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

Optimal. Leaf size=139 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}\right ) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3}{8 b^{3/2}}-\frac{1}{4} x^{3/2} \sqrt{\tanh ^{-1}(\tanh (a+b x))} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )+\frac{1}{3} x^{3/2} \tanh ^{-1}(\tanh (a+b x))^{3/2}+\frac{\sqrt{x} \sqrt{\tanh ^{-1}(\tanh (a+b x))} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}{8 b} \]

[Out]

(ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[ArcTanh[Tanh[a + b*x]]]]*(b*x - ArcTanh[Tanh[a + b*x]])^3)/(8*b^(3/2)) - (x^(3
/2)*(b*x - ArcTanh[Tanh[a + b*x]])*Sqrt[ArcTanh[Tanh[a + b*x]]])/4 + (Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]])^2
*Sqrt[ArcTanh[Tanh[a + b*x]]])/(8*b) + (x^(3/2)*ArcTanh[Tanh[a + b*x]]^(3/2))/3

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Rubi [A]  time = 0.070956, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2169, 2165} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}\right ) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3}{8 b^{3/2}}-\frac{1}{4} x^{3/2} \sqrt{\tanh ^{-1}(\tanh (a+b x))} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )+\frac{1}{3} x^{3/2} \tanh ^{-1}(\tanh (a+b x))^{3/2}+\frac{\sqrt{x} \sqrt{\tanh ^{-1}(\tanh (a+b x))} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}{8 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[ArcTanh[Tanh[a + b*x]]]]*(b*x - ArcTanh[Tanh[a + b*x]])^3)/(8*b^(3/2)) - (x^(3
/2)*(b*x - ArcTanh[Tanh[a + b*x]])*Sqrt[ArcTanh[Tanh[a + b*x]]])/4 + (Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]])^2
*Sqrt[ArcTanh[Tanh[a + b*x]]])/(8*b) + (x^(3/2)*ArcTanh[Tanh[a + b*x]]^(3/2))/3

Rule 2169

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 + n + 1)), x] - Dist[(n*(b*u - a*v))/(a*(m + n + 1)), Int[u^m*v^(n - 1), x], x] /; NeQ[b*u - a*v, 0]]
/; PiecewiseLinearQ[u, v, x] && NeQ[m + n + 2, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !In
tegerQ[n] || LtQ[0, m, n])) &&  !ILtQ[m + n, -2]

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 \sqrt{x} \tanh ^{-1}(\tanh (a+b x))^{3/2} \, dx &=\frac{1}{3} x^{3/2} \tanh ^{-1}(\tanh (a+b x))^{3/2}-\frac{1}{2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \int \sqrt{x} \sqrt{\tanh ^{-1}(\tanh (a+b x))} \, dx\\ &=-\frac{1}{4} x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \sqrt{\tanh ^{-1}(\tanh (a+b x))}+\frac{1}{3} x^{3/2} \tanh ^{-1}(\tanh (a+b x))^{3/2}-\frac{1}{8} \left (\left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )\right ) \int \frac{\sqrt{x}}{\sqrt{\tanh ^{-1}(\tanh (a+b x))}} \, dx\\ &=-\frac{1}{4} x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \sqrt{\tanh ^{-1}(\tanh (a+b x))}+\frac{\sqrt{x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{8 b}+\frac{1}{3} x^{3/2} \tanh ^{-1}(\tanh (a+b x))^{3/2}+\frac{\left (\left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )^2\right ) \int \frac{1}{\sqrt{x} \sqrt{\tanh ^{-1}(\tanh (a+b x))}} \, dx}{16 b}\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}\right ) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3}{8 b^{3/2}}-\frac{1}{4} x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \sqrt{\tanh ^{-1}(\tanh (a+b x))}+\frac{\sqrt{x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{8 b}+\frac{1}{3} x^{3/2} \tanh ^{-1}(\tanh (a+b x))^{3/2}\\ \end{align*}

Mathematica [A]  time = 0.0706793, size = 105, normalized size = 0.76 \[ \frac{\sqrt{x} \sqrt{\tanh ^{-1}(\tanh (a+b x))} \left (8 b x \tanh ^{-1}(\tanh (a+b x))+3 \tanh ^{-1}(\tanh (a+b x))^2-3 b^2 x^2\right )}{24 b}+\frac{\left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3 \log \left (\sqrt{b} \sqrt{\tanh ^{-1}(\tanh (a+b x))}+b \sqrt{x}\right )}{8 b^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x]*Sqrt[ArcTanh[Tanh[a + b*x]]]*(-3*b^2*x^2 + 8*b*x*ArcTanh[Tanh[a + b*x]] + 3*ArcTanh[Tanh[a + b*x]]^2)
)/(24*b) + ((b*x - ArcTanh[Tanh[a + b*x]])^3*Log[b*Sqrt[x] + Sqrt[b]*Sqrt[ArcTanh[Tanh[a + b*x]]]])/(8*b^(3/2)
)

