\(\int \frac {x \sin (c+d x)}{a+b x} \, dx\) [21]

Optimal result

Integrand size = 15, antiderivative size = 69 \[ \int \frac {x \sin (c+d x)}{a+b x} \, dx=-\frac {\cos (c+d x)}{b d}-\frac {a \operatorname {CosIntegral}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{b^2}-\frac {a \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6874, 2718, 3384, 3380, 3383} \[ \int \frac {x \sin (c+d x)}{a+b x} \, dx=-\frac {a \sin \left (c-\frac {a d}{b}\right ) \operatorname {CosIntegral}\left (x d+\frac {a d}{b}\right )}{b^2}-\frac {a \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{b^2}-\frac {\cos (c+d x)}{b d} \]

[In]

Int[(x*Sin[c + d*x])/(a + b*x),x]

[Out]

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

Rule 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

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 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.91 \[ \int \frac {x \sin (c+d x)}{a+b x} \, dx=-\frac {b \cos (c+d x)+a d \operatorname {CosIntegral}\left (d \left (\frac {a}{b}+x\right )\right ) \sin \left (c-\frac {a d}{b}\right )+a d \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )}{b^2 d} \]

[In]

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

[Out]

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.25 (sec) , antiderivative size = 150, normalized size of antiderivative = 2.17

method	result	size
risch	\(-\frac {\cos \left (d x +c \right )}{b d}+\frac {i a \cos \left (\frac {d a -c b}{b}\right ) \operatorname {Ei}_{1}\left (\frac {i d \left (b x +a \right )}{b}\right )}{2 b^{2}}-\frac {i a \cos \left (\frac {d a -c b}{b}\right ) \operatorname {Ei}_{1}\left (-\frac {i d \left (b x +a \right )}{b}\right )}{2 b^{2}}-\frac {a \sin \left (\frac {d a -c b}{b}\right ) \operatorname {Ei}_{1}\left (\frac {i d \left (b x +a \right )}{b}\right )}{2 b^{2}}-\frac {a \sin \left (\frac {d a -c b}{b}\right ) \operatorname {Ei}_{1}\left (-\frac {i d \left (b x +a \right )}{b}\right )}{2 b^{2}}\)	\(150\)
derivativedivides	\(\frac {-\frac {\left (d a -c b \right ) d \left (\frac {\operatorname {Si}\left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{b}-\frac {d \cos \left (d x +c \right )}{b}-d c \left (\frac {\operatorname {Si}\left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{d^{2}}\)	\(180\)
default	\(\frac {-\frac {\left (d a -c b \right ) d \left (\frac {\operatorname {Si}\left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{b}-\frac {d \cos \left (d x +c \right )}{b}-d c \left (\frac {\operatorname {Si}\left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{d^{2}}\)	\(180\)

[In]

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

[Out]

-cos(d*x+c)/b/d+1/2*I*a/b^2*cos((a*d-b*c)/b)*Ei(1,I*d*(b*x+a)/b)-1/2*I*a/b^2*cos((a*d-b*c)/b)*Ei(1,-I*d*(b*x+a
)/b)-1/2*a/b^2*sin((a*d-b*c)/b)*Ei(1,I*d*(b*x+a)/b)-1/2*a/b^2*sin((a*d-b*c)/b)*Ei(1,-I*d*(b*x+a)/b)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.13 \[ \int \frac {x \sin (c+d x)}{a+b x} \, dx=\frac {a d \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) \sin \left (-\frac {b c - a d}{b}\right ) - a d \cos \left (-\frac {b c - a d}{b}\right ) \operatorname {Si}\left (\frac {b d x + a d}{b}\right ) - b \cos \left (d x + c\right )}{b^{2} d} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(x*sin(c + d*x)/(a + b*x), x)

