Integrand size = 15, antiderivative size = 124 \[ \int \frac {x \sin (c+d x)}{(a+b x)^2} \, dx=-\frac {a d \cos \left (c-\frac {a d}{b}\right ) \operatorname {CosIntegral}\left (\frac {a d}{b}+d x\right )}{b^3}+\frac {\operatorname {CosIntegral}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{b^2}+\frac {a \sin (c+d x)}{b^2 (a+b x)}+\frac {\cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^2}+\frac {a d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^3} \]
-a*d*Ci(a*d/b+d*x)*cos(-c+a*d/b)/b^3+cos(-c+a*d/b)*Si(a*d/b+d*x)/b^2-Ci(a* d/b+d*x)*sin(-c+a*d/b)/b^2-a*d*Si(a*d/b+d*x)*sin(-c+a*d/b)/b^3+a*sin(d*x+c )/b^2/(b*x+a)
Time = 0.34 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.77 \[ \int \frac {x \sin (c+d x)}{(a+b x)^2} \, dx=\frac {\operatorname {CosIntegral}\left (d \left (\frac {a}{b}+x\right )\right ) \left (-a d \cos \left (c-\frac {a d}{b}\right )+b \sin \left (c-\frac {a d}{b}\right )\right )+\frac {a b \sin (c+d x)}{a+b x}+\left (b \cos \left (c-\frac {a d}{b}\right )+a d \sin \left (c-\frac {a d}{b}\right )\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )}{b^3} \]
(CosIntegral[d*(a/b + x)]*(-(a*d*Cos[c - (a*d)/b]) + b*Sin[c - (a*d)/b]) + (a*b*Sin[c + d*x])/(a + b*x) + (b*Cos[c - (a*d)/b] + a*d*Sin[c - (a*d)/b] )*SinIntegral[d*(a/b + x)])/b^3
Time = 0.50 (sec) , antiderivative size = 124, 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.
| \(\displaystyle \int \frac {x \sin (c+d x)}{(a+b x)^2} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\sin (c+d x)}{b (a+b x)}-\frac {a \sin (c+d x)}{b (a+b x)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a d \cos \left (c-\frac {a d}{b}\right ) \operatorname {CosIntegral}\left (x d+\frac {a d}{b}\right )}{b^3}+\frac {a d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{b^3}+\frac {\sin \left (c-\frac {a d}{b}\right ) \operatorname {CosIntegral}\left (x d+\frac {a d}{b}\right )}{b^2}+\frac {\cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{b^2}+\frac {a \sin (c+d x)}{b^2 (a+b x)}\) |
-((a*d*Cos[c - (a*d)/b]*CosIntegral[(a*d)/b + d*x])/b^3) + (CosIntegral[(a *d)/b + d*x]*Sin[c - (a*d)/b])/b^2 + (a*Sin[c + d*x])/(b^2*(a + b*x)) + (C os[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/b^2 + (a*d*Sin[c - (a*d)/b]*Si nIntegral[(a*d)/b + d*x])/b^3
3.1.29.3.1 Defintions of rubi rules used
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 313, normalized size of antiderivative = 2.52
| method | result | size |
| risch | \(\frac {\left (-2 a b d x -2 a^{2} d \right ) \sin \left (d x +c \right )}{2 b^{2} \left (b x +a \right ) \left (-d x b -d a \right )}+\frac {\cos \left (\frac {d a -c b}{b}\right ) \operatorname {Ei}_{1}\left (\frac {i d \left (b x +a \right )}{b}\right ) a d}{2 b^{3}}+\frac {\cos \left (\frac {d a -c b}{b}\right ) \operatorname {Ei}_{1}\left (-\frac {i d \left (b x +a \right )}{b}\right ) a d}{2 b^{3}}-\frac {i \cos \left (\frac {d a -c b}{b}\right ) \operatorname {Ei}_{1}\left (\frac {i d \left (b x +a \right )}{b}\right )}{2 b^{2}}+\frac {i \cos \left (\frac {d a -c b}{b}\right ) \operatorname {Ei}_{1}\left (-\frac {i d \left (b x +a \right )}{b}\right )}{2 b^{2}}+\frac {i \sin \left (\frac {d a -c b}{b}\right ) \operatorname {Ei}_{1}\left (\frac {i d \left (b x +a \right )}{b}\right ) a d}{2 b^{3}}-\frac {i \sin \left (\frac {d a -c b}{b}\right ) \operatorname {Ei}_{1}\left (-\frac {i d \left (b x +a \right )}{b}\right ) a d}{2 b^{3}}+\frac {\sin \left (\frac {d a -c b}{b}\right ) \operatorname {Ei}_{1}\left (\frac {i d \left (b x +a \right )}{b}\right )}{2 b^{2}}+\frac {\sin \left (\frac {d a -c b}{b}\right ) \operatorname {Ei}_{1}\left (-\frac {i d \left (b x +a \right )}{b}\right )}{2 b^{2}}\) | \(313\) |
