Integrand size = 15, antiderivative size = 166 \[ \int \frac {(a+b x) \sin (c+d x)}{x^5} \, dx=-\frac {a d \cos (c+d x)}{12 x^3}-\frac {b d \cos (c+d x)}{6 x^2}+\frac {a d^3 \cos (c+d x)}{24 x}-\frac {1}{6} b d^3 \cos (c) \operatorname {CosIntegral}(d x)+\frac {1}{24} a d^4 \operatorname {CosIntegral}(d x) \sin (c)-\frac {a \sin (c+d x)}{4 x^4}-\frac {b \sin (c+d x)}{3 x^3}+\frac {a d^2 \sin (c+d x)}{24 x^2}+\frac {b d^2 \sin (c+d x)}{6 x}+\frac {1}{24} a d^4 \cos (c) \text {Si}(d x)+\frac {1}{6} b d^3 \sin (c) \text {Si}(d x) \] Output:
-1/12*a*d*cos(d*x+c)/x^3-1/6*b*d*cos(d*x+c)/x^2+1/24*a*d^3*cos(d*x+c)/x-1/ 6*b*d^3*cos(c)*Ci(d*x)+1/24*a*d^4*Ci(d*x)*sin(c)-1/4*a*sin(d*x+c)/x^4-1/3* b*sin(d*x+c)/x^3+1/24*a*d^2*sin(d*x+c)/x^2+1/6*b*d^2*sin(d*x+c)/x+1/24*a*d ^4*cos(c)*Si(d*x)+1/6*b*d^3*sin(c)*Si(d*x)
Time = 0.42 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b x) \sin (c+d x)}{x^5} \, dx=\frac {-2 a d x \cos (c+d x)-4 b d x^2 \cos (c+d x)+a d^3 x^3 \cos (c+d x)+d^3 x^4 \operatorname {CosIntegral}(d x) (-4 b \cos (c)+a d \sin (c))-6 a \sin (c+d x)-8 b x \sin (c+d x)+a d^2 x^2 \sin (c+d x)+4 b d^2 x^3 \sin (c+d x)+d^3 x^4 (a d \cos (c)+4 b \sin (c)) \text {Si}(d x)}{24 x^4} \] Input:
Integrate[((a + b*x)*Sin[c + d*x])/x^5,x]
Output:
(-2*a*d*x*Cos[c + d*x] - 4*b*d*x^2*Cos[c + d*x] + a*d^3*x^3*Cos[c + d*x] + d^3*x^4*CosIntegral[d*x]*(-4*b*Cos[c] + a*d*Sin[c]) - 6*a*Sin[c + d*x] - 8*b*x*Sin[c + d*x] + a*d^2*x^2*Sin[c + d*x] + 4*b*d^2*x^3*Sin[c + d*x] + d ^3*x^4*(a*d*Cos[c] + 4*b*Sin[c])*SinIntegral[d*x])/(24*x^4)
Time = 0.59 (sec) , antiderivative size = 166, 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.
