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

Optimal. Leaf size=165 \[ -\frac{4 x^3 \tanh ^{-1}(\tanh (a+b x))^{n+2}}{b^2 (n+1) (n+2)}+\frac{12 x^2 \tanh ^{-1}(\tanh (a+b x))^{n+3}}{b^3 (n+1) (n+2) (n+3)}-\frac{24 x \tanh ^{-1}(\tanh (a+b x))^{n+4}}{b^4 (n+1) (n+2) (n+3) (n+4)}+\frac{24 \tanh ^{-1}(\tanh (a+b x))^{n+5}}{b^5 (n+1) (n+2) (n+3) (n+4) (n+5)}+\frac{x^4 \tanh ^{-1}(\tanh (a+b x))^{n+1}}{b (n+1)} \]

[Out]

(x^4*ArcTanh[Tanh[a + b*x]]^(1 + n))/(b*(1 + n)) - (4*x^3*ArcTanh[Tanh[a + b*x]]^(2 + n))/(b^2*(1 + n)*(2 + n)
) + (12*x^2*ArcTanh[Tanh[a + b*x]]^(3 + n))/(b^3*(1 + n)*(2 + n)*(3 + n)) - (24*x*ArcTanh[Tanh[a + b*x]]^(4 +
n))/(b^4*(1 + n)*(2 + n)*(3 + n)*(4 + n)) + (24*ArcTanh[Tanh[a + b*x]]^(5 + n))/(b^5*(1 + n)*(2 + n)*(3 + n)*(
4 + n)*(5 + n))

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Rubi [A]  time = 0.133738, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2168, 2157, 30} \[ -\frac{4 x^3 \tanh ^{-1}(\tanh (a+b x))^{n+2}}{b^2 (n+1) (n+2)}+\frac{12 x^2 \tanh ^{-1}(\tanh (a+b x))^{n+3}}{b^3 (n+1) (n+2) (n+3)}-\frac{24 x \tanh ^{-1}(\tanh (a+b x))^{n+4}}{b^4 (n+1) (n+2) (n+3) (n+4)}+\frac{24 \tanh ^{-1}(\tanh (a+b x))^{n+5}}{b^5 (n+1) (n+2) (n+3) (n+4) (n+5)}+\frac{x^4 \tanh ^{-1}(\tanh (a+b x))^{n+1}}{b (n+1)} \]

Antiderivative was successfully verified.

[In]

Int[x^4*ArcTanh[Tanh[a + b*x]]^n,x]

[Out]

(x^4*ArcTanh[Tanh[a + b*x]]^(1 + n))/(b*(1 + n)) - (4*x^3*ArcTanh[Tanh[a + b*x]]^(2 + n))/(b^2*(1 + n)*(2 + n)
) + (12*x^2*ArcTanh[Tanh[a + b*x]]^(3 + n))/(b^3*(1 + n)*(2 + n)*(3 + n)) - (24*x*ArcTanh[Tanh[a + b*x]]^(4 +
n))/(b^4*(1 + n)*(2 + n)*(3 + n)*(4 + n)) + (24*ArcTanh[Tanh[a + b*x]]^(5 + n))/(b^5*(1 + n)*(2 + n)*(3 + n)*(
4 + n)*(5 + n))

