3.244 \(\int \frac{1}{\sqrt{x} \sqrt{\tanh ^{-1}(\tanh (a+b x))}} \, dx\)

Optimal. Leaf size=30 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}\right )}{\sqrt{b}} \]

[Out]

(2*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[ArcTanh[Tanh[a + b*x]]]])/Sqrt[b]

________________________________________________________________________________________

Rubi [A]  time = 0.0120342, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {2165} \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[x]*Sqrt[ArcTanh[Tanh[a + b*x]]]),x]

[Out]

(2*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[ArcTanh[Tanh[a + b*x]]]])/Sqrt[b]

Rule 2165

Int[1/(Sqrt[u_]*Sqrt[v_]), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[(2*ArcTanh[(
Rt[a*b, 2]*Sqrt[u])/(a*Sqrt[v])])/Rt[a*b, 2], x] /; NeQ[b*u - a*v, 0] && PosQ[a*b]] /; PiecewiseLinearQ[u, v,
x]

Rubi steps

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

Mathematica [A]  time = 0.0304194, size = 33, normalized size = 1.1 \[ \frac{2 \log \left (\sqrt{b} \sqrt{\tanh ^{-1}(\tanh (a+b x))}+b \sqrt{x}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[x]*Sqrt[ArcTanh[Tanh[a + b*x]]]),x]

[Out]

(2*Log[b*Sqrt[x] + Sqrt[b]*Sqrt[ArcTanh[Tanh[a + b*x]]]])/Sqrt[b]

________________________________________________________________________________________

Maple [A]  time = 0.095, size = 24, normalized size = 0.8 \begin{align*} 2\,{\frac{\ln \left ( \sqrt{b}\sqrt{x}+\sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) } \right ) }{\sqrt{b}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(1/2)/arctanh(tanh(b*x+a))^(1/2),x)

[Out]

2*ln(b^(1/2)*x^(1/2)+arctanh(tanh(b*x+a))^(1/2))/b^(1/2)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x} \sqrt{\operatorname{artanh}\left (\tanh \left (b x + a\right )\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(x)*sqrt(arctanh(tanh(b*x + a)))), x)

________________________________________________________________________________________

Fricas [A]  time = 2.07482, size = 162, normalized size = 5.4 \begin{align*} \left [\frac{\log \left (2 \, b x + 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x} + a\right )}{\sqrt{b}}, -\frac{2 \, \sqrt{-b} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right )}{b}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(1/2)/arctanh(tanh(b*x+a))^(1/2),x, algorithm="fricas")

[Out]

[log(2*b*x + 2*sqrt(b*x + a)*sqrt(b)*sqrt(x) + a)/sqrt(b), -2*sqrt(-b)*arctan(sqrt(b*x + a)*sqrt(-b)/(b*sqrt(x
)))/b]

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x} \sqrt{\operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**(1/2)/atanh(tanh(b*x+a))**(1/2),x)

[Out]

Integral(1/(sqrt(x)*sqrt(atanh(tanh(a + b*x)))), x)

________________________________________________________________________________________

Giac [A]  time = 1.14891, size = 31, normalized size = 1.03 \begin{align*} -\frac{2 \, \log \left ({\left | -\sqrt{b} \sqrt{x} + \sqrt{b x + a} \right |}\right )}{\sqrt{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*log(abs(-sqrt(b)*sqrt(x) + sqrt(b*x + a)))/sqrt(b)