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

Optimal. Leaf size=142 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}\right ) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3}{8 b^{5/2}}-\frac{\sqrt{x} \sqrt{\tanh ^{-1}(\tanh (a+b x))} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}{8 b^2}-\frac{x^{3/2} \sqrt{\tanh ^{-1}(\tanh (a+b x))} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{12 b}+\frac{1}{3} x^{5/2} \sqrt{\tanh ^{-1}(\tanh (a+b x))} \]

[Out]

-(ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[ArcTanh[Tanh[a + b*x]]]]*(b*x - ArcTanh[Tanh[a + b*x]])^3)/(8*b^(5/2)) + (x^(
5/2)*Sqrt[ArcTanh[Tanh[a + b*x]]])/3 - (x^(3/2)*(b*x - ArcTanh[Tanh[a + b*x]])*Sqrt[ArcTanh[Tanh[a + b*x]]])/(
12*b) - (Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]])^2*Sqrt[ArcTanh[Tanh[a + b*x]]])/(8*b^2)

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Rubi [A]  time = 0.0775207, antiderivative size = 142, 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} \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}\right ) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3}{8 b^{5/2}}-\frac{\sqrt{x} \sqrt{\tanh ^{-1}(\tanh (a+b x))} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}{8 b^2}-\frac{x^{3/2} \sqrt{\tanh ^{-1}(\tanh (a+b x))} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{12 b}+\frac{1}{3} x^{5/2} \sqrt{\tanh ^{-1}(\tanh (a+b x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[ArcTanh[Tanh[a + b*x]]]]*(b*x - ArcTanh[Tanh[a + b*x]])^3)/(8*b^(5/2)) + (x^(
5/2)*Sqrt[ArcTanh[Tanh[a + b*x]]])/3 - (x^(3/2)*(b*x - ArcTanh[Tanh[a + b*x]])*Sqrt[ArcTanh[Tanh[a + b*x]]])/(
12*b) - (Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]])^2*Sqrt[ArcTanh[Tanh[a + b*x]]])/(8*b^2)

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 x^{3/2} \sqrt{\tanh ^{-1}(\tanh (a+b x))} \, dx &=\frac{1}{3} x^{5/2} \sqrt{\tanh ^{-1}(\tanh (a+b x))}-\frac{1}{6} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \int \frac{x^{3/2}}{\sqrt{\tanh ^{-1}(\tanh (a+b x))}} \, dx\\ &=\frac{1}{3} x^{5/2} \sqrt{\tanh ^{-1}(\tanh (a+b x))}-\frac{x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{12 b}-\frac{\left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )^2 \int \frac{\sqrt{x}}{\sqrt{\tanh ^{-1}(\tanh (a+b x))}} \, dx}{8 b}\\ &=\frac{1}{3} x^{5/2} \sqrt{\tanh ^{-1}(\tanh (a+b x))}-\frac{x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{12 b}-\frac{\sqrt{x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{8 b^2}+\frac{\left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )^3 \int \frac{1}{\sqrt{x} \sqrt{\tanh ^{-1}(\tanh (a+b x))}} \, dx}{16 b^2}\\ &=-\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^{5/2}}+\frac{1}{3} x^{5/2} \sqrt{\tanh ^{-1}(\tanh (a+b x))}-\frac{x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{12 b}-\frac{\sqrt{x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{8 b^2}\\ \end{align*}

Mathematica [A]  time = 0.0855375, size = 104, normalized size = 0.73 \[ \frac{\sqrt{x} \sqrt{\tanh ^{-1}(\tanh (a+b x))} \left (8 b x \tanh ^{-1}(\tanh (a+b x))-3 \tanh ^{-1}(\tanh (a+b x))^2+3 b^2 x^2\right )}{24 b^2}+\frac{\left (\tanh ^{-1}(\tanh (a+b x))-b x\right )^3 \log \left (\sqrt{b} \sqrt{\tanh ^{-1}(\tanh (a+b x))}+b \sqrt{x}\right )}{8 b^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(3/2)*Sqrt[ArcTanh[Tanh[a + b*x]]],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^2) + ((-(b*x) + ArcTanh[Tanh[a + b*x]])^3*Log[b*Sqrt[x] + Sqrt[b]*Sqrt[ArcTanh[Tanh[a + b*x]]]])/(8*b^(
5/2))

