3.8 \(\int \frac {\text {Si}(b x)}{x^3} \, dx\)

Optimal. Leaf size=46 \[ -\frac {1}{4} b^2 \text {Si}(b x)-\frac {\text {Si}(b x)}{2 x^2}-\frac {\sin (b x)}{4 x^2}-\frac {b \cos (b x)}{4 x} \]

[Out]

-1/4*b*cos(b*x)/x-1/4*b^2*Si(b*x)-1/2*Si(b*x)/x^2-1/4*sin(b*x)/x^2

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6503, 12, 3297, 3299} \[ -\frac {1}{4} b^2 \text {Si}(b x)-\frac {\text {Si}(b x)}{2 x^2}-\frac {\sin (b x)}{4 x^2}-\frac {b \cos (b x)}{4 x} \]

Antiderivative was successfully verified.

[In]

Int[SinIntegral[b*x]/x^3,x]

[Out]

-(b*Cos[b*x])/(4*x) - Sin[b*x]/(4*x^2) - (b^2*SinIntegral[b*x])/4 - SinIntegral[b*x]/(2*x^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 6503

Int[((c_.) + (d_.)*(x_))^(m_.)*SinIntegral[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[((c + d*x)^(m + 1)*SinIntegr
al[a + b*x])/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[((c + d*x)^(m + 1)*Sin[a + b*x])/(a + b*x), x], x] /; F
reeQ[{a, b, c, d, m}, x] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {\text {Si}(b x)}{x^3} \, dx &=-\frac {\text {Si}(b x)}{2 x^2}+\frac {1}{2} b \int \frac {\sin (b x)}{b x^3} \, dx\\ &=-\frac {\text {Si}(b x)}{2 x^2}+\frac {1}{2} \int \frac {\sin (b x)}{x^3} \, dx\\ &=-\frac {\sin (b x)}{4 x^2}-\frac {\text {Si}(b x)}{2 x^2}+\frac {1}{4} b \int \frac {\cos (b x)}{x^2} \, dx\\ &=-\frac {b \cos (b x)}{4 x}-\frac {\sin (b x)}{4 x^2}-\frac {\text {Si}(b x)}{2 x^2}-\frac {1}{4} b^2 \int \frac {\sin (b x)}{x} \, dx\\ &=-\frac {b \cos (b x)}{4 x}-\frac {\sin (b x)}{4 x^2}-\frac {1}{4} b^2 \text {Si}(b x)-\frac {\text {Si}(b x)}{2 x^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 46, normalized size = 1.00 \[ -\frac {1}{4} b^2 \text {Si}(b x)-\frac {\text {Si}(b x)}{2 x^2}-\frac {\sin (b x)}{4 x^2}-\frac {b \cos (b x)}{4 x} \]

Antiderivative was successfully verified.

[In]

Integrate[SinIntegral[b*x]/x^3,x]

[Out]

-1/4*(b*Cos[b*x])/x - Sin[b*x]/(4*x^2) - (b^2*SinIntegral[b*x])/4 - SinIntegral[b*x]/(2*x^2)

________________________________________________________________________________________

fricas [F]  time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {Si}\left (b x\right )}{x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Si(b*x)/x^3,x, algorithm="fricas")

[Out]

integral(sin_integral(b*x)/x^3, x)

________________________________________________________________________________________

giac [C]  time = 1.57, size = 149, normalized size = 3.24 \[ -\frac {b^{2} x^{2} \Im \left (\operatorname {Ci}\left (b x\right ) \right ) \tan \left (\frac {1}{2} \, b x\right )^{2} - b^{2} x^{2} \Im \left (\operatorname {Ci}\left (-b x\right ) \right ) \tan \left (\frac {1}{2} \, b x\right )^{2} + 2 \, b^{2} x^{2} \operatorname {Si}\left (b x\right ) \tan \left (\frac {1}{2} \, b x\right )^{2} + b^{2} x^{2} \Im \left (\operatorname {Ci}\left (b x\right ) \right ) - b^{2} x^{2} \Im \left (\operatorname {Ci}\left (-b x\right ) \right ) + 2 \, b^{2} x^{2} \operatorname {Si}\left (b x\right ) - 2 \, b x \tan \left (\frac {1}{2} \, b x\right )^{2} + 2 \, b x + 4 \, \tan \left (\frac {1}{2} \, b x\right )}{8 \, {\left (x^{2} \tan \left (\frac {1}{2} \, b x\right )^{2} + x^{2}\right )}} - \frac {\operatorname {Si}\left (b x\right )}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Si(b*x)/x^3,x, algorithm="giac")

[Out]

-1/8*(b^2*x^2*imag_part(cos_integral(b*x))*tan(1/2*b*x)^2 - b^2*x^2*imag_part(cos_integral(-b*x))*tan(1/2*b*x)
^2 + 2*b^2*x^2*sin_integral(b*x)*tan(1/2*b*x)^2 + b^2*x^2*imag_part(cos_integral(b*x)) - b^2*x^2*imag_part(cos
_integral(-b*x)) + 2*b^2*x^2*sin_integral(b*x) - 2*b*x*tan(1/2*b*x)^2 + 2*b*x + 4*tan(1/2*b*x))/(x^2*tan(1/2*b
*x)^2 + x^2) - 1/2*sin_integral(b*x)/x^2

________________________________________________________________________________________

maple [A]  time = 0.02, size = 48, normalized size = 1.04 \[ b^{2} \left (-\frac {\Si \left (b x \right )}{2 b^{2} x^{2}}-\frac {\sin \left (b x \right )}{4 b^{2} x^{2}}-\frac {\cos \left (b x \right )}{4 b x}-\frac {\Si \left (b x \right )}{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(Si(b*x)/x^3,x)

[Out]

b^2*(-1/2*Si(b*x)/b^2/x^2-1/4*sin(b*x)/b^2/x^2-1/4*cos(b*x)/b/x-1/4*Si(b*x))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm Si}\left (b x\right )}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Si(b*x)/x^3,x, algorithm="maxima")

[Out]

integrate(Si(b*x)/x^3, x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ -\frac {\frac {\sin \left (b\,x\right )}{2}+\frac {b\,x\,\cos \left (b\,x\right )}{2}}{2\,x^2}-\frac {b^2\,\mathrm {sinint}\left (b\,x\right )}{4}-\frac {\mathrm {sinint}\left (b\,x\right )}{2\,x^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinint(b*x)/x^3,x)

[Out]

- (sin(b*x)/2 + (b*x*cos(b*x))/2)/(2*x^2) - (b^2*sinint(b*x))/4 - sinint(b*x)/(2*x^2)

________________________________________________________________________________________

sympy [A]  time = 0.77, size = 41, normalized size = 0.89 \[ - \frac {b^{2} \operatorname {Si}{\left (b x \right )}}{4} - \frac {b \cos {\left (b x \right )}}{4 x} - \frac {\sin {\left (b x \right )}}{4 x^{2}} - \frac {\operatorname {Si}{\left (b x \right )}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Si(b*x)/x**3,x)

[Out]

-b**2*Si(b*x)/4 - b*cos(b*x)/(4*x) - sin(b*x)/(4*x**2) - Si(b*x)/(2*x**2)

________________________________________________________________________________________