Integrand size = 17, antiderivative size = 56 \[ \int \frac {\left (a+b x^3\right ) \sin (c+d x)}{x^2} \, dx=-\frac {b x \cos (c+d x)}{d}+a d \cos (c) \operatorname {CosIntegral}(d x)+\frac {b \sin (c+d x)}{d^2}-\frac {a \sin (c+d x)}{x}-a d \sin (c) \text {Si}(d x) \]
Time = 0.08 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^3\right ) \sin (c+d x)}{x^2} \, dx=-\frac {b x \cos (c+d x)}{d}+a d \cos (c) \operatorname {CosIntegral}(d x)+\frac {b \sin (c+d x)}{d^2}-\frac {a \sin (c+d x)}{x}-a d \sin (c) \text {Si}(d x) \]
-((b*x*Cos[c + d*x])/d) + a*d*Cos[c]*CosIntegral[d*x] + (b*Sin[c + d*x])/d ^2 - (a*Sin[c + d*x])/x - a*d*Sin[c]*SinIntegral[d*x]
Time = 0.27 (sec) , antiderivative size = 56, 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 = {3820, 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.
| \(\displaystyle \int \frac {\left (a+b x^3\right ) \sin (c+d x)}{x^2} \, dx\) |
| \(\Big \downarrow \) 3820 |
| \(\displaystyle \int \left (\frac {a \sin (c+d x)}{x^2}+b x \sin (c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a d \cos (c) \operatorname {CosIntegral}(d x)-a d \sin (c) \text {Si}(d x)-\frac {a \sin (c+d x)}{x}+\frac {b \sin (c+d x)}{d^2}-\frac {b x \cos (c+d x)}{d}\) |
-((b*x*Cos[c + d*x])/d) + a*d*Cos[c]*CosIntegral[d*x] + (b*Sin[c + d*x])/d ^2 - (a*Sin[c + d*x])/x - a*d*Sin[c]*SinIntegral[d*x]
3.1.84.3.1 Defintions of rubi rules used
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*Sin[(c_.) + (d_.)*(x_ )], x_Symbol] :> Int[ExpandIntegrand[Sin[c + d*x], (e*x)^m*(a + b*x^n)^p, x ], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
Time = 0.21 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.41
| method | result | size |
| derivativedivides | \(d \left (a \left (-\frac {\sin \left (d x +c \right )}{d x}-\operatorname {Si}\left (d x \right ) \sin \left (c \right )+\operatorname {Ci}\left (d x \right ) \cos \left (c \right )\right )+\frac {3 b c \cos \left (d x +c \right )}{d^{3}}+\frac {\left (2 c +1\right ) b \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{3}}\right )\) | \(79\) |
| default | \(d \left (a \left (-\frac {\sin \left (d x +c \right )}{d x}-\operatorname {Si}\left (d x \right ) \sin \left (c \right )+\operatorname {Ci}\left (d x \right ) \cos \left (c \right )\right )+\frac {3 b c \cos \left (d x +c \right )}{d^{3}}+\frac {\left (2 c +1\right ) b \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{3}}\right )\) | \(79\) |
| risch | \(-\frac {-i \operatorname {Ei}_{1}\left (i d x \right ) \sin \left (c \right ) a \,d^{3} x +i \sin \left (c \right ) \operatorname {Ei}_{1}\left (-i d x \right ) a \,d^{3} x +\operatorname {Ei}_{1}\left (i d x \right ) \cos \left (c \right ) a \,d^{3} x +\cos \left (c \right ) \operatorname {Ei}_{1}\left (-i d x \right ) a \,d^{3} x +2 \cos \left (d x +c \right ) b d \,x^{2}+2 \sin \left (d x +c \right ) a \,d^{2}-2 \sin \left (d x +c \right ) b x}{2 d^{2} x}\) | \(109\) |
