3.18 \(\int \frac {x^4 \sin (c+d x)}{a+b x} \, dx\)

Optimal. Leaf size=218 \[ \frac {a^4 \sin \left (c-\frac {a d}{b}\right ) \text {Ci}\left (x d+\frac {a d}{b}\right )}{b^5}+\frac {a^4 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{b^5}+\frac {a^3 \cos (c+d x)}{b^4 d}+\frac {a^2 \sin (c+d x)}{b^3 d^2}-\frac {a^2 x \cos (c+d x)}{b^3 d}-\frac {2 a \cos (c+d x)}{b^2 d^3}-\frac {2 a x \sin (c+d x)}{b^2 d^2}+\frac {a x^2 \cos (c+d x)}{b^2 d}-\frac {6 \sin (c+d x)}{b d^4}+\frac {6 x \cos (c+d x)}{b d^3}+\frac {3 x^2 \sin (c+d x)}{b d^2}-\frac {x^3 \cos (c+d x)}{b d} \]

[Out]

-2*a*cos(d*x+c)/b^2/d^3+a^3*cos(d*x+c)/b^4/d+6*x*cos(d*x+c)/b/d^3-a^2*x*cos(d*x+c)/b^3/d+a*x^2*cos(d*x+c)/b^2/
d-x^3*cos(d*x+c)/b/d+a^4*cos(-c+a*d/b)*Si(a*d/b+d*x)/b^5-a^4*Ci(a*d/b+d*x)*sin(-c+a*d/b)/b^5-6*sin(d*x+c)/b/d^
4+a^2*sin(d*x+c)/b^3/d^2-2*a*x*sin(d*x+c)/b^2/d^2+3*x^2*sin(d*x+c)/b/d^2

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Rubi [A]  time = 0.46, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 15, 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} \[ \frac {a^4 \sin \left (c-\frac {a d}{b}\right ) \text {CosIntegral}\left (\frac {a d}{b}+d x\right )}{b^5}+\frac {a^2 \sin (c+d x)}{b^3 d^2}+\frac {a^4 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{b^5}+\frac {a^3 \cos (c+d x)}{b^4 d}-\frac {a^2 x \cos (c+d x)}{b^3 d}-\frac {2 a x \sin (c+d x)}{b^2 d^2}-\frac {2 a \cos (c+d x)}{b^2 d^3}+\frac {a x^2 \cos (c+d x)}{b^2 d}+\frac {3 x^2 \sin (c+d x)}{b d^2}-\frac {6 \sin (c+d x)}{b d^4}+\frac {6 x \cos (c+d x)}{b d^3}-\frac {x^3 \cos (c+d x)}{b d} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*Sin[c + d*x])/(a + b*x),x]

[Out]

(-2*a*Cos[c + d*x])/(b^2*d^3) + (a^3*Cos[c + d*x])/(b^4*d) + (6*x*Cos[c + d*x])/(b*d^3) - (a^2*x*Cos[c + d*x])
/(b^3*d) + (a*x^2*Cos[c + d*x])/(b^2*d) - (x^3*Cos[c + d*x])/(b*d) + (a^4*CosIntegral[(a*d)/b + d*x]*Sin[c - (
a*d)/b])/b^5 - (6*Sin[c + d*x])/(b*d^4) + (a^2*Sin[c + d*x])/(b^3*d^2) - (2*a*x*Sin[c + d*x])/(b^2*d^2) + (3*x
^2*Sin[c + d*x])/(b*d^2) + (a^4*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/b^5

