\(\int \frac {\text {Si}(a+b x)}{x^3} \, dx\) [24]

Optimal result

Integrand size = 10, antiderivative size = 111 \[ \int \frac {\text {Si}(a+b x)}{x^3} \, dx=\frac {b^2 \cos (a) \operatorname {CosIntegral}(b x)}{2 a}-\frac {b^2 \operatorname {CosIntegral}(b x) \sin (a)}{2 a^2}-\frac {b \sin (a+b x)}{2 a x}-\frac {b^2 \cos (a) \text {Si}(b x)}{2 a^2}-\frac {b^2 \sin (a) \text {Si}(b x)}{2 a}+\frac {b^2 \text {Si}(a+b x)}{2 a^2}-\frac {\text {Si}(a+b x)}{2 x^2} \]

[Out]

1/2*b^2*Ci(b*x)*cos(a)/a-1/2*b^2*cos(a)*Si(b*x)/a^2+1/2*b^2*Si(b*x+a)/a^2-1/2*Si(b*x+a)/x^2-1/2*b^2*Ci(b*x)*si
n(a)/a^2-1/2*b^2*Si(b*x)*sin(a)/a-1/2*b*sin(b*x+a)/a/x

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6638, 6874, 3378, 3384, 3380, 3383} \[ \int \frac {\text {Si}(a+b x)}{x^3} \, dx=-\frac {b^2 \sin (a) \operatorname {CosIntegral}(b x)}{2 a^2}+\frac {b^2 \text {Si}(a+b x)}{2 a^2}-\frac {b^2 \cos (a) \text {Si}(b x)}{2 a^2}+\frac {b^2 \cos (a) \operatorname {CosIntegral}(b x)}{2 a}-\frac {b^2 \sin (a) \text {Si}(b x)}{2 a}-\frac {\text {Si}(a+b x)}{2 x^2}-\frac {b \sin (a+b x)}{2 a x} \]

[In]

Int[SinIntegral[a + b*x]/x^3,x]

[Out]

(b^2*Cos[a]*CosIntegral[b*x])/(2*a) - (b^2*CosIntegral[b*x]*Sin[a])/(2*a^2) - (b*Sin[a + b*x])/(2*a*x) - (b^2*
Cos[a]*SinIntegral[b*x])/(2*a^2) - (b^2*Sin[a]*SinIntegral[b*x])/(2*a) + (b^2*SinIntegral[a + b*x])/(2*a^2) -
SinIntegral[a + b*x]/(2*x^2)

Rule 3378

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 3380

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 3383

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 3384

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 6638

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]

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Si}(a+b x)}{2 x^2}+\frac {1}{2} b \int \frac {\sin (a+b x)}{x^2 (a+b x)} \, dx \\ & = -\frac {\text {Si}(a+b x)}{2 x^2}+\frac {1}{2} b \int \left (\frac {\sin (a+b x)}{a x^2}-\frac {b \sin (a+b x)}{a^2 x}+\frac {b^2 \sin (a+b x)}{a^2 (a+b x)}\right ) \, dx \\ & = -\frac {\text {Si}(a+b x)}{2 x^2}+\frac {b \int \frac {\sin (a+b x)}{x^2} \, dx}{2 a}-\frac {b^2 \int \frac {\sin (a+b x)}{x} \, dx}{2 a^2}+\frac {b^3 \int \frac {\sin (a+b x)}{a+b x} \, dx}{2 a^2} \\ & = -\frac {b \sin (a+b x)}{2 a x}+\frac {b^2 \text {Si}(a+b x)}{2 a^2}-\frac {\text {Si}(a+b x)}{2 x^2}+\frac {b^2 \int \frac {\cos (a+b x)}{x} \, dx}{2 a}-\frac {\left (b^2 \cos (a)\right ) \int \frac {\sin (b x)}{x} \, dx}{2 a^2}-\frac {\left (b^2 \sin (a)\right ) \int \frac {\cos (b x)}{x} \, dx}{2 a^2} \\ & = -\frac {b^2 \operatorname {CosIntegral}(b x) \sin (a)}{2 a^2}-\frac {b \sin (a+b x)}{2 a x}-\frac {b^2 \cos (a) \text {Si}(b x)}{2 a^2}+\frac {b^2 \text {Si}(a+b x)}{2 a^2}-\frac {\text {Si}(a+b x)}{2 x^2}+\frac {\left (b^2 \cos (a)\right ) \int \frac {\cos (b x)}{x} \, dx}{2 a}-\frac {\left (b^2 \sin (a)\right ) \int \frac {\sin (b x)}{x} \, dx}{2 a} \\ & = \frac {b^2 \cos (a) \operatorname {CosIntegral}(b x)}{2 a}-\frac {b^2 \operatorname {CosIntegral}(b x) \sin (a)}{2 a^2}-\frac {b \sin (a+b x)}{2 a x}-\frac {b^2 \cos (a) \text {Si}(b x)}{2 a^2}-\frac {b^2 \sin (a) \text {Si}(b x)}{2 a}+\frac {b^2 \text {Si}(a+b x)}{2 a^2}-\frac {\text {Si}(a+b x)}{2 x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.76 \[ \int \frac {\text {Si}(a+b x)}{x^3} \, dx=-\frac {-b^2 x^2 \operatorname {CosIntegral}(b x) (a \cos (a)-\sin (a))+a b x \sin (a+b x)+b^2 x^2 (\cos (a)+a \sin (a)) \text {Si}(b x)+a^2 \text {Si}(a+b x)-b^2 x^2 \text {Si}(a+b x)}{2 a^2 x^2} \]

