Integrand size = 8, antiderivative size = 32 \[ \int \sin \left (a+\frac {b}{x}\right ) \, dx=-b \cos (a) \operatorname {CosIntegral}\left (\frac {b}{x}\right )+x \sin \left (a+\frac {b}{x}\right )+b \sin (a) \text {Si}\left (\frac {b}{x}\right ) \] Output:
-b*cos(a)*Ci(b/x)+x*sin(a+b/x)+b*sin(a)*Si(b/x)
Time = 0.03 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \sin \left (a+\frac {b}{x}\right ) \, dx=-b \cos (a) \operatorname {CosIntegral}\left (\frac {b}{x}\right )+x \sin \left (a+\frac {b}{x}\right )+b \sin (a) \text {Si}\left (\frac {b}{x}\right ) \] Input:
Integrate[Sin[a + b/x],x]
Output:
-(b*Cos[a]*CosIntegral[b/x]) + x*Sin[a + b/x] + b*Sin[a]*SinIntegral[b/x]
Time = 0.41 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3842, 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.
\(\displaystyle \int \sin \left (a+\frac {b}{x}\right ) \, dx\) |
\(\Big \downarrow \) 3842 |
\(\displaystyle -\int x^2 \sin \left (a+\frac {b}{x}\right )d\frac {1}{x}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\int x^2 \sin \left (a+\frac {b}{x}\right )d\frac {1}{x}\) |
\(\Big \downarrow \) 3778 |
\(\displaystyle x \sin \left (a+\frac {b}{x}\right )-b \int x \cos \left (a+\frac {b}{x}\right )d\frac {1}{x}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle x \sin \left (a+\frac {b}{x}\right )-b \int x \sin \left (a+\frac {b}{x}+\frac {\pi }{2}\right )d\frac {1}{x}\) |
\(\Big \downarrow \) 3784 |
\(\displaystyle x \sin \left (a+\frac {b}{x}\right )-b \left (\cos (a) \int x \cos \left (\frac {b}{x}\right )d\frac {1}{x}-\sin (a) \int x \sin \left (\frac {b}{x}\right )d\frac {1}{x}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle x \sin \left (a+\frac {b}{x}\right )-b \left (\cos (a) \int x \sin \left (\frac {b}{x}+\frac {\pi }{2}\right )d\frac {1}{x}-\sin (a) \int x \sin \left (\frac {b}{x}\right )d\frac {1}{x}\right )\) |
\(\Big \downarrow \) 3780 |
\(\displaystyle x \sin \left (a+\frac {b}{x}\right )-b \left (\cos (a) \int x \sin \left (\frac {b}{x}+\frac {\pi }{2}\right )d\frac {1}{x}-\sin (a) \text {Si}\left (\frac {b}{x}\right )\right )\) |
\(\Big \downarrow \) 3783 |
\(\displaystyle x \sin \left (a+\frac {b}{x}\right )-b \left (\cos (a) \operatorname {CosIntegral}\left (\frac {b}{x}\right )-\sin (a) \text {Si}\left (\frac {b}{x}\right )\right )\) |
Input:
Int[Sin[a + b/x],x]
Output:
x*Sin[a + b/x] - b*(Cos[a]*CosIntegral[b/x] - Sin[a]*SinIntegral[b/x])
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[((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)])^(p_.), x_S ymbol] :> Simp[1/(n*f) Subst[Int[x^(1/n - 1)*(a + b*Sin[c + d*x])^p, x], x, (e + f*x)^n], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && Intege rQ[1/n]
Time = 0.80 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.19
method | result | size |
derivativedivides | \(-b \left (-\frac {\sin \left (a +\frac {b}{x}\right ) x}{b}-\operatorname {Si}\left (\frac {b}{x}\right ) \sin \left (a \right )+\operatorname {Ci}\left (\frac {b}{x}\right ) \cos \left (a \right )\right )\) | \(38\) |
default | \(-b \left (-\frac {\sin \left (a +\frac {b}{x}\right ) x}{b}-\operatorname {Si}\left (\frac {b}{x}\right ) \sin \left (a \right )+\operatorname {Ci}\left (\frac {b}{x}\right ) \cos \left (a \right )\right )\) | \(38\) |
risch | \(\frac {{\mathrm e}^{i a} \operatorname {expIntegral}_{1}\left (-\frac {i b}{x}\right ) b}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {b}{x}\right ) {\mathrm e}^{-i a} b}{2}+i \operatorname {Si}\left (\frac {b}{x}\right ) {\mathrm e}^{-i a} b +\frac {\operatorname {expIntegral}_{1}\left (-\frac {i b}{x}\right ) {\mathrm e}^{-i a} b}{2}+x \sin \left (\frac {a x +b}{x}\right )\) | \(79\) |
