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

Optimal. Leaf size=69 \[ -\frac {a \sin \left (c-\frac {a d}{b}\right ) \text {Ci}\left (x d+\frac {a d}{b}\right )}{b^2}-\frac {a \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{b^2}-\frac {\cos (c+d x)}{b d} \]

[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

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Rubi [A]  time = 0.17, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6742, 2638, 3303, 3299, 3302} \[ -\frac {a \sin \left (c-\frac {a d}{b}\right ) \text {CosIntegral}\left (\frac {a d}{b}+d x\right )}{b^2}-\frac {a \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{b^2}-\frac {\cos (c+d x)}{b d} \]

Antiderivative was successfully verified.

[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 2638

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

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]

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 6742

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

Rubi steps

\begin {align*} \int \frac {x \sin (c+d x)}{a+b x} \, dx &=\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 \text {Ci}\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*}

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Mathematica [A]  time = 0.20, size = 63, normalized size = 0.91 \[ -\frac {a d \sin \left (c-\frac {a d}{b}\right ) \text {Ci}\left (d \left (\frac {a}{b}+x\right )\right )+a d \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )+b \cos (c+d x)}{b^2 d} \]

Antiderivative was successfully verified.

[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))

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fricas [A]  time = 0.64, size = 99, normalized size = 1.43 \[ -\frac {2 \, a d \cos \left (-\frac {b c - a d}{b}\right ) \operatorname {Si}\left (\frac {b d x + a d}{b}\right ) + 2 \, b \cos \left (d x + c\right ) - {\left (a d \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) + a d \operatorname {Ci}\left (-\frac {b d x + a d}{b}\right )\right )} \sin \left (-\frac {b c - a d}{b}\right )}{2 \, b^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [C]  time = 0.84, size = 1647, normalized size = 23.87 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [B]  time = 0.02, size = 180, normalized size = 2.61 \[ \frac {-\frac {\left (d a -c b \right ) d \left (\frac {\Si \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\Ci \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{b}-\frac {d \cos \left (d x +c \right )}{b}-d c \left (\frac {\Si \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\Ci \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [C]  time = 0.53, size = 776, normalized size = 11.25 \[ -\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}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,\sin \left (c+d\,x\right )}{a+b\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*sin(c + d*x))/(a + b*x),x)

[Out]

int((x*sin(c + d*x))/(a + b*x), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \sin {\left (c + d x \right )}}{a + b x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*sin(c + d*x)/(a + b*x), x)

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