3.122 \(\int \frac{\sqrt{d+e x^2} (a+b \text{csch}^{-1}(c x))}{x^3} \, dx\)

Optimal. Leaf size=25 \[ \text{Unintegrable}\left (\frac{\sqrt{d+e x^2} \left (a+b \text{csch}^{-1}(c x)\right )}{x^3},x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.10563, 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 = {} \[ \int \frac{\sqrt{d+e x^2} \left (a+b \text{csch}^{-1}(c x)\right )}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int \frac{\sqrt{d+e x^2} \left (a+b \text{csch}^{-1}(c x)\right )}{x^3} \, dx &=\int \frac{\sqrt{d+e x^2} \left (a+b \text{csch}^{-1}(c x)\right )}{x^3} \, dx\\ \end{align*}

Mathematica [A]  time = 5.63227, size = 0, normalized size = 0. \[ \int \frac{\sqrt{d+e x^2} \left (a+b \text{csch}^{-1}(c x)\right )}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.453, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b{\rm arccsch} \left (cx\right )}{{x}^{3}}\sqrt{e{x}^{2}+d}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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*arccsch(c*x))*(e*x^2+d)^(1/2)/x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsch(c*x))*(e*x^2+d)^(1/2)/x^3,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acsch(c*x))*(e*x**2+d)**(1/2)/x**3,x)

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsch(c*x))*(e*x^2+d)^(1/2)/x^3,x, algorithm="giac")

[Out]

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