meijerg | \(-\frac {\cos \left (a \right ) \sqrt {\pi }\, b \left (\frac {4 \gamma -4-4 \ln \left (x \right )+4 \ln \left (b \right )}{\sqrt {\pi }}+\frac {4}{\sqrt {\pi }}-\frac {4 \gamma }{\sqrt {\pi }}-\frac {4 \ln \left (2\right )}{\sqrt {\pi }}-\frac {4 \ln \left (\frac {b}{2 x}\right )}{\sqrt {\pi }}-\frac {4 x \sin \left (\frac {b}{x}\right )}{\sqrt {\pi }\, b}+\frac {4 \,\operatorname {Ci}\left (\frac {b}{x}\right )}{\sqrt {\pi }}\right )}{4}-\frac {\sin \left (a \right ) \sqrt {\pi }\, \sqrt {b^{2}}\, \left (-\frac {4 x \,b^{2} \cos \left (\frac {\sqrt {b^{2}}}{x}\right )}{\left (b^{2}\right )^{\frac {3}{2}} \sqrt {\pi }}-\frac {4 \,\operatorname {Si}\left (\frac {\sqrt {b^{2}}}{x}\right )}{\sqrt {\pi }}\right )}{4}\) | \(137\) |
Input:
int(sin(a+b/x),x,method=_RETURNVERBOSE)
Output:
-b*(-sin(a+b/x)/b*x-Si(b/x)*sin(a)+Ci(b/x)*cos(a))
Time = 0.08 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \sin \left (a+\frac {b}{x}\right ) \, dx=-b \cos \left (a\right ) \operatorname {Ci}\left (\frac {b}{x}\right ) + b \sin \left (a\right ) \operatorname {Si}\left (\frac {b}{x}\right ) + x \sin \left (\frac {a x + b}{x}\right ) \] Input:
integrate(sin(a+b/x),x, algorithm="fricas")
Output:
-b*cos(a)*cos_integral(b/x) + b*sin(a)*sin_integral(b/x) + x*sin((a*x + b) /x)
\[ \int \sin \left (a+\frac {b}{x}\right ) \, dx=\int \sin {\left (a + \frac {b}{x} \right )}\, dx \] Input:
integrate(sin(a+b/x),x)
Output:
Integral(sin(a + b/x), x)
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.81 \[ \int \sin \left (a+\frac {b}{x}\right ) \, dx=-\frac {1}{2} \, {\left ({\left ({\rm Ei}\left (\frac {i \, b}{x}\right ) + {\rm Ei}\left (-\frac {i \, b}{x}\right )\right )} \cos \left (a\right ) - {\left (-i \, {\rm Ei}\left (\frac {i \, b}{x}\right ) + i \, {\rm Ei}\left (-\frac {i \, b}{x}\right )\right )} \sin \left (a\right )\right )} b + x \sin \left (\frac {a x + b}{x}\right ) \] Input:
integrate(sin(a+b/x),x, algorithm="maxima")
Output:
-1/2*((Ei(I*b/x) + Ei(-I*b/x))*cos(a) - (-I*Ei(I*b/x) + I*Ei(-I*b/x))*sin( a))*b + x*sin((a*x + b)/x)
Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (32) = 64\).
Time = 0.14 (sec) , antiderivative size = 132, normalized size of antiderivative = 4.12 \[ \int \sin \left (a+\frac {b}{x}\right ) \, dx=-\frac {a b^{2} \cos \left (a\right ) \operatorname {Ci}\left (-a + \frac {a x + b}{x}\right ) + a b^{2} \sin \left (a\right ) \operatorname {Si}\left (a - \frac {a x + b}{x}\right ) - \frac {{\left (a x + b\right )} b^{2} \cos \left (a\right ) \operatorname {Ci}\left (-a + \frac {a x + b}{x}\right )}{x} - \frac {{\left (a x + b\right )} b^{2} \sin \left (a\right ) \operatorname {Si}\left (a - \frac {a x + b}{x}\right )}{x} + b^{2} \sin \left (\frac {a x + b}{x}\right )}{{\left (a - \frac {a x + b}{x}\right )} b} \] Input:
integrate(sin(a+b/x),x, algorithm="giac")
Output:
-(a*b^2*cos(a)*cos_integral(-a + (a*x + b)/x) + a*b^2*sin(a)*sin_integral( a - (a*x + b)/x) - (a*x + b)*b^2*cos(a)*cos_integral(-a + (a*x + b)/x)/x - (a*x + b)*b^2*sin(a)*sin_integral(a - (a*x + b)/x)/x + b^2*sin((a*x + b)/ x))/((a - (a*x + b)/x)*b)
Timed out. \[ \int \sin \left (a+\frac {b}{x}\right ) \, dx=\int \sin \left (a+\frac {b}{x}\right ) \,d x \] Input:
int(sin(a + b/x),x)
Output:
int(sin(a + b/x), x)
\[ \int \sin \left (a+\frac {b}{x}\right ) \, dx=\int \sin \left (\frac {a x +b}{x}\right )d x \] Input:
int(sin(a+b/x),x)
Output:
int(sin((a*x + b)/x),x)