\(\int \frac {1}{x^2 (a+b \text {sech}(c+d \sqrt {x}))^2} \, dx\) [51]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {1}{x^2 \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\text {Int}\left (\frac {1}{x^2 \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2},x\right ) \]

[Out]

Unintegrable(1/x^2/(a+b*sech(c+d*x^(1/2)))^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 = {} \[ \int \frac {1}{x^2 \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\int \frac {1}{x^2 \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2} \, dx \]

[In]

Int[1/(x^2*(a + b*Sech[c + d*Sqrt[x]])^2),x]

[Out]

Defer[Int][1/(x^2*(a + b*Sech[c + d*Sqrt[x]])^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x^2 \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 91.96 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{x^2 \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\int \frac {1}{x^2 \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2} \, dx \]

[In]

Integrate[1/(x^2*(a + b*Sech[c + d*Sqrt[x]])^2),x]

[Out]

Integrate[1/(x^2*(a + b*Sech[c + d*Sqrt[x]])^2), x]

Maple [N/A] (verified)

Not integrable

Time = 0.18 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90

\[\int \frac {1}{x^{2} \left (a +b \,\operatorname {sech}\left (c +d \sqrt {x}\right )\right )^{2}}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.20 \[ \int \frac {1}{x^2 \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \operatorname {sech}\left (d \sqrt {x} + c\right ) + a\right )}^{2} x^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 9.90 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\int \frac {1}{x^{2} \left (a + b \operatorname {sech}{\left (c + d \sqrt {x} \right )}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 1.16 (sec) , antiderivative size = 324, normalized size of antiderivative = 16.20 \[ \int \frac {1}{x^2 \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \operatorname {sech}\left (d \sqrt {x} + c\right ) + a\right )}^{2} x^{2}} \,d x } \]

[In]

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

[Out]

-(4*a*b^2*sqrt(x) + (a^3*d*e^(2*c) - a*b^2*d*e^(2*c))*x*e^(2*d*sqrt(x)) + (a^3*d - a*b^2*d)*x + 2*(2*b^3*sqrt(
x)*e^c + (a^2*b*d*e^c - b^3*d*e^c)*x)*e^(d*sqrt(x)))/((a^5*d*e^(2*c) - a^3*b^2*d*e^(2*c))*x^2*e^(2*d*sqrt(x))
+ 2*(a^4*b*d*e^c - a^2*b^3*d*e^c)*x^2*e^(d*sqrt(x)) + (a^5*d - a^3*b^2*d)*x^2) - integrate(2*(3*a*b^2*sqrt(x)
+ (3*b^3*sqrt(x)*e^c + (2*a^2*b*d*e^c - b^3*d*e^c)*x)*e^(d*sqrt(x)))/((a^5*d*e^(2*c) - a^3*b^2*d*e^(2*c))*x^3*
e^(2*d*sqrt(x)) + 2*(a^4*b*d*e^c - a^2*b^3*d*e^c)*x^3*e^(d*sqrt(x)) + (a^5*d - a^3*b^2*d)*x^3), x)

Giac [N/A]

Not integrable

Time = 3.10 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.15 \[ \int \frac {1}{x^2 \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \operatorname {sech}\left (d \sqrt {x} + c\right ) + a\right )}^{2} x^{2}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 2.48 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{x^2 \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\int \frac {1}{x^2\,{\left (a+\frac {b}{\mathrm {cosh}\left (c+d\,\sqrt {x}\right )}\right )}^2} \,d x \]

[In]

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

[Out]

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