[In]

Integrate[SinIntegral[a + b*x]/x^3,x]

[Out]

-1/2*(-(b^2*x^2*CosIntegral[b*x]*(a*Cos[a] - Sin[a])) + a*b*x*Sin[a + b*x] + b^2*x^2*(Cos[a] + a*Sin[a])*SinIn
tegral[b*x] + a^2*SinIntegral[a + b*x] - b^2*x^2*SinIntegral[a + b*x])/(a^2*x^2)

Maple [A] (verified)

Time = 0.41 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.75

method	result	size
parts	\(-\frac {\operatorname {Si}\left (b x +a \right )}{2 x^{2}}+\frac {b^{2} \left (-\frac {\operatorname {Si}\left (b x \right ) \cos \left (a \right )+\operatorname {Ci}\left (b x \right ) \sin \left (a \right )}{a^{2}}+\frac {\operatorname {Si}\left (b x +a \right )}{a^{2}}+\frac {-\frac {\sin \left (b x +a \right )}{b x}-\operatorname {Si}\left (b x \right ) \sin \left (a \right )+\operatorname {Ci}\left (b x \right ) \cos \left (a \right )}{a}\right )}{2}\)	\(83\)
derivativedivides	\(b^{2} \left (-\frac {\operatorname {Si}\left (b x +a \right )}{2 b^{2} x^{2}}-\frac {\operatorname {Si}\left (b x \right ) \cos \left (a \right )+\operatorname {Ci}\left (b x \right ) \sin \left (a \right )}{2 a^{2}}+\frac {\operatorname {Si}\left (b x +a \right )}{2 a^{2}}+\frac {-\frac {\sin \left (b x +a \right )}{b x}-\operatorname {Si}\left (b x \right ) \sin \left (a \right )+\operatorname {Ci}\left (b x \right ) \cos \left (a \right )}{2 a}\right )\)	\(86\)
default	\(b^{2} \left (-\frac {\operatorname {Si}\left (b x +a \right )}{2 b^{2} x^{2}}-\frac {\operatorname {Si}\left (b x \right ) \cos \left (a \right )+\operatorname {Ci}\left (b x \right ) \sin \left (a \right )}{2 a^{2}}+\frac {\operatorname {Si}\left (b x +a \right )}{2 a^{2}}+\frac {-\frac {\sin \left (b x +a \right )}{b x}-\operatorname {Si}\left (b x \right ) \sin \left (a \right )+\operatorname {Ci}\left (b x \right ) \cos \left (a \right )}{2 a}\right )\)	\(86\)

[In]

int(Si(b*x+a)/x^3,x,method=_RETURNVERBOSE)

[Out]

-1/2*Si(b*x+a)/x^2+1/2*b^2*(-1/a^2*(Si(b*x)*cos(a)+Ci(b*x)*sin(a))+1/a^2*Si(b*x+a)+1/a*(-sin(b*x+a)/b/x-Si(b*x
)*sin(a)+Ci(b*x)*cos(a)))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.86 \[ \int \frac {\text {Si}(a+b x)}{x^3} \, dx=-\frac {a b x \sin \left (b x + a\right ) - {\left (a b^{2} x^{2} \operatorname {Ci}\left (b x\right ) - b^{2} x^{2} \operatorname {Si}\left (b x\right )\right )} \cos \left (a\right ) + {\left (a b^{2} x^{2} \operatorname {Si}\left (b x\right ) + b^{2} x^{2} \operatorname {Ci}\left (b x\right )\right )} \sin \left (a\right ) - {\left (b^{2} x^{2} - a^{2}\right )} \operatorname {Si}\left (b x + a\right )}{2 \, a^{2} x^{2}} \]

[In]

integrate(sin_integral(b*x+a)/x^3,x, algorithm="fricas")

[Out]

-1/2*(a*b*x*sin(b*x + a) - (a*b^2*x^2*cos_integral(b*x) - b^2*x^2*sin_integral(b*x))*cos(a) + (a*b^2*x^2*sin_i
ntegral(b*x) + b^2*x^2*cos_integral(b*x))*sin(a) - (b^2*x^2 - a^2)*sin_integral(b*x + a))/(a^2*x^2)

Sympy [F]

\[ \int \frac {\text {Si}(a+b x)}{x^3} \, dx=\int \frac {\operatorname {Si}{\left (a + b x \right )}}{x^{3}}\, dx \]

[In]

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

[Out]

Integral(Si(a + b*x)/x**3, x)

Maxima [F]

\[ \int \frac {\text {Si}(a+b x)}{x^3} \, dx=\int { \frac {\operatorname {Si}\left (b x + a\right )}{x^{3}} \,d x } \]

