\(\int \frac {\cos ^2(a+b x+c x^2)}{x} \, dx\) [19]

Optimal result

Integrand size = 17, antiderivative size = 17 \[ \int \frac {\cos ^2\left (a+b x+c x^2\right )}{x} \, dx=\frac {\log (x)}{2}+\frac {1}{2} \text {Int}\left (\frac {\cos \left (2 a+2 b x+2 c x^2\right )}{x},x\right ) \]

[Out]

1/2*ln(x)+1/2*Unintegrable(cos(2*c*x^2+2*b*x+2*a)/x,x)

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\cos ^2\left (a+b x+c x^2\right )}{x} \, dx=\int \frac {\cos ^2\left (a+b x+c x^2\right )}{x} \, dx \]

[In]

Int[Cos[a + b*x + c*x^2]^2/x,x]

[Out]

Log[x]/2 + Defer[Int][Cos[2*a + 2*b*x + 2*c*x^2]/x, x]/2

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{2 x}+\frac {\cos \left (2 a+2 b x+2 c x^2\right )}{2 x}\right ) \, dx \\ & = \frac {\log (x)}{2}+\frac {1}{2} \int \frac {\cos \left (2 a+2 b x+2 c x^2\right )}{x} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 3.96 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {\cos ^2\left (a+b x+c x^2\right )}{x} \, dx=\int \frac {\cos ^2\left (a+b x+c x^2\right )}{x} \, dx \]

[In]

Integrate[Cos[a + b*x + c*x^2]^2/x,x]

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.43 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {\cos ^2\left (a+b x+c x^2\right )}{x} \, dx=\int { \frac {\cos \left (c x^{2} + b x + a\right )^{2}}{x} \,d x } \]

[In]

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

[Out]

integral(cos(c*x^2 + b*x + a)^2/x, x)

Sympy [N/A]

Not integrable

Time = 0.80 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {\cos ^2\left (a+b x+c x^2\right )}{x} \, dx=\int \frac {\cos ^{2}{\left (a + b x + c x^{2} \right )}}{x}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.65 \[ \int \frac {\cos ^2\left (a+b x+c x^2\right )}{x} \, dx=\int { \frac {\cos \left (c x^{2} + b x + a\right )^{2}}{x} \,d x } \]

[In]

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

[Out]

1/2*integrate(cos(2*c*x^2 + 2*b*x + 2*a)/x, x) + 1/2*log(x)

Giac [N/A]

Not integrable

Time = 0.49 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {\cos ^2\left (a+b x+c x^2\right )}{x} \, dx=\int { \frac {\cos \left (c x^{2} + b x + a\right )^{2}}{x} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 13.75 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {\cos ^2\left (a+b x+c x^2\right )}{x} \, dx=\int \frac {{\cos \left (c\,x^2+b\,x+a\right )}^2}{x} \,d x \]

[In]

int(cos(a + b*x + c*x^2)^2/x,x)

[Out]

int(cos(a + b*x + c*x^2)^2/x, x)