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))^n \, dx &=\frac{x^4 \tanh ^{-1}(\tanh (a+b x))^{1+n}}{b (1+n)}-\frac{4 \int x^3 \tanh ^{-1}(\tanh (a+b x))^{1+n} \, dx}{b (1+n)}\\ &=\frac{x^4 \tanh ^{-1}(\tanh (a+b x))^{1+n}}{b (1+n)}-\frac{4 x^3 \tanh ^{-1}(\tanh (a+b x))^{2+n}}{b^2 (1+n) (2+n)}+\frac{12 \int x^2 \tanh ^{-1}(\tanh (a+b x))^{2+n} \, dx}{b^2 (1+n) (2+n)}\\ &=\frac{x^4 \tanh ^{-1}(\tanh (a+b x))^{1+n}}{b (1+n)}-\frac{4 x^3 \tanh ^{-1}(\tanh (a+b x))^{2+n}}{b^2 (1+n) (2+n)}+\frac{12 x^2 \tanh ^{-1}(\tanh (a+b x))^{3+n}}{b^3 (1+n) (2+n) (3+n)}-\frac{24 \int x \tanh ^{-1}(\tanh (a+b x))^{3+n} \, dx}{b^3 (1+n) (2+n) (3+n)}\\ &=\frac{x^4 \tanh ^{-1}(\tanh (a+b x))^{1+n}}{b (1+n)}-\frac{4 x^3 \tanh ^{-1}(\tanh (a+b x))^{2+n}}{b^2 (1+n) (2+n)}+\frac{12 x^2 \tanh ^{-1}(\tanh (a+b x))^{3+n}}{b^3 (1+n) (2+n) (3+n)}-\frac{24 x \tanh ^{-1}(\tanh (a+b x))^{4+n}}{b^4 (1+n) (2+n) (3+n) (4+n)}+\frac{24 \int \tanh ^{-1}(\tanh (a+b x))^{4+n} \, dx}{b^4 (1+n) (2+n) (3+n) (4+n)}\\ &=\frac{x^4 \tanh ^{-1}(\tanh (a+b x))^{1+n}}{b (1+n)}-\frac{4 x^3 \tanh ^{-1}(\tanh (a+b x))^{2+n}}{b^2 (1+n) (2+n)}+\frac{12 x^2 \tanh ^{-1}(\tanh (a+b x))^{3+n}}{b^3 (1+n) (2+n) (3+n)}-\frac{24 x \tanh ^{-1}(\tanh (a+b x))^{4+n}}{b^4 (1+n) (2+n) (3+n) (4+n)}+\frac{24 \operatorname{Subst}\left (\int x^{4+n} \, dx,x,\tanh ^{-1}(\tanh (a+b x))\right )}{b^5 (1+n) (2+n) (3+n) (4+n)}\\ &=\frac{x^4 \tanh ^{-1}(\tanh (a+b x))^{1+n}}{b (1+n)}-\frac{4 x^3 \tanh ^{-1}(\tanh (a+b x))^{2+n}}{b^2 (1+n) (2+n)}+\frac{12 x^2 \tanh ^{-1}(\tanh (a+b x))^{3+n}}{b^3 (1+n) (2+n) (3+n)}-\frac{24 x \tanh ^{-1}(\tanh (a+b x))^{4+n}}{b^4 (1+n) (2+n) (3+n) (4+n)}+\frac{24 \tanh ^{-1}(\tanh (a+b x))^{5+n}}{b^5 (1+n) (2+n) (3+n) (4+n) (5+n)}\\ \end{align*}

Mathematica [A]  time = 0.10441, size = 146, normalized size = 0.88 \[ \frac{\tanh ^{-1}(\tanh (a+b x))^{n+1} \left (-4 b^3 \left (n^3+12 n^2+47 n+60\right ) x^3 \tanh ^{-1}(\tanh (a+b x))+12 b^2 \left (n^2+9 n+20\right ) x^2 \tanh ^{-1}(\tanh (a+b x))^2-24 b (n+5) x \tanh ^{-1}(\tanh (a+b x))^3+24 \tanh ^{-1}(\tanh (a+b x))^4+b^4 \left (n^4+14 n^3+71 n^2+154 n+120\right ) x^4\right )}{b^5 (n+1) (n+2) (n+3) (n+4) (n+5)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(ArcTanh[Tanh[a + b*x]]^(1 + n)*(b^4*(120 + 154*n + 71*n^2 + 14*n^3 + n^4)*x^4 - 4*b^3*(60 + 47*n + 12*n^2 + n
^3)*x^3*ArcTanh[Tanh[a + b*x]] + 12*b^2*(20 + 9*n + n^2)*x^2*ArcTanh[Tanh[a + b*x]]^2 - 24*b*(5 + n)*x*ArcTanh
[Tanh[a + b*x]]^3 + 24*ArcTanh[Tanh[a + b*x]]^4))/(b^5*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n))