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^4 \sin (c+d x)}{a+b x} \, dx &=\int \left (-\frac {a^3 \sin (c+d x)}{b^4}+\frac {a^2 x \sin (c+d x)}{b^3}-\frac {a x^2 \sin (c+d x)}{b^2}+\frac {x^3 \sin (c+d x)}{b}+\frac {a^4 \sin (c+d x)}{b^4 (a+b x)}\right ) \, dx\\ &=-\frac {a^3 \int \sin (c+d x) \, dx}{b^4}+\frac {a^4 \int \frac {\sin (c+d x)}{a+b x} \, dx}{b^4}+\frac {a^2 \int x \sin (c+d x) \, dx}{b^3}-\frac {a \int x^2 \sin (c+d x) \, dx}{b^2}+\frac {\int x^3 \sin (c+d x) \, dx}{b}\\ &=\frac {a^3 \cos (c+d x)}{b^4 d}-\frac {a^2 x \cos (c+d x)}{b^3 d}+\frac {a x^2 \cos (c+d x)}{b^2 d}-\frac {x^3 \cos (c+d x)}{b d}+\frac {a^2 \int \cos (c+d x) \, dx}{b^3 d}-\frac {(2 a) \int x \cos (c+d x) \, dx}{b^2 d}+\frac {3 \int x^2 \cos (c+d x) \, dx}{b d}+\frac {\left (a^4 \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^4}+\frac {\left (a^4 \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^4}\\ &=\frac {a^3 \cos (c+d x)}{b^4 d}-\frac {a^2 x \cos (c+d x)}{b^3 d}+\frac {a x^2 \cos (c+d x)}{b^2 d}-\frac {x^3 \cos (c+d x)}{b d}+\frac {a^4 \text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{b^5}+\frac {a^2 \sin (c+d x)}{b^3 d^2}-\frac {2 a x \sin (c+d x)}{b^2 d^2}+\frac {3 x^2 \sin (c+d x)}{b d^2}+\frac {a^4 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^5}+\frac {(2 a) \int \sin (c+d x) \, dx}{b^2 d^2}-\frac {6 \int x \sin (c+d x) \, dx}{b d^2}\\ &=-\frac {2 a \cos (c+d x)}{b^2 d^3}+\frac {a^3 \cos (c+d x)}{b^4 d}+\frac {6 x \cos (c+d x)}{b d^3}-\frac {a^2 x \cos (c+d x)}{b^3 d}+\frac {a x^2 \cos (c+d x)}{b^2 d}-\frac {x^3 \cos (c+d x)}{b d}+\frac {a^4 \text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{b^5}+\frac {a^2 \sin (c+d x)}{b^3 d^2}-\frac {2 a x \sin (c+d x)}{b^2 d^2}+\frac {3 x^2 \sin (c+d x)}{b d^2}+\frac {a^4 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^5}-\frac {6 \int \cos (c+d x) \, dx}{b d^3}\\ &=-\frac {2 a \cos (c+d x)}{b^2 d^3}+\frac {a^3 \cos (c+d x)}{b^4 d}+\frac {6 x \cos (c+d x)}{b d^3}-\frac {a^2 x \cos (c+d x)}{b^3 d}+\frac {a x^2 \cos (c+d x)}{b^2 d}-\frac {x^3 \cos (c+d x)}{b d}+\frac {a^4 \text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{b^5}-\frac {6 \sin (c+d x)}{b d^4}+\frac {a^2 \sin (c+d x)}{b^3 d^2}-\frac {2 a x \sin (c+d x)}{b^2 d^2}+\frac {3 x^2 \sin (c+d x)}{b d^2}+\frac {a^4 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^5}\\ \end {align*}

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Mathematica [A]  time = 0.69, size = 158, normalized size = 0.72 \[ \frac {a^4 d^4 \sin \left (c-\frac {a d}{b}\right ) \text {Ci}\left (d \left (\frac {a}{b}+x\right )\right )+a^4 d^4 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )+b \left (b \left (a^2 d^2-2 a b d^2 x+3 b^2 \left (d^2 x^2-2\right )\right ) \sin (c+d x)+d \left (a^3 d^2-a^2 b d^2 x+a b^2 \left (d^2 x^2-2\right )+b^3 x \left (6-d^2 x^2\right )\right ) \cos (c+d x)\right )}{b^5 d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*d^4*CosIntegral[d*(a/b + x)]*Sin[c - (a*d)/b] + b*(d*(a^3*d^2 - a^2*b*d^2*x + b^3*x*(6 - d^2*x^2) + a*b^2
*(-2 + d^2*x^2))*Cos[c + d*x] + b*(a^2*d^2 - 2*a*b*d^2*x + 3*b^2*(-2 + d^2*x^2))*Sin[c + d*x]) + a^4*d^4*Cos[c
- (a*d)/b]*SinIntegral[d*(a/b + x)])/(b^5*d^4)

