3.9 \(\int \frac{(a+b x) \sin (c+d x)}{x^5} \, dx\)

Optimal. Leaf size=166 \[ \frac{1}{24} a d^4 \sin (c) \text{CosIntegral}(d x)+\frac{1}{24} a d^4 \cos (c) \text{Si}(d x)+\frac{a d^2 \sin (c+d x)}{24 x^2}+\frac{a d^3 \cos (c+d x)}{24 x}-\frac{a \sin (c+d x)}{4 x^4}-\frac{a d \cos (c+d x)}{12 x^3}-\frac{1}{6} b d^3 \cos (c) \text{CosIntegral}(d x)+\frac{1}{6} b d^3 \sin (c) \text{Si}(d x)+\frac{b d^2 \sin (c+d x)}{6 x}-\frac{b \sin (c+d x)}{3 x^3}-\frac{b d \cos (c+d x)}{6 x^2} \]

[Out]

-(a*d*Cos[c + d*x])/(12*x^3) - (b*d*Cos[c + d*x])/(6*x^2) + (a*d^3*Cos[c + d*x])/(24*x) - (b*d^3*Cos[c]*CosInt
egral[d*x])/6 + (a*d^4*CosIntegral[d*x]*Sin[c])/24 - (a*Sin[c + d*x])/(4*x^4) - (b*Sin[c + d*x])/(3*x^3) + (a*
d^2*Sin[c + d*x])/(24*x^2) + (b*d^2*Sin[c + d*x])/(6*x) + (a*d^4*Cos[c]*SinIntegral[d*x])/24 + (b*d^3*Sin[c]*S
inIntegral[d*x])/6

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Rubi [A]  time = 0.368483, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6742, 3297, 3303, 3299, 3302} \[ \frac{1}{24} a d^4 \sin (c) \text{CosIntegral}(d x)+\frac{1}{24} a d^4 \cos (c) \text{Si}(d x)+\frac{a d^2 \sin (c+d x)}{24 x^2}+\frac{a d^3 \cos (c+d x)}{24 x}-\frac{a \sin (c+d x)}{4 x^4}-\frac{a d \cos (c+d x)}{12 x^3}-\frac{1}{6} b d^3 \cos (c) \text{CosIntegral}(d x)+\frac{1}{6} b d^3 \sin (c) \text{Si}(d x)+\frac{b d^2 \sin (c+d x)}{6 x}-\frac{b \sin (c+d x)}{3 x^3}-\frac{b d \cos (c+d x)}{6 x^2} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*Sin[c + d*x])/x^5,x]

[Out]

-(a*d*Cos[c + d*x])/(12*x^3) - (b*d*Cos[c + d*x])/(6*x^2) + (a*d^3*Cos[c + d*x])/(24*x) - (b*d^3*Cos[c]*CosInt
egral[d*x])/6 + (a*d^4*CosIntegral[d*x]*Sin[c])/24 - (a*Sin[c + d*x])/(4*x^4) - (b*Sin[c + d*x])/(3*x^3) + (a*
d^2*Sin[c + d*x])/(24*x^2) + (b*d^2*Sin[c + d*x])/(6*x) + (a*d^4*Cos[c]*SinIntegral[d*x])/24 + (b*d^3*Sin[c]*S
inIntegral[d*x])/6

