Integrand size = 12, antiderivative size = 78 \[ \int x^2 \sin \left (a+\frac {b}{x}\right ) \, dx=\frac {1}{6} b x^2 \cos \left (a+\frac {b}{x}\right )+\frac {1}{6} b^3 \cos (a) \operatorname {CosIntegral}\left (\frac {b}{x}\right )-\frac {1}{6} b^2 x \sin \left (a+\frac {b}{x}\right )+\frac {1}{3} x^3 \sin \left (a+\frac {b}{x}\right )-\frac {1}{6} b^3 \sin (a) \text {Si}\left (\frac {b}{x}\right ) \] Output:
1/6*b*x^2*cos(a+b/x)+1/6*b^3*cos(a)*Ci(b/x)-1/6*b^2*x*sin(a+b/x)+1/3*x^3*s in(a+b/x)-1/6*b^3*sin(a)*Si(b/x)
Time = 0.12 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.90 \[ \int x^2 \sin \left (a+\frac {b}{x}\right ) \, dx=\frac {1}{6} \left (b^3 \cos (a) \operatorname {CosIntegral}\left (\frac {b}{x}\right )+x \left (b x \cos \left (a+\frac {b}{x}\right )-b^2 \sin \left (a+\frac {b}{x}\right )+2 x^2 \sin \left (a+\frac {b}{x}\right )\right )-b^3 \sin (a) \text {Si}\left (\frac {b}{x}\right )\right ) \] Input:
Integrate[x^2*Sin[a + b/x],x]
Output:
(b^3*Cos[a]*CosIntegral[b/x] + x*(b*x*Cos[a + b/x] - b^2*Sin[a + b/x] + 2* x^2*Sin[a + b/x]) - b^3*Sin[a]*SinIntegral[b/x])/6
Time = 0.62 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.97, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.083, Rules used = {3860, 3042, 3778, 3042, 3778, 25, 3042, 3778, 3042, 3784, 3042, 3780, 3783}
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.
| \(\displaystyle \int x^2 \sin \left (a+\frac {b}{x}\right ) \, dx\) |
| \(\Big \downarrow \) 3860 |
| \(\displaystyle -\int x^4 \sin \left (a+\frac {b}{x}\right )d\frac {1}{x}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\int x^4 \sin \left (a+\frac {b}{x}\right )d\frac {1}{x}\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle \frac {1}{3} x^3 \sin \left (a+\frac {b}{x}\right )-\frac {1}{3} b \int x^3 \cos \left (a+\frac {b}{x}\right )d\frac {1}{x}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} x^3 \sin \left (a+\frac {b}{x}\right )-\frac {1}{3} b \int x^3 \sin \left (a+\frac {b}{x}+\frac {\pi }{2}\right )d\frac {1}{x}\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle \frac {1}{3} x^3 \sin \left (a+\frac {b}{x}\right )-\frac {1}{3} b \left (\frac {1}{2} b \int -x^2 \sin \left (a+\frac {b}{x}\right )d\frac {1}{x}-\frac {1}{2} x^2 \cos \left (a+\frac {b}{x}\right )\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{3} x^3 \sin \left (a+\frac {b}{x}\right )-\frac {1}{3} b \left (-\frac {1}{2} b \int x^2 \sin \left (a+\frac {b}{x}\right )d\frac {1}{x}-\frac {1}{2} x^2 \cos \left (a+\frac {b}{x}\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} x^3 \sin \left (a+\frac {b}{x}\right )-\frac {1}{3} b \left (-\frac {1}{2} b \int x^2 \sin \left (a+\frac {b}{x}\right )d\frac {1}{x}-\frac {1}{2} x^2 \cos \left (a+\frac {b}{x}\right )\right )\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle \frac {1}{3} x^3 \sin \left (a+\frac {b}{x}\right )-\frac {1}{3} b \left (-\frac {1}{2} b \left (b \int x \cos \left (a+\frac {b}{x}\right )d\frac {1}{x}-x \sin \left (a+\frac {b}{x}\right )\right )-\frac {1}{2} x^2 \cos \left (a+\frac {b}{x}\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} x^3 \sin \left (a+\frac {b}{x}\right )-\frac {1}{3} b \left (-\frac {1}{2} b \left (b \int x \sin \left (a+\frac {b}{x}+\frac {\pi }{2}\right )d\frac {1}{x}-x \sin \left (a+\frac {b}{x}\right )\right )-\frac {1}{2} x^2 \cos \left (a+\frac {b}{x}\right )\right )\) |
| \(\Big \downarrow \) 3784 |
| \(\displaystyle \frac {1}{3} x^3 \sin \left (a+\frac {b}{x}\right )-\frac {1}{3} b \left (-\frac {1}{2} b \left (b \left (\cos (a) \int x \cos \left (\frac {b}{x}\right )d\frac {1}{x}-\sin (a) \int x \sin \left (\frac {b}{x}\right )d\frac {1}{x}\right )-x \sin \left (a+\frac {b}{x}\right )\right )-\frac {1}{2} x^2 \cos \left (a+\frac {b}{x}\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} x^3 \sin \left (a+\frac {b}{x}\right )-\frac {1}{3} b \left (-\frac {1}{2} b \left (b \left (\cos (a) \int x \sin \left (\frac {b}{x}+\frac {\pi }{2}\right )d\frac {1}{x}-\sin (a) \int x \sin \left (\frac {b}{x}\right )d\frac {1}{x}\right )-x \sin \left (a+\frac {b}{x}\right )\right )-\frac {1}{2} x^2 \cos \left (a+\frac {b}{x}\right )\right )\) |
| \(\Big \downarrow \) 3780 |
| \(\displaystyle \frac {1}{3} x^3 \sin \left (a+\frac {b}{x}\right )-\frac {1}{3} b \left (-\frac {1}{2} b \left (b \left (\cos (a) \int x \sin \left (\frac {b}{x}+\frac {\pi }{2}\right )d\frac {1}{x}-\sin (a) \text {Si}\left (\frac {b}{x}\right )\right )-x \sin \left (a+\frac {b}{x}\right )\right )-\frac {1}{2} x^2 \cos \left (a+\frac {b}{x}\right )\right )\) |
| \(\Big \downarrow \) 3783 |
| \(\displaystyle \frac {1}{3} x^3 \sin \left (a+\frac {b}{x}\right )-\frac {1}{3} b \left (-\frac {1}{2} b \left (b \left (\cos (a) \operatorname {CosIntegral}\left (\frac {b}{x}\right )-\sin (a) \text {Si}\left (\frac {b}{x}\right )\right )-x \sin \left (a+\frac {b}{x}\right )\right )-\frac {1}{2} x^2 \cos \left (a+\frac {b}{x}\right )\right )\) |
Input:
Int[x^2*Sin[a + b/x],x]
Output:
(x^3*Sin[a + b/x])/3 - (b*(-1/2*(x^2*Cos[a + b/x]) - (b*(-(x*Sin[a + b/x]) + b*(Cos[a]*CosIntegral[b/x] - Sin[a]*SinIntegral[b/x])))/2))/3
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Sin[e + f*x]/(d*(m + 1))), x] - Simp[f/(d*(m + 1)) Int[( c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[m, - 1]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosInte gral[e - Pi/2 + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* e - c*f)/d] Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* f)/d] Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] && NeQ[d*e - c*f, 0]
Int[(x_)^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol ] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Sin[c + d*x])^ p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simplify[ (m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[ (m + 1)/n], 0]))