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 776, normalized size of antiderivative = 11.25 \[ \int \frac {x \sin (c+d x)}{a+b x} \, dx=-\frac {\frac {{\left (d {\left (-i \, E_{1}\left (\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right ) + i \, E_{1}\left (-\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right )\right )} \cos \left (-\frac {b c - a d}{b}\right ) + d {\left (E_{1}\left (\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right ) + E_{1}\left (-\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right )\right )} \sin \left (-\frac {b c - a d}{b}\right )\right )} c}{b} + \frac {{\left (d x + c\right )} b d \cos \left (d x + c\right )^{3} + {\left (d x + c\right )} b d \cos \left (d x + c\right ) - {\left ({\left (b c d {\left (E_{2}\left (\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right ) + E_{2}\left (-\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right )\right )} - a d^{2} {\left (E_{2}\left (\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right ) + E_{2}\left (-\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right )\right )}\right )} \cos \left (-\frac {b c - a d}{b}\right ) - {\left (a d^{2} {\left (i \, E_{2}\left (\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right ) - i \, E_{2}\left (-\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right )\right )} + b c d {\left (-i \, E_{2}\left (\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right ) + i \, E_{2}\left (-\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right )\right )}\right )} \sin \left (-\frac {b c - a d}{b}\right )\right )} \cos \left (d x + c\right )^{2} + {\left ({\left (d x + c\right )} b d \cos \left (d x + c\right ) - {\left (b c d {\left (E_{2}\left (\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right ) + E_{2}\left (-\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right )\right )} - a d^{2} {\left (E_{2}\left (\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right ) + E_{2}\left (-\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right )\right )}\right )} \cos \left (-\frac {b c - a d}{b}\right ) + {\left (a d^{2} {\left (i \, E_{2}\left (\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right ) - i \, E_{2}\left (-\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right )\right )} + b c d {\left (-i \, E_{2}\left (\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right ) + i \, E_{2}\left (-\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right )\right )}\right )} \sin \left (-\frac {b c - a d}{b}\right )\right )} \sin \left (d x + c\right )^{2}}{{\left ({\left (d x + c\right )} b^{2} - b^{2} c + a b d\right )} \cos \left (d x + c\right )^{2} + {\left ({\left (d x + c\right )} b^{2} - b^{2} c + a b d\right )} \sin \left (d x + c\right )^{2}}}{2 \, d^{2}} \]

[In]

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

[Out]

-1/2*((d*(-I*exp_integral_e(1, (I*(d*x + c)*b - I*b*c + I*a*d)/b) + I*exp_integral_e(1, -(I*(d*x + c)*b - I*b*
c + I*a*d)/b))*cos(-(b*c - a*d)/b) + d*(exp_integral_e(1, (I*(d*x + c)*b - I*b*c + I*a*d)/b) + exp_integral_e(
1, -(I*(d*x + c)*b - I*b*c + I*a*d)/b))*sin(-(b*c - a*d)/b))*c/b + ((d*x + c)*b*d*cos(d*x + c)^3 + (d*x + c)*b
*d*cos(d*x + c) - ((b*c*d*(exp_integral_e(2, (I*(d*x + c)*b - I*b*c + I*a*d)/b) + exp_integral_e(2, -(I*(d*x +
c)*b - I*b*c + I*a*d)/b)) - a*d^2*(exp_integral_e(2, (I*(d*x + c)*b - I*b*c + I*a*d)/b) + exp_integral_e(2, -
(I*(d*x + c)*b - I*b*c + I*a*d)/b)))*cos(-(b*c - a*d)/b) - (a*d^2*(I*exp_integral_e(2, (I*(d*x + c)*b - I*b*c
+ I*a*d)/b) - I*exp_integral_e(2, -(I*(d*x + c)*b - I*b*c + I*a*d)/b)) + b*c*d*(-I*exp_integral_e(2, (I*(d*x +
c)*b - I*b*c + I*a*d)/b) + I*exp_integral_e(2, -(I*(d*x + c)*b - I*b*c + I*a*d)/b)))*sin(-(b*c - a*d)/b))*cos
(d*x + c)^2 + ((d*x + c)*b*d*cos(d*x + c) - (b*c*d*(exp_integral_e(2, (I*(d*x + c)*b - I*b*c + I*a*d)/b) + exp
_integral_e(2, -(I*(d*x + c)*b - I*b*c + I*a*d)/b)) - a*d^2*(exp_integral_e(2, (I*(d*x + c)*b - I*b*c + I*a*d)
/b) + exp_integral_e(2, -(I*(d*x + c)*b - I*b*c + I*a*d)/b)))*cos(-(b*c - a*d)/b) + (a*d^2*(I*exp_integral_e(2
, (I*(d*x + c)*b - I*b*c + I*a*d)/b) - I*exp_integral_e(2, -(I*(d*x + c)*b - I*b*c + I*a*d)/b)) + b*c*d*(-I*ex
p_integral_e(2, (I*(d*x + c)*b - I*b*c + I*a*d)/b) + I*exp_integral_e(2, -(I*(d*x + c)*b - I*b*c + I*a*d)/b)))
*sin(-(b*c - a*d)/b))*sin(d*x + c)^2)/(((d*x + c)*b^2 - b^2*c + a*b*d)*cos(d*x + c)^2 + ((d*x + c)*b^2 - b^2*c
+ a*b*d)*sin(d*x + c)^2))/d^2

