Integrand size = 17, antiderivative size = 91 \[ \int \frac {\left (a+b x^3\right ) \sin (c+d x)}{x^4} \, dx=-\frac {a d \cos (c+d x)}{6 x^2}-\frac {1}{6} a d^3 \cos (c) \operatorname {CosIntegral}(d x)+b \operatorname {CosIntegral}(d x) \sin (c)-\frac {a \sin (c+d x)}{3 x^3}+\frac {a d^2 \sin (c+d x)}{6 x}+b \cos (c) \text {Si}(d x)+\frac {1}{6} a d^3 \sin (c) \text {Si}(d x) \]
-1/6*a*d^3*Ci(d*x)*cos(c)-1/6*a*d*cos(d*x+c)/x^2+b*cos(c)*Si(d*x)+b*Ci(d*x )*sin(c)+1/6*a*d^3*Si(d*x)*sin(c)-1/3*a*sin(d*x+c)/x^3+1/6*a*d^2*sin(d*x+c )/x
Time = 0.12 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+b x^3\right ) \sin (c+d x)}{x^4} \, dx=b \operatorname {CosIntegral}(d x) \sin (c)+\frac {a \cos (d x) \left (-d x \cos (c)-2 \sin (c)+d^2 x^2 \sin (c)\right )}{6 x^3}+\frac {a \left (-2 \cos (c)+d^2 x^2 \cos (c)+d x \sin (c)\right ) \sin (d x)}{6 x^3}+b \cos (c) \text {Si}(d x)-\frac {1}{6} a d^3 (\cos (c) \operatorname {CosIntegral}(d x)-\sin (c) \text {Si}(d x)) \]
b*CosIntegral[d*x]*Sin[c] + (a*Cos[d*x]*(-(d*x*Cos[c]) - 2*Sin[c] + d^2*x^ 2*Sin[c]))/(6*x^3) + (a*(-2*Cos[c] + d^2*x^2*Cos[c] + d*x*Sin[c])*Sin[d*x] )/(6*x^3) + b*Cos[c]*SinIntegral[d*x] - (a*d^3*(Cos[c]*CosIntegral[d*x] - Sin[c]*SinIntegral[d*x]))/6
Time = 0.35 (sec) , antiderivative size = 91, 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.
\(\displaystyle \int \frac {\left (a+b x^3\right ) \sin (c+d x)}{x^4} \, dx\) |
\(\Big \downarrow \) 3820 |
\(\displaystyle \int \left (\frac {a \sin (c+d x)}{x^4}+\frac {b \sin (c+d x)}{x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{6} a d^3 \cos (c) \operatorname {CosIntegral}(d x)+\frac {1}{6} a d^3 \sin (c) \text {Si}(d x)+\frac {a d^2 \sin (c+d x)}{6 x}-\frac {a \sin (c+d x)}{3 x^3}-\frac {a d \cos (c+d x)}{6 x^2}+b \sin (c) \operatorname {CosIntegral}(d x)+b \cos (c) \text {Si}(d x)\) |
-1/6*(a*d*Cos[c + d*x])/x^2 - (a*d^3*Cos[c]*CosIntegral[d*x])/6 + b*CosInt egral[d*x]*Sin[c] - (a*Sin[c + d*x])/(3*x^3) + (a*d^2*Sin[c + d*x])/(6*x) + b*Cos[c]*SinIntegral[d*x] + (a*d^3*Sin[c]*SinIntegral[d*x])/6
3.1.86.3.1 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 = 0.22 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.96
method | result | size |
derivativedivides | \(d^{3} \left (a \left (-\frac {\sin \left (d x +c \right )}{3 d^{3} x^{3}}-\frac {\cos \left (d x +c \right )}{6 d^{2} x^{2}}+\frac {\sin \left (d x +c \right )}{6 d x}+\frac {\operatorname {Si}\left (d x \right ) \sin \left (c \right )}{6}-\frac {\operatorname {Ci}\left (d x \right ) \cos \left (c \right )}{6}\right )+\frac {b \left (\operatorname {Si}\left (d x \right ) \cos \left (c \right )+\operatorname {Ci}\left (d x \right ) \sin \left (c \right )\right )}{d^{3}}\right )\) | \(87\) |
default | \(d^{3} \left (a \left (-\frac {\sin \left (d x +c \right )}{3 d^{3} x^{3}}-\frac {\cos \left (d x +c \right )}{6 d^{2} x^{2}}+\frac {\sin \left (d x +c \right )}{6 d x}+\frac {\operatorname {Si}\left (d x \right ) \sin \left (c \right )}{6}-\frac {\operatorname {Ci}\left (d x \right ) \cos \left (c \right )}{6}\right )+\frac {b \left (\operatorname {Si}\left (d x \right ) \cos \left (c \right )+\operatorname {Ci}\left (d x \right ) \sin \left (c \right )\right )}{d^{3}}\right )\) | \(87\) |
