\(\int \frac {1}{\sqrt {1+b x^2} \text {arccosh}(\sqrt {1+b x^2})} \, dx\) [296]

Optimal result

Integrand size = 26, antiderivative size = 54 \[ \int \frac {1}{\sqrt {1+b x^2} \text {arccosh}\left (\sqrt {1+b x^2}\right )} \, dx=\frac {\sqrt {-1+\sqrt {1+b x^2}} \sqrt {1+\sqrt {1+b x^2}} \log \left (\text {arccosh}\left (\sqrt {1+b x^2}\right )\right )}{b x} \]

[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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {6013, 5891} \[ \int \frac {1}{\sqrt {1+b x^2} \text {arccosh}\left (\sqrt {1+b x^2}\right )} \, dx=\frac {\sqrt {\sqrt {b x^2+1}-1} \sqrt {\sqrt {b x^2+1}+1} \log \left (\text {arccosh}\left (\sqrt {b x^2+1}\right )\right )}{b x} \]

[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 5891

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

Rule 6013

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*} \text {integral}& = \frac {\left (\sqrt {-1+\sqrt {1+b x^2}} \sqrt {1+\sqrt {1+b x^2}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {1+x} \text {arccosh}(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 (\text {arccosh}\left (\sqrt {1+b x^2}\right )\right )}{b x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {1+b x^2} \text {arccosh}\left (\sqrt {1+b x^2}\right )} \, dx=\frac {\sqrt {-1+\sqrt {1+b x^2}} \sqrt {1+\sqrt {1+b x^2}} \log \left (\text {arccosh}\left (\sqrt {1+b x^2}\right )\right )}{b x} \]

[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)

Maple [F]

\[\int \frac {1}{\operatorname {arccosh}\left (\sqrt {b \,x^{2}+1}\right ) \sqrt {b \,x^{2}+1}}d x\]

[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)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.61 \[ \int \frac {1}{\sqrt {1+b x^2} \text {arccosh}\left (\sqrt {1+b x^2}\right )} \, dx=\frac {\sqrt {b x^{2}} \log \left (\log \left (\sqrt {b x^{2} + 1} + \sqrt {b x^{2}}\right )\right )}{b x} \]

[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)

Sympy [F]

\[ \int \frac {1}{\sqrt {1+b x^2} \text {arccosh}\left (\sqrt {1+b x^2}\right )} \, dx=\int \frac {1}{\sqrt {b x^{2} + 1} \operatorname {acosh}{\left (\sqrt {b x^{2} + 1} \right )}}\, dx \]

[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)

Maxima [F]

\[ \int \frac {1}{\sqrt {1+b x^2} \text {arccosh}\left (\sqrt {1+b x^2}\right )} \, dx=\int { \frac {1}{\sqrt {b x^{2} + 1} \operatorname {arcosh}\left (\sqrt {b x^{2} + 1}\right )} \,d x } \]

[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)

Giac [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {1+b x^2} \text {arccosh}\left (\sqrt {1+b x^2}\right )} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {1+b x^2} \text {arccosh}\left (\sqrt {1+b x^2}\right )} \, dx=\int \frac {1}{\mathrm {acosh}\left (\sqrt {b\,x^2+1}\right )\,\sqrt {b\,x^2+1}} \,d x \]

[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)