3.55 \(\int \frac{a+b \text{sech}(c+d \sqrt{x})}{x^{3/2}} \, dx\)

Optimal. Leaf size=29 \[ b \text{Unintegrable}\left (\frac{\text{sech}\left (c+d \sqrt{x}\right )}{x^{3/2}},x\right )-\frac{2 a}{\sqrt{x}} \]

[Out]

(-2*a)/Sqrt[x] + b*Unintegrable[Sech[c + d*Sqrt[x]]/x^(3/2), x]

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Rubi [A]  time = 0.0156501, 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 = {} \[ \int \frac{a+b \text{sech}\left (c+d \sqrt{x}\right )}{x^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*Sech[c + d*Sqrt[x]])/x^(3/2),x]

[Out]

(-2*a)/Sqrt[x] + b*Defer[Int][Sech[c + d*Sqrt[x]]/x^(3/2), x]

Rubi steps

\begin{align*} \int \frac{a+b \text{sech}\left (c+d \sqrt{x}\right )}{x^{3/2}} \, dx &=\int \left (\frac{a}{x^{3/2}}+\frac{b \text{sech}\left (c+d \sqrt{x}\right )}{x^{3/2}}\right ) \, dx\\ &=-\frac{2 a}{\sqrt{x}}+b \int \frac{\text{sech}\left (c+d \sqrt{x}\right )}{x^{3/2}} \, dx\\ \end{align*}

Mathematica [A]  time = 8.43897, size = 0, normalized size = 0. \[ \int \frac{a+b \text{sech}\left (c+d \sqrt{x}\right )}{x^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*Sech[c + d*Sqrt[x]])/x^(3/2),x]

[Out]

Integrate[(a + b*Sech[c + d*Sqrt[x]])/x^(3/2), x]

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Maple [A]  time = 0.072, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b{\rm sech} \left (c+d\sqrt{x}\right ) \right ){x}^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sech(c+d*x^(1/2)))/x^(3/2),x)

[Out]

int((a+b*sech(c+d*x^(1/2)))/x^(3/2),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^{\frac{3}{2}} e^{\left (2 \, d \sqrt{x} + 2 \, c\right )} + x^{\frac{3}{2}}}\,{d x} - \frac{2 \, a}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sech(c+d*x^(1/2)))/x^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \sqrt{x} \operatorname{sech}\left (d \sqrt{x} + c\right ) + a \sqrt{x}}{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^(3/2),x, algorithm="fricas")

[Out]

integral((b*sqrt(x)*sech(d*sqrt(x) + c) + a*sqrt(x))/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^{\frac{3}{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**(3/2),x)

[Out]

Integral((a + b*sech(c + d*sqrt(x)))/x**(3/2), 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^{\frac{3}{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^(3/2),x, algorithm="giac")

[Out]

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