Integrand size = 15, antiderivative size = 48 \[ \int \frac {(a+b x) \sin (c+d x)}{x^2} \, dx=a d \cos (c) \operatorname {CosIntegral}(d x)+b \operatorname {CosIntegral}(d x) \sin (c)-\frac {a \sin (c+d x)}{x}+b \cos (c) \text {Si}(d x)-a d \sin (c) \text {Si}(d x) \] Output:
a*d*cos(c)*Ci(d*x)+b*Ci(d*x)*sin(c)-a*sin(d*x+c)/x+b*cos(c)*Si(d*x)-a*d*si n(c)*Si(d*x)
Time = 0.26 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.25 \[ \int \frac {(a+b x) \sin (c+d x)}{x^2} \, dx=-\frac {a \cos (d x) \sin (c)}{x}+b \operatorname {CosIntegral}(d x) \sin (c)-\frac {a \cos (c) \sin (d x)}{x}+b \cos (c) \text {Si}(d x)+a d (\cos (c) \operatorname {CosIntegral}(d x)-\sin (c) \text {Si}(d x)) \] Input:
Integrate[((a + b*x)*Sin[c + d*x])/x^2,x]
Output:
-((a*Cos[d*x]*Sin[c])/x) + b*CosIntegral[d*x]*Sin[c] - (a*Cos[c]*Sin[d*x]) /x + b*Cos[c]*SinIntegral[d*x] + a*d*(Cos[c]*CosIntegral[d*x] - Sin[c]*Sin Integral[d*x])
Time = 0.39 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {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.
\(\displaystyle \int \frac {(a+b x) \sin (c+d x)}{x^2} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {a \sin (c+d x)}{x^2}+\frac {b \sin (c+d x)}{x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle a d \cos (c) \operatorname {CosIntegral}(d x)-a d \sin (c) \text {Si}(d x)-\frac {a \sin (c+d x)}{x}+b \sin (c) \operatorname {CosIntegral}(d x)+b \cos (c) \text {Si}(d x)\) |
Input:
Int[((a + b*x)*Sin[c + d*x])/x^2,x]
Output:
a*d*Cos[c]*CosIntegral[d*x] + b*CosIntegral[d*x]*Sin[c] - (a*Sin[c + d*x]) /x + b*Cos[c]*SinIntegral[d*x] - a*d*Sin[c]*SinIntegral[d*x]
Time = 0.81 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.17
method | result | size |
derivativedivides | \(d \left (a \left (-\frac {\sin \left (d x +c \right )}{d x}-\operatorname {Si}\left (d x \right ) \sin \left (c \right )+\operatorname {Ci}\left (d x \right ) \cos \left (c \right )\right )+\frac {b \left (\operatorname {Si}\left (d x \right ) \cos \left (c \right )+\operatorname {Ci}\left (d x \right ) \sin \left (c \right )\right )}{d}\right )\) | \(56\) |
default | \(d \left (a \left (-\frac {\sin \left (d x +c \right )}{d x}-\operatorname {Si}\left (d x \right ) \sin \left (c \right )+\operatorname {Ci}\left (d x \right ) \cos \left (c \right )\right )+\frac {b \left (\operatorname {Si}\left (d x \right ) \cos \left (c \right )+\operatorname {Ci}\left (d x \right ) \sin \left (c \right )\right )}{d}\right )\) | \(56\) |
risch | \(\frac {i {\mathrm e}^{i c} \operatorname {expIntegral}_{1}\left (-i d x \right ) b}{2}-\frac {a \,{\mathrm e}^{i c} \operatorname {expIntegral}_{1}\left (-i d x \right ) d}{2}-\frac {i {\mathrm e}^{-i c} \operatorname {expIntegral}_{1}\left (i d x \right ) b}{2}-\frac {a \,{\mathrm e}^{-i c} \operatorname {expIntegral}_{1}\left (i d x \right ) d}{2}-\frac {a \sin \left (d x +c \right )}{x}\) | \(78\) |