| meijerg | \(\frac {2 b \sqrt {\pi }\, \sin \left (c \right ) \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\cos \left (d x \right )}{2 \sqrt {\pi }}+\frac {d x \sin \left (d x \right )}{2 \sqrt {\pi }}\right )}{d^{2}}+\frac {2 b \sqrt {\pi }\, \cos \left (c \right ) \left (-\frac {d x \cos \left (d x \right )}{2 \sqrt {\pi }}+\frac {\sin \left (d x \right )}{2 \sqrt {\pi }}\right )}{d^{2}}+\frac {a \sqrt {\pi }\, \sin \left (c \right ) d^{2} \left (-\frac {4 d^{2} \cos \left (x \sqrt {d^{2}}\right )}{x \left (d^{2}\right )^{\frac {3}{2}} \sqrt {\pi }}-\frac {4 \,\operatorname {Si}\left (x \sqrt {d^{2}}\right )}{\sqrt {\pi }}\right )}{4 \sqrt {d^{2}}}+\frac {a \sqrt {\pi }\, \cos \left (c \right ) d \left (\frac {4 \gamma -4+4 \ln \left (x \right )+4 \ln \left (d \right )}{\sqrt {\pi }}+\frac {4}{\sqrt {\pi }}-\frac {4 \gamma }{\sqrt {\pi }}-\frac {4 \ln \left (2\right )}{\sqrt {\pi }}-\frac {4 \ln \left (\frac {d x}{2}\right )}{\sqrt {\pi }}-\frac {4 \sin \left (d x \right )}{\sqrt {\pi }\, d x}+\frac {4 \,\operatorname {Ci}\left (d x \right )}{\sqrt {\pi }}\right )}{4}\) | \(205\) |
d*(a*(-sin(d*x+c)/d/x-Si(d*x)*sin(c)+Ci(d*x)*cos(c))+3*b/d^3*c*cos(d*x+c)+ (2*c+1)/d^3*b*(sin(d*x+c)-cos(d*x+c)*(d*x+c)))
Time = 0.28 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+b x^3\right ) \sin (c+d x)}{x^2} \, dx=\frac {a d^{3} x \cos \left (c\right ) \operatorname {Ci}\left (d x\right ) - a d^{3} x \sin \left (c\right ) \operatorname {Si}\left (d x\right ) - b d x^{2} \cos \left (d x + c\right ) - {\left (a d^{2} - b x\right )} \sin \left (d x + c\right )}{d^{2} x} \]
(a*d^3*x*cos(c)*cos_integral(d*x) - a*d^3*x*sin(c)*sin_integral(d*x) - b*d *x^2*cos(d*x + c) - (a*d^2 - b*x)*sin(d*x + c))/(d^2*x)
\[ \int \frac {\left (a+b x^3\right ) \sin (c+d x)}{x^2} \, dx=\int \frac {\left (a + b x^{3}\right ) \sin {\left (c + d x \right )}}{x^{2}}\, dx \]
Result contains complex when optimal does not.
Time = 0.54 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.23 \[ \int \frac {\left (a+b x^3\right ) \sin (c+d x)}{x^2} \, dx=\frac {{\left (a {\left (\Gamma \left (-1, i \, d x\right ) + \Gamma \left (-1, -i \, d x\right )\right )} \cos \left (c\right ) + a {\left (-i \, \Gamma \left (-1, i \, d x\right ) + i \, \Gamma \left (-1, -i \, d x\right )\right )} \sin \left (c\right )\right )} d^{3} - 2 \, b d x \cos \left (d x + c\right ) + 2 \, b \sin \left (d x + c\right )}{2 \, d^{2}} \]
1/2*((a*(gamma(-1, I*d*x) + gamma(-1, -I*d*x))*cos(c) + a*(-I*gamma(-1, I* d*x) + I*gamma(-1, -I*d*x))*sin(c))*d^3 - 2*b*d*x*cos(d*x + c) + 2*b*sin(d *x + c))/d^2
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.30 (sec) , antiderivative size = 489, normalized size of antiderivative = 8.73 \[ \int \frac {\left (a+b x^3\right ) \sin (c+d x)}{x^2} \, dx=-\frac {a d^{3} x \Re \left ( \operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + a d^{3} x \Re \left ( \operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + 2 \, a d^{3} x \Im \left ( \operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - 2 \, a d^{3} x \Im \left ( \operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right ) + 4 \, a d^{3} x \operatorname {Si}\left (d x\right ) \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - a d^{3} x \Re \left ( \operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} - a d^{3} x \Re \left ( \operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} + a d^{3} x \Re \left ( \operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, c\right )^{2} + a d^{3} x \Re \left ( \operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, c\right )^{2} + 2 \, b d x^{2} \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + 2 \, a d^{3} x \Im \left ( \operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, c\right ) - 2 \, a d^{3} x \Im \left ( \operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, c\right ) + 4 \, a d^{3} x \operatorname {Si}\left (d x\right ) \tan \left (\frac {1}{2} \, c\right ) - a d^{3} x \Re \left ( \operatorname {Ci}\left (d x\right ) \right ) - a d^{3} x \Re \left ( \operatorname {Ci}\left (-d x\right ) \right ) - 2 \, b d x^{2} \tan \left (\frac {1}{2} \, d x\right )^{2} - 8 \, b d x^{2} \tan \left (\frac {1}{2} \, d x\right ) \tan \left (\frac {1}{2} \, c\right ) - 4 \, a d^{2} \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - 2 \, b d x^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - 4 \, a d^{2} \tan \left (\frac {1}{2} \, d x\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + 4 \, b x \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right ) + 4 \, b x \tan \left (\frac {1}{2} \, d x\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + 2 \, b d x^{2} + 4 \, a d^{2} \tan \left (\frac {1}{2} \, d x\right ) + 4 \, a d^{2} \tan \left (\frac {1}{2} \, c\right ) - 4 \, b x \tan \left (\frac {1}{2} \, d x\right ) - 4 \, b x \tan \left (\frac {1}{2} \, c\right )}{2 \, {\left (d^{2} x \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + d^{2} x \tan \left (\frac {1}{2} \, d x\right )^{2} + d^{2} x \tan \left (\frac {1}{2} \, c\right )^{2} + d^{2} x\right )}} \]
-1/2*(a*d^3*x*real_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 + a *d^3*x*real_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*a*d^3 *x*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c) - 2*a*d^3*x*imag _part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c) + 4*a*d^3*x*sin_integr al(d*x)*tan(1/2*d*x)^2*tan(1/2*c) - a*d^3*x*real_part(cos_integral(d*x))*t an(1/2*d*x)^2 - a*d^3*x*real_part(cos_integral(-d*x))*tan(1/2*d*x)^2 + a*d ^3*x*real_part(cos_integral(d*x))*tan(1/2*c)^2 + a*d^3*x*real_part(cos_int egral(-d*x))*tan(1/2*c)^2 + 2*b*d*x^2*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*a*d^ 3*x*imag_part(cos_integral(d*x))*tan(1/2*c) - 2*a*d^3*x*imag_part(cos_inte gral(-d*x))*tan(1/2*c) + 4*a*d^3*x*sin_integral(d*x)*tan(1/2*c) - a*d^3*x* real_part(cos_integral(d*x)) - a*d^3*x*real_part(cos_integral(-d*x)) - 2*b *d*x^2*tan(1/2*d*x)^2 - 8*b*d*x^2*tan(1/2*d*x)*tan(1/2*c) - 4*a*d^2*tan(1/ 2*d*x)^2*tan(1/2*c) - 2*b*d*x^2*tan(1/2*c)^2 - 4*a*d^2*tan(1/2*d*x)*tan(1/ 2*c)^2 + 4*b*x*tan(1/2*d*x)^2*tan(1/2*c) + 4*b*x*tan(1/2*d*x)*tan(1/2*c)^2 + 2*b*d*x^2 + 4*a*d^2*tan(1/2*d*x) + 4*a*d^2*tan(1/2*c) - 4*b*x*tan(1/2*d *x) - 4*b*x*tan(1/2*c))/(d^2*x*tan(1/2*d*x)^2*tan(1/2*c)^2 + d^2*x*tan(1/2 *d*x)^2 + d^2*x*tan(1/2*c)^2 + d^2*x)
Timed out. \[ \int \frac {\left (a+b x^3\right ) \sin (c+d x)}{x^2} \, dx=\int \frac {\sin \left (c+d\,x\right )\,\left (b\,x^3+a\right )}{x^2} \,d x \]