3.48 \(\int \frac{(a+b x^2) \sin (c+d x)}{x^5} \, dx\)

Optimal. Leaf size=149 \[ \frac{1}{24} a d^4 \sin (c) \text{CosIntegral}(d x)+\frac{1}{24} a d^4 \cos (c) \text{Si}(d x)+\frac{a d^2 \sin (c+d x)}{24 x^2}+\frac{a d^3 \cos (c+d x)}{24 x}-\frac{a \sin (c+d x)}{4 x^4}-\frac{a d \cos (c+d x)}{12 x^3}-\frac{1}{2} b d^2 \sin (c) \text{CosIntegral}(d x)-\frac{1}{2} b d^2 \cos (c) \text{Si}(d x)-\frac{b \sin (c+d x)}{2 x^2}-\frac{b d \cos (c+d x)}{2 x} \]

[Out]

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

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Rubi [A]  time = 0.257703, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {3339, 3297, 3303, 3299, 3302} \[ \frac{1}{24} a d^4 \sin (c) \text{CosIntegral}(d x)+\frac{1}{24} a d^4 \cos (c) \text{Si}(d x)+\frac{a d^2 \sin (c+d x)}{24 x^2}+\frac{a d^3 \cos (c+d x)}{24 x}-\frac{a \sin (c+d x)}{4 x^4}-\frac{a d \cos (c+d x)}{12 x^3}-\frac{1}{2} b d^2 \sin (c) \text{CosIntegral}(d x)-\frac{1}{2} b d^2 \cos (c) \text{Si}(d x)-\frac{b \sin (c+d x)}{2 x^2}-\frac{b d \cos (c+d x)}{2 x} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)*Sin[c + d*x])/x^5,x]

[Out]

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

Rule 3339

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*Sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Int[ExpandIntegran
d[Sin[c + d*x], (e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]

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

Mathematica [A]  time = 0.227678, size = 125, normalized size = 0.84 \[ \frac{d^2 x^4 \sin (c) \left (a d^2-12 b\right ) \text{CosIntegral}(d x)+d^2 x^4 \cos (c) \left (a d^2-12 b\right ) \text{Si}(d x)+a d^2 x^2 \sin (c+d x)+a d^3 x^3 \cos (c+d x)-6 a \sin (c+d x)-2 a d x \cos (c+d x)-12 b x^2 \sin (c+d x)-12 b d x^3 \cos (c+d x)}{24 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)*Sin[c + d*x])/x^5,x]

[Out]

(-2*a*d*x*Cos[c + d*x] - 12*b*d*x^3*Cos[c + d*x] + a*d^3*x^3*Cos[c + d*x] + d^2*(-12*b + a*d^2)*x^4*CosIntegra
l[d*x]*Sin[c] - 6*a*Sin[c + d*x] - 12*b*x^2*Sin[c + d*x] + a*d^2*x^2*Sin[c + d*x] + d^2*(-12*b + a*d^2)*x^4*Co
s[c]*SinIntegral[d*x])/(24*x^4)

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Maple [A]  time = 0.015, size = 131, normalized size = 0.9 \begin{align*}{d}^{4} \left ({\frac{b}{{d}^{2}} \left ( -{\frac{\sin \left ( dx+c \right ) }{2\,{d}^{2}{x}^{2}}}-{\frac{\cos \left ( dx+c \right ) }{2\,dx}}-{\frac{{\it Si} \left ( dx \right ) \cos \left ( c \right ) }{2}}-{\frac{{\it Ci} \left ( dx \right ) \sin \left ( c \right ) }{2}} \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^2+a)*sin(d*x+c)/x^5,x)

[Out]

d^4*(1/d^2*b*(-1/2*sin(d*x+c)/x^2/d^2-1/2*cos(d*x+c)/x/d-1/2*Si(d*x)*cos(c)-1/2*Ci(d*x)*sin(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 = 3.77157, size = 163, normalized size = 1.09 \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^{6} +{\left (b{\left (-12 i \, \Gamma \left (-4, i \, d x\right ) + 12 i \, \Gamma \left (-4, -i \, d x\right )\right )} \cos \left (c\right ) - 12 \, b{\left (\Gamma \left (-4, i \, d x\right ) + \Gamma \left (-4, -i \, d x\right )\right )} \sin \left (c\right )\right )} d^{4}\right )} x^{4} + 2 \, b d x \cos \left (d x + c\right ) + 6 \, b \sin \left (d x + c\right )}{2 \, d^{2} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+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^6 + (b*(-12*I*gamma(-4, I*d*x) + 12*I*gamma(-4, -I*d*x))*cos(c) - 12*b*(gamma(-4, I*d*x) + gamma(-4, -I*d*x
))*sin(c))*d^4)*x^4 + 2*b*d*x*cos(d*x + c) + 6*b*sin(d*x + c))/(d^2*x^4)