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fricas [A]  time = 0.92, size = 213, normalized size = 0.98 \[ \frac {2 \, a^{4} d^{4} \cos \left (-\frac {b c - a d}{b}\right ) \operatorname {Si}\left (\frac {b d x + a d}{b}\right ) - 2 \, {\left (b^{4} d^{3} x^{3} - a b^{3} d^{3} x^{2} - a^{3} b d^{3} + 2 \, a b^{3} d + {\left (a^{2} b^{2} d^{3} - 6 \, b^{4} d\right )} x\right )} \cos \left (d x + c\right ) + 2 \, {\left (3 \, b^{4} d^{2} x^{2} - 2 \, a b^{3} d^{2} x + a^{2} b^{2} d^{2} - 6 \, b^{4}\right )} \sin \left (d x + c\right ) - {\left (a^{4} d^{4} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) + a^{4} d^{4} \operatorname {Ci}\left (-\frac {b d x + a d}{b}\right )\right )} \sin \left (-\frac {b c - a d}{b}\right )}{2 \, b^{5} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*sin(d*x+c)/(b*x+a),x, algorithm="fricas")

[Out]

1/2*(2*a^4*d^4*cos(-(b*c - a*d)/b)*sin_integral((b*d*x + a*d)/b) - 2*(b^4*d^3*x^3 - a*b^3*d^3*x^2 - a^3*b*d^3
+ 2*a*b^3*d + (a^2*b^2*d^3 - 6*b^4*d)*x)*cos(d*x + c) + 2*(3*b^4*d^2*x^2 - 2*a*b^3*d^2*x + a^2*b^2*d^2 - 6*b^4
)*sin(d*x + c) - (a^4*d^4*cos_integral((b*d*x + a*d)/b) + a^4*d^4*cos_integral(-(b*d*x + a*d)/b))*sin(-(b*c -
a*d)/b))/(b^5*d^4)

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giac [C]  time = 2.51, size = 3337, normalized size = 15.31 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*sin(d*x+c)/(b*x+a),x, algorithm="giac")

[Out]

