Integrand size = 12, antiderivative size = 74 \[ \int \frac {\cos \left (a+\frac {b}{x^2}\right )}{x^2} \, dx=-\frac {\sqrt {\frac {\pi }{2}} \cos (a) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{x}\right )}{\sqrt {b}}+\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{x}\right ) \sin (a)}{\sqrt {b}} \]
-1/2*cos(a)*FresnelC(b^(1/2)*2^(1/2)/Pi^(1/2)/x)*2^(1/2)*Pi^(1/2)/b^(1/2)+ 1/2*FresnelS(b^(1/2)*2^(1/2)/Pi^(1/2)/x)*sin(a)*2^(1/2)*Pi^(1/2)/b^(1/2)
Time = 0.12 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.84 \[ \int \frac {\cos \left (a+\frac {b}{x^2}\right )}{x^2} \, dx=-\frac {\sqrt {\frac {\pi }{2}} \left (\cos (a) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{x}\right )-\operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{x}\right ) \sin (a)\right )}{\sqrt {b}} \]
-((Sqrt[Pi/2]*(Cos[a]*FresnelC[(Sqrt[b]*Sqrt[2/Pi])/x] - FresnelS[(Sqrt[b] *Sqrt[2/Pi])/x]*Sin[a]))/Sqrt[b])
Time = 0.27 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3865, 3835, 3832, 3833}
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.
\(\displaystyle \int \frac {\cos \left (a+\frac {b}{x^2}\right )}{x^2} \, dx\) |
\(\Big \downarrow \) 3865 |
\(\displaystyle -\int \cos \left (a+\frac {b}{x^2}\right )d\frac {1}{x}\) |
\(\Big \downarrow \) 3835 |
\(\displaystyle \sin (a) \int \sin \left (\frac {b}{x^2}\right )d\frac {1}{x}-\cos (a) \int \cos \left (\frac {b}{x^2}\right )d\frac {1}{x}\) |
\(\Big \downarrow \) 3832 |
\(\displaystyle \frac {\sqrt {\frac {\pi }{2}} \sin (a) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{x}\right )}{\sqrt {b}}-\cos (a) \int \cos \left (\frac {b}{x^2}\right )d\frac {1}{x}\) |
\(\Big \downarrow \) 3833 |
\(\displaystyle \frac {\sqrt {\frac {\pi }{2}} \sin (a) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{x}\right )}{\sqrt {b}}-\frac {\sqrt {\frac {\pi }{2}} \cos (a) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{x}\right )}{\sqrt {b}}\) |
-((Sqrt[Pi/2]*Cos[a]*FresnelC[(Sqrt[b]*Sqrt[2/Pi])/x])/Sqrt[b]) + (Sqrt[Pi /2]*FresnelS[(Sqrt[b]*Sqrt[2/Pi])/x]*Sin[a])/Sqrt[b]
3.1.42.3.1 Defintions of rubi rules used
Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
Int[Cos[(c_) + (d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[Cos[c] In t[Cos[d*(e + f*x)^2], x], x] - Simp[Sin[c] Int[Sin[d*(e + f*x)^2], x], x] /; FreeQ[{c, d, e, f}, x]
Int[Cos[(a_.) + (b_.)*(x_)^(n_)]*(x_)^(m_.), x_Symbol] :> Simp[2/n Subst[ Int[Cos[a + b*x^2], x], x, x^(n/2)], x] /; FreeQ[{a, b, m, n}, x] && EqQ[m, n/2 - 1]
Time = 0.57 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.65
method | result | size |
derivativedivides | \(-\frac {\sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (a \right ) \operatorname {C}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, x}\right )-\sin \left (a \right ) \operatorname {S}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, x}\right )\right )}{2 \sqrt {b}}\) | \(48\) |
default | \(-\frac {\sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (a \right ) \operatorname {C}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, x}\right )-\sin \left (a \right ) \operatorname {S}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, x}\right )\right )}{2 \sqrt {b}}\) | \(48\) |
