Integrand size = 17, antiderivative size = 149 \[ \int \frac {\sin (c+d x)}{x (a+b x)^2} \, dx=-\frac {d \cos \left (c-\frac {a d}{b}\right ) \operatorname {CosIntegral}\left (\frac {a d}{b}+d x\right )}{a b}+\frac {\operatorname {CosIntegral}(d x) \sin (c)}{a^2}-\frac {\operatorname {CosIntegral}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{a^2}+\frac {\sin (c+d x)}{a (a+b x)}+\frac {\cos (c) \text {Si}(d x)}{a^2}-\frac {\cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{a^2}+\frac {d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{a b} \] Output:
-d*cos(-c+a*d/b)*Ci(a*d/b+d*x)/a/b+Ci(d*x)*sin(c)/a^2+Ci(a*d/b+d*x)*sin(-c +a*d/b)/a^2+sin(d*x+c)/a/(b*x+a)+cos(c)*Si(d*x)/a^2-cos(-c+a*d/b)*Si(a*d/b +d*x)/a^2-d*sin(-c+a*d/b)*Si(a*d/b+d*x)/a/b
Time = 1.14 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.93 \[ \int \frac {\sin (c+d x)}{x (a+b x)^2} \, dx=\frac {\frac {a \cos (d x) \sin (c)}{a+b x}+\operatorname {CosIntegral}(d x) \sin (c)-\frac {\operatorname {CosIntegral}\left (d \left (\frac {a}{b}+x\right )\right ) \left (a d \cos \left (c-\frac {a d}{b}\right )+b \sin \left (c-\frac {a d}{b}\right )\right )}{b}+\frac {a \cos (c) \sin (d x)}{a+b x}+\cos (c) \text {Si}(d x)-\cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )+\frac {a d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )}{b}}{a^2} \] Input:
Integrate[Sin[c + d*x]/(x*(a + b*x)^2),x]
Output:
((a*Cos[d*x]*Sin[c])/(a + b*x) + CosIntegral[d*x]*Sin[c] - (CosIntegral[d* (a/b + x)]*(a*d*Cos[c - (a*d)/b] + b*Sin[c - (a*d)/b]))/b + (a*Cos[c]*Sin[ d*x])/(a + b*x) + Cos[c]*SinIntegral[d*x] - Cos[c - (a*d)/b]*SinIntegral[d *(a/b + x)] + (a*d*Sin[c - (a*d)/b]*SinIntegral[d*(a/b + x)])/b)/a^2
Time = 0.64 (sec) , antiderivative size = 149, 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.
| \(\displaystyle \int \frac {\sin (c+d x)}{x (a+b x)^2} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {b \sin (c+d x)}{a^2 (a+b x)}+\frac {\sin (c+d x)}{a^2 x}-\frac {b \sin (c+d x)}{a (a+b x)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sin \left (c-\frac {a d}{b}\right ) \operatorname {CosIntegral}\left (x d+\frac {a d}{b}\right )}{a^2}-\frac {\cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{a^2}+\frac {\sin (c) \operatorname {CosIntegral}(d x)}{a^2}+\frac {\cos (c) \text {Si}(d x)}{a^2}-\frac {d \cos \left (c-\frac {a d}{b}\right ) \operatorname {CosIntegral}\left (x d+\frac {a d}{b}\right )}{a b}+\frac {d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{a b}+\frac {\sin (c+d x)}{a (a+b x)}\) |
Input:
Int[Sin[c + d*x]/(x*(a + b*x)^2),x]
Output:
-((d*Cos[c - (a*d)/b]*CosIntegral[(a*d)/b + d*x])/(a*b)) + (CosIntegral[d* x]*Sin[c])/a^2 - (CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/a^2 + Sin[c + d*x]/(a*(a + b*x)) + (Cos[c]*SinIntegral[d*x])/a^2 - (Cos[c - (a*d)/b]* SinIntegral[(a*d)/b + d*x])/a^2 + (d*Sin[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/(a*b)
Time = 0.97 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.41
| method | result | size |