| derivativedivides | \(\frac {-\frac {d^{2} \left (d a -c b \right ) \left (-\frac {\sin \left (d x +c \right )}{\left (d a -c b +b \left (d x +c \right )\right ) b}+\frac {\frac {\operatorname {Si}\left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}+\frac {\operatorname {Ci}\left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}}{b}\right )}{b}+\frac {d^{2} \left (\frac {\operatorname {Si}\left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{b}-d^{2} c \left (-\frac {\sin \left (d x +c \right )}{\left (d a -c b +b \left (d x +c \right )\right ) b}+\frac {\frac {\operatorname {Si}\left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}+\frac {\operatorname {Ci}\left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}}{b}\right )}{d^{2}}\) | \(315\) |
| default | \(\frac {-\frac {d^{2} \left (d a -c b \right ) \left (-\frac {\sin \left (d x +c \right )}{\left (d a -c b +b \left (d x +c \right )\right ) b}+\frac {\frac {\operatorname {Si}\left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}+\frac {\operatorname {Ci}\left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}}{b}\right )}{b}+\frac {d^{2} \left (\frac {\operatorname {Si}\left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{b}-d^{2} c \left (-\frac {\sin \left (d x +c \right )}{\left (d a -c b +b \left (d x +c \right )\right ) b}+\frac {\frac {\operatorname {Si}\left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}+\frac {\operatorname {Ci}\left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}}{b}\right )}{d^{2}}\) | \(315\) |
1/2/b^2*(-2*a*b*d*x-2*a^2*d)/(b*x+a)/(-b*d*x-a*d)*sin(d*x+c)+1/2/b^3*cos(( a*d-b*c)/b)*Ei(1,I*d*(b*x+a)/b)*a*d+1/2/b^3*cos((a*d-b*c)/b)*Ei(1,-I*d*(b* x+a)/b)*a*d-1/2*I/b^2*cos((a*d-b*c)/b)*Ei(1,I*d*(b*x+a)/b)+1/2*I/b^2*cos(( a*d-b*c)/b)*Ei(1,-I*d*(b*x+a)/b)+1/2*I/b^3*sin((a*d-b*c)/b)*Ei(1,I*d*(b*x+ a)/b)*a*d-1/2*I/b^3*sin((a*d-b*c)/b)*Ei(1,-I*d*(b*x+a)/b)*a*d+1/2/b^2*sin( (a*d-b*c)/b)*Ei(1,I*d*(b*x+a)/b)+1/2/b^2*sin((a*d-b*c)/b)*Ei(1,-I*d*(b*x+a )/b)
Time = 0.28 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.25 \[ \int \frac {x \sin (c+d x)}{(a+b x)^2} \, dx=\frac {a b \sin \left (d x + c\right ) - {\left ({\left (a b d x + a^{2} d\right )} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) - {\left (b^{2} x + a b\right )} \operatorname {Si}\left (\frac {b d x + a d}{b}\right )\right )} \cos \left (-\frac {b c - a d}{b}\right ) - {\left ({\left (b^{2} x + a b\right )} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) + {\left (a b d x + a^{2} d\right )} \operatorname {Si}\left (\frac {b d x + a d}{b}\right )\right )} \sin \left (-\frac {b c - a d}{b}\right )}{b^{4} x + a b^{3}} \]
(a*b*sin(d*x + c) - ((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_integral((b*d*x + a*d)/b))*sin(-(b*c - a*d)/b))/(b^4*x + a*b^3)
\[ \int \frac {x \sin (c+d x)}{(a+b x)^2} \, dx=\int \frac {x \sin {\left (c + d x \right )}}{\left (a + b x\right )^{2}}\, dx \]
\[ \int \frac {x \sin (c+d x)}{(a+b x)^2} \, dx=\int { \frac {x \sin \left (d x + c\right )}{{\left (b x + a\right )}^{2}} \,d x } \]
-1/2*((b*cos(c)^2 + b*sin(c)^2)*x*cos(d*x + c) + ((a*(exp_integral_e(3, (I *b*d*x + I*a*d)/b) + exp_integral_e(3, -(I*b*d*x + I*a*d)/b))*cos(c)^2 + a *(exp_integral_e(3, (I*b*d*x + I*a*d)/b) + exp_integral_e(3, -(I*b*d*x + I *a*d)/b))*sin(c)^2)*cos(-(b*c - a*d)/b) + (a*(I*exp_integral_e(3, (I*b*d*x + I*a*d)/b) - I*exp_integral_e(3, -(I*b*d*x + I*a*d)/b))*cos(c)^2 + a*(I* exp_integral_e(3, (I*b*d*x + I*a*d)/b) - I*exp_integral_e(3, -(I*b*d*x + I *a*d)/b))*sin(c)^2)*sin(-(b*c - a*d)/b))*cos(d*x + c)^2 + ((a*(exp_integra l_e(3, (I*b*d*x + I*a*d)/b) + exp_integral_e(3, -(I*b*d*x + I*a*d)/b))*cos (c)^2 + a*(exp_integral_e(3, (I*b*d*x + I*a*d)/b) + exp_integral_e(3, -(I* b*d*x + I*a*d)/b))*sin(c)^2)*cos(-(b*c - a*d)/b) + (a*(I*exp_integral_e(3, (I*b*d*x + I*a*d)/b) - I*exp_integral_e(3, -(I*b*d*x + I*a*d)/b))*cos(c)^ 2 + a*(I*exp_integral_e(3, (I*b*d*x + I*a*d)/b) - I*exp_integral_e(3, -(I* b*d*x + I*a*d)/b))*sin(c)^2)*sin(-(b*c - a*d)/b))*sin(d*x + c)^2 + (b*x*co s(d*x + c)^2*cos(c) + b*x*cos(c)*sin(d*x + c)^2)*cos(d*x + 2*c) + 2*(((b^4 *cos(c)^2 + b^4*sin(c)^2)*d*x^2 + 2*(a*b^3*cos(c)^2 + a*b^3*sin(c)^2)*d*x + (a^2*b^2*cos(c)^2 + a^2*b^2*sin(c)^2)*d)*cos(d*x + c)^2 + ((b^4*cos(c)^2 + b^4*sin(c)^2)*d*x^2 + 2*(a*b^3*cos(c)^2 + a*b^3*sin(c)^2)*d*x + (a^2*b^ 2*cos(c)^2 + a^2*b^2*sin(c)^2)*d)*sin(d*x + c)^2)*integrate(1/2*x*cos(d*x + c)/(b^3*d*x^3 + 3*a*b^2*d*x^2 + 3*a^2*b*d*x + a^3*d), x) + 2*(((b^4*cos( c)^2 + b^4*sin(c)^2)*d*x^2 + 2*(a*b^3*cos(c)^2 + a*b^3*sin(c)^2)*d*x + ...
Leaf count of result is larger than twice the leaf count of optimal. 951 vs. \(2 (130) = 260\).
Time = 0.35 (sec) , antiderivative size = 951, normalized size of antiderivative = 7.67 \[ \int \frac {x \sin (c+d x)}{(a+b x)^2} \, dx=-\frac {{\left ({\left (b x + a\right )} a {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} d^{2} \cos \left (-\frac {b c - a d}{b}\right ) \operatorname {Ci}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) - a b c d^{2} \cos \left (-\frac {b c - a d}{b}\right ) \operatorname {Ci}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) + a^{2} d^{3} \cos \left (-\frac {b c - a d}{b}\right ) \operatorname {Ci}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) + {\left (b x + a\right )} a {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} d^{2} \sin \left (-\frac {b c - a d}{b}\right ) \operatorname {Si}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) - a b c d^{2} \sin \left (-\frac {b c - a d}{b}\right ) \operatorname {Si}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) + a^{2} d^{3} \sin \left (-\frac {b c - a d}{b}\right ) \operatorname {Si}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) + {\left (b x + a\right )} b {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} d \operatorname {Ci}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) \sin \left (-\frac {b c - a d}{b}\right ) - b^{2} c d \operatorname {Ci}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) \sin \left (-\frac {b c - a d}{b}\right ) + a b d^{2} \operatorname {Ci}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) \sin \left (-\frac {b c - a d}{b}\right ) - {\left (b x + a\right )} b {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} d \cos \left (-\frac {b c - a d}{b}\right ) \operatorname {Si}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) + b^{2} c d \cos \left (-\frac {b c - a d}{b}\right ) \operatorname {Si}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) - a b d^{2} \cos \left (-\frac {b c - a d}{b}\right ) \operatorname {Si}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) + a b d^{2} \sin \left (-\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )}}{b}\right )\right )} b}{{\left ({\left (b x + a\right )} b^{4} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b^{5} c + a b^{4} d\right )} d} \]
-((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 + a*d)/b)*sin(-(b*c - a*d)/b) - b^2*c*d*co s_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) + 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(-(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^2*c*d*cos(-(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*b*d^2*cos(-(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*b*d^2*sin(-(b*x + a)*(b*c...
Timed out. \[ \int \frac {x \sin (c+d x)}{(a+b x)^2} \, dx=\int \frac {x\,\sin \left (c+d\,x\right )}{{\left (a+b\,x\right )}^2} \,d x \]