Integrand size = 12, antiderivative size = 80 \[ \int \frac {\cos \left (a+b x^2\right )}{x^2} \, dx=-\frac {\cos \left (a+b x^2\right )}{x}-\sqrt {b} \sqrt {2 \pi } \cos (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )-\sqrt {b} \sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right ) \sin (a) \] Output:
-cos(b*x^2+a)/x-b^(1/2)*2^(1/2)*Pi^(1/2)*cos(a)*FresnelS(b^(1/2)*2^(1/2)/P i^(1/2)*x)-b^(1/2)*2^(1/2)*Pi^(1/2)*FresnelC(b^(1/2)*2^(1/2)/Pi^(1/2)*x)*s in(a)
Time = 0.19 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.01 \[ \int \frac {\cos \left (a+b x^2\right )}{x^2} \, dx=-\frac {\cos (a) \cos \left (b x^2\right )}{x}-\sqrt {b} \sqrt {2 \pi } \left (\cos (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )+\operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right ) \sin (a)\right )+\frac {\sin (a) \sin \left (b x^2\right )}{x} \] Input:
Integrate[Cos[a + b*x^2]/x^2,x]
Output:
-((Cos[a]*Cos[b*x^2])/x) - Sqrt[b]*Sqrt[2*Pi]*(Cos[a]*FresnelS[Sqrt[b]*Sqr t[2/Pi]*x] + FresnelC[Sqrt[b]*Sqrt[2/Pi]*x]*Sin[a]) + (Sin[a]*Sin[b*x^2])/ x
Time = 0.29 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3869, 3834, 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.
\(\displaystyle \int \frac {\cos \left (a+b x^2\right )}{x^2} \, dx\) |
\(\Big \downarrow \) 3869 |
\(\displaystyle -2 b \int \sin \left (b x^2+a\right )dx-\frac {\cos \left (a+b x^2\right )}{x}\) |
\(\Big \downarrow \) 3834 |
\(\displaystyle -2 b \left (\sin (a) \int \cos \left (b x^2\right )dx+\cos (a) \int \sin \left (b x^2\right )dx\right )-\frac {\cos \left (a+b x^2\right )}{x}\) |
\(\Big \downarrow \) 3832 |
\(\displaystyle -2 b \left (\sin (a) \int \cos \left (b x^2\right )dx+\frac {\sqrt {\frac {\pi }{2}} \cos (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )}{\sqrt {b}}\right )-\frac {\cos \left (a+b x^2\right )}{x}\) |
\(\Big \downarrow \) 3833 |
\(\displaystyle -2 b \left (\frac {\sqrt {\frac {\pi }{2}} \sin (a) \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )}{\sqrt {b}}+\frac {\sqrt {\frac {\pi }{2}} \cos (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )}{\sqrt {b}}\right )-\frac {\cos \left (a+b x^2\right )}{x}\) |
Input:
Int[Cos[a + b*x^2]/x^2,x]
Output:
-(Cos[a + b*x^2]/x) - 2*b*((Sqrt[Pi/2]*Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]* x])/Sqrt[b] + (Sqrt[Pi/2]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*x]*Sin[a])/Sqrt[b])
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[Sin[(c_) + (d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[Sin[c] In t[Cos[d*(e + f*x)^2], x], x] + Simp[Cos[c] Int[Sin[d*(e + f*x)^2], x], x] /; FreeQ[{c, d, e, f}, x]
Int[Cos[(c_.) + (d_.)*(x_)^(n_)]*((e_.)*(x_))^(m_), x_Symbol] :> Simp[(e*x) ^(m + 1)*(Cos[c + d*x^n]/(e*(m + 1))), x] + Simp[d*(n/(e^n*(m + 1))) Int[ (e*x)^(m + n)*Sin[c + d*x^n], x], x] /; FreeQ[{c, d, e}, x] && IGtQ[n, 0] & & LtQ[m, -1]
Time = 0.64 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.71
method | result | size |
default | \(-\frac {\cos \left (b \,x^{2}+a \right )}{x}-\sqrt {b}\, \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (a \right ) \operatorname {FresnelS}\left (\frac {\sqrt {b}\, \sqrt {2}\, x}{\sqrt {\pi }}\right )+\sin \left (a \right ) \operatorname {FresnelC}\left (\frac {\sqrt {b}\, \sqrt {2}\, x}{\sqrt {\pi }}\right )\right )\) | \(57\) |
