Integrand size = 17, antiderivative size = 188 \[ \int \frac {\sin (c+d x)}{x^2 (a+b x)^2} \, dx=\frac {d \cos (c) \operatorname {CosIntegral}(d x)}{a^2}+\frac {d \cos \left (c-\frac {a d}{b}\right ) \operatorname {CosIntegral}\left (\frac {a d}{b}+d x\right )}{a^2}-\frac {2 b \operatorname {CosIntegral}(d x) \sin (c)}{a^3}+\frac {2 b \operatorname {CosIntegral}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{a^3}-\frac {\sin (c+d x)}{a^2 x}-\frac {b \sin (c+d x)}{a^2 (a+b x)}-\frac {2 b \cos (c) \text {Si}(d x)}{a^3}-\frac {d \sin (c) \text {Si}(d x)}{a^2}+\frac {2 b \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{a^3}-\frac {d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{a^2} \]
d*Ci(d*x)*cos(c)/a^2+d*Ci(a*d/b+d*x)*cos(-c+a*d/b)/a^2-2*b*cos(c)*Si(d*x)/ a^3+2*b*cos(-c+a*d/b)*Si(a*d/b+d*x)/a^3-2*b*Ci(d*x)*sin(c)/a^3-d*Si(d*x)*s in(c)/a^2-2*b*Ci(a*d/b+d*x)*sin(-c+a*d/b)/a^3+d*Si(a*d/b+d*x)*sin(-c+a*d/b )/a^2-sin(d*x+c)/a^2/x-b*sin(d*x+c)/a^2/(b*x+a)
Time = 1.25 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.98 \[ \int \frac {\sin (c+d x)}{x^2 (a+b x)^2} \, dx=-\frac {-a d \cos (c) \operatorname {CosIntegral}(d x)-a d \cos \left (c-\frac {a d}{b}\right ) \operatorname {CosIntegral}\left (d \left (\frac {a}{b}+x\right )\right )+\frac {a (a+2 b x) \cos (d x) \sin (c)}{x (a+b x)}+2 b \operatorname {CosIntegral}(d x) \sin (c)-2 b \operatorname {CosIntegral}\left (d \left (\frac {a}{b}+x\right )\right ) \sin \left (c-\frac {a d}{b}\right )+\frac {a (a+2 b x) \cos (c) \sin (d x)}{x (a+b x)}+2 b \cos (c) \text {Si}(d x)+a d \sin (c) \text {Si}(d x)-2 b \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )+a d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )}{a^3} \]
-((-(a*d*Cos[c]*CosIntegral[d*x]) - a*d*Cos[c - (a*d)/b]*CosIntegral[d*(a/ b + x)] + (a*(a + 2*b*x)*Cos[d*x]*Sin[c])/(x*(a + b*x)) + 2*b*CosIntegral[ d*x]*Sin[c] - 2*b*CosIntegral[d*(a/b + x)]*Sin[c - (a*d)/b] + (a*(a + 2*b* x)*Cos[c]*Sin[d*x])/(x*(a + b*x)) + 2*b*Cos[c]*SinIntegral[d*x] + a*d*Sin[ c]*SinIntegral[d*x] - 2*b*Cos[c - (a*d)/b]*SinIntegral[d*(a/b + x)] + a*d* Sin[c - (a*d)/b]*SinIntegral[d*(a/b + x)])/a^3)
Time = 0.70 (sec) , antiderivative size = 188, 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.
| \(\displaystyle \int \frac {\sin (c+d x)}{x^2 (a+b x)^2} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 b^2 \sin (c+d x)}{a^3 (a+b x)}-\frac {2 b \sin (c+d x)}{a^3 x}+\frac {b^2 \sin (c+d x)}{a^2 (a+b x)^2}+\frac {\sin (c+d x)}{a^2 x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 b \sin (c) \operatorname {CosIntegral}(d x)}{a^3}+\frac {2 b \sin \left (c-\frac {a d}{b}\right ) \operatorname {CosIntegral}\left (x d+\frac {a d}{b}\right )}{a^3}-\frac {2 b \cos (c) \text {Si}(d x)}{a^3}+\frac {2 b \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{a^3}+\frac {d \cos \left (c-\frac {a d}{b}\right ) \operatorname {CosIntegral}\left (x d+\frac {a d}{b}\right )}{a^2}-\frac {d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{a^2}-\frac {b \sin (c+d x)}{a^2 (a+b x)}+\frac {d \cos (c) \operatorname {CosIntegral}(d x)}{a^2}-\frac {d \sin (c) \text {Si}(d x)}{a^2}-\frac {\sin (c+d x)}{a^2 x}\) |
(d*Cos[c]*CosIntegral[d*x])/a^2 + (d*Cos[c - (a*d)/b]*CosIntegral[(a*d)/b + d*x])/a^2 - (2*b*CosIntegral[d*x]*Sin[c])/a^3 + (2*b*CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/a^3 - Sin[c + d*x]/(a^2*x) - (b*Sin[c + d*x])/(a ^2*(a + b*x)) - (2*b*Cos[c]*SinIntegral[d*x])/a^3 - (d*Sin[c]*SinIntegral[ d*x])/a^2 + (2*b*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/a^3 - (d*Sin [c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/a^2
