3.129 \(\int x^4 \tanh ^{-1}(\tanh (a+b x))^{5/2} \, dx\)

Optimal. Leaf size=101 \[ \frac{32 x^2 \tanh ^{-1}(\tanh (a+b x))^{11/2}}{231 b^3}-\frac{16 x^3 \tanh ^{-1}(\tanh (a+b x))^{9/2}}{63 b^2}+\frac{256 \tanh ^{-1}(\tanh (a+b x))^{15/2}}{45045 b^5}-\frac{128 x \tanh ^{-1}(\tanh (a+b x))^{13/2}}{3003 b^4}+\frac{2 x^4 \tanh ^{-1}(\tanh (a+b x))^{7/2}}{7 b} \]

[Out]

(2*x^4*ArcTanh[Tanh[a + b*x]]^(7/2))/(7*b) - (16*x^3*ArcTanh[Tanh[a + b*x]]^(9/2))/(63*b^2) + (32*x^2*ArcTanh[
Tanh[a + b*x]]^(11/2))/(231*b^3) - (128*x*ArcTanh[Tanh[a + b*x]]^(13/2))/(3003*b^4) + (256*ArcTanh[Tanh[a + b*
x]]^(15/2))/(45045*b^5)

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Rubi [A]  time = 0.0647503, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2168, 2157, 30} \[ \frac{32 x^2 \tanh ^{-1}(\tanh (a+b x))^{11/2}}{231 b^3}-\frac{16 x^3 \tanh ^{-1}(\tanh (a+b x))^{9/2}}{63 b^2}+\frac{256 \tanh ^{-1}(\tanh (a+b x))^{15/2}}{45045 b^5}-\frac{128 x \tanh ^{-1}(\tanh (a+b x))^{13/2}}{3003 b^4}+\frac{2 x^4 \tanh ^{-1}(\tanh (a+b x))^{7/2}}{7 b} \]

Antiderivative was successfully verified.

[In]

Int[x^4*ArcTanh[Tanh[a + b*x]]^(5/2),x]

[Out]

(2*x^4*ArcTanh[Tanh[a + b*x]]^(7/2))/(7*b) - (16*x^3*ArcTanh[Tanh[a + b*x]]^(9/2))/(63*b^2) + (32*x^2*ArcTanh[
Tanh[a + b*x]]^(11/2))/(231*b^3) - (128*x*ArcTanh[Tanh[a + b*x]]^(13/2))/(3003*b^4) + (256*ArcTanh[Tanh[a + b*
x]]^(15/2))/(45045*b^5)

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 2157

Int[(u_)^(m_.), x_Symbol] :> With[{c = Simplify[D[u, x]]}, Dist[1/c, Subst[Int[x^m, x], x, u], x]] /; FreeQ[m,
x] && PiecewiseLinearQ[u, x]

