Integrand size = 17, antiderivative size = 121 \[ \int \frac {(a+b x)^2 \sin (c+d x)}{x^3} \, dx=-\frac {a^2 d \cos (c+d x)}{2 x}+2 a b d \cos (c) \operatorname {CosIntegral}(d x)+b^2 \operatorname {CosIntegral}(d x) \sin (c)-\frac {1}{2} a^2 d^2 \operatorname {CosIntegral}(d x) \sin (c)-\frac {a^2 \sin (c+d x)}{2 x^2}-\frac {2 a b \sin (c+d x)}{x}+b^2 \cos (c) \text {Si}(d x)-\frac {1}{2} a^2 d^2 \cos (c) \text {Si}(d x)-2 a b d \sin (c) \text {Si}(d x) \] Output:
-1/2*a^2*d*cos(d*x+c)/x+2*a*b*d*cos(c)*Ci(d*x)+b^2*Ci(d*x)*sin(c)-1/2*a^2* d^2*Ci(d*x)*sin(c)-1/2*a^2*sin(d*x+c)/x^2-2*a*b*sin(d*x+c)/x+b^2*cos(c)*Si (d*x)-1/2*a^2*d^2*cos(c)*Si(d*x)-2*a*b*d*sin(c)*Si(d*x)
Time = 0.45 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.79 \[ \int \frac {(a+b x)^2 \sin (c+d x)}{x^3} \, dx=\frac {1}{2} \left (\operatorname {CosIntegral}(d x) \left (4 a b d \cos (c)+\left (2 b^2-a^2 d^2\right ) \sin (c)\right )-\frac {a (a d x \cos (c+d x)+(a+4 b x) \sin (c+d x))}{x^2}+\left (\left (2 b^2-a^2 d^2\right ) \cos (c)-4 a b d \sin (c)\right ) \text {Si}(d x)\right ) \] Input:
Integrate[((a + b*x)^2*Sin[c + d*x])/x^3,x]
Output:
(CosIntegral[d*x]*(4*a*b*d*Cos[c] + (2*b^2 - a^2*d^2)*Sin[c]) - (a*(a*d*x* Cos[c + d*x] + (a + 4*b*x)*Sin[c + d*x]))/x^2 + ((2*b^2 - a^2*d^2)*Cos[c] - 4*a*b*d*Sin[c])*SinIntegral[d*x])/2
Time = 0.53 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, 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.
| \(\displaystyle \int \frac {(a+b x)^2 \sin (c+d x)}{x^3} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {a^2 \sin (c+d x)}{x^3}+\frac {2 a b \sin (c+d x)}{x^2}+\frac {b^2 \sin (c+d x)}{x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{2} a^2 d^2 \sin (c) \operatorname {CosIntegral}(d x)-\frac {1}{2} a^2 d^2 \cos (c) \text {Si}(d x)-\frac {a^2 \sin (c+d x)}{2 x^2}-\frac {a^2 d \cos (c+d x)}{2 x}+2 a b d \cos (c) \operatorname {CosIntegral}(d x)-2 a b d \sin (c) \text {Si}(d x)-\frac {2 a b \sin (c+d x)}{x}+b^2 \sin (c) \operatorname {CosIntegral}(d x)+b^2 \cos (c) \text {Si}(d x)\) |
Input:
Int[((a + b*x)^2*Sin[c + d*x])/x^3,x]
Output:
-1/2*(a^2*d*Cos[c + d*x])/x + 2*a*b*d*Cos[c]*CosIntegral[d*x] + b^2*CosInt egral[d*x]*Sin[c] - (a^2*d^2*CosIntegral[d*x]*Sin[c])/2 - (a^2*Sin[c + d*x ])/(2*x^2) - (2*a*b*Sin[c + d*x])/x + b^2*Cos[c]*SinIntegral[d*x] - (a^2*d ^2*Cos[c]*SinIntegral[d*x])/2 - 2*a*b*d*Sin[c]*SinIntegral[d*x]
Time = 1.00 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.94
| method | result | size |