3.1.32.3.1 Defintions of rubi rules used
Time = 0.34 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.36
| method | result | size |
| derivativedivides | \(d \left (\frac {-\frac {\sin \left (d x +c \right )}{d x}-\operatorname {Si}\left (d x \right ) \sin \left (c \right )+\operatorname {Ci}\left (d x \right ) \cos \left (c \right )}{a^{2}}+\frac {b^{2} \left (-\frac {\sin \left (d x +c \right )}{\left (d a -c b +b \left (d x +c \right )\right ) b}+\frac {\frac {\operatorname {Si}\left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}+\frac {\operatorname {Ci}\left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}}{b}\right )}{a^{2}}-\frac {2 b \left (\operatorname {Si}\left (d x \right ) \cos \left (c \right )+\operatorname {Ci}\left (d x \right ) \sin \left (c \right )\right )}{d \,a^{3}}+\frac {2 b^{2} \left (\frac {\operatorname {Si}\left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{d \,a^{3}}\right )\) | \(256\) |
| default | \(d \left (\frac {-\frac {\sin \left (d x +c \right )}{d x}-\operatorname {Si}\left (d x \right ) \sin \left (c \right )+\operatorname {Ci}\left (d x \right ) \cos \left (c \right )}{a^{2}}+\frac {b^{2} \left (-\frac {\sin \left (d x +c \right )}{\left (d a -c b +b \left (d x +c \right )\right ) b}+\frac {\frac {\operatorname {Si}\left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}+\frac {\operatorname {Ci}\left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}}{b}\right )}{a^{2}}-\frac {2 b \left (\operatorname {Si}\left (d x \right ) \cos \left (c \right )+\operatorname {Ci}\left (d x \right ) \sin \left (c \right )\right )}{d \,a^{3}}+\frac {2 b^{2} \left (\frac {\operatorname {Si}\left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{d \,a^{3}}\right )\) | \(256\) |
| risch | \(-\frac {i b \,{\mathrm e}^{i c} \operatorname {Ei}_{1}\left (-i d x \right )}{a^{3}}+\frac {i b \,{\mathrm e}^{-\frac {i \left (d a -c b \right )}{b}} \operatorname {Ei}_{1}\left (-i d x -i c -\frac {i a d -i c b}{b}\right )}{a^{3}}-\frac {d \,{\mathrm e}^{i c} \operatorname {Ei}_{1}\left (-i d x \right )}{2 a^{2}}-\frac {d \,{\mathrm e}^{-\frac {i \left (d a -c b \right )}{b}} \operatorname {Ei}_{1}\left (-i d x -i c -\frac {i a d -i c b}{b}\right )}{2 a^{2}}-\frac {d \,{\mathrm e}^{-i c} \operatorname {Ei}_{1}\left (i d x \right )}{2 a^{2}}+\frac {i {\mathrm e}^{-i c} \operatorname {Ei}_{1}\left (i d x \right ) b}{a^{3}}-\frac {d \,{\mathrm e}^{\frac {i \left (d a -c b \right )}{b}} \operatorname {Ei}_{1}\left (i d x +i c +\frac {i \left (d a -c b \right )}{b}\right )}{2 a^{2}}-\frac {i {\mathrm e}^{\frac {i \left (d a -c b \right )}{b}} \operatorname {Ei}_{1}\left (i d x +i c +\frac {i \left (d a -c b \right )}{b}\right ) b}{a^{3}}+\frac {\left (-4 b x -2 a \right ) \sin \left (d x +c \right )}{2 a^{2} x \left (b x +a \right )}\) | \(299\) |
d*(1/a^2*(-sin(d*x+c)/d/x-Si(d*x)*sin(c)+Ci(d*x)*cos(c))+b^2/a^2*(-sin(d*x +c)/(d*a-c*b+b*(d*x+c))/b+(Si(d*x+c+(a*d-b*c)/b)*sin((a*d-b*c)/b)/b+Ci(d*x +c+(a*d-b*c)/b)*cos((a*d-b*c)/b)/b)/b)-2/d/a^3*b*(Si(d*x)*cos(c)+Ci(d*x)*s in(c))+2/d*b^2/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))
