\(\int \frac {(a+b \text {sech}(c+d \sqrt {x}))^2}{x^2} \, dx\) [41]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2}{x^2} \, dx=\text {Int}\left (\frac {\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2}{x^2},x\right ) \]

[Out]

Unintegrable((a+b*sech(c+d*x^(1/2)))^2/x^2,x)

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 20, 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 = {} \[ \int \frac {\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2}{x^2} \, dx=\int \frac {\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2}{x^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 37.84 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2}{x^2} \, dx=\int \frac {\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2}{x^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90

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

[In]

int((a+b*sech(c+d*x^(1/2)))^2/x^2,x)

[Out]

int((a+b*sech(c+d*x^(1/2)))^2/x^2,x)

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.80 \[ \int \frac {\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2}{x^2} \, dx=\int { \frac {{\left (b \operatorname {sech}\left (d \sqrt {x} + c\right ) + a\right )}^{2}}{x^{2}} \,d x } \]

[In]

integrate((a+b*sech(c+d*x^(1/2)))^2/x^2,x, algorithm="fricas")

[Out]

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

Sympy [N/A]

Not integrable

Time = 2.46 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2}{x^2} \, dx=\int \frac {\left (a + b \operatorname {sech}{\left (c + d \sqrt {x} \right )}\right )^{2}}{x^{2}}\, dx \]

[In]

integrate((a+b*sech(c+d*x**(1/2)))**2/x**2,x)

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.57 (sec) , antiderivative size = 110, normalized size of antiderivative = 5.50 \[ \int \frac {\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2}{x^2} \, dx=\int { \frac {{\left (b \operatorname {sech}\left (d \sqrt {x} + c\right ) + a\right )}^{2}}{x^{2}} \,d x } \]

[In]

integrate((a+b*sech(c+d*x^(1/2)))^2/x^2,x, algorithm="maxima")

[Out]

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

Giac [N/A]

Not integrable

Time = 0.49 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2}{x^2} \, dx=\int { \frac {{\left (b \operatorname {sech}\left (d \sqrt {x} + c\right ) + a\right )}^{2}}{x^{2}} \,d x } \]

[In]

integrate((a+b*sech(c+d*x^(1/2)))^2/x^2,x, algorithm="giac")

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.38 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2}{x^2} \, dx=\int \frac {{\left (a+\frac {b}{\mathrm {cosh}\left (c+d\,\sqrt {x}\right )}\right )}^2}{x^2} \,d x \]

[In]

int((a + b/cosh(c + d*x^(1/2)))^2/x^2,x)

[Out]

int((a + b/cosh(c + d*x^(1/2)))^2/x^2, x)