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

Optimal. Leaf size=62 \[ \frac{\sqrt{\sqrt{b x^2+1}-1} \sqrt{\sqrt{b x^2+1}+1} \cosh ^{-1}\left (\sqrt{b x^2+1}\right )^{n+1}}{b (n+1) x} \]

[Out]

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

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Rubi [A]  time = 0.118908, antiderivative size = 62, 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 = {5895, 5676} \[ \frac{\sqrt{\sqrt{b x^2+1}-1} \sqrt{\sqrt{b x^2+1}+1} \cosh ^{-1}\left (\sqrt{b x^2+1}\right )^{n+1}}{b (n+1) x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5895

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

Rule 5676

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(a + b*ArcCosh[c*x])^(n + 1)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n},
x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[d1, 0] && LtQ[d2, 0] && NeQ[n, -1]

Rubi steps

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

Mathematica [A]  time = 0.154314, size = 62, normalized size = 1. \[ \frac{\sqrt{\sqrt{b x^2+1}-1} \sqrt{\sqrt{b x^2+1}+1} \cosh ^{-1}\left (\sqrt{b x^2+1}\right )^{n+1}}{b (n+1) x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.236, size = 0, normalized size = 0. \begin{align*} \int{ \left ({\rm arccosh} \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(arccosh((b*x^2+1)^(1/2))^n/(b*x^2+1)^(1/2),x)

[Out]

int(arccosh((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{arcosh}\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(arccosh((b*x^2+1)^(1/2))^n/(b*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(arccosh(sqrt(b*x^2 + 1))^n/sqrt(b*x^2 + 1), x)

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Fricas [B]  time = 2.22667, size = 282, normalized size = 4.55 \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(arccosh((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(acosh((b*x**2+1)**(1/2))**n/(b*x**2+1)**(1/2),x)

[Out]

Exception raised: TypeError

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Giac [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(arccosh((b*x^2+1)^(1/2))^n/(b*x^2+1)^(1/2),x, algorithm="giac")

[Out]

Timed out