| derivativedivides | \(\frac {\operatorname {Si}\left (d x \right ) \cos \left (c \right )+\operatorname {Ci}\left (d x \right ) \sin \left (c \right )}{a^{2}}-\frac {d b \left (-\frac {\sin \left (d x +c \right )}{\left (a d -b c +b \left (d x +c \right )\right ) b}+\frac {\frac {\operatorname {Si}\left (d x +c +\frac {a d -b c}{b}\right ) \sin \left (\frac {a d -b c}{b}\right )}{b}+\frac {\operatorname {Ci}\left (d x +c +\frac {a d -b c}{b}\right ) \cos \left (\frac {a d -b c}{b}\right )}{b}}{b}\right )}{a}-\frac {b \left (\frac {\operatorname {Si}\left (d x +c +\frac {a d -b c}{b}\right ) \cos \left (\frac {a d -b c}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {a d -b c}{b}\right ) \sin \left (\frac {a d -b c}{b}\right )}{b}\right )}{a^{2}}\) | \(210\) |
| default | \(\frac {\operatorname {Si}\left (d x \right ) \cos \left (c \right )+\operatorname {Ci}\left (d x \right ) \sin \left (c \right )}{a^{2}}-\frac {d b \left (-\frac {\sin \left (d x +c \right )}{\left (a d -b c +b \left (d x +c \right )\right ) b}+\frac {\frac {\operatorname {Si}\left (d x +c +\frac {a d -b c}{b}\right ) \sin \left (\frac {a d -b c}{b}\right )}{b}+\frac {\operatorname {Ci}\left (d x +c +\frac {a d -b c}{b}\right ) \cos \left (\frac {a d -b c}{b}\right )}{b}}{b}\right )}{a}-\frac {b \left (\frac {\operatorname {Si}\left (d x +c +\frac {a d -b c}{b}\right ) \cos \left (\frac {a d -b c}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {a d -b c}{b}\right ) \sin \left (\frac {a d -b c}{b}\right )}{b}\right )}{a^{2}}\) | \(210\) |
| risch | \(\frac {i {\mathrm e}^{i c} \operatorname {expIntegral}_{1}\left (-i d x \right )}{2 a^{2}}-\frac {i {\mathrm e}^{-\frac {i \left (a d -b c \right )}{b}} \operatorname {expIntegral}_{1}\left (-i d x -i c -\frac {i a d -i b c}{b}\right )}{2 a^{2}}+\frac {d \,{\mathrm e}^{-\frac {i \left (a d -b c \right )}{b}} \operatorname {expIntegral}_{1}\left (-i d x -i c -\frac {i a d -i b c}{b}\right )}{2 a b}-\frac {{\mathrm e}^{-i c} \pi \,\operatorname {csgn}\left (d x \right )}{2 a^{2}}+\frac {{\mathrm e}^{-i c} \operatorname {Si}\left (d x \right )}{a^{2}}-\frac {i {\mathrm e}^{-i c} \operatorname {expIntegral}_{1}\left (-i d x \right )}{2 a^{2}}+\frac {{\mathrm e}^{\frac {i \left (a d -b c \right )}{b}} \operatorname {expIntegral}_{1}\left (i d x +i c +\frac {i \left (a d -b c \right )}{b}\right ) d}{2 a b}+\frac {i {\mathrm e}^{\frac {i \left (a d -b c \right )}{b}} \operatorname {expIntegral}_{1}\left (i d x +i c +\frac {i \left (a d -b c \right )}{b}\right )}{2 a^{2}}-\frac {\sin \left (d x +c \right ) d x b}{a \left (b x +a \right ) \left (-d x b -a d \right )}-\frac {\sin \left (d x +c \right ) d}{\left (b x +a \right ) \left (-d x b -a d \right )}\) | \(325\) |
Input:
int(sin(d*x+c)/x/(b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
1/a^2*(Si(d*x)*cos(c)+Ci(d*x)*sin(c))-d*b/a*(-sin(d*x+c)/(a*d-b*c+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)-b/a^2*(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 = 0.08 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.27 \[ \int \frac {\sin (c+d x)}{x (a+b x)^2} \, dx=\frac {a b \sin \left (d x + c\right ) + {\left (b^{2} x + a b\right )} \operatorname {Ci}\left (d x\right ) \sin \left (c\right ) + {\left (b^{2} x + a b\right )} \cos \left (c\right ) \operatorname {Si}\left (d x\right ) - {\left ({\left (a b d x + a^{2} d\right )} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) + {\left (b^{2} x + a b\right )} \operatorname {Si}\left (\frac {b d x + a d}{b}\right )\right )} \cos \left (-\frac {b c - a d}{b}\right ) + {\left ({\left (b^{2} x + a b\right )} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) - {\left (a b d x + a^{2} d\right )} \operatorname {Si}\left (\frac {b d x + a d}{b}\right )\right )} \sin \left (-\frac {b c - a d}{b}\right )}{a^{2} b^{2} x + a^{3} b} \] Input:
integrate(sin(d*x+c)/x/(b*x+a)^2,x, algorithm="fricas")
Output:
(a*b*sin(d*x + c) + (b^2*x + a*b)*cos_integral(d*x)*sin(c) + (b^2*x + a*b) *cos(c)*sin_integral(d*x) - ((a*b*d*x + a^2*d)*cos_integral((b*d*x + a*d)/ b) + (b^2*x + a*b)*sin_integral((b*d*x + a*d)/b))*cos(-(b*c - a*d)/b) + (( b^2*x + a*b)*cos_integral((b*d*x + a*d)/b) - (a*b*d*x + a^2*d)*sin_integra l((b*d*x + a*d)/b))*sin(-(b*c - a*d)/b))/(a^2*b^2*x + a^3*b)
\[ \int \frac {\sin (c+d x)}{x (a+b x)^2} \, dx=\int \frac {\sin {\left (c + d x \right )}}{x \left (a + b x\right )^{2}}\, dx \] Input:
integrate(sin(d*x+c)/x/(b*x+a)**2,x)
Output:
Integral(sin(c + d*x)/(x*(a + b*x)**2), x)
\[ \int \frac {\sin (c+d x)}{x (a+b x)^2} \, dx=\int { \frac {\sin \left (d x + c\right )}{{\left (b x + a\right )}^{2} x} \,d x } \] Input:
integrate(sin(d*x+c)/x/(b*x+a)^2,x, algorithm="maxima")
Output:
integrate(sin(d*x + c)/((b*x + a)^2*x), x)
Leaf count of result is larger than twice the leaf count of optimal. 1281 vs. \(2 (153) = 306\).
Time = 0.20 (sec) , antiderivative size = 1281, normalized size of antiderivative = 8.60 \[ \int \frac {\sin (c+d x)}{x (a+b x)^2} \, dx=\text {Too large to display} \] Input:
integrate(sin(d*x+c)/x/(b*x+a)^2,x, algorithm="giac")
Output:
-((b*x + a)*a*(b*c/(b*x + a) - a*d/(b*x + a) + d)*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*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^2*d^3*cos(-(b*c - a*d)/b)*cos_inte gral(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b) + (b*x + a)*a*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d^2*sin(-(b*c - a*d)/b)*sin_in tegral(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b) - a* b*c*d^2*sin(-(b*c - a*d)/b)*sin_integral(((b*x + a)*(b*c/(b*x + a) - a*d/( b*x + a) + d) - b*c + a*d)/b) + a^2*d^3*sin(-(b*c - a*d)/b)*sin_integral(( (b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b) - (b*x + a)* b*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d*cos_integral((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b - c)*sin(c) + b^2*c*d*cos_integral((b*x + a)*(b *c/(b*x + a) - a*d/(b*x + a) + d)/b - c)*sin(c) - a*b*d^2*cos_integral((b* x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b - c)*sin(c) - (b*x + a)*b*(b* c/(b*x + a) - a*d/(b*x + a) + d)*d*cos_integral(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b)*sin(-(b*c - a*d)/b) + b^2*c*d*cos_int egral(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b)*sin(- (b*c - a*d)/b) - a*b*d^2*cos_integral(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b)*sin(-(b*c - a*d)/b) + (b*x + a)*b*(b*c/(b*x + a ) - a*d/(b*x + a) + d)*d*cos(c)*sin_integral(-(b*x + a)*(b*c/(b*x + a) ...
Timed out. \[ \int \frac {\sin (c+d x)}{x (a+b x)^2} \, dx=\int \frac {\sin \left (c+d\,x\right )}{x\,{\left (a+b\,x\right )}^2} \,d x \] Input:
int(sin(c + d*x)/(x*(a + b*x)^2),x)
Output:
int(sin(c + d*x)/(x*(a + b*x)^2), x)
\[ \int \frac {\sin (c+d x)}{x (a+b x)^2} \, dx=\int \frac {\sin \left (d x +c \right )}{b^{2} x^{3}+2 a b \,x^{2}+a^{2} x}d x \] Input:
int(sin(d*x+c)/x/(b*x+a)^2,x)
Output:
int(sin(c + d*x)/(a**2*x + 2*a*b*x**2 + b**2*x**3),x)