risch | \(\frac {\cos \left (c \right ) \operatorname {Ei}_{1}\left (-i d x \right ) a \,d^{3}}{12}-\frac {i \cos \left (c \right ) \operatorname {Ei}_{1}\left (i d x \right ) b}{2}+\frac {\cos \left (c \right ) \operatorname {Ei}_{1}\left (i d x \right ) a \,d^{3}}{12}+\frac {i \cos \left (c \right ) \operatorname {Ei}_{1}\left (-i d x \right ) b}{2}+\frac {i \sin \left (c \right ) \operatorname {Ei}_{1}\left (-i d x \right ) a \,d^{3}}{12}-\frac {\sin \left (c \right ) \operatorname {Ei}_{1}\left (i d x \right ) b}{2}-\frac {i \sin \left (c \right ) \operatorname {Ei}_{1}\left (i d x \right ) a \,d^{3}}{12}-\frac {\sin \left (c \right ) \operatorname {Ei}_{1}\left (-i d x \right ) b}{2}-\frac {a d \cos \left (d x +c \right )}{6 x^{2}}+\frac {i \left (-2 i a \,d^{8} x^{5}+4 i a \,d^{6} x^{3}\right ) \sin \left (d x +c \right )}{12 d^{6} x^{6}}\) | \(163\) |
meijerg | \(\frac {b \sqrt {\pi }\, \sin \left (c \right ) \left (\frac {2 \gamma +2 \ln \left (x \right )+\ln \left (d^{2}\right )}{\sqrt {\pi }}-\frac {2 \gamma }{\sqrt {\pi }}-\frac {2 \ln \left (2\right )}{\sqrt {\pi }}-\frac {2 \ln \left (\frac {d x}{2}\right )}{\sqrt {\pi }}+\frac {2 \,\operatorname {Ci}\left (d x \right )}{\sqrt {\pi }}\right )}{2}+b \cos \left (c \right ) \operatorname {Si}\left (d x \right )+\frac {a \sqrt {\pi }\, \sin \left (c \right ) d^{4} \left (-\frac {8 \left (-d^{2} x^{2}+2\right ) d^{2} \cos \left (x \sqrt {d^{2}}\right )}{3 x^{3} \left (d^{2}\right )^{\frac {5}{2}} \sqrt {\pi }}+\frac {8 \sin \left (x \sqrt {d^{2}}\right )}{3 d^{2} x^{2} \sqrt {\pi }}+\frac {8 \,\operatorname {Si}\left (x \sqrt {d^{2}}\right )}{3 \sqrt {\pi }}\right )}{16 \sqrt {d^{2}}}+\frac {a \sqrt {\pi }\, \cos \left (c \right ) d^{3} \left (-\frac {8}{\sqrt {\pi }\, x^{2} d^{2}}-\frac {4 \left (2 \gamma -\frac {11}{3}+2 \ln \left (x \right )+2 \ln \left (d \right )\right )}{3 \sqrt {\pi }}+\frac {-\frac {44 d^{2} x^{2}}{9}+8}{d^{2} x^{2} \sqrt {\pi }}+\frac {8 \gamma }{3 \sqrt {\pi }}+\frac {8 \ln \left (2\right )}{3 \sqrt {\pi }}+\frac {8 \ln \left (\frac {d x}{2}\right )}{3 \sqrt {\pi }}-\frac {8 \cos \left (d x \right )}{3 \sqrt {\pi }\, d^{2} x^{2}}-\frac {16 \left (-\frac {5 d^{2} x^{2}}{2}+5\right ) \sin \left (d x \right )}{15 \sqrt {\pi }\, d^{3} x^{3}}-\frac {8 \,\operatorname {Ci}\left (d x \right )}{3 \sqrt {\pi }}\right )}{16}\) | \(284\) |
d^3*(a*(-1/3*sin(d*x+c)/d^3/x^3-1/6*cos(d*x+c)/d^2/x^2+1/6*sin(d*x+c)/d/x+ 1/6*Si(d*x)*sin(c)-1/6*Ci(d*x)*cos(c))+1/d^3*b*(Si(d*x)*cos(c)+Ci(d*x)*sin (c)))
Time = 0.28 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b x^3\right ) \sin (c+d x)}{x^4} \, dx=-\frac {a d x \cos \left (d x + c\right ) + {\left (a d^{3} x^{3} \operatorname {Ci}\left (d x\right ) - 6 \, b x^{3} \operatorname {Si}\left (d x\right )\right )} \cos \left (c\right ) - {\left (a d^{2} x^{2} - 2 \, a\right )} \sin \left (d x + c\right ) - {\left (a d^{3} x^{3} \operatorname {Si}\left (d x\right ) + 6 \, b x^{3} \operatorname {Ci}\left (d x\right )\right )} \sin \left (c\right )}{6 \, x^{3}} \]
-1/6*(a*d*x*cos(d*x + c) + (a*d^3*x^3*cos_integral(d*x) - 6*b*x^3*sin_inte gral(d*x))*cos(c) - (a*d^2*x^2 - 2*a)*sin(d*x + c) - (a*d^3*x^3*sin_integr al(d*x) + 6*b*x^3*cos_integral(d*x))*sin(c))/x^3
\[ \int \frac {\left (a+b x^3\right ) \sin (c+d x)}{x^4} \, dx=\int \frac {\left (a + b x^{3}\right ) \sin {\left (c + d x \right )}}{x^{4}}\, dx \]
Result contains complex when optimal does not.