Time = 0.31 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.39 \[ \int \frac {\sin (c+d x)}{x^2 (a+b x)^2} \, dx=\frac {{\left ({\left (a b d x^{2} + a^{2} d x\right )} \operatorname {Ci}\left (d x\right ) - 2 \, {\left (b^{2} x^{2} + a b x\right )} \operatorname {Si}\left (d x\right )\right )} \cos \left (c\right ) + {\left ({\left (a b d x^{2} + a^{2} d x\right )} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) + 2 \, {\left (b^{2} x^{2} + a b x\right )} \operatorname {Si}\left (\frac {b d x + a d}{b}\right )\right )} \cos \left (-\frac {b c - a d}{b}\right ) - {\left (2 \, a b x + a^{2}\right )} \sin \left (d x + c\right ) - {\left (2 \, {\left (b^{2} x^{2} + a b x\right )} \operatorname {Ci}\left (d x\right ) + {\left (a b d x^{2} + a^{2} d x\right )} \operatorname {Si}\left (d x\right )\right )} \sin \left (c\right ) - {\left (2 \, {\left (b^{2} x^{2} + a b x\right )} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) - {\left (a b d x^{2} + a^{2} d x\right )} \operatorname {Si}\left (\frac {b d x + a d}{b}\right )\right )} \sin \left (-\frac {b c - a d}{b}\right )}{a^{3} b x^{2} + a^{4} x} \]
(((a*b*d*x^2 + a^2*d*x)*cos_integral(d*x) - 2*(b^2*x^2 + a*b*x)*sin_integr al(d*x))*cos(c) + ((a*b*d*x^2 + a^2*d*x)*cos_integral((b*d*x + a*d)/b) + 2 *(b^2*x^2 + a*b*x)*sin_integral((b*d*x + a*d)/b))*cos(-(b*c - a*d)/b) - (2 *a*b*x + a^2)*sin(d*x + c) - (2*(b^2*x^2 + a*b*x)*cos_integral(d*x) + (a*b *d*x^2 + a^2*d*x)*sin_integral(d*x))*sin(c) - (2*(b^2*x^2 + a*b*x)*cos_int egral((b*d*x + a*d)/b) - (a*b*d*x^2 + a^2*d*x)*sin_integral((b*d*x + a*d)/ b))*sin(-(b*c - a*d)/b))/(a^3*b*x^2 + a^4*x)
\[ \int \frac {\sin (c+d x)}{x^2 (a+b x)^2} \, dx=\int \frac {\sin {\left (c + d x \right )}}{x^{2} \left (a + b x\right )^{2}}\, dx \]
\[ \int \frac {\sin (c+d x)}{x^2 (a+b x)^2} \, dx=\int { \frac {\sin \left (d x + c\right )}{{\left (b x + a\right )}^{2} x^{2}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 3180 vs. \(2 (191) = 382\).
Time = 0.38 (sec) , antiderivative size = 3180, normalized size of antiderivative = 16.91 \[ \int \frac {\sin (c+d x)}{x^2 (a+b x)^2} \, dx=\text {Too large to display} \]
((b*x + a)^2*a*(b*c/(b*x + a) - a*d/(b*x + a) + d)^2*d^2*cos(c)*cos_integr al((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b - c)/b - 2*(b*x + a)*a* (b*c/(b*x + a) - a*d/(b*x + a) + d)*c*d^2*cos(c)*cos_integral((b*x + a)*(b *c/(b*x + a) - a*d/(b*x + a) + d)/b - c) + a*b*c^2*d^2*cos(c)*cos_integral ((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b - c) + (b*x + a)*a^2*(b*c /(b*x + a) - a*d/(b*x + a) + d)*d^3*cos(c)*cos_integral((b*x + a)*(b*c/(b* x + a) - a*d/(b*x + a) + d)/b - c)/b - a^2*c*d^3*cos(c)*cos_integral((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b - c) + (b*x + a)^2*a*(b*c/(b*x + a) - a*d/(b*x + a) + d)^2*d^2*cos(-(b*c - a*d)/b)*cos_integral(((b*x + a )*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b)/b - 2*(b*x + a)*a*(b *c/(b*x + a) - a*d/(b*x + a) + d)*c*d^2*cos(-(b*c - a*d)/b)*cos_integral(( (b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b) + a*b*c^2*d^ 2*cos(-(b*c - a*d)/b)*cos_integral(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b) + (b*x + a)*a^2*(b*c/(b*x + a) - a*d/(b*x + a) + d )*d^3*cos(-(b*c - a*d)/b)*cos_integral(((b*x + a)*(b*c/(b*x + a) - a*d/(b* x + a) + d) - b*c + a*d)/b)/b - a^2*c*d^3*cos(-(b*c - a*d)/b)*cos_integral (((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b) + (b*x + a )^2*a*(b*c/(b*x + a) - a*d/(b*x + a) + d)^2*d^2*sin(c)*sin_integral(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b + c)/b - 2*(b*x + a)*a*(b*c/(b* x + a) - a*d/(b*x + a) + d)*c*d^2*sin(c)*sin_integral(-(b*x + a)*(b*c/(...
Timed out. \[ \int \frac {\sin (c+d x)}{x^2 (a+b x)^2} \, dx=\int \frac {\sin \left (c+d\,x\right )}{x^2\,{\left (a+b\,x\right )}^2} \,d x \]