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Maple [B]  time = 0.119, size = 304, normalized size = 2.2 \begin{align*}{\frac{1}{3\,b}\sqrt{x} \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{{\frac{5}{2}}}}-{\frac{a}{12\,b}\sqrt{x} \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{{a}^{2}}{8\,b}\sqrt{x}\sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }}-{\frac{{a}^{3}}{8}\ln \left ( \sqrt{b}\sqrt{x}+\sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) } \right ){b}^{-{\frac{3}{2}}}}-{\frac{3\,{a}^{2} \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx-a \right ) }{8}\ln \left ( \sqrt{b}\sqrt{x}+\sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) } \right ){b}^{-{\frac{3}{2}}}}-{\frac{a \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx-a \right ) }{4\,b}\sqrt{x}\sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }}-{\frac{3\,a \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx-a \right ) ^{2}}{8}\ln \left ( \sqrt{b}\sqrt{x}+\sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) } \right ){b}^{-{\frac{3}{2}}}}-{\frac{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx-a}{12\,b}\sqrt{x} \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx-a \right ) ^{2}}{8\,b}\sqrt{x}\sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }}-{\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx-a \right ) ^{3}}{8}\ln \left ( \sqrt{b}\sqrt{x}+\sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) } \right ){b}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^(1/2)*arctanh(tanh(b*x+a))^(5/2)/b-1/12/b*a*x^(1/2)*arctanh(tanh(b*x+a))^(3/2)-1/8/b*a^2*x^(1/2)*arctanh
(tanh(b*x+a))^(1/2)-1/8/b^(3/2)*ln(b^(1/2)*x^(1/2)+arctanh(tanh(b*x+a))^(1/2))*a^3-3/8/b^(3/2)*a^2*ln(b^(1/2)*
x^(1/2)+arctanh(tanh(b*x+a))^(1/2))*(arctanh(tanh(b*x+a))-b*x-a)-1/4/b*a*(arctanh(tanh(b*x+a))-b*x-a)*x^(1/2)*
arctanh(tanh(b*x+a))^(1/2)-3/8/b^(3/2)*a*ln(b^(1/2)*x^(1/2)+arctanh(tanh(b*x+a))^(1/2))*(arctanh(tanh(b*x+a))-
b*x-a)^2-1/12/b*(arctanh(tanh(b*x+a))-b*x-a)*x^(1/2)*arctanh(tanh(b*x+a))^(3/2)-1/8/b*(arctanh(tanh(b*x+a))-b*
x-a)^2*x^(1/2)*arctanh(tanh(b*x+a))^(1/2)-1/8/b^(3/2)*ln(b^(1/2)*x^(1/2)+arctanh(tanh(b*x+a))^(1/2))*(arctanh(
tanh(b*x+a))-b*x-a)^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{x} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(x)*arctanh(tanh(b*x + a))^(3/2), x)

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Fricas [A]  time = 2.32663, size = 363, normalized size = 2.61 \begin{align*} \left [\frac{3 \, a^{3} \sqrt{b} \log \left (2 \, b x - 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x} + a\right ) + 2 \,{\left (8 \, b^{3} x^{2} + 14 \, a b^{2} x + 3 \, a^{2} b\right )} \sqrt{b x + a} \sqrt{x}}{48 \, b^{2}}, \frac{3 \, a^{3} \sqrt{-b} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) +{\left (8 \, b^{3} x^{2} + 14 \, a b^{2} x + 3 \, a^{2} b\right )} \sqrt{b x + a} \sqrt{x}}{24 \, b^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/48*(3*a^3*sqrt(b)*log(2*b*x - 2*sqrt(b*x + a)*sqrt(b)*sqrt(x) + a) + 2*(8*b^3*x^2 + 14*a*b^2*x + 3*a^2*b)*s
qrt(b*x + a)*sqrt(x))/b^2, 1/24*(3*a^3*sqrt(-b)*arctan(sqrt(b*x + a)*sqrt(-b)/(b*sqrt(x))) + (8*b^3*x^2 + 14*a
*b^2*x + 3*a^2*b)*sqrt(b*x + a)*sqrt(x))/b^2]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{x} \operatorname{atanh}^{\frac{3}{2}}{\left (\tanh{\left (a + b x \right )} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.20014, size = 165, normalized size = 1.19 \begin{align*} \frac{1}{48} \, \sqrt{2}{\left (6 \, \sqrt{2}{\left (\sqrt{b x + a}{\left (2 \, x + \frac{a}{b}\right )} \sqrt{x} + \frac{a^{2} \log \left ({\left | -\sqrt{b} \sqrt{x} + \sqrt{b x + a} \right |}\right )}{b^{\frac{3}{2}}}\right )} a + \sqrt{2}{\left (\sqrt{b x + a}{\left (2 \,{\left (4 \, x + \frac{a}{b}\right )} x - \frac{3 \, a^{2}}{b^{2}}\right )} \sqrt{x} - \frac{3 \, a^{3} \log \left ({\left | -\sqrt{b} \sqrt{x} + \sqrt{b x + a} \right |}\right )}{b^{\frac{5}{2}}}\right )} b\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*sqrt(2)*(6*sqrt(2)*(sqrt(b*x + a)*(2*x + a/b)*sqrt(x) + a^2*log(abs(-sqrt(b)*sqrt(x) + sqrt(b*x + a)))/b^
(3/2))*a + sqrt(2)*(sqrt(b*x + a)*(2*(4*x + a/b)*x - 3*a^2/b^2)*sqrt(x) - 3*a^3*log(abs(-sqrt(b)*sqrt(x) + sqr
t(b*x + a)))/b^(5/2))*b)