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

Optimal. Leaf size=23 \[ \text {Int}\left (\frac {\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2}{x^2},x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2}{x^2} \, dx \]

Verification is Not applicable to the result.

[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*} \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\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 28.30, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2}{x^2} \, dx \]

Verification is Not applicable to the result.

[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]

________________________________________________________________________________________

fricas [A]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \operatorname {sech}\left (d \sqrt {x} + c\right )^{2} + 2 \, a b \operatorname {sech}\left (d \sqrt {x} + c\right ) + a^{2}}{x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {sech}\left (d \sqrt {x} + c\right ) + a\right )}^{2}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

maple [A]  time = 0.58, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \,\mathrm {sech}\left (c +d \sqrt {x}\right )\right )^{2}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {a^{2} d x e^{\left (2 \, d \sqrt {x} + 2 \, c\right )} + a^{2} d x + 4 \, b^{2} \sqrt {x}}{d x^{2} e^{\left (2 \, d \sqrt {x} + 2 \, c\right )} + d x^{2}} + \int \frac {2 \, {\left (2 \, a b d x e^{\left (d \sqrt {x} + c\right )} - 3 \, b^{2} \sqrt {x}\right )}}{d x^{3} e^{\left (2 \, d \sqrt {x} + 2 \, c\right )} + d x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {{\left (a+\frac {b}{\mathrm {cosh}\left (c+d\,\sqrt {x}\right )}\right )}^2}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {sech}{\left (c + d \sqrt {x} \right )}\right )^{2}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________