3.19 \(\int \frac {x^3 \sin (c+d x)}{a+b x} \, dx\)

Optimal. Leaf size=152 \[ -\frac {a^3 \sin \left (c-\frac {a d}{b}\right ) \text {Ci}\left (x d+\frac {a d}{b}\right )}{b^4}-\frac {a^3 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{b^4}-\frac {a^2 \cos (c+d x)}{b^3 d}-\frac {a \sin (c+d x)}{b^2 d^2}+\frac {a x \cos (c+d x)}{b^2 d}+\frac {2 \cos (c+d x)}{b d^3}+\frac {2 x \sin (c+d x)}{b d^2}-\frac {x^2 \cos (c+d x)}{b d} \]

[Out]

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

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Rubi [A]  time = 0.31, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {6742, 2638, 3296, 2637, 3303, 3299, 3302} \[ -\frac {a^3 \sin \left (c-\frac {a d}{b}\right ) \text {CosIntegral}\left (\frac {a d}{b}+d x\right )}{b^4}-\frac {a^3 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{b^4}-\frac {a^2 \cos (c+d x)}{b^3 d}-\frac {a \sin (c+d x)}{b^2 d^2}+\frac {a x \cos (c+d x)}{b^2 d}+\frac {2 x \sin (c+d x)}{b d^2}+\frac {2 \cos (c+d x)}{b d^3}-\frac {x^2 \cos (c+d x)}{b d} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*Sin[c + d*x])/(a + b*x),x]

[Out]

