3.82 \(\int \frac {\tanh ^{-1}(\tanh (a+b x))^4}{x^{10}} \, dx\)

Optimal. Leaf size=80 \[ -\frac {b^3 \tanh ^{-1}(\tanh (a+b x))}{126 x^6}-\frac {b^2 \tanh ^{-1}(\tanh (a+b x))^2}{42 x^7}-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{9 x^9}-\frac {b \tanh ^{-1}(\tanh (a+b x))^3}{18 x^8}-\frac {b^4}{630 x^5} \]

[Out]

-1/630*b^4/x^5-1/126*b^3*arctanh(tanh(b*x+a))/x^6-1/42*b^2*arctanh(tanh(b*x+a))^2/x^7-1/18*b*arctanh(tanh(b*x+
a))^3/x^8-1/9*arctanh(tanh(b*x+a))^4/x^9

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Rubi [A]  time = 0.06, antiderivative size = 80, normalized size of antiderivative = 1.00, 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} \[ -\frac {b^3 \tanh ^{-1}(\tanh (a+b x))}{126 x^6}-\frac {b^2 \tanh ^{-1}(\tanh (a+b x))^2}{42 x^7}-\frac {b \tanh ^{-1}(\tanh (a+b x))^3}{18 x^8}-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{9 x^9}-\frac {b^4}{630 x^5} \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[Tanh[a + b*x]]^4/x^10,x]

[Out]

-b^4/(630*x^5) - (b^3*ArcTanh[Tanh[a + b*x]])/(126*x^6) - (b^2*ArcTanh[Tanh[a + b*x]]^2)/(42*x^7) - (b*ArcTanh
[Tanh[a + b*x]]^3)/(18*x^8) - ArcTanh[Tanh[a + b*x]]^4/(9*x^9)

Rule 30

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

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 \frac {\tanh ^{-1}(\tanh (a+b x))^4}{x^{10}} \, dx &=-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{9 x^9}+\frac {1}{9} (4 b) \int \frac {\tanh ^{-1}(\tanh (a+b x))^3}{x^9} \, dx\\ &=-\frac {b \tanh ^{-1}(\tanh (a+b x))^3}{18 x^8}-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{9 x^9}+\frac {1}{6} b^2 \int \frac {\tanh ^{-1}(\tanh (a+b x))^2}{x^8} \, dx\\ &=-\frac {b^2 \tanh ^{-1}(\tanh (a+b x))^2}{42 x^7}-\frac {b \tanh ^{-1}(\tanh (a+b x))^3}{18 x^8}-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{9 x^9}+\frac {1}{21} b^3 \int \frac {\tanh ^{-1}(\tanh (a+b x))}{x^7} \, dx\\ &=-\frac {b^3 \tanh ^{-1}(\tanh (a+b x))}{126 x^6}-\frac {b^2 \tanh ^{-1}(\tanh (a+b x))^2}{42 x^7}-\frac {b \tanh ^{-1}(\tanh (a+b x))^3}{18 x^8}-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{9 x^9}+\frac {1}{126} b^4 \int \frac {1}{x^6} \, dx\\ &=-\frac {b^4}{630 x^5}-\frac {b^3 \tanh ^{-1}(\tanh (a+b x))}{126 x^6}-\frac {b^2 \tanh ^{-1}(\tanh (a+b x))^2}{42 x^7}-\frac {b \tanh ^{-1}(\tanh (a+b x))^3}{18 x^8}-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{9 x^9}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 71, normalized size = 0.89 \[ -\frac {5 b^3 x^3 \tanh ^{-1}(\tanh (a+b x))+15 b^2 x^2 \tanh ^{-1}(\tanh (a+b x))^2+35 b x \tanh ^{-1}(\tanh (a+b x))^3+70 \tanh ^{-1}(\tanh (a+b x))^4+b^4 x^4}{630 x^9} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTanh[Tanh[a + b*x]]^4/x^10,x]

[Out]

-1/630*(b^4*x^4 + 5*b^3*x^3*ArcTanh[Tanh[a + b*x]] + 15*b^2*x^2*ArcTanh[Tanh[a + b*x]]^2 + 35*b*x*ArcTanh[Tanh
[a + b*x]]^3 + 70*ArcTanh[Tanh[a + b*x]]^4)/x^9