Time = 1.81 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.94
| method | result | size |
| derivativedivides | \(-b^{3} \left (-\frac {\sin \left (a +\frac {b}{x}\right ) x^{3}}{3 b^{3}}-\frac {\cos \left (a +\frac {b}{x}\right ) x^{2}}{6 b^{2}}+\frac {\sin \left (a +\frac {b}{x}\right ) x}{6 b}+\frac {\operatorname {Si}\left (\frac {b}{x}\right ) \sin \left (a \right )}{6}-\frac {\operatorname {Ci}\left (\frac {b}{x}\right ) \cos \left (a \right )}{6}\right )\) | \(73\) |
| default | \(-b^{3} \left (-\frac {\sin \left (a +\frac {b}{x}\right ) x^{3}}{3 b^{3}}-\frac {\cos \left (a +\frac {b}{x}\right ) x^{2}}{6 b^{2}}+\frac {\sin \left (a +\frac {b}{x}\right ) x}{6 b}+\frac {\operatorname {Si}\left (\frac {b}{x}\right ) \sin \left (a \right )}{6}-\frac {\operatorname {Ci}\left (\frac {b}{x}\right ) \cos \left (a \right )}{6}\right )\) | \(73\) |
| risch | \(\frac {i \pi \,\operatorname {csgn}\left (\frac {b}{x}\right ) {\mathrm e}^{-i a} b^{3}}{12}-\frac {i \operatorname {Si}\left (\frac {b}{x}\right ) {\mathrm e}^{-i a} b^{3}}{6}-\frac {\operatorname {expIntegral}_{1}\left (-\frac {i b}{x}\right ) {\mathrm e}^{-i a} b^{3}}{12}-\frac {{\mathrm e}^{i a} \operatorname {expIntegral}_{1}\left (-\frac {i b}{x}\right ) b^{3}}{12}+\frac {x^{2} b \cos \left (\frac {a x +b}{x}\right )}{6}-\frac {\sin \left (\frac {a x +b}{x}\right ) b^{2} x}{6}+\frac {\sin \left (\frac {a x +b}{x}\right ) x^{3}}{3}\) | \(122\) |
| parts | \(b \,x^{2} \operatorname {Si}\left (\frac {b}{x}\right ) \sin \left (a \right )-b \,x^{2} \operatorname {Ci}\left (\frac {b}{x}\right ) \cos \left (a \right )+x^{3} \sin \left (a +\frac {b}{x}\right )+2 b \left (-\cos \left (a \right ) b^{2} \left (-\frac {x^{2} \operatorname {Ci}\left (\frac {b}{x}\right )}{2 b^{2}}-\frac {\cos \left (\frac {b}{x}\right ) x^{2}}{4 b^{2}}+\frac {\sin \left (\frac {b}{x}\right ) x}{4 b}-\frac {\operatorname {Ci}\left (\frac {b}{x}\right )}{4}\right )+\sin \left (a \right ) b^{2} \left (-\frac {x^{2} \operatorname {Si}\left (\frac {b}{x}\right )}{2 b^{2}}-\frac {\sin \left (\frac {b}{x}\right ) x^{2}}{4 b^{2}}-\frac {\cos \left (\frac {b}{x}\right ) x}{4 b}-\frac {\operatorname {Si}\left (\frac {b}{x}\right )}{4}\right )+b^{2} \left (-\frac {\sin \left (a +\frac {b}{x}\right ) x^{3}}{3 b^{3}}-\frac {\cos \left (a +\frac {b}{x}\right ) x^{2}}{6 b^{2}}+\frac {\sin \left (a +\frac {b}{x}\right ) x}{6 b}+\frac {\operatorname {Si}\left (\frac {b}{x}\right ) \sin \left (a \right )}{6}-\frac {\operatorname {Ci}\left (\frac {b}{x}\right ) \cos \left (a \right )}{6}\right )\right )\) | \(227\) |