meijerg | \(-\frac {\cos \left (a \right ) \operatorname {C}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, x}\right ) \sqrt {2}\, \sqrt {\pi }}{2 \sqrt {b}}+\frac {\operatorname {S}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, x}\right ) \sin \left (a \right ) \sqrt {2}\, \sqrt {\pi }}{2 \sqrt {b}}\) | \(56\) |
risch | \(-\frac {{\mathrm e}^{-i a} \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {i b}}{x}\right )}{4 \sqrt {i b}}-\frac {{\mathrm e}^{i a} \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-i b}}{x}\right )}{4 \sqrt {-i b}}\) | \(56\) |
-1/2*2^(1/2)*Pi^(1/2)/b^(1/2)*(cos(a)*FresnelC(b^(1/2)*2^(1/2)/Pi^(1/2)/x) -sin(a)*FresnelS(b^(1/2)*2^(1/2)/Pi^(1/2)/x))
Time = 0.27 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.88 \[ \int \frac {\cos \left (a+\frac {b}{x^2}\right )}{x^2} \, dx=-\frac {\sqrt {2} \pi \sqrt {\frac {b}{\pi }} \cos \left (a\right ) \operatorname {C}\left (\frac {\sqrt {2} \sqrt {\frac {b}{\pi }}}{x}\right ) - \sqrt {2} \pi \sqrt {\frac {b}{\pi }} \operatorname {S}\left (\frac {\sqrt {2} \sqrt {\frac {b}{\pi }}}{x}\right ) \sin \left (a\right )}{2 \, b} \]
-1/2*(sqrt(2)*pi*sqrt(b/pi)*cos(a)*fresnel_cos(sqrt(2)*sqrt(b/pi)/x) - sqr t(2)*pi*sqrt(b/pi)*fresnel_sin(sqrt(2)*sqrt(b/pi)/x)*sin(a))/b
\[ \int \frac {\cos \left (a+\frac {b}{x^2}\right )}{x^2} \, dx=\int \frac {\cos {\left (a + \frac {b}{x^{2}} \right )}}{x^{2}}\, dx \]
Result contains complex when optimal does not.
Time = 0.36 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.32 \[ \int \frac {\cos \left (a+\frac {b}{x^2}\right )}{x^2} \, dx=-\frac {\sqrt {2} \sqrt {x^{4}} {\left ({\left (-\left (i - 1\right ) \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {\frac {i \, b}{x^{2}}}\right ) - 1\right )} + \left (i + 1\right ) \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-\frac {i \, b}{x^{2}}}\right ) - 1\right )}\right )} \cos \left (a\right ) + {\left (-\left (i + 1\right ) \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {\frac {i \, b}{x^{2}}}\right ) - 1\right )} + \left (i - 1\right ) \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-\frac {i \, b}{x^{2}}}\right ) - 1\right )}\right )} \sin \left (a\right )\right )} \left (\frac {b^{2}}{x^{4}}\right )^{\frac {1}{4}}}{8 \, b x} \]
-1/8*sqrt(2)*((-(I - 1)*sqrt(pi)*(erf(sqrt(I*b/x^2)) - 1) + (I + 1)*sqrt(p i)*(erf(sqrt(-I*b/x^2)) - 1))*cos(a) + (-(I + 1)*sqrt(pi)*(erf(sqrt(I*b/x^ 2)) - 1) + (I - 1)*sqrt(pi)*(erf(sqrt(-I*b/x^2)) - 1))*sin(a))*sqrt(x^4)*( b^2/x^4)^(1/4)/(b*x)
\[ \int \frac {\cos \left (a+\frac {b}{x^2}\right )}{x^2} \, dx=\int { \frac {\cos \left (a + \frac {b}{x^{2}}\right )}{x^{2}} \,d x } \]
Time = 13.34 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.74 \[ \int \frac {\cos \left (a+\frac {b}{x^2}\right )}{x^2} \, dx=\frac {\sqrt {2}\,\sqrt {\pi }\,\mathrm {S}\left (\frac {\sqrt {2}\,\sqrt {b}}{x\,\sqrt {\pi }}\right )\,\sin \left (a\right )}{2\,\sqrt {b}}-\frac {\sqrt {2}\,\sqrt {\pi }\,\mathrm {C}\left (\frac {\sqrt {2}\,\sqrt {b}}{x\,\sqrt {\pi }}\right )\,\cos \left (a\right )}{2\,\sqrt {b}} \]