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Maple [B]  time = 0.049, size = 654, normalized size = 4. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(5+n)*x^5*exp(n*ln(arctanh(tanh(b*x+a))))+n/b*(arctanh(tanh(b*x+a))-b*x)/(n^2+9*n+20)*x^4*exp(n*ln(arctanh(t
anh(b*x+a))))-4*n*(a^2+2*a*(arctanh(tanh(b*x+a))-b*x-a)+(arctanh(tanh(b*x+a))-b*x-a)^2)/b^2/(n^3+12*n^2+47*n+6
0)*x^3*exp(n*ln(arctanh(tanh(b*x+a))))+24/b^5/(n^3+12*n^2+47*n+60)/(n^2+3*n+2)*exp(n*ln(arctanh(tanh(b*x+a))))
*a^5+120/b^5/(n^3+12*n^2+47*n+60)/(n^2+3*n+2)*exp(n*ln(arctanh(tanh(b*x+a))))*a^4*(arctanh(tanh(b*x+a))-b*x-a)
+240/b^5/(n^3+12*n^2+47*n+60)/(n^2+3*n+2)*exp(n*ln(arctanh(tanh(b*x+a))))*a^3*(arctanh(tanh(b*x+a))-b*x-a)^2+2
40/b^5/(n^3+12*n^2+47*n+60)/(n^2+3*n+2)*exp(n*ln(arctanh(tanh(b*x+a))))*a^2*(arctanh(tanh(b*x+a))-b*x-a)^3+120
/b^5/(n^3+12*n^2+47*n+60)/(n^2+3*n+2)*exp(n*ln(arctanh(tanh(b*x+a))))*a*(arctanh(tanh(b*x+a))-b*x-a)^4+24/b^5/
(n^3+12*n^2+47*n+60)/(n^2+3*n+2)*exp(n*ln(arctanh(tanh(b*x+a))))*(arctanh(tanh(b*x+a))-b*x-a)^5-24*(a^2+2*a*(a
rctanh(tanh(b*x+a))-b*x-a)+(arctanh(tanh(b*x+a))-b*x-a)^2)^2*n/b^4/(n^3+12*n^2+47*n+60)/(n^2+3*n+2)*x*exp(n*ln
(arctanh(tanh(b*x+a))))+12/b^3*(arctanh(tanh(b*x+a))-b*x)*(a^2+2*a*(arctanh(tanh(b*x+a))-b*x-a)+(arctanh(tanh(
b*x+a))-b*x-a)^2)*n/(2+n)/(n^3+12*n^2+47*n+60)*x^2*exp(n*ln(arctanh(tanh(b*x+a))))

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Maxima [A]  time = 1.79333, size = 188, normalized size = 1.14 \begin{align*} \frac{{\left ({\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{5} x^{5} +{\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a b^{4} x^{4} - 4 \,{\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{2} b^{3} x^{3} + 12 \,{\left (n^{2} + n\right )} a^{3} b^{2} x^{2} - 24 \, a^{4} b n x + 24 \, a^{5}\right )}{\left (b x + a\right )}^{n}}{{\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*b^5*x^5 + (n^4 + 6*n^3 + 11*n^2 + 6*n)*a*b^4*x^4 - 4*(n^3 + 3*n^2 + 2*n)*
a^2*b^3*x^3 + 12*(n^2 + n)*a^3*b^2*x^2 - 24*a^4*b*n*x + 24*a^5)*(b*x + a)^n/((n^5 + 15*n^4 + 85*n^3 + 225*n^2
+ 274*n + 120)*b^5)

