\(\int \frac {(a+b \sec (c+d x^2))^2}{x^2} \, dx\) [14]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int \frac {\left (a+b \sec \left (c+d x^2\right )\right )^2}{x^2} \, dx=\text {Int}\left (\frac {\left (a+b \sec \left (c+d x^2\right )\right )^2}{x^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 18, 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 = {} \[ \int \frac {\left (a+b \sec \left (c+d x^2\right )\right )^2}{x^2} \, dx=\int \frac {\left (a+b \sec \left (c+d x^2\right )\right )^2}{x^2} \, dx \]

[In]

Int[(a + b*Sec[c + d*x^2])^2/x^2,x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 12.10 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a+b \sec \left (c+d x^2\right )\right )^2}{x^2} \, dx=\int \frac {\left (a+b \sec \left (c+d x^2\right )\right )^2}{x^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.30 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 36, normalized size of antiderivative = 2.00 \[ \int \frac {\left (a+b \sec \left (c+d x^2\right )\right )^2}{x^2} \, dx=\int { \frac {{\left (b \sec \left (d x^{2} + c\right ) + a\right )}^{2}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.66 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+b \sec \left (c+d x^2\right )\right )^2}{x^2} \, dx=\int \frac {\left (a + b \sec {\left (c + d x^{2} \right )}\right )^{2}}{x^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.52 (sec) , antiderivative size = 289, normalized size of antiderivative = 16.06 \[ \int \frac {\left (a+b \sec \left (c+d x^2\right )\right )^2}{x^2} \, dx=\int { \frac {{\left (b \sec \left (d x^{2} + c\right ) + a\right )}^{2}}{x^{2}} \,d x } \]

[In]

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

[Out]

-a^2/x + (b^2*sin(2*d*x^2 + 2*c) + (d*x^3*cos(2*d*x^2 + 2*c)^2 + d*x^3*sin(2*d*x^2 + 2*c)^2 + 2*d*x^3*cos(2*d*
x^2 + 2*c) + d*x^3)*integrate((4*a*b*d*x^2*cos(2*d*x^2 + 2*c)*cos(d*x^2 + c) + 4*a*b*d*x^2*cos(d*x^2 + c) + (4
*a*b*d*x^2*sin(d*x^2 + c) + 3*b^2)*sin(2*d*x^2 + 2*c))/(d*x^4*cos(2*d*x^2 + 2*c)^2 + d*x^4*sin(2*d*x^2 + 2*c)^
2 + 2*d*x^4*cos(2*d*x^2 + 2*c) + d*x^4), x))/(d*x^3*cos(2*d*x^2 + 2*c)^2 + d*x^3*sin(2*d*x^2 + 2*c)^2 + 2*d*x^
3*cos(2*d*x^2 + 2*c) + d*x^3)

Giac [N/A]

Not integrable

Time = 1.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a+b \sec \left (c+d x^2\right )\right )^2}{x^2} \, dx=\int { \frac {{\left (b \sec \left (d x^{2} + c\right ) + a\right )}^{2}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 13.36 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {\left (a+b \sec \left (c+d x^2\right )\right )^2}{x^2} \, dx=\int \frac {{\left (a+\frac {b}{\cos \left (d\,x^2+c\right )}\right )}^2}{x^2} \,d x \]

[In]

int((a + b/cos(c + d*x^2))^2/x^2,x)

[Out]

int((a + b/cos(c + d*x^2))^2/x^2, x)