3.218 \(\int \frac {\sqrt {\tanh ^{-1}(\tanh (a+b x))}}{x^{3/2}} \, dx\)

Optimal. Leaf size=49 \[ 2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {\tanh ^{-1}(\tanh (a+b x))}}\right )-\frac {2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{\sqrt {x}} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2168, 2165} \[ 2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {\tanh ^{-1}(\tanh (a+b x))}}\right )-\frac {2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{\sqrt {x}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[ArcTanh[Tanh[a + b*x]]]/x^(3/2),x]

[Out]

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

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]

Rule 2168

Int[(u_)^(m_)*(v_)^(n_.), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[(u^(m + 1)*v^
n)/(a*(m + 1)), x] - Dist[(b*n)/(a*(m + 1)), Int[u^(m + 1)*v^(n - 1), x], x] /; NeQ[b*u - a*v, 0]] /; FreeQ[{m
, n}, x] && PiecewiseLinearQ[u, v, x] && NeQ[m, -1] && ((LtQ[m, -1] && GtQ[n, 0] &&  !(ILtQ[m + n, -2] && (Fra
ctionQ[m] || GeQ[2*n + m + 1, 0]))) || (IGtQ[n, 0] && IGtQ[m, 0] && LeQ[n, m]) || (IGtQ[n, 0] &&  !IntegerQ[m]
) || (ILtQ[m, 0] &&  !IntegerQ[n]))

Rubi steps

\begin {align*} \int \frac {\sqrt {\tanh ^{-1}(\tanh (a+b x))}}{x^{3/2}} \, dx &=-\frac {2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{\sqrt {x}}+b \int \frac {1}{\sqrt {x} \sqrt {\tanh ^{-1}(\tanh (a+b x))}} \, dx\\ &=2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {\tanh ^{-1}(\tanh (a+b x))}}\right )-\frac {2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{\sqrt {x}}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 52, normalized size = 1.06 \[ 2 \sqrt {b} \log \left (\sqrt {b} \sqrt {\tanh ^{-1}(\tanh (a+b x))}+b \sqrt {x}\right )-\frac {2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{\sqrt {x}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[ArcTanh[Tanh[a + b*x]]]/x^(3/2),x]

[Out]

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

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fricas [A]  time = 0.56, size = 89, normalized size = 1.82 \[ \left [\frac {\sqrt {b} x \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, \sqrt {b x + a} \sqrt {x}}{x}, -\frac {2 \, {\left (\sqrt {-b} x \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) + \sqrt {b x + a} \sqrt {x}\right )}}{x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(tanh(b*x+a))^(1/2)/x^(3/2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.23, size = 57, normalized size = 1.16 \[ -\sqrt {b} \log \left ({\left (\sqrt {b} \sqrt {x} - \sqrt {b x + a}\right )}^{2}\right ) + \frac {4 \, a \sqrt {b}}{{\left (\sqrt {b} \sqrt {x} - \sqrt {b x + a}\right )}^{2} - a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(tanh(b*x+a))^(1/2)/x^(3/2),x, algorithm="giac")

[Out]

-sqrt(b)*log((sqrt(b)*sqrt(x) - sqrt(b*x + a))^2) + 4*a*sqrt(b)/((sqrt(b)*sqrt(x) - sqrt(b*x + a))^2 - a)

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maple [B]  time = 0.26, size = 149, normalized size = 3.04 \[ -\frac {2 \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {3}{2}}}{\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right ) \sqrt {x}}+\frac {2 b \sqrt {x}\, \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}{\arctanh \left (\tanh \left (b x +a \right )\right )-b x}+\frac {2 \sqrt {b}\, \ln \left (\sqrt {b}\, \sqrt {x}+\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}\right ) a}{\arctanh \left (\tanh \left (b x +a \right )\right )-b x}+\frac {2 \sqrt {b}\, \ln \left (\sqrt {b}\, \sqrt {x}+\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}\right ) \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )}{\arctanh \left (\tanh \left (b x +a \right )\right )-b x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctanh(tanh(b*x+a))^(1/2)/x^(3/2),x)

[Out]

-2/(arctanh(tanh(b*x+a))-b*x)/x^(1/2)*arctanh(tanh(b*x+a))^(3/2)+2*b/(arctanh(tanh(b*x+a))-b*x)*x^(1/2)*arctan
h(tanh(b*x+a))^(1/2)+2*b^(1/2)/(arctanh(tanh(b*x+a))-b*x)*ln(b^(1/2)*x^(1/2)+arctanh(tanh(b*x+a))^(1/2))*a+2*b
^(1/2)/(arctanh(tanh(b*x+a))-b*x)*ln(b^(1/2)*x^(1/2)+arctanh(tanh(b*x+a))^(1/2))*(arctanh(tanh(b*x+a))-b*x-a)

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maxima [A]  time = 0.46, size = 40, normalized size = 0.82 \[ 2 \, \sqrt {b} \log \left (\frac {b \sqrt {x}}{\sqrt {a b}} + \sqrt {\frac {b x}{a} + 1}\right ) - \frac {2 \, \sqrt {b x + a}}{\sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(tanh(b*x+a))^(1/2)/x^(3/2),x, algorithm="maxima")

[Out]

2*sqrt(b)*log(b*sqrt(x)/sqrt(a*b) + sqrt(b*x/a + 1)) - 2*sqrt(b*x + a)/sqrt(x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}}{x^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atanh(tanh(a + b*x))^(1/2)/x^(3/2),x)

[Out]

int(atanh(tanh(a + b*x))^(1/2)/x^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}}{x^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atanh(tanh(b*x+a))**(1/2)/x**(3/2),x)

[Out]

Integral(sqrt(atanh(tanh(a + b*x)))/x**(3/2), x)

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