[In]

integrate(sin_integral(b*x+a)/x^3,x, algorithm="maxima")

[Out]

integrate(sin_integral(b*x + a)/x^3, x)

Giac [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.29 (sec) , antiderivative size = 809, normalized size of antiderivative = 7.29 \[ \int \frac {\text {Si}(a+b x)}{x^3} \, dx=\text {Too large to display} \]

[In]

integrate(sin_integral(b*x+a)/x^3,x, algorithm="giac")

[Out]

-1/4*(a*b*x*real_part(cos_integral(b*x))*tan(1/2*b*x)^2*tan(1/2*a)^2 + a*b*x*real_part(cos_integral(-b*x))*tan
(1/2*b*x)^2*tan(1/2*a)^2 + 2*a*b*x*imag_part(cos_integral(b*x))*tan(1/2*b*x)^2*tan(1/2*a) - 2*a*b*x*imag_part(
cos_integral(-b*x))*tan(1/2*b*x)^2*tan(1/2*a) + 4*a*b*x*sin_integral(b*x)*tan(1/2*b*x)^2*tan(1/2*a) - b*x*imag
_part(cos_integral(b*x + a))*tan(1/2*b*x)^2*tan(1/2*a)^2 - b*x*imag_part(cos_integral(b*x))*tan(1/2*b*x)^2*tan
(1/2*a)^2 + b*x*imag_part(cos_integral(-b*x - a))*tan(1/2*b*x)^2*tan(1/2*a)^2 + b*x*imag_part(cos_integral(-b*
x))*tan(1/2*b*x)^2*tan(1/2*a)^2 - 2*b*x*sin_integral(b*x + a)*tan(1/2*b*x)^2*tan(1/2*a)^2 - 2*b*x*sin_integral
(b*x)*tan(1/2*b*x)^2*tan(1/2*a)^2 - a*b*x*real_part(cos_integral(b*x))*tan(1/2*b*x)^2 - a*b*x*real_part(cos_in
tegral(-b*x))*tan(1/2*b*x)^2 + 2*b*x*real_part(cos_integral(b*x))*tan(1/2*b*x)^2*tan(1/2*a) + 2*b*x*real_part(
cos_integral(-b*x))*tan(1/2*b*x)^2*tan(1/2*a) + a*b*x*real_part(cos_integral(b*x))*tan(1/2*a)^2 + a*b*x*real_p
art(cos_integral(-b*x))*tan(1/2*a)^2 - b*x*imag_part(cos_integral(b*x + a))*tan(1/2*b*x)^2 + b*x*imag_part(cos
_integral(b*x))*tan(1/2*b*x)^2 + b*x*imag_part(cos_integral(-b*x - a))*tan(1/2*b*x)^2 - b*x*imag_part(cos_inte
gral(-b*x))*tan(1/2*b*x)^2 - 2*b*x*sin_integral(b*x + a)*tan(1/2*b*x)^2 + 2*b*x*sin_integral(b*x)*tan(1/2*b*x)
^2 + 2*a*b*x*imag_part(cos_integral(b*x))*tan(1/2*a) - 2*a*b*x*imag_part(cos_integral(-b*x))*tan(1/2*a) + 4*a*
b*x*sin_integral(b*x)*tan(1/2*a) - b*x*imag_part(cos_integral(b*x + a))*tan(1/2*a)^2 - b*x*imag_part(cos_integ
ral(b*x))*tan(1/2*a)^2 + b*x*imag_part(cos_integral(-b*x - a))*tan(1/2*a)^2 + b*x*imag_part(cos_integral(-b*x)
)*tan(1/2*a)^2 - 2*b*x*sin_integral(b*x + a)*tan(1/2*a)^2 - 2*b*x*sin_integral(b*x)*tan(1/2*a)^2 - a*b*x*real_
part(cos_integral(b*x)) - a*b*x*real_part(cos_integral(-b*x)) + 2*b*x*real_part(cos_integral(b*x))*tan(1/2*a)
+ 2*b*x*real_part(cos_integral(-b*x))*tan(1/2*a) - 4*a*tan(1/2*b*x)^2*tan(1/2*a) - 4*a*tan(1/2*b*x)*tan(1/2*a)
^2 - b*x*imag_part(cos_integral(b*x + a)) + b*x*imag_part(cos_integral(b*x)) + b*x*imag_part(cos_integral(-b*x
- a)) - b*x*imag_part(cos_integral(-b*x)) - 2*b*x*sin_integral(b*x + a) + 2*b*x*sin_integral(b*x) + 4*a*tan(1
/2*b*x) + 4*a*tan(1/2*a))*b/(a^2*x*tan(1/2*b*x)^2*tan(1/2*a)^2 + a^2*x*tan(1/2*b*x)^2 + a^2*x*tan(1/2*a)^2 + a
^2*x) - 1/2*sin_integral(b*x + a)/x^2

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {Si}(a+b x)}{x^3} \, dx=\int \frac {\mathrm {sinint}\left (a+b\,x\right )}{x^3} \,d x \]

[In]

int(sinint(a + b*x)/x^3,x)

[Out]

int(sinint(a + b*x)/x^3, x)