3.42 \(\int \sqrt{c+d x^2} \coth ^{-1}(a x) \, dx\)

Optimal. Leaf size=18 \[ \text{Unintegrable}\left (\coth ^{-1}(a x) \sqrt{c+d x^2},x\right ) \]

[Out]

Unintegrable[Sqrt[c + d*x^2]*ArcCoth[a*x], x]

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Rubi [A]  time = 0.0208016, 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 = {} \[ \int \sqrt{c+d x^2} \coth ^{-1}(a x) \, dx \]

Verification is Not applicable to the result.

[In]

Int[Sqrt[c + d*x^2]*ArcCoth[a*x],x]

[Out]

Defer[Int][Sqrt[c + d*x^2]*ArcCoth[a*x], x]

Rubi steps

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

Mathematica [A]  time = 24.1269, size = 0, normalized size = 0. \[ \int \sqrt{c+d x^2} \coth ^{-1}(a x) \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Sqrt[c + d*x^2]*ArcCoth[a*x],x]

[Out]

Integrate[Sqrt[c + d*x^2]*ArcCoth[a*x], x]

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Maple [A]  time = 0.713, size = 0, normalized size = 0. \begin{align*} \int \sqrt{d{x}^{2}+c}{\rm arccoth} \left (ax\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((d*x^2+c)^(1/2)*arccoth(a*x),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((d*x^2+c)^(1/2)*arccoth(a*x),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 (\sqrt{d x^{2} + c} \operatorname{arcoth}\left (a x\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(d*x^2 + c)*arccoth(a*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(c + d*x**2)*acoth(a*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(d*x^2 + c)*arccoth(a*x), x)