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

Optimal. Leaf size=23 \[ \text{Unintegrable}\left (\frac{a+b \text{csch}^{-1}(c x)}{x^2 \sqrt{d+e x}},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0821906, 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 = {} \[ \int \frac{a+b \text{csch}^{-1}(c x)}{x^2 \sqrt{d+e x}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 8.86307, size = 0, normalized size = 0. \[ \int \frac{a+b \text{csch}^{-1}(c x)}{x^2 \sqrt{d+e x}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 5.133, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b{\rm arccsch} \left (cx\right )}{{x}^{2}}{\frac{1}{\sqrt{ex+d}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a+b*arccsch(c*x))/x^2/(e*x+d)^(1/2),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*arccsch(c*x))/x^2/(e*x+d)^(1/2),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 + d}{\left (b \operatorname{arcsch}\left (c x\right ) + a\right )}}{e x^{3} + d x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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))/x**2/(e*x+d)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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