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

Optimal. Leaf size=25 \[ b \text{Unintegrable}\left (\frac{\text{sech}\left (c+d \sqrt{x}\right )}{x^2},x\right )-\frac{a}{x} \]

[Out]

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

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Rubi [A]  time = 0.0186535, 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 = {} \[ \int \frac{a+b \text{sech}\left (c+d \sqrt{x}\right )}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 7.25293, size = 0, normalized size = 0. \[ \int \frac{a+b \text{sech}\left (c+d \sqrt{x}\right )}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.075, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ( a+b{\rm sech} \left (c+d\sqrt{x}\right ) \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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