Integrand size = 17, antiderivative size = 106 \[ \int \frac {\left (a+b x^2\right ) \sin (c+d x)}{x^4} \, dx=-\frac {a d \cos (c+d x)}{6 x^2}+b d \cos (c) \operatorname {CosIntegral}(d x)-\frac {1}{6} a d^3 \cos (c) \operatorname {CosIntegral}(d x)-\frac {a \sin (c+d x)}{3 x^3}-\frac {b \sin (c+d x)}{x}+\frac {a d^2 \sin (c+d x)}{6 x}-b d \sin (c) \text {Si}(d x)+\frac {1}{6} a d^3 \sin (c) \text {Si}(d x) \] Output:
-1/6*a*d*cos(d*x+c)/x^2+b*d*cos(c)*Ci(d*x)-1/6*a*d^3*cos(c)*Ci(d*x)-1/3*a* sin(d*x+c)/x^3-b*sin(d*x+c)/x+1/6*a*d^2*sin(d*x+c)/x-b*d*sin(c)*Si(d*x)+1/ 6*a*d^3*sin(c)*Si(d*x)
Time = 0.16 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x^2\right ) \sin (c+d x)}{x^4} \, dx=\frac {-a d x \cos (c+d x)+d \left (6 b-a d^2\right ) x^3 \cos (c) \operatorname {CosIntegral}(d x)-2 a \sin (c+d x)-6 b x^2 \sin (c+d x)+a d^2 x^2 \sin (c+d x)+d \left (-6 b+a d^2\right ) x^3 \sin (c) \text {Si}(d x)}{6 x^3} \] Input:
Integrate[((a + b*x^2)*Sin[c + d*x])/x^4,x]
Output:
(-(a*d*x*Cos[c + d*x]) + d*(6*b - a*d^2)*x^3*Cos[c]*CosIntegral[d*x] - 2*a *Sin[c + d*x] - 6*b*x^2*Sin[c + d*x] + a*d^2*x^2*Sin[c + d*x] + d*(-6*b + a*d^2)*x^3*Sin[c]*SinIntegral[d*x])/(6*x^3)
Time = 0.42 (sec) , antiderivative size = 106, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right ) \sin (c+d x)}{x^4} \, dx\) |
| \(\Big \downarrow \) 3820 |
| \(\displaystyle \int \left (\frac {a \sin (c+d x)}{x^4}+\frac {b \sin (c+d x)}{x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{6} a d^3 \cos (c) \operatorname {CosIntegral}(d x)+\frac {1}{6} a d^3 \sin (c) \text {Si}(d x)+\frac {a d^2 \sin (c+d x)}{6 x}-\frac {a \sin (c+d x)}{3 x^3}-\frac {a d \cos (c+d x)}{6 x^2}+b d \cos (c) \operatorname {CosIntegral}(d x)-b d \sin (c) \text {Si}(d x)-\frac {b \sin (c+d x)}{x}\) |
Input:
Int[((a + b*x^2)*Sin[c + d*x])/x^4,x]
Output:
-1/6*(a*d*Cos[c + d*x])/x^2 + b*d*Cos[c]*CosIntegral[d*x] - (a*d^3*Cos[c]* CosIntegral[d*x])/6 - (a*Sin[c + d*x])/(3*x^3) - (b*Sin[c + d*x])/x + (a*d ^2*Sin[c + d*x])/(6*x) - b*d*Sin[c]*SinIntegral[d*x] + (a*d^3*Sin[c]*SinIn tegral[d*x])/6
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 = 0.99 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.96
| method | result | size |
