3.137 \(\int \frac{\sqrt{d+e x^2} (a+b \text{sech}^{-1}(c x))}{x^2} \, dx\)

Optimal. Leaf size=25 \[ \text{Unintegrable}\left (\frac{\sqrt{d+e x^2} \left (a+b \text{sech}^{-1}(c x)\right )}{x^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0874494, 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 = {} \[ \int \frac{\sqrt{d+e x^2} \left (a+b \text{sech}^{-1}(c x)\right )}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.84083, size = 0, normalized size = 0. \[ \int \frac{\sqrt{d+e x^2} \left (a+b \text{sech}^{-1}(c x)\right )}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 1.034, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b{\rm arcsech} \left (cx\right )}{{x}^{2}}\sqrt{e{x}^{2}+d}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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