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

Optimal. Leaf size=177 \[ \frac{3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}\right ) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^4}{64 b^{5/2}}+\frac{3 \sqrt{x} \sqrt{\tanh ^{-1}(\tanh (a+b x))} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3}{64 b^2}+\frac{x^{3/2} \sqrt{\tanh ^{-1}(\tanh (a+b x))} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}{32 b}-\frac{1}{8} x^{5/2} \sqrt{\tanh ^{-1}(\tanh (a+b x))} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )+\frac{1}{4} x^{5/2} \tanh ^{-1}(\tanh (a+b x))^{3/2} \]

[Out]

(3*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[ArcTanh[Tanh[a + b*x]]]]*(b*x - ArcTanh[Tanh[a + b*x]])^4)/(64*b^(5/2)) - (x
^(5/2)*(b*x - ArcTanh[Tanh[a + b*x]])*Sqrt[ArcTanh[Tanh[a + b*x]]])/8 + (x^(3/2)*(b*x - ArcTanh[Tanh[a + b*x]]
)^2*Sqrt[ArcTanh[Tanh[a + b*x]]])/(32*b) + (3*Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]])^3*Sqrt[ArcTanh[Tanh[a + b
*x]]])/(64*b^2) + (x^(5/2)*ArcTanh[Tanh[a + b*x]]^(3/2))/4

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Rubi [A]  time = 0.103711, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2169, 2165} \[ \frac{3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}\right ) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^4}{64 b^{5/2}}+\frac{3 \sqrt{x} \sqrt{\tanh ^{-1}(\tanh (a+b x))} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3}{64 b^2}+\frac{x^{3/2} \sqrt{\tanh ^{-1}(\tanh (a+b x))} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}{32 b}-\frac{1}{8} x^{5/2} \sqrt{\tanh ^{-1}(\tanh (a+b x))} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )+\frac{1}{4} x^{5/2} \tanh ^{-1}(\tanh (a+b x))^{3/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[ArcTanh[Tanh[a + b*x]]]]*(b*x - ArcTanh[Tanh[a + b*x]])^4)/(64*b^(5/2)) - (x
^(5/2)*(b*x - ArcTanh[Tanh[a + b*x]])*Sqrt[ArcTanh[Tanh[a + b*x]]])/8 + (x^(3/2)*(b*x - ArcTanh[Tanh[a + b*x]]
)^2*Sqrt[ArcTanh[Tanh[a + b*x]]])/(32*b) + (3*Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]])^3*Sqrt[ArcTanh[Tanh[a + b
*x]]])/(64*b^2) + (x^(5/2)*ArcTanh[Tanh[a + b*x]]^(3/2))/4

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} \tanh ^{-1}(\tanh (a+b x))^{3/2} \, dx &=\frac{1}{4} x^{5/2} \tanh ^{-1}(\tanh (a+b x))^{3/2}-\frac{1}{8} \left (3 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )\right ) \int x^{3/2} \sqrt{\tanh ^{-1}(\tanh (a+b x))} \, dx\\ &=-\frac{1}{8} x^{5/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \sqrt{\tanh ^{-1}(\tanh (a+b x))}+\frac{1}{4} x^{5/2} \tanh ^{-1}(\tanh (a+b x))^{3/2}+\frac{1}{16} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \int \frac{x^{3/2}}{\sqrt{\tanh ^{-1}(\tanh (a+b x))}} \, dx\\ &=-\frac{1}{8} x^{5/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \sqrt{\tanh ^{-1}(\tanh (a+b x))}+\frac{x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{32 b}+\frac{1}{4} x^{5/2} \tanh ^{-1}(\tanh (a+b x))^{3/2}-\frac{\left (3 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )\right ) \int \frac{\sqrt{x}}{\sqrt{\tanh ^{-1}(\tanh (a+b x))}} \, dx}{64 b}\\ &=-\frac{1}{8} x^{5/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \sqrt{\tanh ^{-1}(\tanh (a+b x))}+\frac{x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{32 b}+\frac{3 \sqrt{x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{64 b^2}+\frac{1}{4} x^{5/2} \tanh ^{-1}(\tanh (a+b x))^{3/2}+\frac{\left (3 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \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}{128 b^2}\\ &=\frac{3 \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 )^4}{64 b^{5/2}}-\frac{1}{8} x^{5/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \sqrt{\tanh ^{-1}(\tanh (a+b x))}+\frac{x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{32 b}+\frac{3 \sqrt{x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{64 b^2}+\frac{1}{4} x^{5/2} \tanh ^{-1}(\tanh (a+b x))^{3/2}\\ \end{align*}