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

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 2638

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

Rule 3296

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

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

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Mathematica [A]  time = 0.58, size = 117, normalized size = 0.77 \[ -\frac {a^3 d^3 \sin \left (c-\frac {a d}{b}\right ) \text {Ci}\left (d \left (\frac {a}{b}+x\right )\right )+a^3 d^3 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )+b \left (\left (a^2 d^2-a b d^2 x+b^2 \left (d^2 x^2-2\right )\right ) \cos (c+d x)+b d (a-2 b x) \sin (c+d x)\right )}{b^4 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.57, size = 167, normalized size = 1.10 \[ -\frac {2 \, a^{3} d^{3} \cos \left (-\frac {b c - a d}{b}\right ) \operatorname {Si}\left (\frac {b d x + a d}{b}\right ) + 2 \, {\left (b^{3} d^{2} x^{2} - a b^{2} d^{2} x + a^{2} b d^{2} - 2 \, b^{3}\right )} \cos \left (d x + c\right ) - 2 \, {\left (2 \, b^{3} d x - a b^{2} d\right )} \sin \left (d x + c\right ) - {\left (a^{3} d^{3} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) + a^{3} d^{3} \operatorname {Ci}\left (-\frac {b d x + a d}{b}\right )\right )} \sin \left (-\frac {b c - a d}{b}\right )}{2 \, b^{4} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [C]  time = 2.12, size = 2709, normalized size = 17.82 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*b^3*d^2*x^2*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - a^3*d^3*imag_part(cos_integral(d*x +
a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + a^3*d^3*imag_part(cos_integral(-d*x - a*d/b))*
tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 2*a^3*d^3*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x + 1
/2*c)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 2*a^3*d^3*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x + 1/2*c)^2*
tan(1/2*c)^2*tan(1/2*a*d/b) - 2*a^3*d^3*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c
)^2*tan(1/2*a*d/b) + 2*a^3*d^3*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)*tan(1/2*
a*d/b)^2 + 2*a^3*d^3*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)*tan(1/2*a*d/b)^2
- 2*a*b^2*d^2*x*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*b^3*d^2*x^2*tan(1/2*d*x + 1/2*c)^2*ta
n(1/2*c)^2 + a^3*d^3*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 - a^3*d^3*imag_p
art(cos_integral(-d*x - a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 + 2*a^3*d^3*sin_integral((b*d*x + a*d)/b)*
tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 - 4*a^3*d^3*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x + 1/2*c)^2*ta
n(1/2*c)*tan(1/2*a*d/b) + 4*a^3*d^3*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)*ta
n(1/2*a*d/b) - 8*a^3*d^3*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)*tan(1/2*a*d/b) + 2*b^
3*d^2*x^2*tan(1/2*d*x + 1/2*c)^2*tan(1/2*a*d/b)^2 + a^3*d^3*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x +
1/2*c)^2*tan(1/2*a*d/b)^2 - a^3*d^3*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*a*d/
b)^2 + 2*a^3*d^3*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x + 1/2*c)^2*tan(1/2*a*d/b)^2 - 2*b^3*d^2*x^2*tan(1/2
*c)^2*tan(1/2*a*d/b)^2 - a^3*d^3*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + a^3*d^3*
imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 2*a^3*d^3*sin_integral((b*d*x + a*d)/b)*
tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a^2*b*d^2*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 2*a^3*d^3*r
eal_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c) - 2*a^3*d^3*real_part(cos_integral(-d*x
- a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c) - 2*a*b^2*d^2*x*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 + 2*a^3*d^3*re
al_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*a*d/b) + 2*a^3*d^3*real_part(cos_integral(-d
*x - a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*a*d/b) - 2*a^3*d^3*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)
^2*tan(1/2*a*d/b) - 2*a^3*d^3*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b) - 2*a*b^2*d^2*
x*tan(1/2*d*x + 1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a^3*d^3*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*
a*d/b)^2 + 2*a^3*d^3*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2 + 2*a*b^2*d^2*x*tan(1/2
*c)^2*tan(1/2*a*d/b)^2 + 8*b^3*d*x*tan(1/2*d*x + 1/2*c)*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*b^3*d^2*x^2*tan(1/2*
d*x + 1/2*c)^2 - a^3*d^3*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x + 1/2*c)^2 + a^3*d^3*imag_part(cos_i
ntegral(-d*x - a*d/b))*tan(1/2*d*x + 1/2*c)^2 - 2*a^3*d^3*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x + 1/2*c)^2
- 2*b^3*d^2*x^2*tan(1/2*c)^2 + a^3*d^3*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2 - a^3*d^3*imag_part(
cos_integral(-d*x - a*d/b))*tan(1/2*c)^2 + 2*a^3*d^3*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)^2 + 2*a^2*b*d^2*
tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 - 4*a^3*d^3*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)
+ 4*a^3*d^3*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) - 8*a^3*d^3*sin_integral((b*d*x +
a*d)/b)*tan(1/2*c)*tan(1/2*a*d/b) - 2*b^3*d^2*x^2*tan(1/2*a*d/b)^2 + a^3*d^3*imag_part(cos_integral(d*x + a*d
/b))*tan(1/2*a*d/b)^2 - a^3*d^3*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b)^2 + 2*a^3*d^3*sin_integra
l((b*d*x + a*d)/b)*tan(1/2*a*d/b)^2 + 2*a^2*b*d^2*tan(1/2*d*x + 1/2*c)^2*tan(1/2*a*d/b)^2 - 2*a^2*b*d^2*tan(1/
2*c)^2*tan(1/2*a*d/b)^2 - 4*a*b^2*d*tan(1/2*d*x + 1/2*c)*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 4*b^3*tan(1/2*d*x + 1
/2*c)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 2*a*b^2*d^2*x*tan(1/2*d*x + 1/2*c)^2 - 2*a^3*d^3*real_part(cos_integra
l(d*x + a*d/b))*tan(1/2*c) - 2*a^3*d^3*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c) + 2*a*b^2*d^2*x*tan(1/
2*c)^2 + 8*b^3*d*x*tan(1/2*d*x + 1/2*c)*tan(1/2*c)^2 + 2*a^3*d^3*real_part(cos_integral(d*x + a*d/b))*tan(1/2*
a*d/b) + 2*a^3*d^3*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b) + 2*a*b^2*d^2*x*tan(1/2*a*d/b)^2 + 8*b
^3*d*x*tan(1/2*d*x + 1/2*c)*tan(1/2*a*d/b)^2 - 2*b^3*d^2*x^2 - a^3*d^3*imag_part(cos_integral(d*x + a*d/b)) +
a^3*d^3*imag_part(cos_integral(-d*x - a*d/b)) - 2*a^3*d^3*sin_integral((b*d*x + a*d)/b) + 2*a^2*b*d^2*tan(1/2*
d*x + 1/2*c)^2 - 2*a^2*b*d^2*tan(1/2*c)^2 - 4*a*b^2*d*tan(1/2*d*x + 1/2*c)*tan(1/2*c)^2 - 4*b^3*tan(1/2*d*x +
1/2*c)^2*tan(1/2*c)^2 - 2*a^2*b*d^2*tan(1/2*a*d/b)^2 - 4*a*b^2*d*tan(1/2*d*x + 1/2*c)*tan(1/2*a*d/b)^2 - 4*b^3
*tan(1/2*d*x + 1/2*c)^2*tan(1/2*a*d/b)^2 + 4*b^3*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a*b^2*d^2*x + 8*b^3*d*x*tan
(1/2*d*x + 1/2*c) - 2*a^2*b*d^2 - 4*a*b^2*d*tan(1/2*d*x + 1/2*c) - 4*b^3*tan(1/2*d*x + 1/2*c)^2 + 4*b^3*tan(1/
2*c)^2 + 4*b^3*tan(1/2*a*d/b)^2 + 4*b^3)/(b^4*d^3*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + b^4*d
^3*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 + b^4*d^3*tan(1/2*d*x + 1/2*c)^2*tan(1/2*a*d/b)^2 + b^4*d^3*tan(1/2*c)^
2*tan(1/2*a*d/b)^2 + b^4*d^3*tan(1/2*d*x + 1/2*c)^2 + b^4*d^3*tan(1/2*c)^2 + b^4*d^3*tan(1/2*a*d/b)^2 + b^4*d^
3)