1/2*(2*b^4*d^3*x^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^3*d^3*x^2*tan(1/2*d*x + 1/2*c)
^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + a^4*d^4*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^4*d^4*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^4*d^4*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*b^4*d^3*x^3*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 + 2*a^4*d^4*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^4*d^4*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*b^4*d^3*x^3*tan(1/2*d*x + 1/2*c)^2*tan(1/2*a*d/b)^2 - 2*a^4*d^4*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^4*d^4*real_part(cos_integr
al(-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*b^4*d^3*x^3*tan(1/2*c)^2*tan(1/2*a*d/
b)^2 + 2*a^2*b^2*d^3*x*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 2*a*b^3*d^3*x^2*tan(1/2*d*x + 1/
2*c)^2*tan(1/2*c)^2 - a^4*d^4*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 + a^4*d
^4*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 - 2*a^4*d^4*sin_integral((b*d*x +
a*d)/b)*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 + 4*a^4*d^4*imag_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) - 4*a^4*d^4*imag_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) + 8*a^4*d^4*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*a*b^3*d^3*x^2*tan(1/2*d*x + 1/2*c)^2*tan(1/2*a*d/b)^2 - a^4*d^4*imag_part(cos_integral(d*x + a*d/b))*ta
n(1/2*d*x + 1/2*c)^2*tan(1/2*a*d/b)^2 + a^4*d^4*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x + 1/2*c)^2*t
an(1/2*a*d/b)^2 - 2*a^4*d^4*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*a*b^3*d^
3*x^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + a^4*d^4*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)
^2 - a^4*d^4*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a^4*d^4*sin_integral((b*d
*x + a*d)/b)*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 12*b^4*d^2*x^2*tan(1/2*d*x + 1/2*c)*tan(1/2*c)^2*tan(1/2*a*d/b)^2
- 2*a^3*b*d^3*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*b^4*d^3*x^3*tan(1/2*d*x + 1/2*c)^2 + 2
*a^4*d^4*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c) + 2*a^4*d^4*real_part(cos_inte
gral(-d*x - a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c) - 2*b^4*d^3*x^3*tan(1/2*c)^2 + 2*a^2*b^2*d^3*x*tan(1/2*d
*x + 1/2*c)^2*tan(1/2*c)^2 - 2*a^4*d^4*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^4*d^4*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^4*d^4*real_p
art(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b) + 2*a^4*d^4*real_part(cos_integral(-d*x - a*d/b))*t
an(1/2*c)^2*tan(1/2*a*d/b) - 2*b^4*d^3*x^3*tan(1/2*a*d/b)^2 + 2*a^2*b^2*d^3*x*tan(1/2*d*x + 1/2*c)^2*tan(1/2*a
*d/b)^2 - 2*a^4*d^4*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*a^4*d^4*real_part(cos
_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*a^2*b^2*d^3*x*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 8*a*b^3
*d^2*x*tan(1/2*d*x + 1/2*c)*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 12*b^4*d*x*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2*tan
(1/2*a*d/b)^2 - 2*a*b^3*d^3*x^2*tan(1/2*d*x + 1/2*c)^2 + a^4*d^4*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*
d*x + 1/2*c)^2 - a^4*d^4*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x + 1/2*c)^2 + 2*a^4*d^4*sin_integral
((b*d*x + a*d)/b)*tan(1/2*d*x + 1/2*c)^2 + 2*a*b^3*d^3*x^2*tan(1/2*c)^2 - a^4*d^4*imag_part(cos_integral(d*x +
a*d/b))*tan(1/2*c)^2 + a^4*d^4*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2 - 2*a^4*d^4*sin_integral((b
*d*x + a*d)/b)*tan(1/2*c)^2 + 12*b^4*d^2*x^2*tan(1/2*d*x + 1/2*c)*tan(1/2*c)^2 - 2*a^3*b*d^3*tan(1/2*d*x + 1/2
*c)^2*tan(1/2*c)^2 + 4*a^4*d^4*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) - 4*a^4*d^4*imag
_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) + 8*a^4*d^4*sin_integral((b*d*x + a*d)/b)*tan(1/2*
c)*tan(1/2*a*d/b) + 2*a*b^3*d^3*x^2*tan(1/2*a*d/b)^2 - a^4*d^4*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*a*
d/b)^2 + a^4*d^4*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b)^2 - 2*a^4*d^4*sin_integral((b*d*x + a*d)
/b)*tan(1/2*a*d/b)^2 + 12*b^4*d^2*x^2*tan(1/2*d*x + 1/2*c)*tan(1/2*a*d/b)^2 - 2*a^3*b*d^3*tan(1/2*d*x + 1/2*c)
^2*tan(1/2*a*d/b)^2 + 2*a^3*b*d^3*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 4*a^2*b^2*d^2*tan(1/2*d*x + 1/2*c)*tan(1/2*c
)^2*tan(1/2*a*d/b)^2 + 4*a*b^3*d*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 2*b^4*d^3*x^3 + 2*a^2*
b^2*d^3*x*tan(1/2*d*x + 1/2*c)^2 + 2*a^4*d^4*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c) + 2*a^4*d^4*real_
part(cos_integral(-d*x - a*d/b))*tan(1/2*c) - 2*a^2*b^2*d^3*x*tan(1/2*c)^2 - 8*a*b^3*d^2*x*tan(1/2*d*x + 1/2*c
)*tan(1/2*c)^2 - 12*b^4*d*x*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 - 2*a^4*d^4*real_part(cos_integral(d*x + a*d/b
))*tan(1/2*a*d/b) - 2*a^4*d^4*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b) - 2*a^2*b^2*d^3*x*tan(1/2*a
*d/b)^2 - 8*a*b^3*d^2*x*tan(1/2*d*x + 1/2*c)*tan(1/2*a*d/b)^2 - 12*b^4*d*x*tan(1/2*d*x + 1/2*c)^2*tan(1/2*a*d/
b)^2 + 12*b^4*d*x*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a*b^3*d^3*x^2 + a^4*d^4*imag_part(cos_integral(d*x + a*d/b
)) - a^4*d^4*imag_part(cos_integral(-d*x - a*d/b)) + 2*a^4*d^4*sin_integral((b*d*x + a*d)/b) + 12*b^4*d^2*x^2*
tan(1/2*d*x + 1/2*c) - 2*a^3*b*d^3*tan(1/2*d*x + 1/2*c)^2 + 2*a^3*b*d^3*tan(1/2*c)^2 + 4*a^2*b^2*d^2*tan(1/2*d
*x + 1/2*c)*tan(1/2*c)^2 + 4*a*b^3*d*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 + 2*a^3*b*d^3*tan(1/2*a*d/b)^2 + 4*a^
2*b^2*d^2*tan(1/2*d*x + 1/2*c)*tan(1/2*a*d/b)^2 + 4*a*b^3*d*tan(1/2*d*x + 1/2*c)^2*tan(1/2*a*d/b)^2 - 4*a*b^3*
d*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 24*b^4*tan(1/2*d*x + 1/2*c)*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 2*a^2*b^2*d^3*x
- 8*a*b^3*d^2*x*tan(1/2*d*x + 1/2*c) - 12*b^4*d*x*tan(1/2*d*x + 1/2*c)^2 + 12*b^4*d*x*tan(1/2*c)^2 + 12*b^4*d*
x*tan(1/2*a*d/b)^2 + 2*a^3*b*d^3 + 4*a^2*b^2*d^2*tan(1/2*d*x + 1/2*c) + 4*a*b^3*d*tan(1/2*d*x + 1/2*c)^2 - 4*a
*b^3*d*tan(1/2*c)^2 - 24*b^4*tan(1/2*d*x + 1/2*c)*tan(1/2*c)^2 - 4*a*b^3*d*tan(1/2*a*d/b)^2 - 24*b^4*tan(1/2*d
*x + 1/2*c)*tan(1/2*a*d/b)^2 + 12*b^4*d*x - 4*a*b^3*d - 24*b^4*tan(1/2*d*x + 1/2*c))/(b^5*d^4*tan(1/2*d*x + 1/
2*c)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + b^5*d^4*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 + b^5*d^4*tan(1/2*d*x + 1/2
*c)^2*tan(1/2*a*d/b)^2 + b^5*d^4*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + b^5*d^4*tan(1/2*d*x + 1/2*c)^2 + b^5*d^4*tan(
1/2*c)^2 + b^5*d^4*tan(1/2*a*d/b)^2 + b^5*d^4)