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fricas [A]  time = 0.55, size = 46, normalized size = 0.58 \[ -\frac {126 \, b^{4} x^{4} + 420 \, a b^{3} x^{3} + 540 \, a^{2} b^{2} x^{2} + 315 \, a^{3} b x + 70 \, a^{4}}{630 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(tanh(b*x+a))^4/x^10,x, algorithm="fricas")

[Out]

-1/630*(126*b^4*x^4 + 420*a*b^3*x^3 + 540*a^2*b^2*x^2 + 315*a^3*b*x + 70*a^4)/x^9

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giac [A]  time = 0.19, size = 46, normalized size = 0.58 \[ -\frac {126 \, b^{4} x^{4} + 420 \, a b^{3} x^{3} + 540 \, a^{2} b^{2} x^{2} + 315 \, a^{3} b x + 70 \, a^{4}}{630 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(tanh(b*x+a))^4/x^10,x, algorithm="giac")

[Out]

-1/630*(126*b^4*x^4 + 420*a*b^3*x^3 + 540*a^2*b^2*x^2 + 315*a^3*b*x + 70*a^4)/x^9

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maple [A]  time = 0.15, size = 74, normalized size = 0.92 \[ -\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{4}}{9 x^{9}}+\frac {4 b \left (-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{3}}{8 x^{8}}+\frac {3 b \left (-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{2}}{7 x^{7}}+\frac {2 b \left (-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )}{6 x^{6}}-\frac {b}{30 x^{5}}\right )}{7}\right )}{8}\right )}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctanh(tanh(b*x+a))^4/x^10,x)

[Out]

-1/9*arctanh(tanh(b*x+a))^4/x^9+4/9*b*(-1/8/x^8*arctanh(tanh(b*x+a))^3+3/8*b*(-1/7/x^7*arctanh(tanh(b*x+a))^2+
2/7*b*(-1/6/x^6*arctanh(tanh(b*x+a))-1/30/x^5*b)))

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maxima [A]  time = 0.61, size = 72, normalized size = 0.90 \[ -\frac {1}{630} \, {\left (b {\left (\frac {b^{2}}{x^{5}} + \frac {5 \, b \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )}{x^{6}}\right )} + \frac {15 \, b \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2}}{x^{7}}\right )} b - \frac {b \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{3}}{18 \, x^{8}} - \frac {\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{4}}{9 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(tanh(b*x+a))^4/x^10,x, algorithm="maxima")

[Out]

-1/630*(b*(b^2/x^5 + 5*b*arctanh(tanh(b*x + a))/x^6) + 15*b*arctanh(tanh(b*x + a))^2/x^7)*b - 1/18*b*arctanh(t
anh(b*x + a))^3/x^8 - 1/9*arctanh(tanh(b*x + a))^4/x^9

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mupad [B]  time = 1.10, size = 70, normalized size = 0.88 \[ -\frac {{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^4}{9\,x^9}-\frac {b^4}{630\,x^5}-\frac {b^2\,{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^2}{42\,x^7}-\frac {b^3\,\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}{126\,x^6}-\frac {b\,{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^3}{18\,x^8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atanh(tanh(a + b*x))^4/x^10,x)

[Out]

- atanh(tanh(a + b*x))^4/(9*x^9) - b^4/(630*x^5) - (b^2*atanh(tanh(a + b*x))^2)/(42*x^7) - (b^3*atanh(tanh(a +
b*x)))/(126*x^6) - (b*atanh(tanh(a + b*x))^3)/(18*x^8)

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sympy [A]  time = 13.00, size = 76, normalized size = 0.95 \[ - \frac {b^{4}}{630 x^{5}} - \frac {b^{3} \operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}{126 x^{6}} - \frac {b^{2} \operatorname {atanh}^{2}{\left (\tanh {\left (a + b x \right )} \right )}}{42 x^{7}} - \frac {b \operatorname {atanh}^{3}{\left (\tanh {\left (a + b x \right )} \right )}}{18 x^{8}} - \frac {\operatorname {atanh}^{4}{\left (\tanh {\left (a + b x \right )} \right )}}{9 x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atanh(tanh(b*x+a))**4/x**10,x)

[Out]

-b**4/(630*x**5) - b**3*atanh(tanh(a + b*x))/(126*x**6) - b**2*atanh(tanh(a + b*x))**2/(42*x**7) - b*atanh(tan
h(a + b*x))**3/(18*x**8) - atanh(tanh(a + b*x))**4/(9*x**9)

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