risch | \(-\frac {i {\mathrm e}^{-i a} b \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {i b}\, x \right )}{2 \sqrt {i b}}+\frac {i {\mathrm e}^{i a} b \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-i b}\, x \right )}{2 \sqrt {-i b}}-\frac {\cos \left (b \,x^{2}+a \right )}{x}\) | \(69\) |
meijerg | \(\frac {\cos \left (a \right ) \sqrt {\pi }\, \left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (-\frac {4 \sqrt {2}\, \cos \left (b \,x^{2}\right )}{\sqrt {\pi }\, x \left (b^{2}\right )^{\frac {1}{4}}}-\frac {8 \sqrt {b}\, \operatorname {FresnelS}\left (\frac {\sqrt {b}\, \sqrt {2}\, x}{\sqrt {\pi }}\right )}{\left (b^{2}\right )^{\frac {1}{4}}}\right )}{8}-\frac {\sin \left (a \right ) \sqrt {\pi }\, \sqrt {b}\, \sqrt {2}\, \left (-\frac {4 \sqrt {2}\, \sin \left (b \,x^{2}\right )}{\sqrt {b}\, \sqrt {\pi }\, x}+8 \,\operatorname {FresnelC}\left (\frac {\sqrt {b}\, \sqrt {2}\, x}{\sqrt {\pi }}\right )\right )}{8}\) | \(110\) |
Input:
int(cos(b*x^2+a)/x^2,x,method=_RETURNVERBOSE)
Output:
-cos(b*x^2+a)/x-b^(1/2)*2^(1/2)*Pi^(1/2)*(cos(a)*FresnelS(b^(1/2)*2^(1/2)/ Pi^(1/2)*x)+sin(a)*FresnelC(b^(1/2)*2^(1/2)/Pi^(1/2)*x))
Time = 0.08 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.88 \[ \int \frac {\cos \left (a+b x^2\right )}{x^2} \, dx=-\frac {\sqrt {2} \pi x \sqrt {\frac {b}{\pi }} \cos \left (a\right ) \operatorname {S}\left (\sqrt {2} x \sqrt {\frac {b}{\pi }}\right ) + \sqrt {2} \pi x \sqrt {\frac {b}{\pi }} \operatorname {C}\left (\sqrt {2} x \sqrt {\frac {b}{\pi }}\right ) \sin \left (a\right ) + \cos \left (b x^{2} + a\right )}{x} \] Input:
integrate(cos(b*x^2+a)/x^2,x, algorithm="fricas")
Output:
-(sqrt(2)*pi*x*sqrt(b/pi)*cos(a)*fresnel_sin(sqrt(2)*x*sqrt(b/pi)) + sqrt( 2)*pi*x*sqrt(b/pi)*fresnel_cos(sqrt(2)*x*sqrt(b/pi))*sin(a) + cos(b*x^2 + a))/x
\[ \int \frac {\cos \left (a+b x^2\right )}{x^2} \, dx=\int \frac {\cos {\left (a + b x^{2} \right )}}{x^{2}}\, dx \] Input:
integrate(cos(b*x**2+a)/x**2,x)
Output:
Integral(cos(a + b*x**2)/x**2, x)
Result contains complex when optimal does not.
Time = 0.18 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.91 \[ \int \frac {\cos \left (a+b x^2\right )}{x^2} \, dx=\frac {\sqrt {b x^{2}} {\left ({\left (-\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, i \, b x^{2}\right ) + \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, -i \, b x^{2}\right )\right )} \cos \left (a\right ) + {\left (\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, i \, b x^{2}\right ) - \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, -i \, b x^{2}\right )\right )} \sin \left (a\right )\right )}}{8 \, x} \] Input:
integrate(cos(b*x^2+a)/x^2,x, algorithm="maxima")
Output:
1/8*sqrt(b*x^2)*((-(I + 1)*sqrt(2)*gamma(-1/2, I*b*x^2) + (I - 1)*sqrt(2)* gamma(-1/2, -I*b*x^2))*cos(a) + ((I - 1)*sqrt(2)*gamma(-1/2, I*b*x^2) - (I + 1)*sqrt(2)*gamma(-1/2, -I*b*x^2))*sin(a))/x
\[ \int \frac {\cos \left (a+b x^2\right )}{x^2} \, dx=\int { \frac {\cos \left (b x^{2} + a\right )}{x^{2}} \,d x } \] Input:
integrate(cos(b*x^2+a)/x^2,x, algorithm="giac")
Output:
integrate(cos(b*x^2 + a)/x^2, x)
Timed out. \[ \int \frac {\cos \left (a+b x^2\right )}{x^2} \, dx=\int \frac {\cos \left (b\,x^2+a\right )}{x^2} \,d x \] Input:
int(cos(a + b*x^2)/x^2,x)
Output:
int(cos(a + b*x^2)/x^2, x)
\[ \int \frac {\cos \left (a+b x^2\right )}{x^2} \, dx=\frac {\left (\int \frac {\cos \left (b \,x^{2}+a \right )}{x^{2}}d x \right ) x +\left (\int \frac {1}{x^{2}}d x \right ) x +1}{x} \] Input:
int(cos(b*x^2+a)/x^2,x)
Output:
(int(cos(a + b*x**2)/x**2,x)*x + int(1/x**2,x)*x + 1)/x