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maple [B]  time = 0.03, size = 777, normalized size = 3.56 \[ \frac {\frac {\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\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^{4}}+\frac {\left (-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}+a^{2} b \,d^{2}-2 a \,b^{2} c d +b^{3} c^{2}-a \,b^{2} d +b^{3} c +b^{3}\right ) d \left (-\left (d x +c \right )^{3} \cos \left (d x +c \right )+3 \left (d x +c \right )^{2} \sin \left (d x +c \right )-6 \sin \left (d x +c \right )+6 \left (d x +c \right ) \cos \left (d x +c \right )\right )}{b^{4}}+\frac {4 \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 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^{3}}-\frac {4 d c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}-a b d +b^{2} c +b^{2}\right ) \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 {6 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\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^{2}}+\frac {6 \left (-d a +c b +b \right ) d \,c^{2} \left (\sin \left (d x +c \right )-\left (d x +c \right ) \cos \left (d x +c \right )\right )}{b^{2}}+\frac {4 \left (d a -c b \right ) 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 )}{b}+\frac {4 d \,c^{3} \cos \left (d x +c \right )}{b}+d \,c^{4} \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^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d^5*((a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)*d/b^4*(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^3*d^3+3*a^2*b*c*d^2-3*a*b^2*c^2*d+b^3*c^3+a^2*b*d^2-2*a
*b^2*c*d+b^3*c^2-a*b^2*d+b^3*c+b^3)*d/b^4*(-(d*x+c)^3*cos(d*x+c)+3*(d*x+c)^2*sin(d*x+c)-6*sin(d*x+c)+6*(d*x+c)
*cos(d*x+c))+4*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*d*c/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)-4*d*c*(a^2*d^2-2*a*b*c*d+b^2*c^2-a*b*d+b^2*c+b^2)/b^3*(-(d*x+c)^2*c
os(d*x+c)+2*cos(d*x+c)+2*(d*x+c)*sin(d*x+c))+6*(a^2*d^2-2*a*b*c*d+b^2*c^2)*d*c^2/b^2*(Si(d*x+c+(a*d-b*c)/b)*co
s((a*d-b*c)/b)/b-Ci(d*x+c+(a*d-b*c)/b)*sin((a*d-b*c)/b)/b)+6*(-a*d+b*c+b)*d*c^2/b^2*(sin(d*x+c)-(d*x+c)*cos(d*
x+c))+4*(a*d-b*c)*d*c^3/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)+
4*d*c^3/b*cos(d*x+c)+d*c^4*(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^4*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 = -0.00 \[ \int \frac {x^4\,\sin \left (c+d\,x\right )}{a+b\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4*sin(c + d*x))/(a + b*x),x)

[Out]

int((x^4*sin(c + d*x))/(a + b*x), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4} \sin {\left (c + d x \right )}}{a + b x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*sin(d*x+c)/(b*x+a),x)

[Out]

Integral(x**4*sin(c + d*x)/(a + b*x), x)

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