| meijerg | \(-\frac {b^{3} \sqrt {\pi }\, \cos \left (a \right ) \left (-\frac {8 x^{2}}{\sqrt {\pi }\, b^{2}}-\frac {4 \left (2 \gamma -\frac {11}{3}-2 \ln \left (x \right )+2 \ln \left (b \right )\right )}{3 \sqrt {\pi }}+\frac {8 x^{2} \left (-\frac {55 b^{2}}{2 x^{2}}+45\right )}{45 \sqrt {\pi }\, b^{2}}+\frac {8 \gamma }{3 \sqrt {\pi }}+\frac {8 \ln \left (2\right )}{3 \sqrt {\pi }}+\frac {8 \ln \left (\frac {b}{2 x}\right )}{3 \sqrt {\pi }}-\frac {8 x^{2} \cos \left (\frac {b}{x}\right )}{3 \sqrt {\pi }\, b^{2}}-\frac {16 x^{3} \left (-\frac {5 b^{2}}{2 x^{2}}+5\right ) \sin \left (\frac {b}{x}\right )}{15 \sqrt {\pi }\, b^{3}}-\frac {8 \,\operatorname {Ci}\left (\frac {b}{x}\right )}{3 \sqrt {\pi }}\right )}{16}-\frac {b^{2} \sqrt {\pi }\, \sin \left (a \right ) \sqrt {b^{2}}\, \left (-\frac {8 \left (-\frac {b^{2}}{x^{2}}+2\right ) x^{3} b^{2} \cos \left (\frac {\sqrt {b^{2}}}{x}\right )}{3 \left (b^{2}\right )^{\frac {5}{2}} \sqrt {\pi }}+\frac {8 x^{2} \sin \left (\frac {\sqrt {b^{2}}}{x}\right )}{3 b^{2} \sqrt {\pi }}+\frac {8 \,\operatorname {Si}\left (\frac {\sqrt {b^{2}}}{x}\right )}{3 \sqrt {\pi }}\right )}{16}\) | \(231\) |
Input:
int(x^2*sin(a+b/x),x,method=_RETURNVERBOSE)
Output:
-b^3*(-1/3*sin(a+b/x)/b^3*x^3-1/6*cos(a+b/x)/b^2*x^2+1/6*sin(a+b/x)/b*x+1/ 6*Si(b/x)*sin(a)-1/6*Ci(b/x)*cos(a))
Time = 0.08 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.85 \[ \int x^2 \sin \left (a+\frac {b}{x}\right ) \, dx=\frac {1}{6} \, b^{3} \cos \left (a\right ) \operatorname {Ci}\left (\frac {b}{x}\right ) - \frac {1}{6} \, b^{3} \sin \left (a\right ) \operatorname {Si}\left (\frac {b}{x}\right ) + \frac {1}{6} \, b x^{2} \cos \left (\frac {a x + b}{x}\right ) - \frac {1}{6} \, {\left (b^{2} x - 2 \, x^{3}\right )} \sin \left (\frac {a x + b}{x}\right ) \] Input:
integrate(x^2*sin(a+b/x),x, algorithm="fricas")
Output:
1/6*b^3*cos(a)*cos_integral(b/x) - 1/6*b^3*sin(a)*sin_integral(b/x) + 1/6* b*x^2*cos((a*x + b)/x) - 1/6*(b^2*x - 2*x^3)*sin((a*x + b)/x)
\[ \int x^2 \sin \left (a+\frac {b}{x}\right ) \, dx=\int x^{2} \sin {\left (a + \frac {b}{x} \right )}\, dx \] Input:
integrate(x**2*sin(a+b/x),x)
Output:
Integral(x**2*sin(a + b/x), x)
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.10 \[ \int x^2 \sin \left (a+\frac {b}{x}\right ) \, dx=\frac {1}{12} \, {\left ({\left ({\rm Ei}\left (\frac {i \, b}{x}\right ) + {\rm Ei}\left (-\frac {i \, b}{x}\right )\right )} \cos \left (a\right ) + {\left (i \, {\rm Ei}\left (\frac {i \, b}{x}\right ) - i \, {\rm Ei}\left (-\frac {i \, b}{x}\right )\right )} \sin \left (a\right )\right )} b^{3} + \frac {1}{6} \, b x^{2} \cos \left (\frac {a x + b}{x}\right ) - \frac {1}{6} \, {\left (b^{2} x - 2 \, x^{3}\right )} \sin \left (\frac {a x + b}{x}\right ) \] Input:
integrate(x^2*sin(a+b/x),x, algorithm="maxima")
Output:
1/12*((Ei(I*b/x) + Ei(-I*b/x))*cos(a) + (I*Ei(I*b/x) - I*Ei(-I*b/x))*sin(a ))*b^3 + 1/6*b*x^2*cos((a*x + b)/x) - 1/6*(b^2*x - 2*x^3)*sin((a*x + b)/x)
Leaf count of result is larger than twice the leaf count of optimal. 400 vs. \(2 (68) = 136\).