| derivativedivides | \(d^{2} \left (a^{2} \left (-\frac {\sin \left (d x +c \right )}{2 d^{2} x^{2}}-\frac {\cos \left (d x +c \right )}{2 d x}-\frac {\operatorname {Si}\left (d x \right ) \cos \left (c \right )}{2}-\frac {\operatorname {Ci}\left (d x \right ) \sin \left (c \right )}{2}\right )+\frac {2 a b \left (-\frac {\sin \left (d x +c \right )}{d x}-\operatorname {Si}\left (d x \right ) \sin \left (c \right )+\operatorname {Ci}\left (d x \right ) \cos \left (c \right )\right )}{d}+\frac {b^{2} \left (\operatorname {Si}\left (d x \right ) \cos \left (c \right )+\operatorname {Ci}\left (d x \right ) \sin \left (c \right )\right )}{d^{2}}\right )\) | \(114\) |
| default | \(d^{2} \left (a^{2} \left (-\frac {\sin \left (d x +c \right )}{2 d^{2} x^{2}}-\frac {\cos \left (d x +c \right )}{2 d x}-\frac {\operatorname {Si}\left (d x \right ) \cos \left (c \right )}{2}-\frac {\operatorname {Ci}\left (d x \right ) \sin \left (c \right )}{2}\right )+\frac {2 a b \left (-\frac {\sin \left (d x +c \right )}{d x}-\operatorname {Si}\left (d x \right ) \sin \left (c \right )+\operatorname {Ci}\left (d x \right ) \cos \left (c \right )\right )}{d}+\frac {b^{2} \left (\operatorname {Si}\left (d x \right ) \cos \left (c \right )+\operatorname {Ci}\left (d x \right ) \sin \left (c \right )\right )}{d^{2}}\right )\) | \(114\) |
| risch | \(-\cos \left (c \right ) \operatorname {expIntegral}_{1}\left (-i d x \right ) a b d -\cos \left (c \right ) \operatorname {expIntegral}_{1}\left (i d x \right ) a b d -\frac {i \cos \left (c \right ) \operatorname {expIntegral}_{1}\left (-i d x \right ) a^{2} d^{2}}{4}+\frac {i \cos \left (c \right ) \operatorname {expIntegral}_{1}\left (i d x \right ) a^{2} d^{2}}{4}+\frac {i \cos \left (c \right ) \operatorname {expIntegral}_{1}\left (-i d x \right ) b^{2}}{2}-\frac {i \cos \left (c \right ) \operatorname {expIntegral}_{1}\left (i d x \right ) b^{2}}{2}-i \sin \left (c \right ) \operatorname {expIntegral}_{1}\left (-i d x \right ) a b d +i \sin \left (c \right ) \operatorname {expIntegral}_{1}\left (i d x \right ) a b d +\frac {\sin \left (c \right ) \operatorname {expIntegral}_{1}\left (-i d x \right ) a^{2} d^{2}}{4}+\frac {\sin \left (c \right ) \operatorname {expIntegral}_{1}\left (i d x \right ) a^{2} d^{2}}{4}-\frac {\sin \left (c \right ) \operatorname {expIntegral}_{1}\left (-i d x \right ) b^{2}}{2}-\frac {\sin \left (c \right ) \operatorname {expIntegral}_{1}\left (i d x \right ) b^{2}}{2}-\frac {a^{2} d \cos \left (d x +c \right )}{2 x}+\frac {\left (-8 a b \,d^{4} x^{3}-2 a^{2} d^{4} x^{2}\right ) \sin \left (d x +c \right )}{4 d^{4} x^{4}}\) | \(239\) |
