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

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

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

[In]

Int[(a + b*Sech[c + d*x^2])^2/x^2,x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 10.62 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a+b \text {sech}\left (c+d x^2\right )\right )^2}{x^2} \, dx=\int \frac {\left (a+b \text {sech}\left (c+d x^2\right )\right )^2}{x^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.07 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 2.00 \[ \int \frac {\left (a+b \text {sech}\left (c+d x^2\right )\right )^2}{x^2} \, dx=\int { \frac {{\left (b \operatorname {sech}\left (d x^{2} + c\right ) + a\right )}^{2}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+b \text {sech}\left (c+d x^2\right )\right )^2}{x^2} \, dx=\int \frac {\left (a + b \operatorname {sech}{\left (c + d x^{2} \right )}\right )^{2}}{x^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.43 (sec) , antiderivative size = 87, normalized size of antiderivative = 4.83 \[ \int \frac {\left (a+b \text {sech}\left (c+d x^2\right )\right )^2}{x^2} \, dx=\int { \frac {{\left (b \operatorname {sech}\left (d x^{2} + c\right ) + a\right )}^{2}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.82 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a+b \text {sech}\left (c+d x^2\right )\right )^2}{x^2} \, dx=\int { \frac {{\left (b \operatorname {sech}\left (d x^{2} + c\right ) + a\right )}^{2}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.15 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {\left (a+b \text {sech}\left (c+d x^2\right )\right )^2}{x^2} \, dx=\int \frac {{\left (a+\frac {b}{\mathrm {cosh}\left (d\,x^2+c\right )}\right )}^2}{x^2} \,d x \]

[In]

int((a + b/cosh(c + d*x^2))^2/x^2,x)

[Out]

int((a + b/cosh(c + d*x^2))^2/x^2, x)