Rule 6742

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

Rule 3297

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

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{(a+b x) \sin (c+d x)}{x^5} \, dx &=\int \left (\frac{a \sin (c+d x)}{x^5}+\frac{b \sin (c+d x)}{x^4}\right ) \, dx\\ &=a \int \frac{\sin (c+d x)}{x^5} \, dx+b \int \frac{\sin (c+d x)}{x^4} \, dx\\ &=-\frac{a \sin (c+d x)}{4 x^4}-\frac{b \sin (c+d x)}{3 x^3}+\frac{1}{4} (a d) \int \frac{\cos (c+d x)}{x^4} \, dx+\frac{1}{3} (b d) \int \frac{\cos (c+d x)}{x^3} \, dx\\ &=-\frac{a d \cos (c+d x)}{12 x^3}-\frac{b d \cos (c+d x)}{6 x^2}-\frac{a \sin (c+d x)}{4 x^4}-\frac{b \sin (c+d x)}{3 x^3}-\frac{1}{12} \left (a d^2\right ) \int \frac{\sin (c+d x)}{x^3} \, dx-\frac{1}{6} \left (b d^2\right ) \int \frac{\sin (c+d x)}{x^2} \, dx\\ &=-\frac{a d \cos (c+d x)}{12 x^3}-\frac{b d \cos (c+d x)}{6 x^2}-\frac{a \sin (c+d x)}{4 x^4}-\frac{b \sin (c+d x)}{3 x^3}+\frac{a d^2 \sin (c+d x)}{24 x^2}+\frac{b d^2 \sin (c+d x)}{6 x}-\frac{1}{24} \left (a d^3\right ) \int \frac{\cos (c+d x)}{x^2} \, dx-\frac{1}{6} \left (b d^3\right ) \int \frac{\cos (c+d x)}{x} \, dx\\ &=-\frac{a d \cos (c+d x)}{12 x^3}-\frac{b d \cos (c+d x)}{6 x^2}+\frac{a d^3 \cos (c+d x)}{24 x}-\frac{a \sin (c+d x)}{4 x^4}-\frac{b \sin (c+d x)}{3 x^3}+\frac{a d^2 \sin (c+d x)}{24 x^2}+\frac{b d^2 \sin (c+d x)}{6 x}+\frac{1}{24} \left (a d^4\right ) \int \frac{\sin (c+d x)}{x} \, dx-\frac{1}{6} \left (b d^3 \cos (c)\right ) \int \frac{\cos (d x)}{x} \, dx+\frac{1}{6} \left (b d^3 \sin (c)\right ) \int \frac{\sin (d x)}{x} \, dx\\ &=-\frac{a d \cos (c+d x)}{12 x^3}-\frac{b d \cos (c+d x)}{6 x^2}+\frac{a d^3 \cos (c+d x)}{24 x}-\frac{1}{6} b d^3 \cos (c) \text{Ci}(d x)-\frac{a \sin (c+d x)}{4 x^4}-\frac{b \sin (c+d x)}{3 x^3}+\frac{a d^2 \sin (c+d x)}{24 x^2}+\frac{b d^2 \sin (c+d x)}{6 x}+\frac{1}{6} b d^3 \sin (c) \text{Si}(d x)+\frac{1}{24} \left (a d^4 \cos (c)\right ) \int \frac{\sin (d x)}{x} \, dx+\frac{1}{24} \left (a d^4 \sin (c)\right ) \int \frac{\cos (d x)}{x} \, dx\\ &=-\frac{a d \cos (c+d x)}{12 x^3}-\frac{b d \cos (c+d x)}{6 x^2}+\frac{a d^3 \cos (c+d x)}{24 x}-\frac{1}{6} b d^3 \cos (c) \text{Ci}(d x)+\frac{1}{24} a d^4 \text{Ci}(d x) \sin (c)-\frac{a \sin (c+d x)}{4 x^4}-\frac{b \sin (c+d x)}{3 x^3}+\frac{a d^2 \sin (c+d x)}{24 x^2}+\frac{b d^2 \sin (c+d x)}{6 x}+\frac{1}{24} a d^4 \cos (c) \text{Si}(d x)+\frac{1}{6} b d^3 \sin (c) \text{Si}(d x)\\ \end{align*}

Mathematica [A]  time = 0.268251, size = 138, normalized size = 0.83 \[ \frac{d^3 x^4 \text{CosIntegral}(d x) (a d \sin (c)-4 b \cos (c))+d^3 x^4 \text{Si}(d x) (a d \cos (c)+4 b \sin (c))+a d^2 x^2 \sin (c+d x)+a d^3 x^3 \cos (c+d x)-6 a \sin (c+d x)-2 a d x \cos (c+d x)+4 b d^2 x^3 \sin (c+d x)-4 b d x^2 \cos (c+d x)-8 b x \sin (c+d x)}{24 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*Sin[c + d*x])/x^5,x]

[Out]

(-2*a*d*x*Cos[c + d*x] - 4*b*d*x^2*Cos[c + d*x] + a*d^3*x^3*Cos[c + d*x] + d^3*x^4*CosIntegral[d*x]*(-4*b*Cos[
c] + a*d*Sin[c]) - 6*a*Sin[c + d*x] - 8*b*x*Sin[c + d*x] + a*d^2*x^2*Sin[c + d*x] + 4*b*d^2*x^3*Sin[c + d*x] +
d^3*x^4*(a*d*Cos[c] + 4*b*Sin[c])*SinIntegral[d*x])/(24*x^4)

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Maple [A]  time = 0.013, size = 145, normalized size = 0.9 \begin{align*}{d}^{4} \left ({\frac{b}{d} \left ( -{\frac{\sin \left ( dx+c \right ) }{3\,{d}^{3}{x}^{3}}}-{\frac{\cos \left ( dx+c \right ) }{6\,{d}^{2}{x}^{2}}}+{\frac{\sin \left ( dx+c \right ) }{6\,dx}}+{\frac{{\it Si} \left ( dx \right ) \sin \left ( c \right ) }{6}}-{\frac{{\it Ci} \left ( dx \right ) \cos \left ( c \right ) }{6}} \right ) }+a \left ( -{\frac{\sin \left ( dx+c \right ) }{4\,{x}^{4}{d}^{4}}}-{\frac{\cos \left ( dx+c \right ) }{12\,{d}^{3}{x}^{3}}}+{\frac{\sin \left ( dx+c \right ) }{24\,{d}^{2}{x}^{2}}}+{\frac{\cos \left ( dx+c \right ) }{24\,dx}}+{\frac{{\it Si} \left ( dx \right ) \cos \left ( c \right ) }{24}}+{\frac{{\it Ci} \left ( dx \right ) \sin \left ( c \right ) }{24}} \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*sin(d*x+c)/x^5,x)

