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

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

[Out]

x^4*arctanh(tanh(b*x+a))^(1+n)/b/(1+n)-4*x^3*arctanh(tanh(b*x+a))^(2+n)/b^2/(1+n)/(2+n)+12*x^2*arctanh(tanh(b*
x+a))^(3+n)/b^3/(3+n)/(n^2+3*n+2)-24*x*arctanh(tanh(b*x+a))^(4+n)/b^4/(n^2+5*n+4)/(n^2+5*n+6)+24*arctanh(tanh(
b*x+a))^(5+n)/b^5/(n^2+7*n+12)/(n^3+8*n^2+17*n+10)

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Rubi [A]  time = 0.13, antiderivative size = 165, normalized size of antiderivative = 1.00, 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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 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]))

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*}

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Mathematica [A]  time = 0.11, 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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fricas [B]  time = 0.40, size = 374, normalized size = 2.27 \[ -\frac {{\left (24 \, a^{4} b n x - {\left (b^{5} n^{4} + 10 \, b^{5} n^{3} + 35 \, b^{5} n^{2} + 50 \, b^{5} n + 24 \, b^{5}\right )} x^{5} - 24 \, a^{5} - {\left (a b^{4} n^{4} + 6 \, a b^{4} n^{3} + 11 \, a b^{4} n^{2} + 6 \, a b^{4} n\right )} x^{4} + 4 \, {\left (a^{2} b^{3} n^{3} + 3 \, a^{2} b^{3} n^{2} + 2 \, a^{2} b^{3} n\right )} x^{3} - 12 \, {\left (a^{3} b^{2} n^{2} + a^{3} b^{2} n\right )} x^{2}\right )} \cosh \left (n \log \left (b x + a\right )\right ) + {\left (24 \, a^{4} b n x - {\left (b^{5} n^{4} + 10 \, b^{5} n^{3} + 35 \, b^{5} n^{2} + 50 \, b^{5} n + 24 \, b^{5}\right )} x^{5} - 24 \, a^{5} - {\left (a b^{4} n^{4} + 6 \, a b^{4} n^{3} + 11 \, a b^{4} n^{2} + 6 \, a b^{4} n\right )} x^{4} + 4 \, {\left (a^{2} b^{3} n^{3} + 3 \, a^{2} b^{3} n^{2} + 2 \, a^{2} b^{3} n\right )} x^{3} - 12 \, {\left (a^{3} b^{2} n^{2} + a^{3} b^{2} n\right )} x^{2}\right )} \sinh \left (n \log \left (b x + a\right )\right )}{b^{5} n^{5} + 15 \, b^{5} n^{4} + 85 \, b^{5} n^{3} + 225 \, b^{5} n^{2} + 274 \, b^{5} n + 120 \, b^{5}} \]

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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giac [B]  time = 0.14, size = 332, normalized size = 2.01 \[ \frac {{\left (b x + a\right )}^{n} b^{5} n^{4} x^{5} + {\left (b x + a\right )}^{n} a b^{4} n^{4} x^{4} + 10 \, {\left (b x + a\right )}^{n} b^{5} n^{3} x^{5} + 6 \, {\left (b x + a\right )}^{n} a b^{4} n^{3} x^{4} + 35 \, {\left (b x + a\right )}^{n} b^{5} n^{2} x^{5} - 4 \, {\left (b x + a\right )}^{n} a^{2} b^{3} n^{3} x^{3} + 11 \, {\left (b x + a\right )}^{n} a b^{4} n^{2} x^{4} + 50 \, {\left (b x + a\right )}^{n} b^{5} n x^{5} - 12 \, {\left (b x + a\right )}^{n} a^{2} b^{3} n^{2} x^{3} + 6 \, {\left (b x + a\right )}^{n} a b^{4} n x^{4} + 24 \, {\left (b x + a\right )}^{n} b^{5} x^{5} + 12 \, {\left (b x + a\right )}^{n} a^{3} b^{2} n^{2} x^{2} - 8 \, {\left (b x + a\right )}^{n} a^{2} b^{3} n x^{3} + 12 \, {\left (b x + a\right )}^{n} a^{3} b^{2} n x^{2} - 24 \, {\left (b x + a\right )}^{n} a^{4} b n x + 24 \, {\left (b x + a\right )}^{n} a^{5}}{b^{5} n^{5} + 15 \, b^{5} n^{4} + 85 \, b^{5} n^{3} + 225 \, b^{5} n^{2} + 274 \, b^{5} n + 120 \, b^{5}} \]