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^4 \tanh ^{-1}(\tanh (a+b x))^{5/2} \, dx &=\frac{2 x^4 \tanh ^{-1}(\tanh (a+b x))^{7/2}}{7 b}-\frac{8 \int x^3 \tanh ^{-1}(\tanh (a+b x))^{7/2} \, dx}{7 b}\\ &=\frac{2 x^4 \tanh ^{-1}(\tanh (a+b x))^{7/2}}{7 b}-\frac{16 x^3 \tanh ^{-1}(\tanh (a+b x))^{9/2}}{63 b^2}+\frac{16 \int x^2 \tanh ^{-1}(\tanh (a+b x))^{9/2} \, dx}{21 b^2}\\ &=\frac{2 x^4 \tanh ^{-1}(\tanh (a+b x))^{7/2}}{7 b}-\frac{16 x^3 \tanh ^{-1}(\tanh (a+b x))^{9/2}}{63 b^2}+\frac{32 x^2 \tanh ^{-1}(\tanh (a+b x))^{11/2}}{231 b^3}-\frac{64 \int x \tanh ^{-1}(\tanh (a+b x))^{11/2} \, dx}{231 b^3}\\ &=\frac{2 x^4 \tanh ^{-1}(\tanh (a+b x))^{7/2}}{7 b}-\frac{16 x^3 \tanh ^{-1}(\tanh (a+b x))^{9/2}}{63 b^2}+\frac{32 x^2 \tanh ^{-1}(\tanh (a+b x))^{11/2}}{231 b^3}-\frac{128 x \tanh ^{-1}(\tanh (a+b x))^{13/2}}{3003 b^4}+\frac{128 \int \tanh ^{-1}(\tanh (a+b x))^{13/2} \, dx}{3003 b^4}\\ &=\frac{2 x^4 \tanh ^{-1}(\tanh (a+b x))^{7/2}}{7 b}-\frac{16 x^3 \tanh ^{-1}(\tanh (a+b x))^{9/2}}{63 b^2}+\frac{32 x^2 \tanh ^{-1}(\tanh (a+b x))^{11/2}}{231 b^3}-\frac{128 x \tanh ^{-1}(\tanh (a+b x))^{13/2}}{3003 b^4}+\frac{128 \operatorname{Subst}\left (\int x^{13/2} \, dx,x,\tanh ^{-1}(\tanh (a+b x))\right )}{3003 b^5}\\ &=\frac{2 x^4 \tanh ^{-1}(\tanh (a+b x))^{7/2}}{7 b}-\frac{16 x^3 \tanh ^{-1}(\tanh (a+b x))^{9/2}}{63 b^2}+\frac{32 x^2 \tanh ^{-1}(\tanh (a+b x))^{11/2}}{231 b^3}-\frac{128 x \tanh ^{-1}(\tanh (a+b x))^{13/2}}{3003 b^4}+\frac{256 \tanh ^{-1}(\tanh (a+b x))^{15/2}}{45045 b^5}\\ \end{align*}

Mathematica [A]  time = 0.0381337, size = 83, normalized size = 0.82 \[ \frac{2 \tanh ^{-1}(\tanh (a+b x))^{7/2} \left (-5720 b^3 x^3 \tanh ^{-1}(\tanh (a+b x))+3120 b^2 x^2 \tanh ^{-1}(\tanh (a+b x))^2-960 b x \tanh ^{-1}(\tanh (a+b x))^3+128 \tanh ^{-1}(\tanh (a+b x))^4+6435 b^4 x^4\right )}{45045 b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*ArcTanh[Tanh[a + b*x]]^(5/2),x]

[Out]

(2*ArcTanh[Tanh[a + b*x]]^(7/2)*(6435*b^4*x^4 - 5720*b^3*x^3*ArcTanh[Tanh[a + b*x]] + 3120*b^2*x^2*ArcTanh[Tan
h[a + b*x]]^2 - 960*b*x*ArcTanh[Tanh[a + b*x]]^3 + 128*ArcTanh[Tanh[a + b*x]]^4))/(45045*b^5)

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Maple [A]  time = 0.035, size = 154, normalized size = 1.5 \begin{align*} 2\,{\frac{1/15\, \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{15/2}+1/13\, \left ( -4\,{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) +4\,bx \right ) \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{13/2}+1/11\, \left ( 2\, \left ( bx-{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2}+ \left ( -2\,{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) +2\,bx \right ) ^{2} \right ) \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{11/2}+2/9\, \left ( bx-{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2} \left ( -2\,{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) +2\,bx \right ) \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{9/2}+1/7\, \left ( bx-{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{4} \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{7/2}}{{b}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*arctanh(tanh(b*x+a))^(5/2),x)

[Out]

2/b^5*(1/15*arctanh(tanh(b*x+a))^(15/2)+1/13*(-4*arctanh(tanh(b*x+a))+4*b*x)*arctanh(tanh(b*x+a))^(13/2)+1/11*
(2*(b*x-arctanh(tanh(b*x+a)))^2+(-2*arctanh(tanh(b*x+a))+2*b*x)^2)*arctanh(tanh(b*x+a))^(11/2)+2/9*(b*x-arctan
h(tanh(b*x+a)))^2*(-2*arctanh(tanh(b*x+a))+2*b*x)*arctanh(tanh(b*x+a))^(9/2)+1/7*(b*x-arctanh(tanh(b*x+a)))^4*
arctanh(tanh(b*x+a))^(7/2))

