^2*c^2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(b*c/d)^2 - 2*b^2*c^2*si
n_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(b*c/d)^2 - 4*b^2*c*d*x*real_part(cos_integral(2*b*x + 2*b*c/d))*t
an(a)*tan(b*c/d)^2 - 4*b^2*c*d*x*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)*tan(b*c/d)^2 + b^2*c^2*imag_
part(cos_integral(2*b*x + 2*b*c/d))*tan(a)^2*tan(b*c/d)^2 - b^2*c^2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*
tan(a)^2*tan(b*c/d)^2 + 2*b^2*c^2*sin_integral(2*(b*d*x + b*c)/d)*tan(a)^2*tan(b*c/d)^2 + b*c*d*tan(b*x)^2*tan
(a)^2*tan(b*c/d)^2 + 2*b^2*c*d*x*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2 - 2*b^2*c*d*x*imag_part(c
os_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2 + 4*b^2*c*d*x*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2 + 2*b^2*d^2
*x^2*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(a) + 2*b^2*d^2*x^2*real_part(cos_integral(-2*b*x - 2*b*c/d))
*tan(a) + 2*b^2*c^2*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(a) + 2*b^2*c^2*real_part(cos_integ
ral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a) - 2*b^2*c*d*x*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)^2 + 2*b
^2*c*d*x*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2 - 4*b^2*c*d*x*sin_integral(2*(b*d*x + b*c)/d)*tan(
a)^2 + b*d^2*x*tan(b*x)^2*tan(a)^2 - 2*b^2*d^2*x^2*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*c/d) - 2*b^2
*d^2*x^2*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*c/d) - 2*b^2*c^2*real_part(cos_integral(2*b*x + 2*b*c
/d))*tan(b*x)^2*tan(b*c/d) - 2*b^2*c^2*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(b*c/d) + 8*b^2
*c*d*x*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)*tan(b*c/d) - 8*b^2*c*d*x*imag_part(cos_integral(-2*b*x
- 2*b*c/d))*tan(a)*tan(b*c/d) + 16*b^2*c*d*x*sin_integral(2*(b*d*x + b*c)/d)*tan(a)*tan(b*c/d) + 2*b^2*c^2*rea
l_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)^2*tan(b*c/d) + 2*b^2*c^2*real_part(cos_integral(-2*b*x - 2*b*c/d)
)*tan(a)^2*tan(b*c/d) - 2*b^2*c*d*x*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*c/d)^2 + 2*b^2*c*d*x*imag_p
art(cos_integral(-2*b*x - 2*b*c/d))*tan(b*c/d)^2 - 4*b^2*c*d*x*sin_integral(2*(b*d*x + b*c)/d)*tan(b*c/d)^2 -
b*d^2*x*tan(b*x)^2*tan(b*c/d)^2 - 2*b^2*c^2*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)*tan(b*c/d)^2 - 2*b
^2*c^2*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)*tan(b*c/d)^2 - 4*b*d^2*x*tan(b*x)*tan(a)*tan(b*c/d)^2
- b*d^2*x*tan(a)^2*tan(b*c/d)^2 + b^2*d^2*x^2*imag_part(cos_integral(2*b*x + 2*b*c/d)) - b^2*d^2*x^2*imag_part
(cos_integral(-2*b*x - 2*b*c/d)) + 2*b^2*d^2*x^2*sin_integral(2*(b*d*x + b*c)/d) + b^2*c^2*imag_part(cos_integ
ral(2*b*x + 2*b*c/d))*tan(b*x)^2 - b^2*c^2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2 + 2*b^2*c^2*si
n_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2 + 4*b^2*c*d*x*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(a) + 4*b^2
