Integrand size = 10, antiderivative size = 111 \[ \int \frac {\text {Si}(a+b x)}{x^3} \, dx=\frac {b^2 \cos (a) \operatorname {CosIntegral}(b x)}{2 a}-\frac {b^2 \operatorname {CosIntegral}(b x) \sin (a)}{2 a^2}-\frac {b \sin (a+b x)}{2 a x}-\frac {b^2 \cos (a) \text {Si}(b x)}{2 a^2}-\frac {b^2 \sin (a) \text {Si}(b x)}{2 a}+\frac {b^2 \text {Si}(a+b x)}{2 a^2}-\frac {\text {Si}(a+b x)}{2 x^2} \]
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)*sin(a)/a^2-1/2*b^2*Si(b*x)*sin(a)/a-1/2* b*sin(b*x+a)/a/x
Time = 0.23 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.76 \[ \int \frac {\text {Si}(a+b x)}{x^3} \, dx=-\frac {-b^2 x^2 \operatorname {CosIntegral}(b x) (a \cos (a)-\sin (a))+a b x \sin (a+b x)+b^2 x^2 (\cos (a)+a \sin (a)) \text {Si}(b x)+a^2 \text {Si}(a+b x)-b^2 x^2 \text {Si}(a+b x)}{2 a^2 x^2} \]
-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])*SinIntegral[b*x] + a^2*SinIntegral[a + b*x] - b^2*x^2*SinIntegral[a + b*x])/(a^2*x^2)
Time = 0.58 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.83, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {7057, 7293, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\text {Si}(a+b x)}{x^3} \, dx\) |
| \(\Big \downarrow \) 7057 |
| \(\displaystyle \frac {1}{2} b \int \frac {\sin (a+b x)}{x^2 (a+b x)}dx-\frac {\text {Si}(a+b x)}{2 x^2}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} b \int \left (\frac {\sin (a+b x) b^2}{a^2 (a+b x)}-\frac {\sin (a+b x) b}{a^2 x}+\frac {\sin (a+b x)}{a x^2}\right )dx-\frac {\text {Si}(a+b x)}{2 x^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} b \left (-\frac {b \sin (a) \operatorname {CosIntegral}(b x)}{a^2}+\frac {b \text {Si}(a+b x)}{a^2}-\frac {b \cos (a) \text {Si}(b x)}{a^2}+\frac {b \cos (a) \operatorname {CosIntegral}(b x)}{a}-\frac {b \sin (a) \text {Si}(b x)}{a}-\frac {\sin (a+b x)}{a x}\right )-\frac {\text {Si}(a+b x)}{2 x^2}\) |
-1/2*SinIntegral[a + b*x]/x^2 + (b*((b*Cos[a]*CosIntegral[b*x])/a - (b*Cos Integral[b*x]*Sin[a])/a^2 - Sin[a + b*x]/(a*x) - (b*Cos[a]*SinIntegral[b*x ])/a^2 - (b*Sin[a]*SinIntegral[b*x])/a + (b*SinIntegral[a + b*x])/a^2))/2
3.1.24.3.1 Defintions of rubi rules used
Int[((c_.) + (d_.)*(x_))^(m_.)*SinIntegral[(a_.) + (b_.)*(x_)], x_Symbol] : > Simp[(c + d*x)^(m + 1)*(SinIntegral[a + b*x]/(d*(m + 1))), x] - Simp[b/(d *(m + 1)) Int[(c + d*x)^(m + 1)*(Sin[a + b*x]/(a + b*x)), x], x] /; FreeQ [{a, b, c, d, m}, x] && NeQ[m, -1]
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 | \(b^{2} \left (-\frac {\operatorname {Si}\left (b x +a \right )}{2 b^{2} x^{2}}-\frac {\operatorname {Si}\left (b x \right ) \cos \left (a \right )+\operatorname {Ci}\left (b x \right ) \sin \left (a \right )}{2 a^{2}}+\frac {\operatorname {Si}\left (b x +a \right )}{2 a^{2}}+\frac {-\frac {\sin \left (b x +a \right )}{b x}-\operatorname {Si}\left (b x \right ) \sin \left (a \right )+\operatorname {Ci}\left (b x \right ) \cos \left (a \right )}{2 a}\right )\) | \(86\) |
-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*S i(b*x+a)+1/a*(-sin(b*x+a)/b/x-Si(b*x)*sin(a)+Ci(b*x)*cos(a)))
Time = 0.28 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.86 \[ \int \frac {\text {Si}(a+b x)}{x^3} \, dx=-\frac {a b x \sin \left (b x + a\right ) - {\left (a b^{2} x^{2} \operatorname {Ci}\left (b x\right ) - b^{2} x^{2} \operatorname {Si}\left (b x\right )\right )} \cos \left (a\right ) + {\left (a b^{2} x^{2} \operatorname {Si}\left (b x\right ) + b^{2} x^{2} \operatorname {Ci}\left (b x\right )\right )} \sin \left (a\right ) - {\left (b^{2} x^{2} - a^{2}\right )} \operatorname {Si}\left (b x + a\right )}{2 \, a^{2} x^{2}} \]
-1/2*(a*b*x*sin(b*x + a) - (a*b^2*x^2*cos_integral(b*x) - b^2*x^2*sin_inte gral(b*x))*cos(a) + (a*b^2*x^2*sin_integral(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)
\[ \int \frac {\text {Si}(a+b x)}{x^3} \, dx=\int \frac {\operatorname {Si}{\left (a + b x \right )}}{x^{3}}\, dx \]
\[ \int \frac {\text {Si}(a+b x)}{x^3} \, dx=\int { \frac {\operatorname {Si}\left (b x + a\right )}{x^{3}} \,d x } \]
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 \[ \int \frac {\text {Si}(a+b x)}{x^3} \, dx=\text {Too large to display} \]
-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*ima g_part(cos_integral(b*x))*tan(1/2*b*x)^2*tan(1/2*a) - 2*a*b*x*imag_part(co s_integral(-b*x))*tan(1/2*b*x)^2*tan(1/2*a) + 4*a*b*x*sin_integral(b*x)*ta n(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*si n_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_integral(-b*x))*tan(1/2*b*x)^2 + 2*b*x*real_pa rt(cos_integral(b*x))*tan(1/2*b*x)^2*tan(1/2*a) + 2*b*x*real_part(cos_inte gral(-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_part(cos_integral(-b*x))*tan(1/2*a)^2 - b*x*ima g_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_integral(-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_par...
Timed out. \[ \int \frac {\text {Si}(a+b x)}{x^3} \, dx=\int \frac {\mathrm {sinint}\left (a+b\,x\right )}{x^3} \,d x \]