Integrand size = 17, antiderivative size = 189 \[ \int \frac {\sin (c+d x)}{x^3 (a+b x)} \, dx=-\frac {d \cos (c+d x)}{2 a x}-\frac {b d \cos (c) \operatorname {CosIntegral}(d x)}{a^2}+\frac {b^2 \operatorname {CosIntegral}(d x) \sin (c)}{a^3}-\frac {d^2 \operatorname {CosIntegral}(d x) \sin (c)}{2 a}-\frac {b^2 \operatorname {CosIntegral}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{a^3}-\frac {\sin (c+d x)}{2 a x^2}+\frac {b \sin (c+d x)}{a^2 x}+\frac {b^2 \cos (c) \text {Si}(d x)}{a^3}-\frac {d^2 \cos (c) \text {Si}(d x)}{2 a}+\frac {b d \sin (c) \text {Si}(d x)}{a^2}-\frac {b^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{a^3} \]
-b*d*Ci(d*x)*cos(c)/a^2-1/2*d*cos(d*x+c)/a/x+b^2*cos(c)*Si(d*x)/a^3-1/2*d^ 2*cos(c)*Si(d*x)/a-b^2*cos(-c+a*d/b)*Si(a*d/b+d*x)/a^3+b^2*Ci(d*x)*sin(c)/ a^3-1/2*d^2*Ci(d*x)*sin(c)/a+b*d*Si(d*x)*sin(c)/a^2+b^2*Ci(a*d/b+d*x)*sin( -c+a*d/b)/a^3-1/2*sin(d*x+c)/a/x^2+b*sin(d*x+c)/a^2/x
Time = 0.43 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.93 \[ \int \frac {\sin (c+d x)}{x^3 (a+b x)} \, dx=-\frac {a^2 d x \cos (c+d x)+x^2 \operatorname {CosIntegral}(d x) \left (2 a b d \cos (c)+\left (-2 b^2+a^2 d^2\right ) \sin (c)\right )+2 b^2 x^2 \operatorname {CosIntegral}\left (d \left (\frac {a}{b}+x\right )\right ) \sin \left (c-\frac {a d}{b}\right )+a^2 \sin (c+d x)-2 a b x \sin (c+d x)-2 b^2 x^2 \cos (c) \text {Si}(d x)+a^2 d^2 x^2 \cos (c) \text {Si}(d x)-2 a b d x^2 \sin (c) \text {Si}(d x)+2 b^2 x^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )}{2 a^3 x^2} \]
-1/2*(a^2*d*x*Cos[c + d*x] + x^2*CosIntegral[d*x]*(2*a*b*d*Cos[c] + (-2*b^ 2 + a^2*d^2)*Sin[c]) + 2*b^2*x^2*CosIntegral[d*(a/b + x)]*Sin[c - (a*d)/b] + a^2*Sin[c + d*x] - 2*a*b*x*Sin[c + d*x] - 2*b^2*x^2*Cos[c]*SinIntegral[ d*x] + a^2*d^2*x^2*Cos[c]*SinIntegral[d*x] - 2*a*b*d*x^2*Sin[c]*SinIntegra l[d*x] + 2*b^2*x^2*Cos[c - (a*d)/b]*SinIntegral[d*(a/b + x)])/(a^3*x^2)
Time = 0.67 (sec) , antiderivative size = 189, 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.
