3.66 \(\int x^6 \tanh ^{-1}(\tanh (a+b x))^4 \, dx\)

Optimal. Leaf size=80 \[ -\frac{1}{210} b^3 x^{10} \tanh ^{-1}(\tanh (a+b x))+\frac{1}{42} b^2 x^9 \tanh ^{-1}(\tanh (a+b x))^2-\frac{1}{14} b x^8 \tanh ^{-1}(\tanh (a+b x))^3+\frac{1}{7} x^7 \tanh ^{-1}(\tanh (a+b x))^4+\frac{b^4 x^{11}}{2310} \]

[Out]

(b^4*x^11)/2310 - (b^3*x^10*ArcTanh[Tanh[a + b*x]])/210 + (b^2*x^9*ArcTanh[Tanh[a + b*x]]^2)/42 - (b*x^8*ArcTa
nh[Tanh[a + b*x]]^3)/14 + (x^7*ArcTanh[Tanh[a + b*x]]^4)/7

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Rubi [A]  time = 0.0632376, 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} \[ -\frac{1}{210} b^3 x^{10} \tanh ^{-1}(\tanh (a+b x))+\frac{1}{42} b^2 x^9 \tanh ^{-1}(\tanh (a+b x))^2-\frac{1}{14} b x^8 \tanh ^{-1}(\tanh (a+b x))^3+\frac{1}{7} x^7 \tanh ^{-1}(\tanh (a+b x))^4+\frac{b^4 x^{11}}{2310} \]

Antiderivative was successfully verified.

[In]

Int[x^6*ArcTanh[Tanh[a + b*x]]^4,x]

[Out]

(b^4*x^11)/2310 - (b^3*x^10*ArcTanh[Tanh[a + b*x]])/210 + (b^2*x^9*ArcTanh[Tanh[a + b*x]]^2)/42 - (b*x^8*ArcTa
nh[Tanh[a + b*x]]^3)/14 + (x^7*ArcTanh[Tanh[a + b*x]]^4)/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^6 \tanh ^{-1}(\tanh (a+b x))^4 \, dx &=\frac{1}{7} x^7 \tanh ^{-1}(\tanh (a+b x))^4-\frac{1}{7} (4 b) \int x^7 \tanh ^{-1}(\tanh (a+b x))^3 \, dx\\ &=-\frac{1}{14} b x^8 \tanh ^{-1}(\tanh (a+b x))^3+\frac{1}{7} x^7 \tanh ^{-1}(\tanh (a+b x))^4+\frac{1}{14} \left (3 b^2\right ) \int x^8 \tanh ^{-1}(\tanh (a+b x))^2 \, dx\\ &=\frac{1}{42} b^2 x^9 \tanh ^{-1}(\tanh (a+b x))^2-\frac{1}{14} b x^8 \tanh ^{-1}(\tanh (a+b x))^3+\frac{1}{7} x^7 \tanh ^{-1}(\tanh (a+b x))^4-\frac{1}{21} b^3 \int x^9 \tanh ^{-1}(\tanh (a+b x)) \, dx\\ &=-\frac{1}{210} b^3 x^{10} \tanh ^{-1}(\tanh (a+b x))+\frac{1}{42} b^2 x^9 \tanh ^{-1}(\tanh (a+b x))^2-\frac{1}{14} b x^8 \tanh ^{-1}(\tanh (a+b x))^3+\frac{1}{7} x^7 \tanh ^{-1}(\tanh (a+b x))^4+\frac{1}{210} b^4 \int x^{10} \, dx\\ &=\frac{b^4 x^{11}}{2310}-\frac{1}{210} b^3 x^{10} \tanh ^{-1}(\tanh (a+b x))+\frac{1}{42} b^2 x^9 \tanh ^{-1}(\tanh (a+b x))^2-\frac{1}{14} b x^8 \tanh ^{-1}(\tanh (a+b x))^3+\frac{1}{7} x^7 \tanh ^{-1}(\tanh (a+b x))^4\\ \end{align*}

Mathematica [A]  time = 0.0622043, size = 71, normalized size = 0.89 \[ \frac{x^7 \left (-11 b^3 x^3 \tanh ^{-1}(\tanh (a+b x))+55 b^2 x^2 \tanh ^{-1}(\tanh (a+b x))^2-165 b x \tanh ^{-1}(\tanh (a+b x))^3+330 \tanh ^{-1}(\tanh (a+b x))^4+b^4 x^4\right )}{2310} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6*ArcTanh[Tanh[a + b*x]]^4,x]