| derivativedivides | \(d^{3} \left (a \left (-\frac {\sin \left (d x +c \right )}{3 d^{3} x^{3}}-\frac {\cos \left (d x +c \right )}{6 d^{2} x^{2}}+\frac {\sin \left (d x +c \right )}{6 d x}+\frac {\operatorname {Si}\left (d x \right ) \sin \left (c \right )}{6}-\frac {\operatorname {Ci}\left (d x \right ) \cos \left (c \right )}{6}\right )+\frac {b \left (-\frac {\sin \left (d x +c \right )}{d x}-\operatorname {Si}\left (d x \right ) \sin \left (c \right )+\operatorname {Ci}\left (d x \right ) \cos \left (c \right )\right )}{d^{2}}\right )\) | \(102\) |
| default | \(d^{3} \left (a \left (-\frac {\sin \left (d x +c \right )}{3 d^{3} x^{3}}-\frac {\cos \left (d x +c \right )}{6 d^{2} x^{2}}+\frac {\sin \left (d x +c \right )}{6 d x}+\frac {\operatorname {Si}\left (d x \right ) \sin \left (c \right )}{6}-\frac {\operatorname {Ci}\left (d x \right ) \cos \left (c \right )}{6}\right )+\frac {b \left (-\frac {\sin \left (d x +c \right )}{d x}-\operatorname {Si}\left (d x \right ) \sin \left (c \right )+\operatorname {Ci}\left (d x \right ) \cos \left (c \right )\right )}{d^{2}}\right )\) | \(102\) |
| risch | \(\frac {\operatorname {expIntegral}_{1}\left (-i d x \right ) \cos \left (c \right ) a \,d^{3}}{12}+\frac {\operatorname {expIntegral}_{1}\left (i d x \right ) \cos \left (c \right ) a \,d^{3}}{12}-\frac {\operatorname {expIntegral}_{1}\left (-i d x \right ) \cos \left (c \right ) b d}{2}-\frac {\operatorname {expIntegral}_{1}\left (i d x \right ) \cos \left (c \right ) b d}{2}+\frac {i \operatorname {expIntegral}_{1}\left (-i d x \right ) \sin \left (c \right ) a \,d^{3}}{12}-\frac {i \operatorname {expIntegral}_{1}\left (i d x \right ) \sin \left (c \right ) a \,d^{3}}{12}-\frac {i \operatorname {expIntegral}_{1}\left (-i d x \right ) \sin \left (c \right ) b d}{2}+\frac {i \operatorname {expIntegral}_{1}\left (i d x \right ) \sin \left (c \right ) b d}{2}-\frac {a d \cos \left (d x +c \right )}{6 x^{2}}+\frac {i \left (-2 i a \,d^{7} x^{5}+12 i b \,d^{5} x^{5}+4 i a \,d^{5} x^{3}\right ) \sin \left (d x +c \right )}{12 d^{5} x^{6}}\) | \(177\) |
| meijerg | \(\frac {d^{2} b \sin \left (c \right ) \sqrt {\pi }\, \left (-\frac {4 d^{2} \cos \left (x \sqrt {d^{2}}\right )}{x \left (d^{2}\right )^{\frac {3}{2}} \sqrt {\pi }}-\frac {4 \,\operatorname {Si}\left (x \sqrt {d^{2}}\right )}{\sqrt {\pi }}\right )}{4 \sqrt {d^{2}}}+\frac {d b \cos \left (c \right ) \sqrt {\pi }\, \left (\frac {4 \gamma -4+4 \ln \left (x \right )+4 \ln \left (d \right )}{\sqrt {\pi }}+\frac {4}{\sqrt {\pi }}-\frac {4 \gamma }{\sqrt {\pi }}-\frac {4 \ln \left (2\right )}{\sqrt {\pi }}-\frac {4 \ln \left (\frac {d