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Fricas [B]  time = 2.18292, size = 788, normalized size = 4.78 \begin{align*} -\frac{{\left (24 \, a^{4} b n x -{\left (b^{5} n^{4} + 10 \, b^{5} n^{3} + 35 \, b^{5} n^{2} + 50 \, b^{5} n + 24 \, b^{5}\right )} x^{5} - 24 \, a^{5} -{\left (a b^{4} n^{4} + 6 \, a b^{4} n^{3} + 11 \, a b^{4} n^{2} + 6 \, a b^{4} n\right )} x^{4} + 4 \,{\left (a^{2} b^{3} n^{3} + 3 \, a^{2} b^{3} n^{2} + 2 \, a^{2} b^{3} n\right )} x^{3} - 12 \,{\left (a^{3} b^{2} n^{2} + a^{3} b^{2} n\right )} x^{2}\right )} \cosh \left (n \log \left (b x + a\right )\right ) +{\left (24 \, a^{4} b n x -{\left (b^{5} n^{4} + 10 \, b^{5} n^{3} + 35 \, b^{5} n^{2} + 50 \, b^{5} n + 24 \, b^{5}\right )} x^{5} - 24 \, a^{5} -{\left (a b^{4} n^{4} + 6 \, a b^{4} n^{3} + 11 \, a b^{4} n^{2} + 6 \, a b^{4} n\right )} x^{4} + 4 \,{\left (a^{2} b^{3} n^{3} + 3 \, a^{2} b^{3} n^{2} + 2 \, a^{2} b^{3} n\right )} x^{3} - 12 \,{\left (a^{3} b^{2} n^{2} + a^{3} b^{2} n\right )} x^{2}\right )} \sinh \left (n \log \left (b x + a\right )\right )}{b^{5} n^{5} + 15 \, b^{5} n^{4} + 85 \, b^{5} n^{3} + 225 \, b^{5} n^{2} + 274 \, b^{5} n + 120 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((24*a^4*b*n*x - (b^5*n^4 + 10*b^5*n^3 + 35*b^5*n^2 + 50*b^5*n + 24*b^5)*x^5 - 24*a^5 - (a*b^4*n^4 + 6*a*b^4*
n^3 + 11*a*b^4*n^2 + 6*a*b^4*n)*x^4 + 4*(a^2*b^3*n^3 + 3*a^2*b^3*n^2 + 2*a^2*b^3*n)*x^3 - 12*(a^3*b^2*n^2 + a^
3*b^2*n)*x^2)*cosh(n*log(b*x + a)) + (24*a^4*b*n*x - (b^5*n^4 + 10*b^5*n^3 + 35*b^5*n^2 + 50*b^5*n + 24*b^5)*x
^5 - 24*a^5 - (a*b^4*n^4 + 6*a*b^4*n^3 + 11*a*b^4*n^2 + 6*a*b^4*n)*x^4 + 4*(a^2*b^3*n^3 + 3*a^2*b^3*n^2 + 2*a^
2*b^3*n)*x^3 - 12*(a^3*b^2*n^2 + a^3*b^2*n)*x^2)*sinh(n*log(b*x + a)))/(b^5*n^5 + 15*b^5*n^4 + 85*b^5*n^3 + 22
5*b^5*n^2 + 274*b^5*n + 120*b^5)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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Giac [B]  time = 1.17808, size = 448, normalized size = 2.72 \begin{align*} \frac{{\left (b x + a\right )}^{n} b^{5} n^{4} x^{5} +{\left (b x + a\right )}^{n} a b^{4} n^{4} x^{4} + 10 \,{\left (b x + a\right )}^{n} b^{5} n^{3} x^{5} + 6 \,{\left (b x + a\right )}^{n} a b^{4} n^{3} x^{4} + 35 \,{\left (b x + a\right )}^{n} b^{5} n^{2} x^{5} - 4 \,{\left (b x + a\right )}^{n} a^{2} b^{3} n^{3} x^{3} + 11 \,{\left (b x + a\right )}^{n} a b^{4} n^{2} x^{4} + 50 \,{\left (b x + a\right )}^{n} b^{5} n x^{5} - 12 \,{\left (b x + a\right )}^{n} a^{2} b^{3} n^{2} x^{3} + 6 \,{\left (b x + a\right )}^{n} a b^{4} n x^{4} + 24 \,{\left (b x + a\right )}^{n} b^{5} x^{5} + 12 \,{\left (b x + a\right )}^{n} a^{3} b^{2} n^{2} x^{2} - 8 \,{\left (b x + a\right )}^{n} a^{2} b^{3} n x^{3} + 12 \,{\left (b x + a\right )}^{n} a^{3} b^{2} n x^{2} - 24 \,{\left (b x + a\right )}^{n} a^{4} b n x + 24 \,{\left (b x + a\right )}^{n} a^{5}}{b^{5} n^{5} + 15 \, b^{5} n^{4} + 85 \, b^{5} n^{3} + 225 \, b^{5} n^{2} + 274 \, b^{5} n + 120 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b*x + a)^n*b^5*n^4*x^5 + (b*x + a)^n*a*b^4*n^4*x^4 + 10*(b*x + a)^n*b^5*n^3*x^5 + 6*(b*x + a)^n*a*b^4*n^3*x^
4 + 35*(b*x + a)^n*b^5*n^2*x^5 - 4*(b*x + a)^n*a^2*b^3*n^3*x^3 + 11*(b*x + a)^n*a*b^4*n^2*x^4 + 50*(b*x + a)^n
*b^5*n*x^5 - 12*(b*x + a)^n*a^2*b^3*n^2*x^3 + 6*(b*x + a)^n*a*b^4*n*x^4 + 24*(b*x + a)^n*b^5*x^5 + 12*(b*x + a
)^n*a^3*b^2*n^2*x^2 - 8*(b*x + a)^n*a^2*b^3*n*x^3 + 12*(b*x + a)^n*a^3*b^2*n*x^2 - 24*(b*x + a)^n*a^4*b*n*x +
24*(b*x + a)^n*a^5)/(b^5*n^5 + 15*b^5*n^4 + 85*b^5*n^3 + 225*b^5*n^2 + 274*b^5*n + 120*b^5)