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

Optimal. Leaf size=54 \[ \frac {\sqrt {\sqrt {b x^2+1}-1} \sqrt {\sqrt {b x^2+1}+1} \log \left (\cosh ^{-1}\left (\sqrt {b x^2+1}\right )\right )}{b x} \]

[Out]

ln(arccosh((b*x^2+1)^(1/2)))*(-1+(b*x^2+1)^(1/2))^(1/2)*(1+(b*x^2+1)^(1/2))^(1/2)/b/x

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Rubi [A]  time = 0.11, antiderivative size = 54, normalized size of antiderivative = 1.00, 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, 5674} \[ \frac {\sqrt {\sqrt {b x^2+1}-1} \sqrt {\sqrt {b x^2+1}+1} \log \left (\cosh ^{-1}\left (\sqrt {b x^2+1}\right )\right )}{b x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5674

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

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]

Rubi steps

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

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Mathematica [A]  time = 0.09, size = 54, normalized size = 1.00 \[ \frac {\sqrt {\sqrt {b x^2+1}-1} \sqrt {\sqrt {b x^2+1}+1} \log \left (\cosh ^{-1}\left (\sqrt {b x^2+1}\right )\right )}{b x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.59, size = 33, normalized size = 0.61 \[ \frac {\sqrt {b x^{2}} \log \left (\log \left (\sqrt {b x^{2} + 1} + \sqrt {b x^{2}}\right )\right )}{b x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arccosh((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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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [F]  time = 0.28, size = 0, normalized size = 0.00 \[ \int \frac {1}{\mathrm {arccosh}\left (\sqrt {b \,x^{2}+1}\right ) \sqrt {b \,x^{2}+1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/arccosh((b*x^2+1)^(1/2))/(b*x^2+1)^(1/2),x)

[Out]

int(1/arccosh((b*x^2+1)^(1/2))/(b*x^2+1)^(1/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b x^{2} + 1} \operatorname {arcosh}\left (\sqrt {b x^{2} + 1}\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arccosh((b*x^2+1)^(1/2))/(b*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\mathrm {acosh}\left (\sqrt {b\,x^2+1}\right )\,\sqrt {b\,x^2+1}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(acosh((b*x^2 + 1)^(1/2))*(b*x^2 + 1)^(1/2)),x)

[Out]

int(1/(acosh((b*x^2 + 1)^(1/2))*(b*x^2 + 1)^(1/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b x^{2} + 1} \operatorname {acosh}{\left (\sqrt {b x^{2} + 1} \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/acosh((b*x**2+1)**(1/2))/(b*x**2+1)**(1/2),x)

[Out]

Integral(1/(sqrt(b*x**2 + 1)*acosh(sqrt(b*x**2 + 1))), x)

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