\(\int (\frac {\cos (a+b x+c x^2)}{x^2}+\frac {b \sin (a+b x+c x^2)}{x}) \, dx\) [5]

Optimal result

Integrand size = 32, antiderivative size = 111 \[ \int \left (\frac {\cos \left (a+b x+c x^2\right )}{x^2}+\frac {b \sin \left (a+b x+c x^2\right )}{x}\right ) \, dx=-\frac {\cos \left (a+b x+c x^2\right )}{x}-\sqrt {c} \sqrt {2 \pi } \cos \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )-\sqrt {c} \sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right ) \sin \left (a-\frac {b^2}{4 c}\right ) \] Output:

-cos(c*x^2+b*x+a)/x-c^(1/2)*2^(1/2)*Pi^(1/2)*cos(a-1/4*b^2/c)*FresnelS(1/2 
*(2*c*x+b)/c^(1/2)*2^(1/2)/Pi^(1/2))-c^(1/2)*2^(1/2)*Pi^(1/2)*FresnelC(1/2 
*(2*c*x+b)/c^(1/2)*2^(1/2)/Pi^(1/2))*sin(a-1/4*b^2/c)
 

Mathematica [A] (verified)

Time = 2.96 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.99 \[ \int \left (\frac {\cos \left (a+b x+c x^2\right )}{x^2}+\frac {b \sin \left (a+b x+c x^2\right )}{x}\right ) \, dx=-\frac {\cos (a+x (b+c x))+\sqrt {c} \sqrt {2 \pi } x \cos \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )+\sqrt {c} \sqrt {2 \pi } x \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right ) \sin \left (a-\frac {b^2}{4 c}\right )}{x} \] Input:

Integrate[Cos[a + b*x + c*x^2]/x^2 + (b*Sin[a + b*x + c*x^2])/x,x]
 

Output:

-((Cos[a + x*(b + c*x)] + Sqrt[c]*Sqrt[2*Pi]*x*Cos[a - b^2/(4*c)]*FresnelS 
[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])] + Sqrt[c]*Sqrt[2*Pi]*x*FresnelC[(b + 2* 
c*x)/(Sqrt[c]*Sqrt[2*Pi])]*Sin[a - b^2/(4*c)])/x)
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {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.

\(\displaystyle \int \left (\frac {b \sin \left (a+b x+c x^2\right )}{x}+\frac {\cos \left (a+b x+c x^2\right )}{x^2}\right ) \, dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\sqrt {2 \pi } \sqrt {c} \sin \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )-\sqrt {2 \pi } \sqrt {c} \cos \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )-\frac {\cos \left (a+b x+c x^2\right )}{x}\)

Input:

Int[Cos[a + b*x + c*x^2]/x^2 + (b*Sin[a + b*x + c*x^2])/x,x]
 

Output:

-(Cos[a + b*x + c*x^2]/x) - Sqrt[c]*Sqrt[2*Pi]*Cos[a - b^2/(4*c)]*FresnelS 
[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])] - Sqrt[c]*Sqrt[2*Pi]*FresnelC[(b + 2*c* 
x)/(Sqrt[c]*Sqrt[2*Pi])]*Sin[a - b^2/(4*c)]
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [F]

\[\int \left (\frac {\cos \left (c \,x^{2}+b x +a \right )}{x^{2}}+\frac {b \sin \left (c \,x^{2}+b x +a \right )}{x}\right )d x\]

Input:

int(cos(c*x^2+b*x+a)/x^2+b*sin(c*x^2+b*x+a)/x,x)
 

Output:

int(cos(c*x^2+b*x+a)/x^2+b*sin(c*x^2+b*x+a)/x,x)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.04 \[ \int \left (\frac {\cos \left (a+b x+c x^2\right )}{x^2}+\frac {b \sin \left (a+b x+c x^2\right )}{x}\right ) \, dx=-\frac {\sqrt {2} \pi x \sqrt {\frac {c}{\pi }} \cos \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) \operatorname {S}\left (\frac {\sqrt {2} {\left (2 \, c x + b\right )} \sqrt {\frac {c}{\pi }}}{2 \, c}\right ) + \sqrt {2} \pi x \sqrt {\frac {c}{\pi }} \operatorname {C}\left (\frac {\sqrt {2} {\left (2 \, c x + b\right )} \sqrt {\frac {c}{\pi }}}{2 \, c}\right ) \sin \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + \cos \left (c x^{2} + b x + a\right )}{x} \] Input:

integrate(cos(c*x^2+b*x+a)/x^2+b*sin(c*x^2+b*x+a)/x,x, algorithm="fricas")
 