Time = 0.90 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.45 \[ \int \frac {\left (a+b x^3\right ) \sin (c+d x)}{x^4} \, dx=-\frac {{\left ({\left (a {\left (\Gamma \left (-3, i \, d x\right ) + \Gamma \left (-3, -i \, d x\right )\right )} \cos \left (c\right ) + a {\left (-i \, \Gamma \left (-3, i \, d x\right ) + i \, \Gamma \left (-3, -i \, d x\right )\right )} \sin \left (c\right )\right )} d^{6} - 6 \, {\left (b {\left (i \, \Gamma \left (-3, i \, d x\right ) - i \, \Gamma \left (-3, -i \, d x\right )\right )} \cos \left (c\right ) + b {\left (\Gamma \left (-3, i \, d x\right ) + \Gamma \left (-3, -i \, d x\right )\right )} \sin \left (c\right )\right )} d^{3}\right )} x^{3} + 2 \, b d x \sin \left (d x + c\right ) + 2 \, {\left (b d^{2} x^{2} - 2 \, b\right )} \cos \left (d x + c\right )}{2 \, d^{3} x^{3}} \]
-1/2*(((a*(gamma(-3, I*d*x) + gamma(-3, -I*d*x))*cos(c) + a*(-I*gamma(-3, I*d*x) + I*gamma(-3, -I*d*x))*sin(c))*d^6 - 6*(b*(I*gamma(-3, I*d*x) - I*g amma(-3, -I*d*x))*cos(c) + b*(gamma(-3, I*d*x) + gamma(-3, -I*d*x))*sin(c) )*d^3)*x^3 + 2*b*d*x*sin(d*x + c) + 2*(b*d^2*x^2 - 2*b)*cos(d*x + c))/(d^3 *x^3)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.30 (sec) , antiderivative size = 796, normalized size of antiderivative = 8.75 \[ \int \frac {\left (a+b x^3\right ) \sin (c+d x)}{x^4} \, dx=\text {Too large to display} \]
1/12*(a*d^3*x^3*real_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 + a*d^3*x^3*real_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*a *d^3*x^3*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c) - 2*a*d^3* x^3*imag_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c) + 4*a*d^3*x^3* sin_integral(d*x)*tan(1/2*d*x)^2*tan(1/2*c) - a*d^3*x^3*real_part(cos_inte gral(d*x))*tan(1/2*d*x)^2 - a*d^3*x^3*real_part(cos_integral(-d*x))*tan(1/ 2*d*x)^2 + a*d^3*x^3*real_part(cos_integral(d*x))*tan(1/2*c)^2 + a*d^3*x^3 *real_part(cos_integral(-d*x))*tan(1/2*c)^2 + 2*a*d^3*x^3*imag_part(cos_in tegral(d*x))*tan(1/2*c) - 2*a*d^3*x^3*imag_part(cos_integral(-d*x))*tan(1/ 2*c) + 4*a*d^3*x^3*sin_integral(d*x)*tan(1/2*c) - 6*b*x^3*imag_part(cos_in tegral(d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 + 6*b*x^3*imag_part(cos_integral( -d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 - 12*b*x^3*sin_integral(d*x)*tan(1/2*d* x)^2*tan(1/2*c)^2 - a*d^3*x^3*real_part(cos_integral(d*x)) - a*d^3*x^3*rea l_part(cos_integral(-d*x)) - 4*a*d^2*x^2*tan(1/2*d*x)^2*tan(1/2*c) + 12*b* x^3*real_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c) + 12*b*x^3*real _part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c) - 4*a*d^2*x^2*tan(1/2* d*x)*tan(1/2*c)^2 + 6*b*x^3*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2 - 6*b*x^3*imag_part(cos_integral(-d*x))*tan(1/2*d*x)^2 + 12*b*x^3*sin_integr al(d*x)*tan(1/2*d*x)^2 - 6*b*x^3*imag_part(cos_integral(d*x))*tan(1/2*c)^2 + 6*b*x^3*imag_part(cos_integral(-d*x))*tan(1/2*c)^2 - 12*b*x^3*sin_in...
Timed out. \[ \int \frac {\left (a+b x^3\right ) \sin (c+d x)}{x^4} \, dx=\int \frac {\sin \left (c+d\,x\right )\,\left (b\,x^3+a\right )}{x^4} \,d x \]