| \(\displaystyle \int \frac {(a+b x) \sin (c+d x)}{x^5} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {a \sin (c+d x)}{x^5}+\frac {b \sin (c+d x)}{x^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{24} a d^4 \sin (c) \operatorname {CosIntegral}(d x)+\frac {1}{24} a d^4 \cos (c) \text {Si}(d x)+\frac {a d^3 \cos (c+d x)}{24 x}+\frac {a d^2 \sin (c+d x)}{24 x^2}-\frac {a \sin (c+d x)}{4 x^4}-\frac {a d \cos (c+d x)}{12 x^3}-\frac {1}{6} b d^3 \cos (c) \operatorname {CosIntegral}(d x)+\frac {1}{6} b d^3 \sin (c) \text {Si}(d x)+\frac {b d^2 \sin (c+d x)}{6 x}-\frac {b \sin (c+d x)}{3 x^3}-\frac {b d \cos (c+d x)}{6 x^2}\) |
Input:
Int[((a + b*x)*Sin[c + d*x])/x^5,x]
Output:
-1/12*(a*d*Cos[c + d*x])/x^3 - (b*d*Cos[c + d*x])/(6*x^2) + (a*d^3*Cos[c + d*x])/(24*x) - (b*d^3*Cos[c]*CosIntegral[d*x])/6 + (a*d^4*CosIntegral[d*x ]*Sin[c])/24 - (a*Sin[c + d*x])/(4*x^4) - (b*Sin[c + d*x])/(3*x^3) + (a*d^ 2*Sin[c + d*x])/(24*x^2) + (b*d^2*Sin[c + d*x])/(6*x) + (a*d^4*Cos[c]*SinI ntegral[d*x])/24 + (b*d^3*Sin[c]*SinIntegral[d*x])/6
Time = 0.97 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.87
| method | result | size |
| derivativedivides | \(d^{4} \left (a \left (-\frac {\sin \left (d x +c \right )}{4 d^{4} x^{4}}-\frac {\cos \left (d x +c \right )}{12 d^{3} x^{3}}+\frac {\sin \left (d x +c \right )}{24 d^{2} x^{2}}+\frac {\cos \left (d x +c \right )}{24 d x}+\frac {\operatorname {Si}\left (d x \right ) \cos \left (c \right )}{24}+\frac {\operatorname {Ci}\left (d x \right ) \sin \left (c \right )}{24}\right )+\frac {b \left (-\frac {\sin \left (d x +c \right )}{3 d^{3} x^{3}}-\frac {\cos \left (d x +c \right )}{6 d^{2} x^{2}}+\frac {\sin \left (d x +c \right )}{6 d x}+\frac {\operatorname {Si}\left (d x \right ) \sin \left (c \right )}{6}-\frac {\operatorname {Ci}\left (d x \right ) \cos \left (c \right )}{6}\right )}{d}\right )\) | \(145\) |
| default | \(d^{4} \left (a \left (-\frac {\sin \left (d x +c \right )}{4 d^{4} x^{4}}-\frac {\cos \left (d x +c \right )}{12 d^{3} x^{3}}+\frac {\sin \left (d x +c \right )}{24 d^{2} x^{2}}+\frac {\cos \left (d x +c \right )}{24 d x}+\frac {\operatorname {Si}\left (d x \right ) \cos \left (c \right )}{24}+\frac {\operatorname {Ci}\left (d x \right ) \sin \left (c \right )}{24}\right )+\frac {b \left (-\frac {\sin \left (d x +c \right )}{3 d^{3} x^{3}}-\frac {\cos \left (d x +c \right )}{6 d^{2} x^{2}}+\frac {\sin \left (d x +c \right )}{6 d x}+\frac {\operatorname {Si}\left (d x \right ) \sin \left (c \right )}{6}-\frac {\operatorname {Ci}\left (d x \right ) \cos \left (c \right )}{6}\right )}{d}\right )\) | \(145\) |