*c*d*x*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a) - b^2*c^2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan
(a)^2 + b^2*c^2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2 - 2*b^2*c^2*sin_integral(2*(b*d*x + b*c)/d)
*tan(a)^2 + b*c*d*tan(b*x)^2*tan(a)^2 - 4*b^2*c*d*x*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*c/d) - 4*b^
2*c*d*x*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*c/d) + 4*b^2*c^2*imag_part(cos_integral(2*b*x + 2*b*c/
d))*tan(a)*tan(b*c/d) - 4*b^2*c^2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)*tan(b*c/d) + 8*b^2*c^2*sin_
integral(2*(b*d*x + b*c)/d)*tan(a)*tan(b*c/d) - b^2*c^2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*c/d)^2
+ b^2*c^2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*c/d)^2 - 2*b^2*c^2*sin_integral(2*(b*d*x + b*c)/d)*t
an(b*c/d)^2 - b*c*d*tan(b*x)^2*tan(b*c/d)^2 - 4*b*c*d*tan(b*x)*tan(a)*tan(b*c/d)^2 - d^2*tan(b*x)^2*tan(a)*tan
(b*c/d)^2 - b*c*d*tan(a)^2*tan(b*c/d)^2 - d^2*tan(b*x)*tan(a)^2*tan(b*c/d)^2 + 2*b^2*c*d*x*imag_part(cos_integ
ral(2*b*x + 2*b*c/d)) - 2*b^2*c*d*x*imag_part(cos_integral(-2*b*x - 2*b*c/d)) + 4*b^2*c*d*x*sin_integral(2*(b*
d*x + b*c)/d) - b*d^2*x*tan(b*x)^2 + 2*b^2*c^2*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(a) + 2*b^2*c^2*rea
l_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a) - 4*b*d^2*x*tan(b*x)*tan(a) - b*d^2*x*tan(a)^2 - 2*b^2*c^2*real_
part(cos_integral(2*b*x + 2*b*c/d))*tan(b*c/d) - 2*b^2*c^2*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*c/d
) + b*d^2*x*tan(b*c/d)^2 + b^2*c^2*imag_part(cos_integral(2*b*x + 2*b*c/d)) - b^2*c^2*imag_part(cos_integral(-
2*b*x - 2*b*c/d)) + 2*b^2*c^2*sin_integral(2*(b*d*x + b*c)/d) - b*c*d*tan(b*x)^2 - 4*b*c*d*tan(b*x)*tan(a) - d
^2*tan(b*x)^2*tan(a) - b*c*d*tan(a)^2 - d^2*tan(b*x)*tan(a)^2 + b*c*d*tan(b*c/d)^2 + d^2*tan(b*x)*tan(b*c/d)^2
+ d^2*tan(a)*tan(b*c/d)^2 + b*d^2*x + b*c*d + d^2*tan(b*x) + d^2*tan(a))/(d^5*x^2*tan(b*x)^2*tan(a)^2*tan(b*c
/d)^2 + 2*c*d^4*x*tan(b*x)^2*tan(a)^2*tan(b*c/d)^2 + d^5*x^2*tan(b*x)^2*tan(a)^2 + d^5*x^2*tan(b*x)^2*tan(b*c/
d)^2 + d^5*x^2*tan(a)^2*tan(b*c/d)^2 + c^2*d^3*tan(b*x)^2*tan(a)^2*tan(b*c/d)^2 + 2*c*d^4*x*tan(b*x)^2*tan(a)^
2 + 2*c*d^4*x*tan(b*x)^2*tan(b*c/d)^2 + 2*c*d^4*x*tan(a)^2*tan(b*c/d)^2 + d^5*x^2*tan(b*x)^2 + d^5*x^2*tan(a)^
2 + c^2*d^3*tan(b*x)^2*tan(a)^2 + d^5*x^2*tan(b*c/d)^2 + c^2*d^3*tan(b*x)^2*tan(b*c/d)^2 + c^2*d^3*tan(a)^2*ta
n(b*c/d)^2 + 2*c*d^4*x*tan(b*x)^2 + 2*c*d^4*x*tan(a)^2 + 2*c*d^4*x*tan(b*c/d)^2 + d^5*x^2 + c^2*d^3*tan(b*x)^2
+ c^2*d^3*tan(a)^2 + c^2*d^3*tan(b*c/d)^2 + 2*c*d^4*x + c^2*d^3)

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos (a+b x) \sin (a+b x)}{(c+d x)^3} \, dx=\int \frac {\cos \left (a+b\,x\right )\,\sin \left (a+b\,x\right )}{{\left (c+d\,x\right )}^3} \,d x \]

[In]

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

[Out]

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