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Fricas [A]  time = 1.72718, size = 340, normalized size = 2.28 \begin{align*} \frac{2 \,{\left (a d^{4} - 12 \, b d^{2}\right )} x^{4} \cos \left (c\right ) \operatorname{Si}\left (d x\right ) + 2 \,{\left ({\left (a d^{3} - 12 \, b d\right )} x^{3} - 2 \, a d x\right )} \cos \left (d x + c\right ) + 2 \,{\left ({\left (a d^{2} - 12 \, b\right )} x^{2} - 6 \, a\right )} \sin \left (d x + c\right ) +{\left ({\left (a d^{4} - 12 \, b d^{2}\right )} x^{4} \operatorname{Ci}\left (d x\right ) +{\left (a d^{4} - 12 \, b d^{2}\right )} x^{4} \operatorname{Ci}\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^2+a)*sin(d*x+c)/x^5,x, algorithm="fricas")

[Out]

1/48*(2*(a*d^4 - 12*b*d^2)*x^4*cos(c)*sin_integral(d*x) + 2*((a*d^3 - 12*b*d)*x^3 - 2*a*d*x)*cos(d*x + c) + 2*
((a*d^2 - 12*b)*x^2 - 6*a)*sin(d*x + c) + ((a*d^4 - 12*b*d^2)*x^4*cos_integral(d*x) + (a*d^4 - 12*b*d^2)*x^4*c
os_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^{2}\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**2+a)*sin(d*x+c)/x**5,x)

[Out]

Integral((a + b*x**2)*sin(c + d*x)/x**5, x)

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Giac [C]  time = 1.15804, size = 1466, normalized size = 9.84 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+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) - 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 + 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 - 1
2*b*d^2*x^4*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 + 12*b*d^2*x^4*imag_part(cos_integral(-d*
x))*tan(1/2*d*x)^2*tan(1/2*c)^2 - 24*b*d^2*x^4*sin_integral(d*x)*tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*a*d^4*x^4*rea
l_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) + 24*b*d^2*x^4*rea
l_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c) + 24*b*d^2*x^4*real_part(cos_integral(-d*x))*tan(1/2*d*x)^
2*tan(1/2*c) - 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*im
ag_part(cos_integral(-d*x)) - 2*a*d^4*x^4*sin_integral(d*x) + 12*b*d^2*x^4*imag_part(cos_integral(d*x))*tan(1/
2*d*x)^2 - 12*b*d^2*x^4*imag_part(cos_integral(-d*x))*tan(1/2*d*x)^2 + 24*b*d^2*x^4*sin_integral(d*x)*tan(1/2*
d*x)^2 - 12*b*d^2*x^4*imag_part(cos_integral(d*x))*tan(1/2*c)^2 + 12*b*d^2*x^4*imag_part(cos_integral(-d*x))*t
an(1/2*c)^2 - 24*b*d^2*x^4*sin_integral(d*x)*tan(1/2*c)^2 + 2*a*d^3*x^3*tan(1/2*d*x)^2 + 24*b*d^2*x^4*real_par
t(cos_integral(d*x))*tan(1/2*c) + 24*b*d^2*x^4*real_part(cos_integral(-d*x))*tan(1/2*c) + 8*a*d^3*x^3*tan(1/2*
d*x)*tan(1/2*c) + 2*a*d^3*x^3*tan(1/2*c)^2 + 24*b*d*x^3*tan(1/2*d*x)^2*tan(1/2*c)^2 + 12*b*d^2*x^4*imag_part(c
os_integral(d*x)) - 12*b*d^2*x^4*imag_part(cos_integral(-d*x)) + 24*b*d^2*x^4*sin_integral(d*x) + 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 - 2*a*d^3*x^3 - 24*b*d*x^3*tan(1/2*d*x)^2 -
96*b*d*x^3*tan(1/2*d*x)*tan(1/2*c) - 24*b*d*x^3*tan(1/2*c)^2 + 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) - 4*a*d^2*x^2*tan(1/2*c) - 48*b*x^2*tan(1/2*d*x)^2*tan(1/2*c) - 48*b*x^2*tan(1/2*d*x)*tan(1/2*
c)^2 + 24*b*d*x^3 - 4*a*d*x*tan(1/2*d*x)^2 - 16*a*d*x*tan(1/2*d*x)*tan(1/2*c) - 4*a*d*x*tan(1/2*c)^2 + 48*b*x^
2*tan(1/2*d*x) + 48*b*x^2*tan(1/2*c) - 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 + 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*tan(1/2*d*x)^2 + x^4*tan(1/2*
c)^2 + x^4)