Time = 0.12 (sec) , antiderivative size = 400, normalized size of antiderivative = 5.13 \[ \int x^2 \sin \left (a+\frac {b}{x}\right ) \, dx=\frac {a^{3} b^{4} \cos \left (a\right ) \operatorname {Ci}\left (-a + \frac {a x + b}{x}\right ) + a^{3} b^{4} \sin \left (a\right ) \operatorname {Si}\left (a - \frac {a x + b}{x}\right ) - \frac {3 \, {\left (a x + b\right )} a^{2} b^{4} \cos \left (a\right ) \operatorname {Ci}\left (-a + \frac {a x + b}{x}\right )}{x} - \frac {3 \, {\left (a x + b\right )} a^{2} b^{4} \sin \left (a\right ) \operatorname {Si}\left (a - \frac {a x + b}{x}\right )}{x} + \frac {3 \, {\left (a x + b\right )}^{2} a b^{4} \cos \left (a\right ) \operatorname {Ci}\left (-a + \frac {a x + b}{x}\right )}{x^{2}} + a^{2} b^{4} \sin \left (\frac {a x + b}{x}\right ) + \frac {3 \, {\left (a x + b\right )}^{2} a b^{4} \sin \left (a\right ) \operatorname {Si}\left (a - \frac {a x + b}{x}\right )}{x^{2}} + a b^{4} \cos \left (\frac {a x + b}{x}\right ) - \frac {{\left (a x + b\right )}^{3} b^{4} \cos \left (a\right ) \operatorname {Ci}\left (-a + \frac {a x + b}{x}\right )}{x^{3}} - \frac {2 \, {\left (a x + b\right )} a b^{4} \sin \left (\frac {a x + b}{x}\right )}{x} - \frac {{\left (a x + b\right )}^{3} b^{4} \sin \left (a\right ) \operatorname {Si}\left (a - \frac {a x + b}{x}\right )}{x^{3}} - \frac {{\left (a x + b\right )} b^{4} \cos \left (\frac {a x + b}{x}\right )}{x} - 2 \, b^{4} \sin \left (\frac {a x + b}{x}\right ) + \frac {{\left (a x + b\right )}^{2} b^{4} \sin \left (\frac {a x + b}{x}\right )}{x^{2}}}{6 \, {\left (a^{3} - \frac {3 \, {\left (a x + b\right )} a^{2}}{x} + \frac {3 \, {\left (a x + b\right )}^{2} a}{x^{2}} - \frac {{\left (a x + b\right )}^{3}}{x^{3}}\right )} b} \] Input:
integrate(x^2*sin(a+b/x),x, algorithm="giac")
Output:
1/6*(a^3*b^4*cos(a)*cos_integral(-a + (a*x + b)/x) + a^3*b^4*sin(a)*sin_in tegral(a - (a*x + b)/x) - 3*(a*x + b)*a^2*b^4*cos(a)*cos_integral(-a + (a* x + b)/x)/x - 3*(a*x + b)*a^2*b^4*sin(a)*sin_integral(a - (a*x + b)/x)/x + 3*(a*x + b)^2*a*b^4*cos(a)*cos_integral(-a + (a*x + b)/x)/x^2 + a^2*b^4*s in((a*x + b)/x) + 3*(a*x + b)^2*a*b^4*sin(a)*sin_integral(a - (a*x + b)/x) /x^2 + a*b^4*cos((a*x + b)/x) - (a*x + b)^3*b^4*cos(a)*cos_integral(-a + ( a*x + b)/x)/x^3 - 2*(a*x + b)*a*b^4*sin((a*x + b)/x)/x - (a*x + b)^3*b^4*s in(a)*sin_integral(a - (a*x + b)/x)/x^3 - (a*x + b)*b^4*cos((a*x + b)/x)/x - 2*b^4*sin((a*x + b)/x) + (a*x + b)^2*b^4*sin((a*x + b)/x)/x^2)/((a^3 - 3*(a*x + b)*a^2/x + 3*(a*x + b)^2*a/x^2 - (a*x + b)^3/x^3)*b)
Timed out. \[ \int x^2 \sin \left (a+\frac {b}{x}\right ) \, dx=\int x^2\,\sin \left (a+\frac {b}{x}\right ) \,d x \] Input:
int(x^2*sin(a + b/x),x)
Output:
int(x^2*sin(a + b/x), x)
\[ \int x^2 \sin \left (a+\frac {b}{x}\right ) \, dx=\int \sin \left (\frac {a x +b}{x}\right ) x^{2}d x \] Input:
int(x^2*sin(a+b/x),x)
Output:
int(sin((a*x + b)/x)*x**2,x)