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

Optimal. Leaf size=37 \[ \frac{\sqrt{b x^2} \sinh ^{-1}\left (\sqrt{b x^2-1}\right )^{n+1}}{b (n+1) x} \]

[Out]

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

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Rubi [A]  time = 0.0665338, antiderivative size = 37, 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, 5675} \[ \frac{\sqrt{b x^2} \sinh ^{-1}\left (\sqrt{b x^2-1}\right )^{n+1}}{b (n+1) x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[b*x^2]*ArcSinh[Sqrt[-1 + b*x^2]]^(1 + n))/(b*(1 + n)*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 5675

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

Rubi steps

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

Mathematica [A]  time = 0.0396998, size = 37, normalized size = 1. \[ \frac{\sqrt{b x^2} \sinh ^{-1}\left (\sqrt{b x^2-1}\right )^{n+1}}{b (n+1) x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.177, size = 0, normalized size = 0. \begin{align*} \int{ \left ({\it Arcsinh} \left ( \sqrt{b{x}^{2}-1} \right ) \right ) ^{n}{\frac{1}{\sqrt{b{x}^{2}-1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.78902, size = 282, normalized size = 7.62 \begin{align*} \frac{\sqrt{b x^{2}} \cosh \left (n \log \left (\log \left (\sqrt{b x^{2} - 1} + \sqrt{b x^{2}}\right )\right )\right ) \log \left (\sqrt{b x^{2} - 1} + \sqrt{b x^{2}}\right ) + \sqrt{b x^{2}} \log \left (\sqrt{b x^{2} - 1} + \sqrt{b x^{2}}\right ) \sinh \left (n \log \left (\log \left (\sqrt{b x^{2} - 1} + \sqrt{b x^{2}}\right )\right )\right )}{{\left (b n + b\right )} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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