\(\int \frac {(a+b \text {sech}(c+d x^2))^2}{x} \, dx\) [13]

Optimal result

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

[Out]

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

[In]

Int[(a + b*Sech[c + d*x^2])^2/x,x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 27.46 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a+b \text {sech}\left (c+d x^2\right )\right )^2}{x} \, dx=\int \frac {\left (a+b \text {sech}\left (c+d x^2\right )\right )^2}{x} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

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

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 2.00 \[ \int \frac {\left (a+b \text {sech}\left (c+d x^2\right )\right )^2}{x} \, dx=\int { \frac {{\left (b \operatorname {sech}\left (d x^{2} + c\right ) + a\right )}^{2}}{x} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 6.61 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a+b \text {sech}\left (c+d x^2\right )\right )^2}{x} \, dx=\int \frac {\left (a + b \operatorname {sech}{\left (c + d x^{2} \right )}\right )^{2}}{x}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 86, normalized size of antiderivative = 4.78 \[ \int \frac {\left (a+b \text {sech}\left (c+d x^2\right )\right )^2}{x} \, dx=\int { \frac {{\left (b \operatorname {sech}\left (d x^{2} + c\right ) + a\right )}^{2}}{x} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a+b \text {sech}\left (c+d x^2\right )\right )^2}{x} \, dx=\int { \frac {{\left (b \operatorname {sech}\left (d x^{2} + c\right ) + a\right )}^{2}}{x} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.16 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {\left (a+b \text {sech}\left (c+d x^2\right )\right )^2}{x} \, dx=\int \frac {{\left (a+\frac {b}{\mathrm {cosh}\left (d\,x^2+c\right )}\right )}^2}{x} \,d x \]

[In]

int((a + b/cosh(c + d*x^2))^2/x,x)

[Out]

int((a + b/cosh(c + d*x^2))^2/x, x)