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

Optimal. Leaf size=80 \[ -\frac{1}{70} b^3 x^8 \tanh ^{-1}(\tanh (a+b x))+\frac{2}{35} b^2 x^7 \tanh ^{-1}(\tanh (a+b x))^2-\frac{2}{15} b x^6 \tanh ^{-1}(\tanh (a+b x))^3+\frac{1}{5} x^5 \tanh ^{-1}(\tanh (a+b x))^4+\frac{b^4 x^9}{630} \]

[Out]

(b^4*x^9)/630 - (b^3*x^8*ArcTanh[Tanh[a + b*x]])/70 + (2*b^2*x^7*ArcTanh[Tanh[a + b*x]]^2)/35 - (2*b*x^6*ArcTa
nh[Tanh[a + b*x]]^3)/15 + (x^5*ArcTanh[Tanh[a + b*x]]^4)/5

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Rubi [A]  time = 0.0547148, 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} \[ -\frac{1}{70} b^3 x^8 \tanh ^{-1}(\tanh (a+b x))+\frac{2}{35} b^2 x^7 \tanh ^{-1}(\tanh (a+b x))^2-\frac{2}{15} b x^6 \tanh ^{-1}(\tanh (a+b x))^3+\frac{1}{5} x^5 \tanh ^{-1}(\tanh (a+b x))^4+\frac{b^4 x^9}{630} \]

Antiderivative was successfully verified.

[In]

Int[x^4*ArcTanh[Tanh[a + b*x]]^4,x]

[Out]

(b^4*x^9)/630 - (b^3*x^8*ArcTanh[Tanh[a + b*x]])/70 + (2*b^2*x^7*ArcTanh[Tanh[a + b*x]]^2)/35 - (2*b*x^6*ArcTa
nh[Tanh[a + b*x]]^3)/15 + (x^5*ArcTanh[Tanh[a + b*x]]^4)/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 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))^4 \, dx &=\frac{1}{5} x^5 \tanh ^{-1}(\tanh (a+b x))^4-\frac{1}{5} (4 b) \int x^5 \tanh ^{-1}(\tanh (a+b x))^3 \, dx\\ &=-\frac{2}{15} b x^6 \tanh ^{-1}(\tanh (a+b x))^3+\frac{1}{5} x^5 \tanh ^{-1}(\tanh (a+b x))^4+\frac{1}{5} \left (2 b^2\right ) \int x^6 \tanh ^{-1}(\tanh (a+b x))^2 \, dx\\ &=\frac{2}{35} b^2 x^7 \tanh ^{-1}(\tanh (a+b x))^2-\frac{2}{15} b x^6 \tanh ^{-1}(\tanh (a+b x))^3+\frac{1}{5} x^5 \tanh ^{-1}(\tanh (a+b x))^4-\frac{1}{35} \left (4 b^3\right ) \int x^7 \tanh ^{-1}(\tanh (a+b x)) \, dx\\ &=-\frac{1}{70} b^3 x^8 \tanh ^{-1}(\tanh (a+b x))+\frac{2}{35} b^2 x^7 \tanh ^{-1}(\tanh (a+b x))^2-\frac{2}{15} b x^6 \tanh ^{-1}(\tanh (a+b x))^3+\frac{1}{5} x^5 \tanh ^{-1}(\tanh (a+b x))^4+\frac{1}{70} b^4 \int x^8 \, dx\\ &=\frac{b^4 x^9}{630}-\frac{1}{70} b^3 x^8 \tanh ^{-1}(\tanh (a+b x))+\frac{2}{35} b^2 x^7 \tanh ^{-1}(\tanh (a+b x))^2-\frac{2}{15} b x^6 \tanh ^{-1}(\tanh (a+b x))^3+\frac{1}{5} x^5 \tanh ^{-1}(\tanh (a+b x))^4\\ \end{align*}

