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

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

[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

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Rubi [A]  time = 0.464447, antiderivative size = 218, normalized size of antiderivative = 1., 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 6742

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

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 2637

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

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 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]

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*}

Mathematica [A]  time = 0.676357, size = 158, normalized size = 0.72 \[ \frac{b \left (b \left (a^2 d^2-2 a b d^2 x+3 b^2 \left (d^2 x^2-2\right )\right ) \sin (c+d x)+d \left (-a^2 b d^2 x+a^3 d^2+a b^2 \left (d^2 x^2-2\right )+b^3 x \left (6-d^2 x^2\right )\right ) \cos (c+d x)\right )+a^4 d^4 \sin \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (d \left (\frac{a}{b}+x\right )\right )+a^4 d^4 \cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (d \left (\frac{a}{b}+x\right )\right )}{b^5 d^4} \]

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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Maple [B]  time = 0.013, size = 777, normalized size = 3.6 \begin{align*} \text{result too large to display} \end{align*}

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^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))+(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)-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*cos(d*x+c)+2*cos(d*x+c)+2*(d*x+
c)*sin(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)+6*(-a*d+b*c+b)*d*c^2/b^2*(sin(d*x+c)-(d*x+c)*cos(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)*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)+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)+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., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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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Fricas [A]  time = 1.73244, size = 473, normalized size = 2.17 \begin{align*} \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}} \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4} \sin{\left (c + d x \right )}}{a + b x}\, dx \end{align*}

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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Giac [C]  time = 1.15967, size = 911, normalized size = 4.18 \begin{align*} \text{result too large to display} \end{align*}

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*(a^4*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - a^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*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)^2*tan(1/2*a*d/b)^2 +
2*a^4*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b) + 2*a^4*real_part(cos_integral(-d*x - a
*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b) - 2*a^4*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2 -
2*a^4*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2 - a^4*imag_part(cos_integral(d*x + a*d
/b))*tan(1/2*c)^2 + a^4*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2 - 2*a^4*sin_integral((b*d*x + a*d)/
b)*tan(1/2*c)^2 + 4*a^4*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) - 4*a^4*imag_part(cos_i
ntegral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) + 8*a^4*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)*tan(1/2*a*d/
b) - a^4*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*a*d/b)^2 + a^4*imag_part(cos_integral(-d*x - a*d/b))*tan
(1/2*a*d/b)^2 - 2*a^4*sin_integral((b*d*x + a*d)/b)*tan(1/2*a*d/b)^2 + 2*a^4*real_part(cos_integral(d*x + a*d/
b))*tan(1/2*c) + 2*a^4*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c) - 2*a^4*real_part(cos_integral(d*x + a
*d/b))*tan(1/2*a*d/b) - 2*a^4*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b) + a^4*imag_part(cos_integra
l(d*x + a*d/b)) - a^4*imag_part(cos_integral(-d*x - a*d/b)) + 2*a^4*sin_integral((b*d*x + a*d)/b))/(b^5*tan(1/
2*c)^2*tan(1/2*a*d/b)^2 + b^5*tan(1/2*c)^2 + b^5*tan(1/2*a*d/b)^2 + b^5)