| meijerg | \(\frac {b^{2} \sin \left (c \right ) \sqrt {\pi }\, \left (\frac {2 \gamma +2 \ln \left (x \right )+\ln \left (d^{2}\right )}{\sqrt {\pi }}-\frac {2 \gamma }{\sqrt {\pi }}-\frac {2 \ln \left (2\right )}{\sqrt {\pi }}-\frac {2 \ln \left (\frac {d x}{2}\right )}{\sqrt {\pi }}+\frac {2 \,\operatorname {Ci}\left (d x \right )}{\sqrt {\pi }}\right )}{2}+b^{2} \cos \left (c \right ) \operatorname {Si}\left (d x \right )+\frac {d^{2} a b \sin \left (c \right ) \sqrt {\pi }\, \left (-\frac {4 d^{2} \cos \left (x \sqrt {d^{2}}\right )}{x \left (d^{2}\right )^{\frac {3}{2}} \sqrt {\pi }}-\frac {4 \,\operatorname {Si}\left (x \sqrt {d^{2}}\right )}{\sqrt {\pi }}\right )}{2 \sqrt {d^{2}}}+\frac {d a b \cos \left (c \right ) \sqrt {\pi }\, \left (\frac {4 \gamma -4+4 \ln \left (x \right )+4 \ln \left (d \right )}{\sqrt {\pi }}+\frac {4}{\sqrt {\pi }}-\frac {4 \gamma }{\sqrt {\pi }}-\frac {4 \ln \left (2\right )}{\sqrt {\pi }}-\frac {4 \ln \left (\frac {d x}{2}\right )}{\sqrt {\pi }}-\frac {4 \sin \left (d x \right )}{\sqrt {\pi }\, x d}+\frac {4 \,\operatorname {Ci}\left (d x \right )}{\sqrt {\pi }}\right )}{2}+\frac {a^{2} \sin \left (c \right ) \sqrt {\pi }\, d^{2} \left (-\frac {4}{\sqrt {\pi }\, x^{2} d^{2}}-\frac {2 \left (2 \gamma -3+2 \ln \left (x \right )+\ln \left (d^{2}\right )\right )}{\sqrt {\pi }}+\frac {-6 x^{2} d^{2}+4}{\sqrt {\pi }\, x^{2} d^{2}}+\frac {4 \gamma }{\sqrt {\pi }}+\frac {4 \ln \left (2\right )}{\sqrt {\pi }}+\frac {4 \ln \left (\frac {d x}{2}\right )}{\sqrt {\pi }}-\frac {4 \cos \left (d x \right )}{\sqrt {\pi }\, x^{2} d^{2}}+\frac {4 \sin \left (d x \right )}{\sqrt {\pi }\, x d}-\frac {4 \,\operatorname {Ci}\left (d x \right )}{\sqrt {\pi }}\right )}{8}+\frac {a^{2} \cos \left (c \right ) \sqrt {\pi }\, d^{2} \left (-\frac {4 \cos \left (d x \right )}{d x \sqrt {\pi }}-\frac {4 \sin \left (d x \right )}{x^{2} d^{2} \sqrt {\pi }}-\frac {4 \,\operatorname {Si}\left (d x \right )}{\sqrt {\pi }}\right )}{8}\) | \(386\) |
Input:
int((b*x+a)^2*sin(d*x+c)/x^3,x,method=_RETURNVERBOSE)
Output:
d^2*(a^2*(-1/2*sin(d*x+c)/d^2/x^2-1/2*cos(d*x+c)/d/x-1/2*Si(d*x)*cos(c)-1/ 2*Ci(d*x)*sin(c))+2/d*a*b*(-sin(d*x+c)/d/x-Si(d*x)*sin(c)+Ci(d*x)*cos(c))+ 1/d^2*b^2*(Si(d*x)*cos(c)+Ci(d*x)*sin(c)))
Time = 0.08 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b x)^2 \sin (c+d x)}{x^3} \, dx=-\frac {a^{2} d x \cos \left (d x + c\right ) - {\left (4 \, a b d x^{2} \operatorname {Ci}\left (d x\right ) - {\left (a^{2} d^{2} - 2 \, b^{2}\right )} x^{2} \operatorname {Si}\left (d x\right )\right )} \cos \left (c\right ) + {\left (4 \, a b x + a^{2}\right )} \sin \left (d x + c\right ) + {\left (4 \, a b d x^{2} \operatorname {Si}\left (d x\right ) + {\left (a^{2} d^{2} - 2 \, b^{2}\right )} x^{2} \operatorname {Ci}\left (d x\right )\right )} \sin \left (c\right )}{2 \, x^{2}} \] Input:
integrate((b*x+a)^2*sin(d*x+c)/x^3,x, algorithm="fricas")
Output:
-1/2*(a^2*d*x*cos(d*x + c) - (4*a*b*d*x^2*cos_integral(d*x) - (a^2*d^2 - 2 *b^2)*x^2*sin_integral(d*x))*cos(c) + (4*a*b*x + a^2)*sin(d*x + c) + (4*a* b*d*x^2*sin_integral(d*x) + (a^2*d^2 - 2*b^2)*x^2*cos_integral(d*x))*sin(c ))/x^2
\[ \int \frac {(a+b x)^2 \sin (c+d x)}{x^3} \, dx=\int \frac {\left (a + b x\right )^{2} \sin {\left (c + d x \right )}}{x^{3}}\, dx \] Input:
integrate((b*x+a)**2*sin(d*x+c)/x**3,x)
Output:
Integral((a + b*x)**2*sin(c + d*x)/x**3, x)
Result contains complex when optimal does not.
Time = 0.96 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.56 \[ \int \frac {(a+b x)^2 \sin (c+d x)}{x^3} \, dx=-\frac {{\left ({\left (a^{2} {\left (-i \, \Gamma \left (-2, i \, d x\right ) + i \, \Gamma \left (-2, -i \, d x\right )\right )} \cos \left (c\right ) - a^{2} {\left (\Gamma \left (-2, i \, d x\right ) + \Gamma \left (-2, -i \, d x\right )\right )} \sin \left (c\right )\right )} d^{4} + 4 \, {\left (a b {\left (\Gamma \left (-2, i \, d x\right ) + \Gamma \left (-2, -i \, d x\right )\right )} \cos \left (c\right ) - a b {\left (i \, \Gamma \left (-2, i \, d x\right ) - i \, \Gamma \left (-2, -i \, d x\right )\right )} \sin \left (c\right )\right )} d^{3} - 2 \, {\left (b^{2} {\left (-i \, \Gamma \left (-2, i \, d x\right ) + i \, \Gamma \left (-2, -i \, d x\right )\right )} \cos \left (c\right ) - b^{2} {\left (\Gamma \left (-2, i \, d x\right ) + \Gamma \left (-2, -i \, d x\right )\right )} \sin \left (c\right )\right )} d^{2}\right )} x^{2} + 2 \, b^{2} \sin \left (d x + c\right ) + 2 \, {\left (b^{2} d x + 2 \, a b d\right )} \cos \left (d x + c\right )}{2 \, d^{2} x^{2}} \] Input:
integrate((b*x+a)^2*sin(d*x+c)/x^3,x, algorithm="maxima")
Output:
-1/2*(((a^2*(-I*gamma(-2, I*d*x) + I*gamma(-2, -I*d*x))*cos(c) - a^2*(gamm a(-2, I*d*x) + gamma(-2, -I*d*x))*sin(c))*d^4 + 4*(a*b*(gamma(-2, I*d*x) + gamma(-2, -I*d*x))*cos(c) - a*b*(I*gamma(-2, I*d*x) - I*gamma(-2, -I*d*x) )*sin(c))*d^3 - 2*(b^2*(-I*gamma(-2, I*d*x) + I*gamma(-2, -I*d*x))*cos(c) - b^2*(gamma(-2, I*d*x) + gamma(-2, -I*d*x))*sin(c))*d^2)*x^2 + 2*b^2*sin( d*x + c) + 2*(b^2*d*x + 2*a*b*d)*cos(d*x + c))/(d^2*x^2)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.15 (sec) , antiderivative size = 1182, normalized size of antiderivative = 9.77 \[ \int \frac {(a+b x)^2 \sin (c+d x)}{x^3} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^2*sin(d*x+c)/x^3,x, algorithm="giac")
Output:
1/4*(a^2*d^2*x^2*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 - a^2*d^2*x^2*imag_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*a^2*d^2*x^2*sin_integral(d*x)*tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*a^2*d^2*x^ 2*real_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c) - 2*a^2*d^2*x^2*r eal_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c) - 4*a*b*d*x^2*real_ part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 - 4*a*b*d*x^2*real_par t(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 - a^2*d^2*x^2*imag_part( cos_integral(d*x))*tan(1/2*d*x)^2 + a^2*d^2*x^2*imag_part(cos_integral(-d* x))*tan(1/2*d*x)^2 - 2*a^2*d^2*x^2*sin_integral(d*x)*tan(1/2*d*x)^2 - 8*a* b*d*x^2*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c) + 8*a*b*d*x ^2*imag_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c) - 16*a*b*d*x^2* sin_integral(d*x)*tan(1/2*d*x)^2*tan(1/2*c) + a^2*d^2*x^2*imag_part(cos_in tegral(d*x))*tan(1/2*c)^2 - a^2*d^2*x^2*imag_part(cos_integral(-d*x))*tan( 1/2*c)^2 + 2*a^2*d^2*x^2*sin_integral(d*x)*tan(1/2*c)^2 - 2*b^2*x^2*imag_p art(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*b^2*x^2*imag_part(c os_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 - 4*b^2*x^2*sin_integral(d* x)*tan(1/2*d*x)^2*tan(1/2*c)^2 + 4*a*b*d*x^2*real_part(cos_integral(d*x))* tan(1/2*d*x)^2 + 4*a*b*d*x^2*real_part(cos_integral(-d*x))*tan(1/2*d*x)^2 - 2*a^2*d^2*x^2*real_part(cos_integral(d*x))*tan(1/2*c) - 2*a^2*d^2*x^2*re al_part(cos_integral(-d*x))*tan(1/2*c) + 4*b^2*x^2*real_part(cos_integr...
Timed out. \[ \int \frac {(a+b x)^2 \sin (c+d x)}{x^3} \, dx=\int \frac {\sin \left (c+d\,x\right )\,{\left (a+b\,x\right )}^2}{x^3} \,d x \] Input:
int((sin(c + d*x)*(a + b*x)^2)/x^3,x)
Output:
int((sin(c + d*x)*(a + b*x)^2)/x^3, x)
\[ \int \frac {(a+b x)^2 \sin (c+d x)}{x^3} \, dx=\frac {-4 \cos \left (d x +c \right ) a b -2 \cos \left (d x +c \right ) b^{2} x -2 \left (\int \frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} x^{2}+x^{2}}d x \right ) a^{2} d^{2} x^{2}+4 \left (\int \frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} x^{2}+x^{2}}d x \right ) b^{2} x^{2}-16 \left (\int \frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} x^{3}+x^{3}}d x \right ) a b \,x^{2}-\sin \left (d x +c \right ) a^{2} d -a^{2} d^{2} x -4 a b +2 b^{2} x}{2 d \,x^{2}} \] Input:
int((b*x+a)^2*sin(d*x+c)/x^3,x)
Output:
( - 4*cos(c + d*x)*a*b - 2*cos(c + d*x)*b**2*x - 2*int(tan((c + d*x)/2)**2 /(tan((c + d*x)/2)**2*x**2 + x**2),x)*a**2*d**2*x**2 + 4*int(tan((c + d*x) /2)**2/(tan((c + d*x)/2)**2*x**2 + x**2),x)*b**2*x**2 - 16*int(1/(tan((c + d*x)/2)**2*x**3 + x**3),x)*a*b*x**2 - sin(c + d*x)*a**2*d - a**2*d**2*x - 4*a*b + 2*b**2*x)/(2*d*x**2)