[Out]

d^4*(b/d*(-1/3*sin(d*x+c)/x^3/d^3-1/6*cos(d*x+c)/x^2/d^2+1/6*sin(d*x+c)/x/d+1/6*Si(d*x)*sin(c)-1/6*Ci(d*x)*cos
(c))+a*(-1/4*sin(d*x+c)/x^4/d^4-1/12*cos(d*x+c)/x^3/d^3+1/24*sin(d*x+c)/x^2/d^2+1/24*cos(d*x+c)/x/d+1/24*Si(d*
x)*cos(c)+1/24*Ci(d*x)*sin(c)))

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Maxima [C]  time = 2.23723, size = 151, normalized size = 0.91 \begin{align*} -\frac{{\left ({\left (a{\left (i \, \Gamma \left (-4, i \, d x\right ) - i \, \Gamma \left (-4, -i \, d x\right )\right )} \cos \left (c\right ) + a{\left (\Gamma \left (-4, i \, d x\right ) + \Gamma \left (-4, -i \, d x\right )\right )} \sin \left (c\right )\right )} d^{5} -{\left (4 \, b{\left (\Gamma \left (-4, i \, d x\right ) + \Gamma \left (-4, -i \, d x\right )\right )} \cos \left (c\right ) - b{\left (4 i \, \Gamma \left (-4, i \, d x\right ) - 4 i \, \Gamma \left (-4, -i \, d x\right )\right )} \sin \left (c\right )\right )} d^{4}\right )} x^{4} + 2 \, b \cos \left (d x + c\right )}{2 \, d x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*sin(d*x+c)/x^5,x, algorithm="maxima")

[Out]

-1/2*(((a*(I*gamma(-4, I*d*x) - I*gamma(-4, -I*d*x))*cos(c) + a*(gamma(-4, I*d*x) + gamma(-4, -I*d*x))*sin(c))
*d^5 - (4*b*(gamma(-4, I*d*x) + gamma(-4, -I*d*x))*cos(c) - b*(4*I*gamma(-4, I*d*x) - 4*I*gamma(-4, -I*d*x))*s
in(c))*d^4)*x^4 + 2*b*cos(d*x + c))/(d*x^4)

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Fricas [A]  time = 1.68891, size = 440, normalized size = 2.65 \begin{align*} \frac{2 \,{\left (a d^{3} x^{3} - 4 \, b d x^{2} - 2 \, a d x\right )} \cos \left (d x + c\right ) + 2 \,{\left (a d^{4} x^{4} \operatorname{Si}\left (d x\right ) - 2 \, b d^{3} x^{4} \operatorname{Ci}\left (d x\right ) - 2 \, b d^{3} x^{4} \operatorname{Ci}\left (-d x\right )\right )} \cos \left (c\right ) + 2 \,{\left (4 \, b d^{2} x^{3} + a d^{2} x^{2} - 8 \, b x - 6 \, a\right )} \sin \left (d x + c\right ) +{\left (a d^{4} x^{4} \operatorname{Ci}\left (d x\right ) + a d^{4} x^{4} \operatorname{Ci}\left (-d x\right ) + 8 \, b d^{3} x^{4} \operatorname{Si}\left (d x\right )\right )} \sin \left (c\right )}{48 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*sin(d*x+c)/x^5,x, algorithm="fricas")

[Out]

1/48*(2*(a*d^3*x^3 - 4*b*d*x^2 - 2*a*d*x)*cos(d*x + c) + 2*(a*d^4*x^4*sin_integral(d*x) - 2*b*d^3*x^4*cos_inte
gral(d*x) - 2*b*d^3*x^4*cos_integral(-d*x))*cos(c) + 2*(4*b*d^2*x^3 + a*d^2*x^2 - 8*b*x - 6*a)*sin(d*x + c) +
(a*d^4*x^4*cos_integral(d*x) + a*d^4*x^4*cos_integral(-d*x) + 8*b*d^3*x^4*sin_integral(d*x))*sin(c))/x^4

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x\right ) \sin{\left (c + d x \right )}}{x^{5}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*sin(d*x+c)/x**5,x)

[Out]

Integral((a + b*x)*sin(c + d*x)/x**5, x)

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Giac [C]  time = 1.15808, size = 1496, normalized size = 9.01 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*sin(d*x+c)/x^5,x, algorithm="giac")

[Out]