meijerg | \(\frac {b \sin \left (c \right ) \sqrt {\pi }\, \left (\frac {2 \gamma +2 \ln \left (x \right )+\ln \left (d^{2}\right )}{\sqrt {\pi }}-\frac {2 \gamma }{\sqrt {\pi }}-\frac {2 \ln \left (2\right )}{\sqrt {\pi }}-\frac {2 \ln \left (\frac {d x}{2}\right )}{\sqrt {\pi }}+\frac {2 \,\operatorname {Ci}\left (d x \right )}{\sqrt {\pi }}\right )}{2}+b \cos \left (c \right ) \operatorname {Si}\left (d x \right )+\frac {a \sin \left (c \right ) \sqrt {\pi }\, d^{2} \left (-\frac {4 d^{2} \cos \left (x \sqrt {d^{2}}\right )}{x \left (d^{2}\right )^{\frac {3}{2}} \sqrt {\pi }}-\frac {4 \,\operatorname {Si}\left (x \sqrt {d^{2}}\right )}{\sqrt {\pi }}\right )}{4 \sqrt {d^{2}}}+\frac {a \cos \left (c \right ) \sqrt {\pi }\, d \left (\frac {4 \gamma -4+4 \ln \left (x \right )+4 \ln \left (d \right )}{\sqrt {\pi }}+\frac {4}{\sqrt {\pi }}-\frac {4 \gamma }{\sqrt {\pi }}-\frac {4 \ln \left (2\right )}{\sqrt {\pi }}-\frac {4 \ln \left (\frac {d x}{2}\right )}{\sqrt {\pi }}-\frac {4 \sin \left (d x \right )}{\sqrt {\pi }\, x d}+\frac {4 \,\operatorname {Ci}\left (d x \right )}{\sqrt {\pi }}\right )}{4}\) | \(201\) |
Input:
int((b*x+a)*sin(d*x+c)/x^2,x,method=_RETURNVERBOSE)
Output:
d*(a*(-sin(d*x+c)/d/x-Si(d*x)*sin(c)+Ci(d*x)*cos(c))+1/d*b*(Si(d*x)*cos(c) +Ci(d*x)*sin(c)))
Time = 0.07 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.12 \[ \int \frac {(a+b x) \sin (c+d x)}{x^2} \, dx=\frac {{\left (a d x \operatorname {Ci}\left (d x\right ) + b x \operatorname {Si}\left (d x\right )\right )} \cos \left (c\right ) - a \sin \left (d x + c\right ) - {\left (a d x \operatorname {Si}\left (d x\right ) - b x \operatorname {Ci}\left (d x\right )\right )} \sin \left (c\right )}{x} \] Input:
integrate((b*x+a)*sin(d*x+c)/x^2,x, algorithm="fricas")
Output:
((a*d*x*cos_integral(d*x) + b*x*sin_integral(d*x))*cos(c) - a*sin(d*x + c) - (a*d*x*sin_integral(d*x) - b*x*cos_integral(d*x))*sin(c))/x
\[ \int \frac {(a+b x) \sin (c+d x)}{x^2} \, dx=\int \frac {\left (a + b x\right ) \sin {\left (c + d x \right )}}{x^{2}}\, dx \] Input:
integrate((b*x+a)*sin(d*x+c)/x**2,x)
Output:
Integral((a + b*x)*sin(c + d*x)/x**2, x)
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 108, normalized size of antiderivative = 2.25 \[ \int \frac {(a+b x) \sin (c+d x)}{x^2} \, dx=\frac {{\left ({\left (a {\left (\Gamma \left (-1, i \, d x\right ) + \Gamma \left (-1, -i \, d x\right )\right )} \cos \left (c\right ) - a {\left (i \, \Gamma \left (-1, i \, d x\right ) - i \, \Gamma \left (-1, -i \, d x\right )\right )} \sin \left (c\right )\right )} d^{2} - {\left (b {\left (-i \, \Gamma \left (-1, i \, d x\right ) + i \, \Gamma \left (-1, -i \, d x\right )\right )} \cos \left (c\right ) - b {\left (\Gamma \left (-1, i \, d x\right ) + \Gamma \left (-1, -i \, d x\right )\right )} \sin \left (c\right )\right )} d\right )} x - 2 \, b \cos \left (d x + c\right )}{2 \, d x} \] Input:
integrate((b*x+a)*sin(d*x+c)/x^2,x, algorithm="maxima")
