3.184 \(\int x^{5/2} \tanh ^{-1}(\tanh (a+b x))^3 \, dx\)

Optimal. Leaf size=69 \[ \frac{16}{231} b^2 x^{11/2} \tanh ^{-1}(\tanh (a+b x))-\frac{4}{21} b x^{9/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac{2}{7} x^{7/2} \tanh ^{-1}(\tanh (a+b x))^3-\frac{32 b^3 x^{13/2}}{3003} \]

[Out]

(-32*b^3*x^(13/2))/3003 + (16*b^2*x^(11/2)*ArcTanh[Tanh[a + b*x]])/231 - (4*b*x^(9/2)*ArcTanh[Tanh[a + b*x]]^2
)/21 + (2*x^(7/2)*ArcTanh[Tanh[a + b*x]]^3)/7

________________________________________________________________________________________

Rubi [A]  time = 0.0377565, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2168, 30} \[ \frac{16}{231} b^2 x^{11/2} \tanh ^{-1}(\tanh (a+b x))-\frac{4}{21} b x^{9/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac{2}{7} x^{7/2} \tanh ^{-1}(\tanh (a+b x))^3-\frac{32 b^3 x^{13/2}}{3003} \]

Antiderivative was successfully verified.

[In]

Int[x^(5/2)*ArcTanh[Tanh[a + b*x]]^3,x]

[Out]

(-32*b^3*x^(13/2))/3003 + (16*b^2*x^(11/2)*ArcTanh[Tanh[a + b*x]])/231 - (4*b*x^(9/2)*ArcTanh[Tanh[a + b*x]]^2
)/21 + (2*x^(7/2)*ArcTanh[Tanh[a + b*x]]^3)/7

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]))

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int x^{5/2} \tanh ^{-1}(\tanh (a+b x))^3 \, dx &=\frac{2}{7} x^{7/2} \tanh ^{-1}(\tanh (a+b x))^3-\frac{1}{7} (6 b) \int x^{7/2} \tanh ^{-1}(\tanh (a+b x))^2 \, dx\\ &=-\frac{4}{21} b x^{9/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac{2}{7} x^{7/2} \tanh ^{-1}(\tanh (a+b x))^3+\frac{1}{21} \left (8 b^2\right ) \int x^{9/2} \tanh ^{-1}(\tanh (a+b x)) \, dx\\ &=\frac{16}{231} b^2 x^{11/2} \tanh ^{-1}(\tanh (a+b x))-\frac{4}{21} b x^{9/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac{2}{7} x^{7/2} \tanh ^{-1}(\tanh (a+b x))^3-\frac{1}{231} \left (16 b^3\right ) \int x^{11/2} \, dx\\ &=-\frac{32 b^3 x^{13/2}}{3003}+\frac{16}{231} b^2 x^{11/2} \tanh ^{-1}(\tanh (a+b x))-\frac{4}{21} b x^{9/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac{2}{7} x^{7/2} \tanh ^{-1}(\tanh (a+b x))^3\\ \end{align*}

Mathematica [A]  time = 0.0393816, size = 57, normalized size = 0.83 \[ \frac{2 x^{7/2} \left (104 b^2 x^2 \tanh ^{-1}(\tanh (a+b x))-286 b x \tanh ^{-1}(\tanh (a+b x))^2+429 \tanh ^{-1}(\tanh (a+b x))^3-16 b^3 x^3\right )}{3003} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(7/2)*(-16*b^3*x^3 + 104*b^2*x^2*ArcTanh[Tanh[a + b*x]] - 286*b*x*ArcTanh[Tanh[a + b*x]]^2 + 429*ArcTanh[
Tanh[a + b*x]]^3))/3003

________________________________________________________________________________________

Maple [A]  time = 0.044, size = 56, normalized size = 0.8 \begin{align*}{\frac{2\, \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{3}}{7}{x}^{{\frac{7}{2}}}}-{\frac{12\,b}{7} \left ({\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2}}{9}{x}^{{\frac{9}{2}}}}-{\frac{4\,b}{9} \left ({\frac{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }{11}{x}^{{\frac{11}{2}}}}-{\frac{2\,b}{143}{x}^{{\frac{13}{2}}}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*arctanh(tanh(b*x+a))^3,x)

[Out]

2/7*x^(7/2)*arctanh(tanh(b*x+a))^3-12/7*b*(1/9*x^(9/2)*arctanh(tanh(b*x+a))^2-4/9*b*(1/11*x^(11/2)*arctanh(tan
h(b*x+a))-2/143*x^(13/2)*b))

________________________________________________________________________________________

Maxima [A]  time = 1.06479, size = 74, normalized size = 1.07 \begin{align*} -\frac{4}{21} \, b x^{\frac{9}{2}} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{2} + \frac{2}{7} \, x^{\frac{7}{2}} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{3} - \frac{16}{3003} \,{\left (2 \, b^{2} x^{\frac{13}{2}} - 13 \, b x^{\frac{11}{2}} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )\right )} b \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*arctanh(tanh(b*x+a))^3,x, algorithm="maxima")

[Out]

-4/21*b*x^(9/2)*arctanh(tanh(b*x + a))^2 + 2/7*x^(7/2)*arctanh(tanh(b*x + a))^3 - 16/3003*(2*b^2*x^(13/2) - 13
*b*x^(11/2)*arctanh(tanh(b*x + a)))*b

________________________________________________________________________________________

Fricas [A]  time = 2.04076, size = 104, normalized size = 1.51 \begin{align*} \frac{2}{3003} \,{\left (231 \, b^{3} x^{6} + 819 \, a b^{2} x^{5} + 1001 \, a^{2} b x^{4} + 429 \, a^{3} x^{3}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*arctanh(tanh(b*x+a))^3,x, algorithm="fricas")

[Out]

2/3003*(231*b^3*x^6 + 819*a*b^2*x^5 + 1001*a^2*b*x^4 + 429*a^3*x^3)*sqrt(x)

________________________________________________________________________________________

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**(5/2)*atanh(tanh(b*x+a))**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.13728, size = 47, normalized size = 0.68 \begin{align*} \frac{2}{13} \, b^{3} x^{\frac{13}{2}} + \frac{6}{11} \, a b^{2} x^{\frac{11}{2}} + \frac{2}{3} \, a^{2} b x^{\frac{9}{2}} + \frac{2}{7} \, a^{3} x^{\frac{7}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*arctanh(tanh(b*x+a))^3,x, algorithm="giac")

[Out]

2/13*b^3*x^(13/2) + 6/11*a*b^2*x^(11/2) + 2/3*a^2*b*x^(9/2) + 2/7*a^3*x^(7/2)