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maple [B]  time = 0.03, size = 514, normalized size = 3.38 \[ \frac {-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) d \left (\frac {\Si \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\Ci \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{b^{3}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}-a b d +b^{2} c +b^{2}\right ) d \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{b^{3}}-\frac {3 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) d c \left (\frac {\Si \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\Ci \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{b^{2}}-\frac {3 d c \left (-d a +c b +b \right ) \left (\sin \left (d x +c \right )-\left (d x +c \right ) \cos \left (d x +c \right )\right )}{b^{2}}-\frac {3 \left (d a -c b \right ) d \,c^{2} \left (\frac {\Si \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\Ci \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{b}-\frac {3 d \,c^{2} \cos \left (d x +c \right )}{b}-d \,c^{3} \left (\frac {\Si \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\Ci \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d^4*(-(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*d/b^3*(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)+(a^2*d^2-2*a*b*c*d+b^2*c^2-a*b*d+b^2*c+b^2)*d/b^3*(-(d*x+c)^2*cos(d*x+c)+2*c
os(d*x+c)+2*(d*x+c)*sin(d*x+c))-3*(a^2*d^2-2*a*b*c*d+b^2*c^2)*d*c/b^2*(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)-3*d*c*(-a*d+b*c+b)/b^2*(sin(d*x+c)-(d*x+c)*cos(d*x+c))-3*(a*d-b*c)
*d*c^2/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)-3*d*c^2/b*cos(d*x
+c)-d*c^3*(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 [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^3\,\sin \left (c+d\,x\right )}{a+b\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*sin(c + d*x))/(a + b*x),x)

[Out]

int((x^3*sin(c + d*x))/(a + b*x), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \sin {\left (c + d x \right )}}{a + b x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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