Output:

-(sqrt(2)*pi*x*sqrt(c/pi)*cos(-1/4*(b^2 - 4*a*c)/c)*fresnel_sin(1/2*sqrt(2 
)*(2*c*x + b)*sqrt(c/pi)/c) + sqrt(2)*pi*x*sqrt(c/pi)*fresnel_cos(1/2*sqrt 
(2)*(2*c*x + b)*sqrt(c/pi)/c)*sin(-1/4*(b^2 - 4*a*c)/c) + cos(c*x^2 + b*x 
+ a))/x
 

Sympy [F]

\[ \int \left (\frac {\cos \left (a+b x+c x^2\right )}{x^2}+\frac {b \sin \left (a+b x+c x^2\right )}{x}\right ) \, dx=\int \frac {b x \sin {\left (a + b x + c x^{2} \right )} + \cos {\left (a + b x + c x^{2} \right )}}{x^{2}}\, dx \] Input:

integrate(cos(c*x**2+b*x+a)/x**2+b*sin(c*x**2+b*x+a)/x,x)
 

Output:

Integral((b*x*sin(a + b*x + c*x**2) + cos(a + b*x + c*x**2))/x**2, x)
 

Maxima [F]

\[ \int \left (\frac {\cos \left (a+b x+c x^2\right )}{x^2}+\frac {b \sin \left (a+b x+c x^2\right )}{x}\right ) \, dx=\int { \frac {b \sin \left (c x^{2} + b x + a\right )}{x} + \frac {\cos \left (c x^{2} + b x + a\right )}{x^{2}} \,d x } \] Input:

integrate(cos(c*x^2+b*x+a)/x^2+b*sin(c*x^2+b*x+a)/x,x, algorithm="maxima")
 

Output:

integrate(b*sin(c*x^2 + b*x + a)/x + cos(c*x^2 + b*x + a)/x^2, x)
 

Giac [F]

\[ \int \left (\frac {\cos \left (a+b x+c x^2\right )}{x^2}+\frac {b \sin \left (a+b x+c x^2\right )}{x}\right ) \, dx=\int { \frac {b \sin \left (c x^{2} + b x + a\right )}{x} + \frac {\cos \left (c x^{2} + b x + a\right )}{x^{2}} \,d x } \] Input:

integrate(cos(c*x^2+b*x+a)/x^2+b*sin(c*x^2+b*x+a)/x,x, algorithm="giac")
 

Output:

integrate(b*sin(c*x^2 + b*x + a)/x + cos(c*x^2 + b*x + a)/x^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \left (\frac {\cos \left (a+b x+c x^2\right )}{x^2}+\frac {b \sin \left (a+b x+c x^2\right )}{x}\right ) \, dx=\int \frac {\cos \left (c\,x^2+b\,x+a\right )}{x^2}+\frac {b\,\sin \left (c\,x^2+b\,x+a\right )}{x} \,d x \] Input:

int(cos(a + b*x + c*x^2)/x^2 + (b*sin(a + b*x + c*x^2))/x,x)
 

Output:

int(cos(a + b*x + c*x^2)/x^2 + (b*sin(a + b*x + c*x^2))/x, x)
 

Reduce [F]

\[ \int \left (\frac {\cos \left (a+b x+c x^2\right )}{x^2}+\frac {b \sin \left (a+b x+c x^2\right )}{x}\right ) \, dx=\frac {\left (\int \frac {\cos \left (c \,x^{2}+b x +a \right )}{x^{2}}d x \right ) x +\left (\int \frac {\sin \left (c \,x^{2}+b x +a \right )}{x}d x \right ) b x +\left (\int \frac {1}{x^{2}}d x \right ) x +1}{x} \] Input:

int(cos(c*x^2+b*x+a)/x^2+b*sin(c*x^2+b*x+a)/x,x)
 

Output:

(int(cos(a + b*x + c*x**2)/x**2,x)*x + int(sin(a + b*x + c*x**2)/x,x)*b*x 
+ int(1/x**2,x)*x + 1)/x