x}{2}\right )}{\sqrt {\pi }}-\frac {4 \sin \left (d x \right )}{\sqrt {\pi }\, x d}+\frac {4 \,\operatorname {Ci}\left (d x \right )}{\sqrt {\pi }}\right )}{4}+\frac {a \sin \left (c \right ) \sqrt {\pi }\, d^{4} \left (-\frac {8 \left (-x^{2} d^{2}+2\right ) d^{2} \cos \left (x \sqrt {d^{2}}\right )}{3 x^{3} \left (d^{2}\right )^{\frac {5}{2}} \sqrt {\pi }}+\frac {8 \sin \left (x \sqrt {d^{2}}\right )}{3 x^{2} d^{2} \sqrt {\pi }}+\frac {8 \,\operatorname {Si}\left (x \sqrt {d^{2}}\right )}{3 \sqrt {\pi }}\right )}{16 \sqrt {d^{2}}}+\frac {a \cos \left (c \right ) \sqrt {\pi }\, d^{3} \left (-\frac {8}{\sqrt {\pi }\, x^{2} d^{2}}-\frac {4 \left (2 \gamma -\frac {11}{3}+2 \ln \left (x \right )+2 \ln \left (d \right )\right )}{3 \sqrt {\pi }}+\frac {-\frac {44 x^{2} d^{2}}{9}+8}{\sqrt {\pi }\, x^{2} d^{2}}+\frac {8 \gamma }{3 \sqrt {\pi }}+\frac {8 \ln \left (2\right )}{3 \sqrt {\pi }}+\frac {8 \ln \left (\frac {d x}{2}\right )}{3 \sqrt {\pi }}-\frac {8 \cos \left (d x \right )}{3 \sqrt {\pi }\, x^{2} d^{2}}-\frac {16 \left (-\frac {5 x^{2} d^{2}}{2}+5\right ) \sin \left (d x \right )}{15 \sqrt {\pi }\, x^{3} d^{3}}-\frac {8 \,\operatorname {Ci}\left (d x \right )}{3 \sqrt {\pi }}\right )}{16}\) | \(353\) |
Input:
int((b*x^2+a)*sin(d*x+c)/x^4,x,method=_RETURNVERBOSE)
Output:
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))+b/d^2*(-sin(d*x+c)/d/x-Si(d*x)*sin( c)+Ci(d*x)*cos(c)))
Time = 0.08 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.77 \[ \int \frac {\left (a+b x^2\right ) \sin (c+d x)}{x^4} \, dx=-\frac {{\left (a d^{3} - 6 \, b d\right )} x^{3} \cos \left (c\right ) \operatorname {Ci}\left (d x\right ) - {\left (a d^{3} - 6 \, b d\right )} x^{3} \sin \left (c\right ) \operatorname {Si}\left (d x\right ) + a d x \cos \left (d x + c\right ) - {\left ({\left (a d^{2} - 6 \, b\right )} x^{2} - 2 \, a\right )} \sin \left (d x + c\right )}{6 \, x^{3}} \] Input:
integrate((b*x^2+a)*sin(d*x+c)/x^4,x, algorithm="fricas")
Output:
-1/6*((a*d^3 - 6*b*d)*x^3*cos(c)*cos_integral(d*x) - (a*d^3 - 6*b*d)*x^3*s in(c)*sin_integral(d*x) + a*d*x*cos(d*x + c) - ((a*d^2 - 6*b)*x^2 - 2*a)*s in(d*x + c))/x^3
\[ \int \frac {\left (a+b x^2\right ) \sin (c+d x)}{x^4} \, dx=\int \frac {\left (a + b x^{2}\right ) \sin {\left (c + d x \right )}}{x^{4}}\, dx \] Input:
integrate((b*x**2+a)*sin(d*x+c)/x**4,x)
Output:
Integral((a + b*x**2)*sin(c + d*x)/x**4, x)
Result contains complex when optimal does not.
