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

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

[Out]

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

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Rubi [A]  time = 0.37, antiderivative size = 166, normalized size of antiderivative = 1.00, 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 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 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 {(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*}

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Mathematica [A]  time = 0.29, size = 138, normalized size = 0.83 \[ \frac {d^3 x^4 \text {Ci}(d x) (a d \sin (c)-4 b \cos (c))+d^3 x^4 \text {Si}(d x) (a d \cos (c)+4 b \sin (c))+a d^3 x^3 \cos (c+d x)+a d^2 x^2 \sin (c+d x)-6 a \sin (c+d x)-2 a d x \cos (c+d x)+4 b d^2 x^3 \sin (c+d x)-4 b d x^2 \cos (c+d x)-8 b x \sin (c+d x)}{24 x^4} \]

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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fricas [A]  time = 0.75, size = 154, normalized size = 0.93 \[ \frac {2 \, {\left (a d^{3} x^{3} - 4 \, b d x^{2} - 2 \, a d x\right )} \cos \left (d x + c\right ) + 2 \, {\left (a d^{4} x^{4} \operatorname {Si}\left (d x\right ) - 2 \, b d^{3} x^{4} \operatorname {Ci}\left (d x\right ) - 2 \, b d^{3} x^{4} \operatorname {Ci}\left (-d x\right )\right )} \cos \relax (c) + 2 \, {\left (4 \, b d^{2} x^{3} + a d^{2} x^{2} - 8 \, b x - 6 \, a\right )} \sin \left (d x + c\right ) + {\left (a d^{4} x^{4} \operatorname {Ci}\left (d x\right ) + a d^{4} x^{4} \operatorname {Ci}\left (-d x\right ) + 8 \, b d^{3} x^{4} \operatorname {Si}\left (d x\right )\right )} \sin \relax (c)}{48 \, x^{4}} \]

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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giac [C]  time = 0.85, size = 1108, normalized size = 6.67 \[ \text {result too large to display} \]

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)

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maple [A]  time = 0.03, size = 145, normalized size = 0.87 \[ d^{4} \left (\frac {b \left (-\frac {\sin \left (d x +c \right )}{3 x^{3} d^{3}}-\frac {\cos \left (d x +c \right )}{6 x^{2} d^{2}}+\frac {\sin \left (d x +c \right )}{6 x d}+\frac {\Si \left (d x \right ) \sin \relax (c )}{6}-\frac {\Ci \left (d x \right ) \cos \relax (c )}{6}\right )}{d}+a \left (-\frac {\sin \left (d x +c \right )}{4 x^{4} d^{4}}-\frac {\cos \left (d x +c \right )}{12 x^{3} d^{3}}+\frac {\sin \left (d x +c \right )}{24 x^{2} d^{2}}+\frac {\cos \left (d x +c \right )}{24 x d}+\frac {\Si \left (d x \right ) \cos \relax (c )}{24}+\frac {\Ci \left (d x \right ) \sin \relax (c )}{24}\right )\right ) \]

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 = 1.92, size = 112, normalized size = 0.67 \[ -\frac {{\left ({\left (a {\left (i \, \Gamma \left (-4, i \, d x\right ) - i \, \Gamma \left (-4, -i \, d x\right )\right )} \cos \relax (c) + a {\left (\Gamma \left (-4, i \, d x\right ) + \Gamma \left (-4, -i \, d x\right )\right )} \sin \relax (c)\right )} d^{5} - {\left (4 \, b {\left (\Gamma \left (-4, i \, d x\right ) + \Gamma \left (-4, -i \, d x\right )\right )} \cos \relax (c) - b {\left (4 i \, \Gamma \left (-4, i \, d x\right ) - 4 i \, \Gamma \left (-4, -i \, d x\right )\right )} \sin \relax (c)\right )} d^{4}\right )} x^{4} + 2 \, b \cos \left (d x + c\right )}{2 \, d x^{4}} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sin \left (c+d\,x\right )\,\left (a+b\,x\right )}{x^5} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((sin(c + d*x)*(a + b*x))/x^5,x)

[Out]

int((sin(c + d*x)*(a + b*x))/x^5, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b x\right ) \sin {\left (c + d x \right )}}{x^{5}}\, dx \]

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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