| risch | \(\frac {\operatorname {expIntegral}_{1}\left (-i d x \right ) \cos \left (c \right ) b \,d^{3}}{12}+\frac {\operatorname {expIntegral}_{1}\left (i d x \right ) \cos \left (c \right ) b \,d^{3}}{12}+\frac {i \operatorname {expIntegral}_{1}\left (-i d x \right ) \cos \left (c \right ) a \,d^{4}}{48}-\frac {i \operatorname {expIntegral}_{1}\left (i d x \right ) \cos \left (c \right ) a \,d^{4}}{48}+\frac {i \operatorname {expIntegral}_{1}\left (-i d x \right ) \sin \left (c \right ) b \,d^{3}}{12}-\frac {i \operatorname {expIntegral}_{1}\left (i d x \right ) \sin \left (c \right ) b \,d^{3}}{12}-\frac {\operatorname {expIntegral}_{1}\left (-i d x \right ) \sin \left (c \right ) a \,d^{4}}{48}-\frac {\operatorname {expIntegral}_{1}\left (i d x \right ) \sin \left (c \right ) a \,d^{4}}{48}-\frac {i \left (2 i a \,d^{8} x^{7}-8 i b \,d^{6} x^{6}-4 i a \,d^{6} x^{5}\right ) \cos \left (d x +c \right )}{48 d^{5} x^{8}}-\frac {\left (-8 b \,d^{7} x^{7}-2 a \,d^{7} x^{6}+16 b \,d^{5} x^{5}+12 a \,d^{5} x^{4}\right ) \sin \left (d x +c \right )}{48 d^{5} x^{8}}\) | \(223\) |
| meijerg | \(\frac {d^{4} b \sin \left (c \right ) \sqrt {\pi }\, \left (-\frac {8 \left (-x^{2} d^{2}+2\right ) d^{2} \cos \left (x \sqrt {d^{2}}\right )}{3 x^{3} \left (d^{2}\right )^{\frac {5}{2}} \sqrt {\pi }}+\frac {8 \sin \left (x \sqrt {d^{2}}\right )}{3 x^{2} d^{2} \sqrt {\pi }}+\frac {8 \,\operatorname {Si}\left (x \sqrt {d^{2}}\right )}{3 \sqrt {\pi }}\right )}{16 \sqrt {d^{2}}}+\frac {d^{3} b \cos \left (c \right ) \sqrt {\pi }\, \left (-\frac {8}{\sqrt {\pi }\, x^{2} d^{2}}-\frac {4 \left (2 \gamma -\frac {11}{3}+2 \ln \left (x \right )+2 \ln \left (d \right )\right )}{3 \sqrt {\pi }}+\frac {-\frac {44 x^{2} d^{2}}{9}+8}{\sqrt {\pi }\, x^{2} d^{2}}+\frac {8 \gamma }{3 \sqrt {\pi }}+\frac {8 \ln \left (2\right )}{3 \sqrt {\pi }}+\frac {8 \ln \left (\frac {d x}{2}\right )}{3 \sqrt {\pi }}-\frac {8 \cos \left (d x \right )}{3 \sqrt {\pi }\, x^{2} d^{2}}-\frac {16 \left (-\frac {5 x^{2} d^{2}}{2}+5\right ) \sin \left (d x \right )}{15 \sqrt {\pi }\, x^{3} d^{3}}-\frac {8 \,\operatorname {Ci}\left (d x \right )}{3 \sqrt {\pi }}\right )}{16}+\frac {a \sin \left (c \right ) \sqrt {\pi }\, d^{4} \left (-\frac {8}{\sqrt {\pi }\, x^{4} d^{4}}+\frac {8}{\sqrt {\pi }\, x^{2} d^{2}}+\frac {\frac {4 \gamma }{3}-\frac {25}{9}+\frac {4 \ln \left (x \right )}{3}+\frac {2 \ln \left (d^{2}\right )}{3}}{\sqrt {\pi }}+\frac {\frac {25}{9} x^{4} d^{4}-8 x^{2} d^{2}+8}{\sqrt {\pi }\, x^{4} d^{4}}-\frac {4 \gamma }{3 \sqrt {\pi }}-\frac {4 \ln \left (2\right )}{3 \sqrt {\pi }}-\frac {4 \ln \left (\frac {d x}{2}\right )}{3 \sqrt {\pi }}-\frac {8 \left (-\frac {15 x^{2} d^{2}}{2}+45\right ) \cos \left (d x \right )}{45 \sqrt {\pi }\, x^{4} d^{4}}+\frac {8 \left (-\frac {15 x^{2} d^{2}}{2}+15\right ) \sin \left (d x \right )}{45 \sqrt {\pi }\, x^{3} d^{3}}+\frac {4 \,\operatorname {Ci}\left (d x \right )}{3 \sqrt {\pi }}\right )}{32}+\frac {a \cos \left (c \right ) \sqrt {\pi }\, d^{4} \left (-\frac {8 \left (-\frac {x^{2} d^{2}}{2}+1\right ) \cos \left (d x \right )}{3 d^{3} x^{3} \sqrt {\pi }}-\frac {8 \left (-\frac {x^{2} d^{2}}{2}+3\right ) \sin \left (d x \right )}{3 d^{4} x^{4} \sqrt {\pi }}+\frac {4 \,\operatorname {Si}\left (d x \right )}{3 \sqrt {\pi }}\right )}{32}\) | \(453\) |