-1/48*(a*d^4*x^4*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 - a*d^4*x^4*imag_part(cos_integral(-
d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*a*d^4*x^4*sin_integral(d*x)*tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*a*d^4*x^4*re
al_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c) - 2*a*d^4*x^4*real_part(cos_integral(-d*x))*tan(1/2*d*x)^
2*tan(1/2*c) - 4*b*d^3*x^4*real_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 - 4*b*d^3*x^4*real_part(co
s_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 - a*d^4*x^4*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2 + a*d^4*
x^4*imag_part(cos_integral(-d*x))*tan(1/2*d*x)^2 - 2*a*d^4*x^4*sin_integral(d*x)*tan(1/2*d*x)^2 - 8*b*d^3*x^4*
imag_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c) + 8*b*d^3*x^4*imag_part(cos_integral(-d*x))*tan(1/2*d*x
)^2*tan(1/2*c) - 16*b*d^3*x^4*sin_integral(d*x)*tan(1/2*d*x)^2*tan(1/2*c) + a*d^4*x^4*imag_part(cos_integral(d
*x))*tan(1/2*c)^2 - a*d^4*x^4*imag_part(cos_integral(-d*x))*tan(1/2*c)^2 + 2*a*d^4*x^4*sin_integral(d*x)*tan(1
/2*c)^2 + 4*b*d^3*x^4*real_part(cos_integral(d*x))*tan(1/2*d*x)^2 + 4*b*d^3*x^4*real_part(cos_integral(-d*x))*
tan(1/2*d*x)^2 - 2*a*d^4*x^4*real_part(cos_integral(d*x))*tan(1/2*c) - 2*a*d^4*x^4*real_part(cos_integral(-d*x
))*tan(1/2*c) - 4*b*d^3*x^4*real_part(cos_integral(d*x))*tan(1/2*c)^2 - 4*b*d^3*x^4*real_part(cos_integral(-d*
x))*tan(1/2*c)^2 - 2*a*d^3*x^3*tan(1/2*d*x)^2*tan(1/2*c)^2 - a*d^4*x^4*imag_part(cos_integral(d*x)) + a*d^4*x^
4*imag_part(cos_integral(-d*x)) - 2*a*d^4*x^4*sin_integral(d*x) - 8*b*d^3*x^4*imag_part(cos_integral(d*x))*tan
(1/2*c) + 8*b*d^3*x^4*imag_part(cos_integral(-d*x))*tan(1/2*c) - 16*b*d^3*x^4*sin_integral(d*x)*tan(1/2*c) + 4
*b*d^3*x^4*real_part(cos_integral(d*x)) + 4*b*d^3*x^4*real_part(cos_integral(-d*x)) + 2*a*d^3*x^3*tan(1/2*d*x)
^2 + 8*a*d^3*x^3*tan(1/2*d*x)*tan(1/2*c) + 16*b*d^2*x^3*tan(1/2*d*x)^2*tan(1/2*c) + 2*a*d^3*x^3*tan(1/2*c)^2 +
16*b*d^2*x^3*tan(1/2*d*x)*tan(1/2*c)^2 + 4*a*d^2*x^2*tan(1/2*d*x)^2*tan(1/2*c) + 4*a*d^2*x^2*tan(1/2*d*x)*tan
(1/2*c)^2 + 8*b*d*x^2*tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*a*d^3*x^3 - 16*b*d^2*x^3*tan(1/2*d*x) - 16*b*d^2*x^3*tan
(1/2*c) + 4*a*d*x*tan(1/2*d*x)^2*tan(1/2*c)^2 - 4*a*d^2*x^2*tan(1/2*d*x) - 8*b*d*x^2*tan(1/2*d*x)^2 - 4*a*d^2*
x^2*tan(1/2*c) - 32*b*d*x^2*tan(1/2*d*x)*tan(1/2*c) - 8*b*d*x^2*tan(1/2*c)^2 - 4*a*d*x*tan(1/2*d*x)^2 - 16*a*d
*x*tan(1/2*d*x)*tan(1/2*c) - 32*b*x*tan(1/2*d*x)^2*tan(1/2*c) - 4*a*d*x*tan(1/2*c)^2 - 32*b*x*tan(1/2*d*x)*tan
(1/2*c)^2 + 8*b*d*x^2 - 24*a*tan(1/2*d*x)^2*tan(1/2*c) - 24*a*tan(1/2*d*x)*tan(1/2*c)^2 + 4*a*d*x + 32*b*x*tan
(1/2*d*x) + 32*b*x*tan(1/2*c) + 24*a*tan(1/2*d*x) + 24*a*tan(1/2*c))/(x^4*tan(1/2*d*x)^2*tan(1/2*c)^2 + x^4*ta
n(1/2*d*x)^2 + x^4*tan(1/2*c)^2 + x^4)