Giac [C] (verification not implemented)

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

Time = 0.32 (sec) , antiderivative size = 1647, normalized size of antiderivative = 23.87 \[ \int \frac {x \sin (c+d x)}{a+b x} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/2*(a*d*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - a*d*imag_part(co
s_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a*d*sin_integral((b*d*x + a*d)/b)*t
an(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a*d*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/
2*c)^2*tan(1/2*a*d/b) + 2*a*d*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)
- 2*a*d*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*a*d*real_part(cos
_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 - a*d*imag_part(cos_integral(d*x + a*d/b))
*tan(1/2*d*x)^2*tan(1/2*c)^2 + a*d*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*a*d*s
in_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)^2 + 4*a*d*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*
d*x)^2*tan(1/2*c)*tan(1/2*a*d/b) - 4*a*d*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1
/2*a*d/b) + 8*a*d*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b) - a*d*imag_part(cos_i
ntegral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + a*d*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)
^2*tan(1/2*a*d/b)^2 - 2*a*d*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + a*d*imag_part(cos_
integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - a*d*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2*
tan(1/2*a*d/b)^2 + 2*a*d*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*b*tan(1/2*d*x)^2*tan(
1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a*d*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) + 2*a*d*real_
part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) - 2*a*d*real_part(cos_integral(d*x + a*d/b))*tan(1/
2*d*x)^2*tan(1/2*a*d/b) - 2*a*d*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b) + 2*a*d*re
al_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b) + 2*a*d*real_part(cos_integral(-d*x - a*d/b))*t
an(1/2*c)^2*tan(1/2*a*d/b) - 2*a*d*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*a*d*re
al_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2 + a*d*imag_part(cos_integral(d*x + a*d/b))*tan
(1/2*d*x)^2 - a*d*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2 + 2*a*d*sin_integral((b*d*x + a*d)/b)*t
an(1/2*d*x)^2 - a*d*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2 + a*d*imag_part(cos_integral(-d*x - a*d/
b))*tan(1/2*c)^2 - 2*a*d*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)^2 + 2*b*tan(1/2*d*x)^2*tan(1/2*c)^2 + 4*a*d*
imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) - 4*a*d*imag_part(cos_integral(-d*x - a*d/b))*t
an(1/2*c)*tan(1/2*a*d/b) + 8*a*d*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)*tan(1/2*a*d/b) - a*d*imag_part(cos_i
ntegral(d*x + a*d/b))*tan(1/2*a*d/b)^2 + a*d*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b)^2 - 2*a*d*si
n_integral((b*d*x + a*d)/b)*tan(1/2*a*d/b)^2 - 2*b*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 - 8*b*tan(1/2*d*x)*tan(1/2*
c)*tan(1/2*a*d/b)^2 - 2*b*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a*d*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c
) + 2*a*d*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c) - 2*a*d*real_part(cos_integral(d*x + a*d/b))*tan(1/
2*a*d/b) - 2*a*d*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b) + a*d*imag_part(cos_integral(d*x + a*d/b
)) - a*d*imag_part(cos_integral(-d*x - a*d/b)) + 2*a*d*sin_integral((b*d*x + a*d)/b) - 2*b*tan(1/2*d*x)^2 - 8*
b*tan(1/2*d*x)*tan(1/2*c) - 2*b*tan(1/2*c)^2 + 2*b*tan(1/2*a*d/b)^2 + 2*b)/(b^2*d*tan(1/2*d*x)^2*tan(1/2*c)^2*
tan(1/2*a*d/b)^2 + b^2*d*tan(1/2*d*x)^2*tan(1/2*c)^2 + b^2*d*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + b^2*d*tan(1/2*c
)^2*tan(1/2*a*d/b)^2 + b^2*d*tan(1/2*d*x)^2 + b^2*d*tan(1/2*c)^2 + b^2*d*tan(1/2*a*d/b)^2 + b^2*d)

Mupad [F(-1)]

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

[In]

int((x*sin(c + d*x))/(a + b*x),x)

[Out]

int((x*sin(c + d*x))/(a + b*x), x)