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

Optimal. Leaf size=20 \[ \text{Unintegrable}\left (\frac{\left (a+b \text{sech}\left (c+d x^2\right )\right )^2}{x^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0240201, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\left (a+b \text{sech}\left (c+d x^2\right )\right )^2}{x^2} \, dx \]

Verification is Not applicable to the result.

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

Mathematica [A]  time = 9.13065, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \text{sech}\left (c+d x^2\right )\right )^2}{x^2} \, dx \]

Verification is Not applicable to the result.

[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]

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Maple [A]  time = 0.088, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\rm sech} \left (d{x}^{2}+c\right ) \right ) ^{2}}{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{b^{2}}{d x^{3} e^{\left (2 \, d x^{2} + 2 \, c\right )} + d x^{3}} - \frac{a^{2}}{x} + \int \frac{4 \, a b d x^{2} e^{\left (d x^{2} + c\right )} - 3 \, b^{2}}{d x^{4} e^{\left (2 \, d x^{2} + 2 \, c\right )} + d x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} \operatorname{sech}\left (d x^{2} + c\right )^{2} + 2 \, a b \operatorname{sech}\left (d x^{2} + c\right ) + a^{2}}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{sech}{\left (c + d x^{2} \right )}\right )^{2}}{x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{sech}\left (d x^{2} + c\right ) + a\right )}^{2}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)