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)

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maple [B]  time = 0.14, size = 654, normalized size = 3.96 \[ \frac {x^{5} {\mathrm e}^{n \ln \left (\arctanh \left (\tanh \left (b x +a \right )\right )\right )}}{5+n}+\frac {n \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right ) x^{4} {\mathrm e}^{n \ln \left (\arctanh \left (\tanh \left (b x +a \right )\right )\right )}}{b \left (n^{2}+9 n +20\right )}-\frac {4 n \left (a^{2}+2 a \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )+\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2}\right ) x^{3} {\mathrm e}^{n \ln \left (\arctanh \left (\tanh \left (b x +a \right )\right )\right )}}{b^{2} \left (n^{3}+12 n^{2}+47 n +60\right )}+\frac {24 \,{\mathrm e}^{n \ln \left (\arctanh \left (\tanh \left (b x +a \right )\right )\right )} a^{5}}{b^{5} \left (n^{3}+12 n^{2}+47 n +60\right ) \left (n^{2}+3 n +2\right )}+\frac {120 \,{\mathrm e}^{n \ln \left (\arctanh \left (\tanh \left (b x +a \right )\right )\right )} a^{4} \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )}{b^{5} \left (n^{3}+12 n^{2}+47 n +60\right ) \left (n^{2}+3 n +2\right )}+\frac {240 \,{\mathrm e}^{n \ln \left (\arctanh \left (\tanh \left (b x +a \right )\right )\right )} a^{3} \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2}}{b^{5} \left (n^{3}+12 n^{2}+47 n +60\right ) \left (n^{2}+3 n +2\right )}+\frac {240 \,{\mathrm e}^{n \ln \left (\arctanh \left (\tanh \left (b x +a \right )\right )\right )} a^{2} \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{3}}{b^{5} \left (n^{3}+12 n^{2}+47 n +60\right ) \left (n^{2}+3 n +2\right )}+\frac {120 \,{\mathrm e}^{n \ln \left (\arctanh \left (\tanh \left (b x +a \right )\right )\right )} a \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{4}}{b^{5} \left (n^{3}+12 n^{2}+47 n +60\right ) \left (n^{2}+3 n +2\right )}+\frac {24 \,{\mathrm e}^{n \ln \left (\arctanh \left (\tanh \left (b x +a \right )\right )\right )} \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{5}}{b^{5} \left (n^{3}+12 n^{2}+47 n +60\right ) \left (n^{2}+3 n +2\right )}-\frac {24 \left (a^{2}+2 a \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )+\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2}\right )^{2} n x \,{\mathrm e}^{n \ln \left (\arctanh \left (\tanh \left (b x +a \right )\right )\right )}}{b^{4} \left (n^{3}+12 n^{2}+47 n +60\right ) \left (n^{2}+3 n +2\right )}+\frac {12 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right ) \left (a^{2}+2 a \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )+\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2}\right ) n \,x^{2} {\mathrm e}^{n \ln \left (\arctanh \left (\tanh \left (b x +a \right )\right )\right )}}{b^{3} \left (2+n \right ) \left (n^{3}+12 n^{2}+47 n +60\right )} \]

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 = 0.53, size = 139, normalized size = 0.84 \[ \frac {{\left ({\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{5} x^{5} + {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a b^{4} x^{4} - 4 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{2} b^{3} x^{3} + 12 \, {\left (n^{2} + n\right )} a^{3} b^{2} x^{2} - 24 \, a^{4} b n x + 24 \, a^{5}\right )} {\left (b x + a\right )}^{n}}{{\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{5}} \]

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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mupad [B]  time = 1.67, size = 546, normalized size = 3.31 \[ -{\left (\frac {\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{2}-\frac {\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{2}\right )}^n\,\left (\frac {3\,{\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^5}{4\,b^5\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}-\frac {x^5\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}{n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120}+\frac {3\,n\,x\,{\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^4}{2\,b^4\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}+\frac {n\,x^4\,\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )\,\left (n^3+6\,n^2+11\,n+6\right )}{2\,b\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}+\frac {3\,n\,x^2\,\left (n+1\right )\,{\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^3}{2\,b^3\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}+\frac {n\,x^3\,{\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^2\,\left (n^2+3\,n+2\right )}{b^2\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*atanh(tanh(a + b*x))^n,x)