Mathematica [A]  time = 0.0891484, size = 122, normalized size = 0.69 \[ \frac{\sqrt{b} \sqrt{x} \sqrt{\tanh ^{-1}(\tanh (a+b x))} \left (11 b^2 x^2 \tanh ^{-1}(\tanh (a+b x))+11 b x \tanh ^{-1}(\tanh (a+b x))^2-3 \tanh ^{-1}(\tanh (a+b x))^3-3 b^3 x^3\right )+3 \left (\tanh ^{-1}(\tanh (a+b x))-b x\right )^4 \log \left (\sqrt{b} \sqrt{\tanh ^{-1}(\tanh (a+b x))}+b \sqrt{x}\right )}{64 b^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[b]*Sqrt[x]*Sqrt[ArcTanh[Tanh[a + b*x]]]*(-3*b^3*x^3 + 11*b^2*x^2*ArcTanh[Tanh[a + b*x]] + 11*b*x*ArcTanh
[Tanh[a + b*x]]^2 - 3*ArcTanh[Tanh[a + b*x]]^3) + 3*(-(b*x) + ArcTanh[Tanh[a + b*x]])^4*Log[b*Sqrt[x] + Sqrt[b
]*Sqrt[ArcTanh[Tanh[a + b*x]]]])/(64*b^(5/2))

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Maple [B]  time = 0.115, size = 471, normalized size = 2.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^(3/2)*arctanh(tanh(b*x+a))^(5/2)/b-1/8/b^2*a*x^(1/2)*arctanh(tanh(b*x+a))^(5/2)+1/32/b^2*a^2*x^(1/2)*arc
tanh(tanh(b*x+a))^(3/2)+3/64/b^2*a^3*x^(1/2)*arctanh(tanh(b*x+a))^(1/2)+3/64/b^(5/2)*ln(b^(1/2)*x^(1/2)+arctan
h(tanh(b*x+a))^(1/2))*a^4+3/16/b^(5/2)*a^3*ln(b^(1/2)*x^(1/2)+arctanh(tanh(b*x+a))^(1/2))*(arctanh(tanh(b*x+a)
)-b*x-a)+9/64/b^2*a^2*(arctanh(tanh(b*x+a))-b*x-a)*x^(1/2)*arctanh(tanh(b*x+a))^(1/2)+9/32/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)^2+1/16/b^2*a*(arctanh(tanh(b*x+a))-b*x-a)
*x^(1/2)*arctanh(tanh(b*x+a))^(3/2)+9/64/b^2*a*(arctanh(tanh(b*x+a))-b*x-a)^2*x^(1/2)*arctanh(tanh(b*x+a))^(1/
2)+3/16/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)^3-1/8/b^2*(arcta
nh(tanh(b*x+a))-b*x-a)*x^(1/2)*arctanh(tanh(b*x+a))^(5/2)+1/32/b^2*(arctanh(tanh(b*x+a))-b*x-a)^2*x^(1/2)*arct
anh(tanh(b*x+a))^(3/2)+3/64/b^2*(arctanh(tanh(b*x+a))-b*x-a)^3*x^(1/2)*arctanh(tanh(b*x+a))^(1/2)+3/64/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)^4

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{\frac{3}{2}} \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^(3/2)*arctanh(tanh(b*x+a))^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.30879, size = 412, normalized size = 2.33 \begin{align*} \left [\frac{3 \, a^{4} \sqrt{b} \log \left (2 \, b x + 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x} + a\right ) + 2 \,{\left (16 \, b^{4} x^{3} + 24 \, a b^{3} x^{2} + 2 \, a^{2} b^{2} x - 3 \, a^{3} b\right )} \sqrt{b x + a} \sqrt{x}}{128 \, b^{3}}, -\frac{3 \, a^{4} \sqrt{-b} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) -{\left (16 \, b^{4} x^{3} + 24 \, a b^{3} x^{2} + 2 \, a^{2} b^{2} x - 3 \, a^{3} b\right )} \sqrt{b x + a} \sqrt{x}}{64 \, 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))^(3/2),x, algorithm="fricas")

[Out]

[1/128*(3*a^4*sqrt(b)*log(2*b*x + 2*sqrt(b*x + a)*sqrt(b)*sqrt(x) + a) + 2*(16*b^4*x^3 + 24*a*b^3*x^2 + 2*a^2*
b^2*x - 3*a^3*b)*sqrt(b*x + a)*sqrt(x))/b^3, -1/64*(3*a^4*sqrt(-b)*arctan(sqrt(b*x + a)*sqrt(-b)/(b*sqrt(x)))
- (16*b^4*x^3 + 24*a*b^3*x^2 + 2*a^2*b^2*x - 3*a^3*b)*sqrt(b*x + a)*sqrt(x))/b^3]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.20397, size = 198, normalized size = 1.12 \begin{align*} \frac{1}{384} \, \sqrt{2}{\left (8 \, \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 )} a + \sqrt{2}{\left ({\left (2 \,{\left (4 \,{\left (6 \, x + \frac{a}{b}\right )} x - \frac{5 \, a^{2}}{b^{2}}\right )} x + \frac{15 \, a^{3}}{b^{3}}\right )} \sqrt{b x + a} \sqrt{x} + \frac{15 \, a^{4} \log \left ({\left | -\sqrt{b} \sqrt{x} + \sqrt{b x + a} \right |}\right )}{b^{\frac{7}{2}}}\right )} b\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/384*sqrt(2)*(8*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)
+ sqrt(b*x + a)))/b^(5/2))*a + sqrt(2)*((2*(4*(6*x + a/b)*x - 5*a^2/b^2)*x + 15*a^3/b^3)*sqrt(b*x + a)*sqrt(x
) + 15*a^4*log(abs(-sqrt(b)*sqrt(x) + sqrt(b*x + a)))/b^(7/2))*b)