Output:
1/2*(((a*(gamma(-1, I*d*x) + gamma(-1, -I*d*x))*cos(c) - a*(I*gamma(-1, I* d*x) - I*gamma(-1, -I*d*x))*sin(c))*d^2 - (b*(-I*gamma(-1, I*d*x) + I*gamm a(-1, -I*d*x))*cos(c) - b*(gamma(-1, I*d*x) + gamma(-1, -I*d*x))*sin(c))*d )*x - 2*b*cos(d*x + c))/(d*x)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 569, normalized size of antiderivative = 11.85 \[ \int \frac {(a+b x) \sin (c+d x)}{x^2} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)*sin(d*x+c)/x^2,x, algorithm="giac")
Output:
-1/2*(a*d*x*real_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 + a*d *x*real_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*a*d*x*ima g_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c) - 2*a*d*x*imag_part(co s_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c) + 4*a*d*x*sin_integral(d*x)*ta n(1/2*d*x)^2*tan(1/2*c) + b*x*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2* tan(1/2*c)^2 - b*x*imag_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c) ^2 + 2*b*x*sin_integral(d*x)*tan(1/2*d*x)^2*tan(1/2*c)^2 - a*d*x*real_part (cos_integral(d*x))*tan(1/2*d*x)^2 - a*d*x*real_part(cos_integral(-d*x))*t an(1/2*d*x)^2 - 2*b*x*real_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2* c) - 2*b*x*real_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c) + a*d*x *real_part(cos_integral(d*x))*tan(1/2*c)^2 + a*d*x*real_part(cos_integral( -d*x))*tan(1/2*c)^2 - b*x*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2 + b* x*imag_part(cos_integral(-d*x))*tan(1/2*d*x)^2 - 2*b*x*sin_integral(d*x)*t an(1/2*d*x)^2 + 2*a*d*x*imag_part(cos_integral(d*x))*tan(1/2*c) - 2*a*d*x* imag_part(cos_integral(-d*x))*tan(1/2*c) + 4*a*d*x*sin_integral(d*x)*tan(1 /2*c) + b*x*imag_part(cos_integral(d*x))*tan(1/2*c)^2 - b*x*imag_part(cos_ integral(-d*x))*tan(1/2*c)^2 + 2*b*x*sin_integral(d*x)*tan(1/2*c)^2 - a*d* x*real_part(cos_integral(d*x)) - a*d*x*real_part(cos_integral(-d*x)) - 2*b *x*real_part(cos_integral(d*x))*tan(1/2*c) - 2*b*x*real_part(cos_integral( -d*x))*tan(1/2*c) - 4*a*tan(1/2*d*x)^2*tan(1/2*c) - 4*a*tan(1/2*d*x)*ta...
Timed out. \[ \int \frac {(a+b x) \sin (c+d x)}{x^2} \, dx=\int \frac {\sin \left (c+d\,x\right )\,\left (a+b\,x\right )}{x^2} \,d x \] Input:
int((sin(c + d*x)*(a + b*x))/x^2,x)
Output:
int((sin(c + d*x)*(a + b*x))/x^2, x)
\[ \int \frac {(a+b x) \sin (c+d x)}{x^2} \, dx=\frac {-\cos \left (d x +c \right ) b +2 \left (\int \frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} x^{2}+x^{2}}d x \right ) b x +2 \left (\int \frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} x +x}d x \right ) a \,d^{2} x -\mathrm {log}\left (x \right ) a \,d^{2} x -\sin \left (d x +c \right ) a d +b}{d x} \] Input:
int((b*x+a)*sin(d*x+c)/x^2,x)
Output:
( - cos(c + d*x)*b + 2*int(tan((c + d*x)/2)**2/(tan((c + d*x)/2)**2*x**2 + x**2),x)*b*x + 2*int(1/(tan((c + d*x)/2)**2*x + x),x)*a*d**2*x - log(x)*a *d**2*x - sin(c + d*x)*a*d + b)/(d*x)