Input:
int((b*x+a)*sin(d*x+c)/x^5,x,method=_RETURNVERBOSE)
Output:
d^4*(a*(-1/4*sin(d*x+c)/d^4/x^4-1/12*cos(d*x+c)/d^3/x^3+1/24*sin(d*x+c)/d^ 2/x^2+1/24*cos(d*x+c)/d/x+1/24*Si(d*x)*cos(c)+1/24*Ci(d*x)*sin(c))+1/d*b*( -1/3*sin(d*x+c)/d^3/x^3-1/6*cos(d*x+c)/d^2/x^2+1/6*sin(d*x+c)/d/x+1/6*Si(d *x)*sin(c)-1/6*Ci(d*x)*cos(c)))
Time = 0.07 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.75 \[ \int \frac {(a+b x) \sin (c+d x)}{x^5} \, dx=\frac {{\left (a d^{3} x^{3} - 4 \, b d x^{2} - 2 \, a d x\right )} \cos \left (d x + c\right ) + {\left (a d^{4} x^{4} \operatorname {Si}\left (d x\right ) - 4 \, b d^{3} x^{4} \operatorname {Ci}\left (d x\right )\right )} \cos \left (c\right ) + {\left (4 \, b d^{2} x^{3} + a d^{2} x^{2} - 8 \, b x - 6 \, a\right )} \sin \left (d x + c\right ) + {\left (a d^{4} x^{4} \operatorname {Ci}\left (d x\right ) + 4 \, b d^{3} x^{4} \operatorname {Si}\left (d x\right )\right )} \sin \left (c\right )}{24 \, x^{4}} \] Input:
integrate((b*x+a)*sin(d*x+c)/x^5,x, algorithm="fricas")
Output:
1/24*((a*d^3*x^3 - 4*b*d*x^2 - 2*a*d*x)*cos(d*x + c) + (a*d^4*x^4*sin_inte gral(d*x) - 4*b*d^3*x^4*cos_integral(d*x))*cos(c) + (4*b*d^2*x^3 + a*d^2*x ^2 - 8*b*x - 6*a)*sin(d*x + c) + (a*d^4*x^4*cos_integral(d*x) + 4*b*d^3*x^ 4*sin_integral(d*x))*sin(c))/x^4
\[ \int \frac {(a+b x) \sin (c+d x)}{x^5} \, dx=\int \frac {\left (a + b x\right ) \sin {\left (c + d x \right )}}{x^{5}}\, dx \] Input:
integrate((b*x+a)*sin(d*x+c)/x**5,x)
Output:
Integral((a + b*x)*sin(c + d*x)/x**5, x)
Result contains complex when optimal does not.
Time = 0.35 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.66 \[ \int \frac {(a+b x) \sin (c+d x)}{x^5} \, dx=-\frac {{\left ({\left (a {\left (i \, \Gamma \left (-4, i \, d x\right ) - i \, \Gamma \left (-4, -i \, d x\right )\right )} \cos \left (c\right ) + a {\left (\Gamma \left (-4, i \, d x\right ) + \Gamma \left (-4, -i \, d x\right )\right )} \sin \left (c\right )\right )} d^{5} - 4 \, {\left (b {\left (\Gamma \left (-4, i \, d x\right ) + \Gamma \left (-4, -i \, d x\right )\right )} \cos \left (c\right ) + b {\left (-i \, \Gamma \left (-4, i \, d x\right ) + i \, \Gamma \left (-4, -i \, d x\right )\right )} \sin \left (c\right )\right )} d^{4}\right )} x^{4} + 2 \, b \cos \left (d x + c\right )}{2 \, d x^{4}} \] Input:
integrate((b*x+a)*sin(d*x+c)/x^5,x, algorithm="maxima")
Output:
-1/2*(((a*(I*gamma(-4, I*d*x) - I*gamma(-4, -I*d*x))*cos(c) + a*(gamma(-4, I*d*x) + gamma(-4, -I*d*x))*sin(c))*d^5 - 4*(b*(gamma(-4, I*d*x) + gamma( -4, -I*d*x))*cos(c) + b*(-I*gamma(-4, I*d*x) + I*gamma(-4, -I*d*x))*sin(c) )*d^4)*x^4 + 2*b*cos(d*x + c))/(d*x^4)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 1108, normalized size of antiderivative = 6.67 \[ \int \frac {(a+b x) \sin (c+d x)}{x^5} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)*sin(d*x+c)/x^5,x, algorithm="giac")
Output:
-1/48*(a*d^4*x^4*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 - a*d^4*x^4*imag_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2* a*d^4*x^4*sin_integral(d*x)*tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*a*d^4*x^4*real _part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c) - 2*a*d^4*x^4*real_part (cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c) - 4*b*d^3*x^4*real_part(cos _integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 - 4*b*d^3*x^4*real_part(cos_in tegral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 - a*d^4*x^4*imag_part(cos_integr al(d*x))*tan(1/2*d*x)^2 + a*d^4*x^4*imag_part(cos_integral(-d*x))*tan(1/2* d*x)^2 - 2*a*d^4*x^4*sin_integral(d*x)*tan(1/2*d*x)^2 - 8*b*d^3*x^4*imag_p art(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c) + 8*b*d^3*x^4*imag_part(c os_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c) - 16*b*d^3*x^4*sin_integral(d *x)*tan(1/2*d*x)^2*tan(1/2*c) + a*d^4*x^4*imag_part(cos_integral(d*x))*tan (1/2*c)^2 - a*d^4*x^4*imag_part(cos_integral(-d*x))*tan(1/2*c)^2 + 2*a*d^4 *x^4*sin_integral(d*x)*tan(1/2*c)^2 + 4*b*d^3*x^4*real_part(cos_integral(d *x))*tan(1/2*d*x)^2 + 4*b*d^3*x^4*real_part(cos_integral(-d*x))*tan(1/2*d* x)^2 - 2*a*d^4*x^4*real_part(cos_integral(d*x))*tan(1/2*c) - 2*a*d^4*x^4*r eal_part(cos_integral(-d*x))*tan(1/2*c) - 4*b*d^3*x^4*real_part(cos_integr al(d*x))*tan(1/2*c)^2 - 4*b*d^3*x^4*real_part(cos_integral(-d*x))*tan(1/2* c)^2 - 2*a*d^3*x^3*tan(1/2*d*x)^2*tan(1/2*c)^2 - a*d^4*x^4*imag_part(cos_i ntegral(d*x)) + a*d^4*x^4*imag_part(cos_integral(-d*x)) - 2*a*d^4*x^4*s...
Timed out. \[ \int \frac {(a+b x) \sin (c+d x)}{x^5} \, dx=\int \frac {\sin \left (c+d\,x\right )\,\left (a+b\,x\right )}{x^5} \,d x \] Input:
int((sin(c + d*x)*(a + b*x))/x^5,x)
Output:
int((sin(c + d*x)*(a + b*x))/x^5, x)
\[ \int \frac {(a+b x) \sin (c+d x)}{x^5} \, dx=\frac {-12 \cos \left (d x +c \right ) b -6 \left (\int \frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} x^{4}+x^{4}}d x \right ) a \,d^{2} x^{4}-96 \left (\int \frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} x^{5}+x^{5}}d x \right ) b \,x^{4}-3 \sin \left (d x +c \right ) a d -a \,d^{2} x -12 b}{12 d \,x^{4}} \] Input:
int((b*x+a)*sin(d*x+c)/x^5,x)
Output:
( - 12*cos(c + d*x)*b - 6*int(tan((c + d*x)/2)**2/(tan((c + d*x)/2)**2*x** 4 + x**4),x)*a*d**2*x**4 - 96*int(1/(tan((c + d*x)/2)**2*x**5 + x**5),x)*b *x**4 - 3*sin(c + d*x)*a*d - a*d**2*x - 12*b)/(12*d*x**4)