3.371 \(\int \frac{1}{\sqrt{-1+b x^2} \sinh ^{-1}(\sqrt{-1+b x^2})} \, dx\)

Optimal. Leaf size=29 \[ \frac{\sqrt{b x^2} \log \left (\sinh ^{-1}\left (\sqrt{b x^2-1}\right )\right )}{b x} \]

[Out]

(Sqrt[b*x^2]*Log[ArcSinh[Sqrt[-1 + b*x^2]]])/(b*x)

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Rubi [A]  time = 0.0615313, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {5894, 5673} \[ \frac{\sqrt{b x^2} \log \left (\sinh ^{-1}\left (\sqrt{b x^2-1}\right )\right )}{b x} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[-1 + b*x^2]*ArcSinh[Sqrt[-1 + b*x^2]]),x]

[Out]

(Sqrt[b*x^2]*Log[ArcSinh[Sqrt[-1 + b*x^2]]])/(b*x)

Rule 5894

Int[ArcSinh[Sqrt[-1 + (b_.)*(x_)^2]]^(n_.)/Sqrt[-1 + (b_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[b*x^2]/(b*x), Subst
[Int[ArcSinh[x]^n/Sqrt[1 + x^2], x], x, Sqrt[-1 + b*x^2]], x] /; FreeQ[{b, n}, x]

Rule 5673

Int[1/(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*Sqrt[(d_) + (e_.)*(x_)^2]), x_Symbol] :> Simp[Log[a + b*ArcSinh[c*x
]]/(b*c*Sqrt[d]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[d, 0]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{-1+b x^2} \sinh ^{-1}\left (\sqrt{-1+b x^2}\right )} \, dx &=\frac{\sqrt{b x^2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^2} \sinh ^{-1}(x)} \, dx,x,\sqrt{-1+b x^2}\right )}{b x}\\ &=\frac{\sqrt{b x^2} \log \left (\sinh ^{-1}\left (\sqrt{-1+b x^2}\right )\right )}{b x}\\ \end{align*}

Mathematica [A]  time = 0.0208927, size = 24, normalized size = 0.83 \[ \frac{x \log \left (\sinh ^{-1}\left (\sqrt{b x^2-1}\right )\right )}{\sqrt{b x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[-1 + b*x^2]*ArcSinh[Sqrt[-1 + b*x^2]]),x]

[Out]

(x*Log[ArcSinh[Sqrt[-1 + b*x^2]]])/Sqrt[b*x^2]

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Maple [F]  time = 0.146, size = 0, normalized size = 0. \begin{align*} \int{ \left ({\it Arcsinh} \left ( \sqrt{b{x}^{2}-1} \right ) \right ) ^{-1}{\frac{1}{\sqrt{b{x}^{2}-1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/arcsinh((b*x^2-1)^(1/2))/(b*x^2-1)^(1/2),x)

[Out]

int(1/arcsinh((b*x^2-1)^(1/2))/(b*x^2-1)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b x^{2} - 1} \operatorname{arsinh}\left (\sqrt{b x^{2} - 1}\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arcsinh((b*x^2-1)^(1/2))/(b*x^2-1)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(b*x^2 - 1)*arcsinh(sqrt(b*x^2 - 1))), x)

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Fricas [A]  time = 2.4263, size = 80, normalized size = 2.76 \begin{align*} \frac{\sqrt{b x^{2}} \log \left (\log \left (\sqrt{b x^{2} - 1} + \sqrt{b x^{2}}\right )\right )}{b x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arcsinh((b*x^2-1)^(1/2))/(b*x^2-1)^(1/2),x, algorithm="fricas")

[Out]

sqrt(b*x^2)*log(log(sqrt(b*x^2 - 1) + sqrt(b*x^2)))/(b*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b x^{2} - 1} \operatorname{asinh}{\left (\sqrt{b x^{2} - 1} \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/asinh((b*x**2-1)**(1/2))/(b*x**2-1)**(1/2),x)

[Out]

Integral(1/(sqrt(b*x**2 - 1)*asinh(sqrt(b*x**2 - 1))), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b x^{2} - 1} \operatorname{arsinh}\left (\sqrt{b x^{2} - 1}\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arcsinh((b*x^2-1)^(1/2))/(b*x^2-1)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(b*x^2 - 1)*arcsinh(sqrt(b*x^2 - 1))), x)