[Out]

-(log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1))/2 - log(2/(exp(2*a)*exp(2*b*x) + 1))/2)^n*((3*(log(2/
(exp(2*a)*exp(2*b*x) + 1)) - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1)) + 2*b*x)^5)/(4*b^5*(274*n
+ 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120)) - (x^5*(50*n + 35*n^2 + 10*n^3 + n^4 + 24))/(274*n + 225*n^2 + 85*n^
3 + 15*n^4 + n^5 + 120) + (3*n*x*(log(2/(exp(2*a)*exp(2*b*x) + 1)) - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp
(2*b*x) + 1)) + 2*b*x)^4)/(2*b^4*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120)) + (n*x^4*(log(2/(exp(2*a)*ex
p(2*b*x) + 1)) - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1)) + 2*b*x)*(11*n + 6*n^2 + n^3 + 6))/(2*
b*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120)) + (3*n*x^2*(n + 1)*(log(2/(exp(2*a)*exp(2*b*x) + 1)) - log(
(2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1)) + 2*b*x)^3)/(2*b^3*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5
+ 120)) + (n*x^3*(log(2/(exp(2*a)*exp(2*b*x) + 1)) - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1)) +
2*b*x)^2*(3*n + n^2 + 2))/(b^2*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120)))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x**5*atanh(tanh(a))**n/5, Eq(b, 0)), (-x**4/(4*b*atanh(tanh(a + b*x))**4) - x**3/(3*b**2*atanh(tanh
(a + b*x))**3) - x**2/(2*b**3*atanh(tanh(a + b*x))**2) - x/(b**4*atanh(tanh(a + b*x))) + log(atanh(tanh(a + b*
x)))/b**5, Eq(n, -5)), (Integral(x**4/atanh(tanh(a + b*x))**4, x), Eq(n, -4)), (Integral(x**4/atanh(tanh(a + b
*x))**3, x), Eq(n, -3)), (Integral(x**4/atanh(tanh(a + b*x))**2, x), Eq(n, -2)), (Integral(x**4/atanh(tanh(a +
b*x)), x), Eq(n, -1)), (b**4*n**4*x**4*atanh(tanh(a + b*x))*atanh(tanh(a + b*x))**n/(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) + 14*b**4*n**3*x**4*atanh(tanh(a + b*x))*atanh(tanh(a
+ b*x))**n/(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) + 71*b**4*n**2*x
**4*atanh(tanh(a + b*x))*atanh(tanh(a + b*x))**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 27
4*b**5*n + 120*b**5) + 154*b**4*n*x**4*atanh(tanh(a + b*x))*atanh(tanh(a + b*x))**n/(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) + 120*b**4*x**4*atanh(tanh(a + b*x))*atanh(tanh(a + b*
x))**n/(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) - 4*b**3*n**3*x**3*at
anh(tanh(a + b*x))**2*atanh(tanh(a + b*x))**n/(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) - 48*b**3*n**2*x**3*atanh(tanh(a + b*x))**2*atanh(tanh(a + b*x))**n/(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) - 188*b**3*n*x**3*atanh(tanh(a + b*x))**2*atanh(tanh
(a + b*x))**n/(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) - 240*b**3*x**
3*atanh(tanh(a + b*x))**2*atanh(tanh(a + b*x))**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 2
74*b**5*n + 120*b**5) + 12*b**2*n**2*x**2*atanh(tanh(a + b*x))**3*atanh(tanh(a + b*x))**n/(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) + 108*b**2*n*x**2*atanh(tanh(a + b*x))**3*atanh(
tanh(a + b*x))**n/(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) + 240*b**2
*x**2*atanh(tanh(a + b*x))**3*atanh(tanh(a + b*x))**n/(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) - 24*b*n*x*atanh(tanh(a + b*x))**4*atanh(tanh(a + b*x))**n/(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) - 120*b*x*atanh(tanh(a + b*x))**4*atanh(tanh(a + b*x)
)**n/(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) + 24*atanh(tanh(a + b*x
))**5*atanh(tanh(a + b*x))**n/(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
), True))

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