| \(\displaystyle \int \frac {\sin (c+d x)}{x^3 (a+b x)} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {b^3 \sin (c+d x)}{a^3 (a+b x)}+\frac {b^2 \sin (c+d x)}{a^3 x}-\frac {b \sin (c+d x)}{a^2 x^2}+\frac {\sin (c+d x)}{a x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b^2 \sin (c) \operatorname {CosIntegral}(d x)}{a^3}-\frac {b^2 \sin \left (c-\frac {a d}{b}\right ) \operatorname {CosIntegral}\left (x d+\frac {a d}{b}\right )}{a^3}+\frac {b^2 \cos (c) \text {Si}(d x)}{a^3}-\frac {b^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{a^3}-\frac {b d \cos (c) \operatorname {CosIntegral}(d x)}{a^2}+\frac {b d \sin (c) \text {Si}(d x)}{a^2}+\frac {b \sin (c+d x)}{a^2 x}-\frac {d^2 \sin (c) \operatorname {CosIntegral}(d x)}{2 a}-\frac {d^2 \cos (c) \text {Si}(d x)}{2 a}-\frac {\sin (c+d x)}{2 a x^2}-\frac {d \cos (c+d x)}{2 a x}\) |
-1/2*(d*Cos[c + d*x])/(a*x) - (b*d*Cos[c]*CosIntegral[d*x])/a^2 + (b^2*Cos Integral[d*x]*Sin[c])/a^3 - (d^2*CosIntegral[d*x]*Sin[c])/(2*a) - (b^2*Cos Integral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/a^3 - Sin[c + d*x]/(2*a*x^2) + ( b*Sin[c + d*x])/(a^2*x) + (b^2*Cos[c]*SinIntegral[d*x])/a^3 - (d^2*Cos[c]* SinIntegral[d*x])/(2*a) + (b*d*Sin[c]*SinIntegral[d*x])/a^2 - (b^2*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/a^3
3.1.25.3.1 Defintions of rubi rules used
Time = 0.36 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.07
| method | result | size |
| derivativedivides | \(d^{2} \left (-\frac {b^{3} \left (\frac {\operatorname {Si}\left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{d^{2} a^{3}}-\frac {b \left (-\frac {\sin \left (d x +c \right )}{d x}-\operatorname {Si}\left (d x \right ) \sin \left (c \right )+\operatorname {Ci}\left (d x \right ) \cos \left (c \right )\right )}{a^{2} d}+\frac {b^{2} \left (\operatorname {Si}\left (d x \right ) \cos \left (c \right )+\operatorname {Ci}\left (d x \right ) \sin \left (c \right )\right )}{a^{3} d^{2}}+\frac {-\frac {\sin \left (d x +c \right )}{2 d^{2} x^{2}}-\frac {\cos \left (d x +c \right )}{2 d x}-\frac {\operatorname {Si}\left (d x \right ) \cos \left (c \right )}{2}-\frac {\operatorname {Ci}\left (d x \right ) \sin \left (c \right )}{2}}{a}\right )\) | \(202\) |
| default | \(d^{2} \left (-\frac {b^{3} \left (\frac {\operatorname {Si}\left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{d^{2} a^{3}}-\frac {b \left (-\frac {\sin \left (d x +c \right )}{d x}-\operatorname {Si}\left (d x \right ) \sin \left (c \right )+\operatorname {Ci}\left (d x \right ) \cos \left (c \right )\right )}{a^{2} d}+\frac {b^{2} \left (\operatorname {Si}\left (d x \right ) \cos \left (c \right )+\operatorname {Ci}\left (d x \right ) \sin \left (c \right )\right )}{a^{3} d^{2}}+\frac {-\frac {\sin \left (d x +c \right )}{2 d^{2} x^{2}}-\frac {\cos \left (d x +c \right )}{2 d x}-\frac {\operatorname {Si}\left (d x \right ) \cos \left (c \right )}{2}-\frac {\operatorname {Ci}\left (d x \right ) \sin \left (c \right )}{2}}{a}\right )\) | \(202\) |
| risch | \(-\frac {i d^{2} {\mathrm e}^{i c} \operatorname {Ei}_{1}\left (-i d x \right )}{4 a}+\frac {i b^{2} {\mathrm e}^{i c} \operatorname {Ei}_{1}\left (-i d x \right )}{2 a^{3}}-\frac {i b^{2} {\mathrm e}^{-\frac {i \left (d a -c b \right )}{b}} \operatorname {Ei}_{1}\left (-i d x -i c -\frac {i a d -i c b}{b}\right )}{2 a^{3}}+\frac {d b \,{\mathrm e}^{i c} \operatorname {Ei}_{1}\left (-i d x \right )}{2 a^{2}}+\frac {d \,{\mathrm e}^{-i c} \operatorname {Ei}_{1}\left (i d x \right ) b}{2 a^{2}}+\frac {i d^{2} {\mathrm e}^{-i c} \operatorname {Ei}_{1}\left (i d x \right )}{4 a}-\frac {i {\mathrm e}^{-i c} \operatorname {Ei}_{1}\left (i d x \right ) b^{2}}{2 a^{3}}+\frac {i b^{2} {\mathrm e}^{\frac {i \left (d a -c b \right )}{b}} \operatorname {Ei}_{1}\left (i d x +i c +\frac {i \left (d a -c b \right )}{b}\right )}{2 a^{3}}-\frac {d \cos \left (d x +c \right )}{2 a x}-\frac {\left (-4 b x +2 a \right ) \sin \left (d x +c \right )}{4 a^{2} x^{2}}\) | \(263\) |