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Maple [B]  time = 0.167, size = 304, normalized size = 2.1 \begin{align*}{\frac{1}{3\,b}{x}^{{\frac{3}{2}}} \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{a}{4\,{b}^{2}}\sqrt{x} \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{{a}^{2}}{8\,{b}^{2}}\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{5}{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{5}{2}}}}+{\frac{a \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx-a \right ) }{4\,{b}^{2}}\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{5}{2}}}}-{\frac{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx-a}{4\,{b}^{2}}\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}^{2}}\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{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^(3/2)*arctanh(tanh(b*x+a))^(3/2)/b-1/4/b^2*a*x^(1/2)*arctanh(tanh(b*x+a))^(3/2)+1/8/b^2*a^2*x^(1/2)*arct
anh(tanh(b*x+a))^(1/2)+1/8/b^(5/2)*ln(b^(1/2)*x^(1/2)+arctanh(tanh(b*x+a))^(1/2))*a^3+3/8/b^(5/2)*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^2*a*(arctanh(tanh(b*x+a))-b*x-a)*x^(
1/2)*arctanh(tanh(b*x+a))^(1/2)+3/8/b^(5/2)*a*ln(b^(1/2)*x^(1/2)+arctanh(tanh(b*x+a))^(1/2))*(arctanh(tanh(b*x
+a))-b*x-a)^2-1/4/b^2*(arctanh(tanh(b*x+a))-b*x-a)*x^(1/2)*arctanh(tanh(b*x+a))^(3/2)+1/8/b^2*(arctanh(tanh(b*
x+a))-b*x-a)^2*x^(1/2)*arctanh(tanh(b*x+a))^(1/2)+1/8/b^(5/2)*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 x^{\frac{3}{2}} \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(x^(3/2)*arctanh(tanh(b*x+a))^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.12963, size = 362, normalized size = 2.55 \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} + 2 \, a b^{2} x - 3 \, a^{2} b\right )} \sqrt{b x + a} \sqrt{x}}{48 \, b^{3}}, -\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} + 2 \, a b^{2} x - 3 \, a^{2} b\right )} \sqrt{b x + a} \sqrt{x}}{24 \, b^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)*arctanh(tanh(b*x+a))^(1/2),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 + 2*a*b^2*x - 3*a^2*b)*sq
rt(b*x + a)*sqrt(x))/b^3, -1/24*(3*a^3*sqrt(-b)*arctan(sqrt(b*x + a)*sqrt(-b)/(b*sqrt(x))) - (8*b^3*x^2 + 2*a*
b^2*x - 3*a^2*b)*sqrt(b*x + a)*sqrt(x))/b^3]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{\frac{3}{2}} \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(x**(3/2)*atanh(tanh(b*x+a))**(1/2),x)

[Out]

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

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Giac [A]  time = 1.16849, size = 81, normalized size = 0.57 \begin{align*} \frac{1}{24} \, \sqrt{b x + a}{\left (2 \,{\left (4 \, x + \frac{a}{b}\right )} x - \frac{3 \, a^{2}}{b^{2}}\right )} \sqrt{x} - \frac{a^{3} \log \left ({\left | -\sqrt{b} \sqrt{x} + \sqrt{b x + a} \right |}\right )}{8 \, b^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*sqrt(b*x + a)*(2*(4*x + a/b)*x - 3*a^2/b^2)*sqrt(x) - 1/8*a^3*log(abs(-sqrt(b)*sqrt(x) + sqrt(b*x + a)))/
b^(5/2)