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

Optimal. Leaf size=80 \[ -\frac{1}{126} b^3 x^9 \tanh ^{-1}(\tanh (a+b x))+\frac{1}{28} b^2 x^8 \tanh ^{-1}(\tanh (a+b x))^2-\frac{2}{21} b x^7 \tanh ^{-1}(\tanh (a+b x))^3+\frac{1}{6} x^6 \tanh ^{-1}(\tanh (a+b x))^4+\frac{b^4 x^{10}}{1260} \]

[Out]

(b^4*x^10)/1260 - (b^3*x^9*ArcTanh[Tanh[a + b*x]])/126 + (b^2*x^8*ArcTanh[Tanh[a + b*x]]^2)/28 - (2*b*x^7*ArcT
anh[Tanh[a + b*x]]^3)/21 + (x^6*ArcTanh[Tanh[a + b*x]]^4)/6

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Rubi [A]  time = 0.0564928, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2168, 30} \[ -\frac{1}{126} b^3 x^9 \tanh ^{-1}(\tanh (a+b x))+\frac{1}{28} b^2 x^8 \tanh ^{-1}(\tanh (a+b x))^2-\frac{2}{21} b x^7 \tanh ^{-1}(\tanh (a+b x))^3+\frac{1}{6} x^6 \tanh ^{-1}(\tanh (a+b x))^4+\frac{b^4 x^{10}}{1260} \]

Antiderivative was successfully verified.

[In]

Int[x^5*ArcTanh[Tanh[a + b*x]]^4,x]

[Out]

(b^4*x^10)/1260 - (b^3*x^9*ArcTanh[Tanh[a + b*x]])/126 + (b^2*x^8*ArcTanh[Tanh[a + b*x]]^2)/28 - (2*b*x^7*ArcT
anh[Tanh[a + b*x]]^3)/21 + (x^6*ArcTanh[Tanh[a + b*x]]^4)/6

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 \tanh ^{-1}(\tanh (a+b x))^4 \, dx &=\frac{1}{6} x^6 \tanh ^{-1}(\tanh (a+b x))^4-\frac{1}{3} (2 b) \int x^6 \tanh ^{-1}(\tanh (a+b x))^3 \, dx\\ &=-\frac{2}{21} b x^7 \tanh ^{-1}(\tanh (a+b x))^3+\frac{1}{6} x^6 \tanh ^{-1}(\tanh (a+b x))^4+\frac{1}{7} \left (2 b^2\right ) \int x^7 \tanh ^{-1}(\tanh (a+b x))^2 \, dx\\ &=\frac{1}{28} b^2 x^8 \tanh ^{-1}(\tanh (a+b x))^2-\frac{2}{21} b x^7 \tanh ^{-1}(\tanh (a+b x))^3+\frac{1}{6} x^6 \tanh ^{-1}(\tanh (a+b x))^4-\frac{1}{14} b^3 \int x^8 \tanh ^{-1}(\tanh (a+b x)) \, dx\\ &=-\frac{1}{126} b^3 x^9 \tanh ^{-1}(\tanh (a+b x))+\frac{1}{28} b^2 x^8 \tanh ^{-1}(\tanh (a+b x))^2-\frac{2}{21} b x^7 \tanh ^{-1}(\tanh (a+b x))^3+\frac{1}{6} x^6 \tanh ^{-1}(\tanh (a+b x))^4+\frac{1}{126} b^4 \int x^9 \, dx\\ &=\frac{b^4 x^{10}}{1260}-\frac{1}{126} b^3 x^9 \tanh ^{-1}(\tanh (a+b x))+\frac{1}{28} b^2 x^8 \tanh ^{-1}(\tanh (a+b x))^2-\frac{2}{21} b x^7 \tanh ^{-1}(\tanh (a+b x))^3+\frac{1}{6} x^6 \tanh ^{-1}(\tanh (a+b x))^4\\ \end{align*}

Mathematica [A]  time = 0.0308633, size = 71, normalized size = 0.89 \[ \frac{x^6 \left (-10 b^3 x^3 \tanh ^{-1}(\tanh (a+b x))+45 b^2 x^2 \tanh ^{-1}(\tanh (a+b x))^2-120 b x \tanh ^{-1}(\tanh (a+b x))^3+210 \tanh ^{-1}(\tanh (a+b x))^4+b^4 x^4\right )}{1260} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*ArcTanh[Tanh[a + b*x]]^4,x]