[Out]

(x^7*(b^4*x^4 - 11*b^3*x^3*ArcTanh[Tanh[a + b*x]] + 55*b^2*x^2*ArcTanh[Tanh[a + b*x]]^2 - 165*b*x*ArcTanh[Tanh
[a + b*x]]^3 + 330*ArcTanh[Tanh[a + b*x]]^4))/2310

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Maple [A]  time = 0.04, size = 74, normalized size = 0.9 \begin{align*}{\frac{{x}^{7} \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{4}}{7}}-{\frac{4\,b}{7} \left ({\frac{{x}^{8} \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{3}}{8}}-{\frac{3\,b}{8} \left ({\frac{{x}^{9} \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2}}{9}}-{\frac{2\,b}{9} \left ({\frac{{x}^{10}{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }{10}}-{\frac{{x}^{11}b}{110}} \right ) } \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*arctanh(tanh(b*x+a))^4,x)

[Out]

1/7*x^7*arctanh(tanh(b*x+a))^4-4/7*b*(1/8*x^8*arctanh(tanh(b*x+a))^3-3/8*b*(1/9*x^9*arctanh(tanh(b*x+a))^2-2/9
*b*(1/10*x^10*arctanh(tanh(b*x+a))-1/110*x^11*b)))

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Maxima [A]  time = 1.73779, size = 97, normalized size = 1.21 \begin{align*} -\frac{1}{14} \, b x^{8} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{3} + \frac{1}{7} \, x^{7} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{4} + \frac{1}{2310} \,{\left (55 \, b x^{9} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{2} +{\left (b^{2} x^{11} - 11 \, b x^{10} \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^6*arctanh(tanh(b*x+a))^4,x, algorithm="maxima")

[Out]

-1/14*b*x^8*arctanh(tanh(b*x + a))^3 + 1/7*x^7*arctanh(tanh(b*x + a))^4 + 1/2310*(55*b*x^9*arctanh(tanh(b*x +
a))^2 + (b^2*x^11 - 11*b*x^10*arctanh(tanh(b*x + a)))*b)*b

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Fricas [A]  time = 1.55855, size = 108, normalized size = 1.35 \begin{align*} \frac{1}{11} \, b^{4} x^{11} + \frac{2}{5} \, a b^{3} x^{10} + \frac{2}{3} \, a^{2} b^{2} x^{9} + \frac{1}{2} \, a^{3} b x^{8} + \frac{1}{7} \, a^{4} x^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*arctanh(tanh(b*x+a))^4,x, algorithm="fricas")

[Out]

1/11*b^4*x^11 + 2/5*a*b^3*x^10 + 2/3*a^2*b^2*x^9 + 1/2*a^3*b*x^8 + 1/7*a^4*x^7

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Sympy [A]  time = 32.5487, size = 75, normalized size = 0.94 \begin{align*} \frac{b^{4} x^{11}}{2310} - \frac{b^{3} x^{10} \operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )}}{210} + \frac{b^{2} x^{9} \operatorname{atanh}^{2}{\left (\tanh{\left (a + b x \right )} \right )}}{42} - \frac{b x^{8} \operatorname{atanh}^{3}{\left (\tanh{\left (a + b x \right )} \right )}}{14} + \frac{x^{7} \operatorname{atanh}^{4}{\left (\tanh{\left (a + b x \right )} \right )}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*atanh(tanh(b*x+a))**4,x)

[Out]

b**4*x**11/2310 - b**3*x**10*atanh(tanh(a + b*x))/210 + b**2*x**9*atanh(tanh(a + b*x))**2/42 - b*x**8*atanh(ta
nh(a + b*x))**3/14 + x**7*atanh(tanh(a + b*x))**4/7

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Giac [A]  time = 1.12612, size = 62, normalized size = 0.78 \begin{align*} \frac{1}{11} \, b^{4} x^{11} + \frac{2}{5} \, a b^{3} x^{10} + \frac{2}{3} \, a^{2} b^{2} x^{9} + \frac{1}{2} \, a^{3} b x^{8} + \frac{1}{7} \, a^{4} x^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*arctanh(tanh(b*x+a))^4,x, algorithm="giac")

[Out]

1/11*b^4*x^11 + 2/5*a*b^3*x^10 + 2/3*a^2*b^2*x^9 + 1/2*a^3*b*x^8 + 1/7*a^4*x^7