\(\int \frac {a+b \sec (c+d x^2)}{x^2} \, dx\) [22]

Optimal result

Integrand size = 16, antiderivative size = 16 \[ \int \frac {a+b \sec \left (c+d x^2\right )}{x^2} \, dx=-\frac {a}{x}+b \text {Int}\left (\frac {\sec \left (c+d x^2\right )}{x^2},x\right ) \]

[Out]

-a/x+b*Unintegrable(sec(d*x^2+c)/x^2,x)

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 16, 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 = {} \[ \int \frac {a+b \sec \left (c+d x^2\right )}{x^2} \, dx=\int \frac {a+b \sec \left (c+d x^2\right )}{x^2} \, dx \]

[In]

Int[(a + b*Sec[c + d*x^2])/x^2,x]

[Out]

-(a/x) + b*Defer[Int][Sec[c + d*x^2]/x^2, x]

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

Mathematica [N/A]

Not integrable

Time = 0.07 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {a+b \sec \left (c+d x^2\right )}{x^2} \, dx=\int \frac {a+b \sec \left (c+d x^2\right )}{x^2} \, dx \]

[In]

Integrate[(a + b*Sec[c + d*x^2])/x^2,x]

[Out]

Integrate[(a + b*Sec[c + d*x^2])/x^2, x]

Maple [N/A] (verified)

Not integrable

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {a+b \sec \left (c+d x^2\right )}{x^2} \, dx=\int { \frac {b \sec \left (d x^{2} + c\right ) + a}{x^{2}} \,d x } \]

[In]

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

[Out]

integral((b*sec(d*x^2 + c) + a)/x^2, x)

Sympy [N/A]

Not integrable

Time = 0.44 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int \frac {a+b \sec \left (c+d x^2\right )}{x^2} \, dx=\int \frac {a + b \sec {\left (c + d x^{2} \right )}}{x^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.44 (sec) , antiderivative size = 118, normalized size of antiderivative = 7.38 \[ \int \frac {a+b \sec \left (c+d x^2\right )}{x^2} \, dx=\int { \frac {b \sec \left (d x^{2} + c\right ) + a}{x^{2}} \,d x } \]

[In]

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

[Out]

2*b*integrate((cos(2*d*x^2 + 2*c)*cos(d*x^2 + c) + sin(2*d*x^2 + 2*c)*sin(d*x^2 + c) + cos(d*x^2 + c))/(x^2*co
s(2*d*x^2 + 2*c)^2 + x^2*sin(2*d*x^2 + 2*c)^2 + 2*x^2*cos(2*d*x^2 + 2*c) + x^2), x) - a/x

Giac [N/A]

Not integrable

Time = 0.43 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {a+b \sec \left (c+d x^2\right )}{x^2} \, dx=\int { \frac {b \sec \left (d x^{2} + c\right ) + a}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int \frac {a+b \sec \left (c+d x^2\right )}{x^2} \, dx=\int \frac {a+\frac {b}{\cos \left (d\,x^2+c\right )}}{x^2} \,d x \]

[In]

int((a + b/cos(c + d*x^2))/x^2,x)

[Out]

int((a + b/cos(c + d*x^2))/x^2, x)