Time = 0.64 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+b x^2\right ) \sin (c+d x)}{x^4} \, dx=-\frac {{\left ({\left (a {\left (\Gamma \left (-3, i \, d x\right ) + \Gamma \left (-3, -i \, d x\right )\right )} \cos \left (c\right ) + a {\left (-i \, \Gamma \left (-3, i \, d x\right ) + i \, \Gamma \left (-3, -i \, d x\right )\right )} \sin \left (c\right )\right )} d^{5} - 6 \, {\left (b {\left (\Gamma \left (-3, i \, d x\right ) + \Gamma \left (-3, -i \, d x\right )\right )} \cos \left (c\right ) + b {\left (-i \, \Gamma \left (-3, i \, d x\right ) + i \, \Gamma \left (-3, -i \, d x\right )\right )} \sin \left (c\right )\right )} d^{3}\right )} x^{3} + 2 \, b d x \cos \left (d x + c\right ) + 4 \, b \sin \left (d x + c\right )}{2 \, d^{2} x^{3}} \] Input:
integrate((b*x^2+a)*sin(d*x+c)/x^4,x, algorithm="maxima")
Output:
-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^5 - 6*(b*(gamma(-3, I*d*x) + gamma (-3, -I*d*x))*cos(c) + b*(-I*gamma(-3, I*d*x) + I*gamma(-3, -I*d*x))*sin(c ))*d^3)*x^3 + 2*b*d*x*cos(d*x + c) + 4*b*sin(d*x + c))/(d^2*x^3)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 834, normalized size of antiderivative = 7.87 \[ \int \frac {\left (a+b x^2\right ) \sin (c+d x)}{x^4} \, dx=\text {Too large to display} \] Input:
integrate((b*x^2+a)*sin(d*x+c)/x^4,x, algorithm="giac")
Output:
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 - 6*b*d*x^3*real_part(cos_inte gral(d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 - 6*b*d*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*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) - 12*b*d*x^3*imag_part(cos_integral (d*x))*tan(1/2*d*x)^2*tan(1/2*c) + 12*b*d*x^3*imag_part(cos_integral(-d*x) )*tan(1/2*d*x)^2*tan(1/2*c) - 24*b*d*x^3*sin_integral(d*x)*tan(1/2*d*x)^2* tan(1/2*c) - a*d^3*x^3*real_part(cos_integral(d*x)) - a*d^3*x^3*real_part( cos_integral(-d*x)) + 6*b*d*x^3*real_part(cos_integral(d*x))*tan(1/2*d*x)^ 2 + 6*b*d*x^3*real_part(cos_integral(-d*x))*tan(1/2*d*x)^2 - 4*a*d^2*x^2*t an(1/2*d*x)^2*tan(1/2*c) - 6*b*d*x^3*real_part(cos_integral(d*x))*tan(1/2* c)^2 - 6*b*d*x^3*real_part(cos_integral(-d*x))*tan(1/2*c)^2 - 4*a*d^2*x^2* tan(1/2*d*x)*tan(1/2*c)^2 - 12*b*d*x^3*imag_part(cos_integral(d*x))*tan...
Timed out. \[ \int \frac {\left (a+b x^2\right ) \sin (c+d x)}{x^4} \, dx=\int \frac {\sin \left (c+d\,x\right )\,\left (b\,x^2+a\right )}{x^4} \,d x \] Input:
int((sin(c + d*x)*(a + b*x^2))/x^4,x)
Output:
int((sin(c + d*x)*(a + b*x^2))/x^4, x)
\[ \int \frac {\left (a+b x^2\right ) \sin (c+d x)}{x^4} \, dx=\frac {-6 \cos \left (d x +c \right ) b x +4 \left (\int \frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} x^{3}+x^{3}}d x \right ) a \,d^{2} x^{3}-24 \left (\int \frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} x^{3}+x^{3}}d x \right ) b \,x^{3}-2 \sin \left (d x +c \right ) a d +a \,d^{2} x -6 b x}{6 d \,x^{3}} \] Input:
int((b*x^2+a)*sin(d*x+c)/x^4,x)
Output:
( - 6*cos(c + d*x)*b*x + 4*int(1/(tan((c + d*x)/2)**2*x**3 + x**3),x)*a*d* *2*x**3 - 24*int(1/(tan((c + d*x)/2)**2*x**3 + x**3),x)*b*x**3 - 2*sin(c + d*x)*a*d + a*d**2*x - 6*b*x)/(6*d*x**3)