Mathematica [A]  time = 0.0329736, size = 71, normalized size = 0.89 \[ \frac{1}{630} x^5 \left (-9 b^3 x^3 \tanh ^{-1}(\tanh (a+b x))+36 b^2 x^2 \tanh ^{-1}(\tanh (a+b x))^2-84 b x \tanh ^{-1}(\tanh (a+b x))^3+126 \tanh ^{-1}(\tanh (a+b x))^4+b^4 x^4\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^5*(b^4*x^4 - 9*b^3*x^3*ArcTanh[Tanh[a + b*x]] + 36*b^2*x^2*ArcTanh[Tanh[a + b*x]]^2 - 84*b*x*ArcTanh[Tanh[a
+ b*x]]^3 + 126*ArcTanh[Tanh[a + b*x]]^4))/630

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Maple [A]  time = 0.04, size = 74, normalized size = 0.9 \begin{align*}{\frac{{x}^{5} \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{4}}{5}}-{\frac{4\,b}{5} \left ({\frac{{x}^{6} \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{3}}{6}}-{\frac{b}{2} \left ({\frac{{x}^{7} \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2}}{7}}-{\frac{2\,b}{7} \left ({\frac{{x}^{8}{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }{8}}-{\frac{{x}^{9}b}{72}} \right ) } \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*x^5*arctanh(tanh(b*x+a))^4-4/5*b*(1/6*x^6*arctanh(tanh(b*x+a))^3-1/2*b*(1/7*x^7*arctanh(tanh(b*x+a))^2-2/7
*b*(1/8*x^8*arctanh(tanh(b*x+a))-1/72*x^9*b)))

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Maxima [A]  time = 1.77846, size = 97, normalized size = 1.21 \begin{align*} -\frac{2}{15} \, b x^{6} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{3} + \frac{1}{5} \, x^{5} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{4} + \frac{1}{630} \,{\left (36 \, b x^{7} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{2} +{\left (b^{2} x^{9} - 9 \, b x^{8} \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^4*arctanh(tanh(b*x+a))^4,x, algorithm="maxima")

[Out]

-2/15*b*x^6*arctanh(tanh(b*x + a))^3 + 1/5*x^5*arctanh(tanh(b*x + a))^4 + 1/630*(36*b*x^7*arctanh(tanh(b*x + a
))^2 + (b^2*x^9 - 9*b*x^8*arctanh(tanh(b*x + a)))*b)*b

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Fricas [A]  time = 1.46481, size = 104, normalized size = 1.3 \begin{align*} \frac{1}{9} \, b^{4} x^{9} + \frac{1}{2} \, a b^{3} x^{8} + \frac{6}{7} \, a^{2} b^{2} x^{7} + \frac{2}{3} \, a^{3} b x^{6} + \frac{1}{5} \, a^{4} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*b^4*x^9 + 1/2*a*b^3*x^8 + 6/7*a^2*b^2*x^7 + 2/3*a^3*b*x^6 + 1/5*a^4*x^5

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Sympy [A]  time = 11.2878, size = 78, normalized size = 0.98 \begin{align*} \frac{b^{4} x^{9}}{630} - \frac{b^{3} x^{8} \operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )}}{70} + \frac{2 b^{2} x^{7} \operatorname{atanh}^{2}{\left (\tanh{\left (a + b x \right )} \right )}}{35} - \frac{2 b x^{6} \operatorname{atanh}^{3}{\left (\tanh{\left (a + b x \right )} \right )}}{15} + \frac{x^{5} \operatorname{atanh}^{4}{\left (\tanh{\left (a + b x \right )} \right )}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b**4*x**9/630 - b**3*x**8*atanh(tanh(a + b*x))/70 + 2*b**2*x**7*atanh(tanh(a + b*x))**2/35 - 2*b*x**6*atanh(ta
nh(a + b*x))**3/15 + x**5*atanh(tanh(a + b*x))**4/5

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Giac [A]  time = 1.14128, size = 62, normalized size = 0.78 \begin{align*} \frac{1}{9} \, b^{4} x^{9} + \frac{1}{2} \, a b^{3} x^{8} + \frac{6}{7} \, a^{2} b^{2} x^{7} + \frac{2}{3} \, a^{3} b x^{6} + \frac{1}{5} \, a^{4} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*b^4*x^9 + 1/2*a*b^3*x^8 + 6/7*a^2*b^2*x^7 + 2/3*a^3*b*x^6 + 1/5*a^4*x^5