[Out]

(x^6*(b^4*x^4 - 10*b^3*x^3*ArcTanh[Tanh[a + b*x]] + 45*b^2*x^2*ArcTanh[Tanh[a + b*x]]^2 - 120*b*x*ArcTanh[Tanh
[a + b*x]]^3 + 210*ArcTanh[Tanh[a + b*x]]^4))/1260

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Maple [A]  time = 0.04, size = 74, normalized size = 0.9 \begin{align*}{\frac{{x}^{6} \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{4}}{6}}-{\frac{2\,b}{3} \left ({\frac{{x}^{7} \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{3}}{7}}-{\frac{3\,b}{7} \left ({\frac{{x}^{8} \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2}}{8}}-{\frac{b}{4} \left ({\frac{{x}^{9}{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }{9}}-{\frac{{x}^{10}b}{90}} \right ) } \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*x^6*arctanh(tanh(b*x+a))^4-2/3*b*(1/7*x^7*arctanh(tanh(b*x+a))^3-3/7*b*(1/8*x^8*arctanh(tanh(b*x+a))^2-1/4
*b*(1/9*x^9*arctanh(tanh(b*x+a))-1/90*x^10*b)))

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Maxima [A]  time = 1.73772, size = 97, normalized size = 1.21 \begin{align*} -\frac{2}{21} \, b x^{7} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{3} + \frac{1}{6} \, x^{6} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{4} + \frac{1}{1260} \,{\left (45 \, b x^{8} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{2} +{\left (b^{2} x^{10} - 10 \, b x^{9} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )\right )} b\right )} b \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/21*b*x^7*arctanh(tanh(b*x + a))^3 + 1/6*x^6*arctanh(tanh(b*x + a))^4 + 1/1260*(45*b*x^8*arctanh(tanh(b*x +
a))^2 + (b^2*x^10 - 10*b*x^9*arctanh(tanh(b*x + a)))*b)*b

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Fricas [A]  time = 1.51244, size = 107, normalized size = 1.34 \begin{align*} \frac{1}{10} \, b^{4} x^{10} + \frac{4}{9} \, a b^{3} x^{9} + \frac{3}{4} \, a^{2} b^{2} x^{8} + \frac{4}{7} \, a^{3} b x^{7} + \frac{1}{6} \, a^{4} x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*b^4*x^10 + 4/9*a*b^3*x^9 + 3/4*a^2*b^2*x^8 + 4/7*a^3*b*x^7 + 1/6*a^4*x^6

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Sympy [A]  time = 18.9303, size = 76, normalized size = 0.95 \begin{align*} \frac{b^{4} x^{10}}{1260} - \frac{b^{3} x^{9} \operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )}}{126} + \frac{b^{2} x^{8} \operatorname{atanh}^{2}{\left (\tanh{\left (a + b x \right )} \right )}}{28} - \frac{2 b x^{7} \operatorname{atanh}^{3}{\left (\tanh{\left (a + b x \right )} \right )}}{21} + \frac{x^{6} \operatorname{atanh}^{4}{\left (\tanh{\left (a + b x \right )} \right )}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b**4*x**10/1260 - b**3*x**9*atanh(tanh(a + b*x))/126 + b**2*x**8*atanh(tanh(a + b*x))**2/28 - 2*b*x**7*atanh(t
anh(a + b*x))**3/21 + x**6*atanh(tanh(a + b*x))**4/6

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Giac [A]  time = 1.12196, size = 62, normalized size = 0.78 \begin{align*} \frac{1}{10} \, b^{4} x^{10} + \frac{4}{9} \, a b^{3} x^{9} + \frac{3}{4} \, a^{2} b^{2} x^{8} + \frac{4}{7} \, a^{3} b x^{7} + \frac{1}{6} \, a^{4} x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*b^4*x^10 + 4/9*a*b^3*x^9 + 3/4*a^2*b^2*x^8 + 4/7*a^3*b*x^7 + 1/6*a^4*x^6