\(\int \frac {\cos ((a+b x)^2)}{x^2} \, dx\) [89]

Optimal result

Integrand size = 12, antiderivative size = 12 \[ \int \frac {\cos \left ((a+b x)^2\right )}{x^2} \, dx=\text {Int}\left (\frac {\cos \left ((a+b x)^2\right )}{x^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 12, 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 = {} \[ \int \frac {\cos \left ((a+b x)^2\right )}{x^2} \, dx=\int \frac {\cos \left ((a+b x)^2\right )}{x^2} \, dx \]

[In]

Int[Cos[(a + b*x)^2]/x^2,x]

[Out]

Defer[Int][Cos[(a + b*x)^2]/x^2, x]

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

Mathematica [N/A]

Not integrable

Time = 3.93 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\cos \left ((a+b x)^2\right )}{x^2} \, dx=\int \frac {\cos \left ((a+b x)^2\right )}{x^2} \, dx \]

[In]

Integrate[Cos[(a + b*x)^2]/x^2,x]

[Out]

Integrate[Cos[(a + b*x)^2]/x^2, x]

Maple [N/A] (verified)

Not integrable

Time = 0.16 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.92 \[ \int \frac {\cos \left ((a+b x)^2\right )}{x^2} \, dx=\int { \frac {\cos \left ({\left (b x + a\right )}^{2}\right )}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.95 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.83 \[ \int \frac {\cos \left ((a+b x)^2\right )}{x^2} \, dx=\int \frac {\cos {\left (a^{2} + 2 a b x + b^{2} x^{2} \right )}}{x^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.42 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\cos \left ((a+b x)^2\right )}{x^2} \, dx=\int { \frac {\cos \left ({\left (b x + a\right )}^{2}\right )}{x^{2}} \,d x } \]

[In]

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

[Out]

integrate(cos((b*x + a)^2)/x^2, x)

Giac [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\cos \left ((a+b x)^2\right )}{x^2} \, dx=\int { \frac {\cos \left ({\left (b x + a\right )}^{2}\right )}{x^{2}} \,d x } \]

[In]

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

[Out]

integrate(cos((b*x + a)^2)/x^2, x)

Mupad [N/A]

Not integrable

Time = 13.09 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\cos \left ((a+b x)^2\right )}{x^2} \, dx=\int \frac {\cos \left ({\left (a+b\,x\right )}^2\right )}{x^2} \,d x \]

[In]

int(cos((a + b*x)^2)/x^2,x)

[Out]

int(cos((a + b*x)^2)/x^2, x)