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Maxima [A]  time = 1.83062, size = 86, normalized size = 0.85 \begin{align*} \frac{2 \,{\left (3003 \, b^{5} x^{5} + 1155 \, a b^{4} x^{4} - 840 \, a^{2} b^{3} x^{3} + 560 \, a^{3} b^{2} x^{2} - 320 \, a^{4} b x + 128 \, a^{5}\right )}{\left (b x + a\right )}^{\frac{5}{2}}}{45045 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arctanh(tanh(b*x+a))^(5/2),x, algorithm="maxima")

[Out]

2/45045*(3003*b^5*x^5 + 1155*a*b^4*x^4 - 840*a^2*b^3*x^3 + 560*a^3*b^2*x^2 - 320*a^4*b*x + 128*a^5)*(b*x + a)^
(5/2)/b^5

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Fricas [A]  time = 2.04585, size = 205, normalized size = 2.03 \begin{align*} \frac{2 \,{\left (3003 \, b^{7} x^{7} + 7161 \, a b^{6} x^{6} + 4473 \, a^{2} b^{5} x^{5} + 35 \, a^{3} b^{4} x^{4} - 40 \, a^{4} b^{3} x^{3} + 48 \, a^{5} b^{2} x^{2} - 64 \, a^{6} b x + 128 \, a^{7}\right )} \sqrt{b x + a}}{45045 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arctanh(tanh(b*x+a))^(5/2),x, algorithm="fricas")

[Out]

2/45045*(3003*b^7*x^7 + 7161*a*b^6*x^6 + 4473*a^2*b^5*x^5 + 35*a^3*b^4*x^4 - 40*a^4*b^3*x^3 + 48*a^5*b^2*x^2 -
64*a^6*b*x + 128*a^7)*sqrt(b*x + a)/b^5

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

[Out]

Timed out

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Giac [B]  time = 1.17011, size = 324, normalized size = 3.21 \begin{align*} \frac{\sqrt{2}{\left (\frac{13 \, \sqrt{2}{\left (315 \,{\left (b x + a\right )}^{\frac{11}{2}} - 1540 \,{\left (b x + a\right )}^{\frac{9}{2}} a + 2970 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{2} - 2772 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{3} + 1155 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{4}\right )} a^{2}}{b^{4}} + \frac{10 \, \sqrt{2}{\left (693 \,{\left (b x + a\right )}^{\frac{13}{2}} - 4095 \,{\left (b x + a\right )}^{\frac{11}{2}} a + 10010 \,{\left (b x + a\right )}^{\frac{9}{2}} a^{2} - 12870 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{3} + 9009 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{4} - 3003 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{5}\right )} a}{b^{4}} + \frac{\sqrt{2}{\left (3003 \,{\left (b x + a\right )}^{\frac{15}{2}} - 20790 \,{\left (b x + a\right )}^{\frac{13}{2}} a + 61425 \,{\left (b x + a\right )}^{\frac{11}{2}} a^{2} - 100100 \,{\left (b x + a\right )}^{\frac{9}{2}} a^{3} + 96525 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{4} - 54054 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{5} + 15015 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{6}\right )}}{b^{4}}\right )}}{45045 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arctanh(tanh(b*x+a))^(5/2),x, algorithm="giac")

[Out]

1/45045*sqrt(2)*(13*sqrt(2)*(315*(b*x + a)^(11/2) - 1540*(b*x + a)^(9/2)*a + 2970*(b*x + a)^(7/2)*a^2 - 2772*(
b*x + a)^(5/2)*a^3 + 1155*(b*x + a)^(3/2)*a^4)*a^2/b^4 + 10*sqrt(2)*(693*(b*x + a)^(13/2) - 4095*(b*x + a)^(11
/2)*a + 10010*(b*x + a)^(9/2)*a^2 - 12870*(b*x + a)^(7/2)*a^3 + 9009*(b*x + a)^(5/2)*a^4 - 3003*(b*x + a)^(3/2
)*a^5)*a/b^4 + sqrt(2)*(3003*(b*x + a)^(15/2) - 20790*(b*x + a)^(13/2)*a + 61425*(b*x + a)^(11/2)*a^2 - 100100
*(b*x + a)^(9/2)*a^3 + 96525*(b*x + a)^(7/2)*a^4 - 54054*(b*x + a)^(5/2)*a^5 + 15015*(b*x + a)^(3/2)*a^6)/b^4)
/b