d^2*(-1/d^2*b^3/a^3*(Si(d*x+c+(a*d-b*c)/b)*cos((a*d-b*c)/b)/b-Ci(d*x+c+(a* d-b*c)/b)*sin((a*d-b*c)/b)/b)-b/a^2/d*(-sin(d*x+c)/d/x-Si(d*x)*sin(c)+Ci(d *x)*cos(c))+b^2/a^3/d^2*(Si(d*x)*cos(c)+Ci(d*x)*sin(c))+1/a*(-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)))
Time = 0.28 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.98 \[ \int \frac {\sin (c+d x)}{x^3 (a+b x)} \, dx=\frac {2 \, b^{2} x^{2} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) \sin \left (-\frac {b c - a d}{b}\right ) - 2 \, b^{2} x^{2} \cos \left (-\frac {b c - a d}{b}\right ) \operatorname {Si}\left (\frac {b d x + a d}{b}\right ) - a^{2} d x \cos \left (d x + c\right ) - {\left (2 \, a b d x^{2} \operatorname {Ci}\left (d x\right ) + {\left (a^{2} d^{2} - 2 \, b^{2}\right )} x^{2} \operatorname {Si}\left (d x\right )\right )} \cos \left (c\right ) + {\left (2 \, a b x - a^{2}\right )} \sin \left (d x + c\right ) + {\left (2 \, a b d x^{2} \operatorname {Si}\left (d x\right ) - {\left (a^{2} d^{2} - 2 \, b^{2}\right )} x^{2} \operatorname {Ci}\left (d x\right )\right )} \sin \left (c\right )}{2 \, a^{3} x^{2}} \]
1/2*(2*b^2*x^2*cos_integral((b*d*x + a*d)/b)*sin(-(b*c - a*d)/b) - 2*b^2*x ^2*cos(-(b*c - a*d)/b)*sin_integral((b*d*x + a*d)/b) - a^2*d*x*cos(d*x + c ) - (2*a*b*d*x^2*cos_integral(d*x) + (a^2*d^2 - 2*b^2)*x^2*sin_integral(d* x))*cos(c) + (2*a*b*x - a^2)*sin(d*x + c) + (2*a*b*d*x^2*sin_integral(d*x) - (a^2*d^2 - 2*b^2)*x^2*cos_integral(d*x))*sin(c))/(a^3*x^2)
\[ \int \frac {\sin (c+d x)}{x^3 (a+b x)} \, dx=\int \frac {\sin {\left (c + d x \right )}}{x^{3} \left (a + b x\right )}\, dx \]
\[ \int \frac {\sin (c+d x)}{x^3 (a+b x)} \, dx=\int { \frac {\sin \left (d x + c\right )}{{\left (b x + a\right )} x^{3}} \,d x } \]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.35 (sec) , antiderivative size = 4565, normalized size of antiderivative = 24.15 \[ \int \frac {\sin (c+d x)}{x^3 (a+b x)} \, dx=\text {Too large to display} \]
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* tan(1/2*a*d/b)^2 - a^2*d^2*x^2*imag_part(cos_integral(-d*x))*tan(1/2*d*x)^ 2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a^2*d^2*x^2*sin_integral(d*x)*tan(1/2* d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 2*a^2*d^2*x^2*real_part(cos_integra l(d*x))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*a^2*d^2*x^2*real_pa rt(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 + 2*a*b* d*x^2*real_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d /b)^2 + 2*a*b*d*x^2*real_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c )^2*tan(1/2*a*d/b)^2 + 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 - a^2*d^2*x^2*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*a*d/ b)^2 + a^2*d^2*x^2*imag_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*a* d/b)^2 - 2*a^2*d^2*x^2*sin_integral(d*x)*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + 4*a*b*d*x^2*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/ 2*a*d/b)^2 - 4*a*b*d*x^2*imag_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan( 1/2*c)*tan(1/2*a*d/b)^2 + 8*a*b*d*x^2*sin_integral(d*x)*tan(1/2*d*x)^2*tan (1/2*c)*tan(1/2*a*d/b)^2 + a^2*d^2*x^2*imag_part(cos_integral(d*x))*tan(1/ 2*c)^2*tan(1/2*a*d/b)^2 - a^2*d^2*x^2*imag_part(cos_integral(-d*x))*tan(1/ 2*c)^2*tan(1/2*a*d/b)^2 + 2*a^2*d^2*x^2*sin_integral(d*x)*tan(1/2*c)^2*...
Timed out. \[ \int \frac {\sin (c+d x)}{x^3 (a+b x)} \, dx=\int \frac {\sin \left (c+d\,x\right )}{x^3\,\left (a+b\,x\right )} \,d x \]