3.1.55 \(\int x^3 \tanh ^{-1}(\tanh (a+b x))^3 \, dx\) [55]

Optimal. Leaf size=61 \[ -\frac {1}{140} b^3 x^7+\frac {1}{20} b^2 x^6 \tanh ^{-1}(\tanh (a+b x))-\frac {3}{20} b x^5 \tanh ^{-1}(\tanh (a+b x))^2+\frac {1}{4} x^4 \tanh ^{-1}(\tanh (a+b x))^3 \]

[Out]

-1/140*b^3*x^7+1/20*b^2*x^6*arctanh(tanh(b*x+a))-3/20*b*x^5*arctanh(tanh(b*x+a))^2+1/4*x^4*arctanh(tanh(b*x+a)
)^3

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Rubi [A]
time = 0.03, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2199, 30} \begin {gather*} \frac {1}{20} b^2 x^6 \tanh ^{-1}(\tanh (a+b x))-\frac {3}{20} b x^5 \tanh ^{-1}(\tanh (a+b x))^2+\frac {1}{4} x^4 \tanh ^{-1}(\tanh (a+b x))^3-\frac {1}{140} b^3 x^7 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*ArcTanh[Tanh[a + b*x]]^3,x]

[Out]

-1/140*(b^3*x^7) + (b^2*x^6*ArcTanh[Tanh[a + b*x]])/20 - (3*b*x^5*ArcTanh[Tanh[a + b*x]]^2)/20 + (x^4*ArcTanh[
Tanh[a + b*x]]^3)/4

Rule 30

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

Rule 2199

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^3 \tanh ^{-1}(\tanh (a+b x))^3 \, dx &=\frac {1}{4} x^4 \tanh ^{-1}(\tanh (a+b x))^3-\frac {1}{4} (3 b) \int x^4 \tanh ^{-1}(\tanh (a+b x))^2 \, dx\\ &=-\frac {3}{20} b x^5 \tanh ^{-1}(\tanh (a+b x))^2+\frac {1}{4} x^4 \tanh ^{-1}(\tanh (a+b x))^3+\frac {1}{10} \left (3 b^2\right ) \int x^5 \tanh ^{-1}(\tanh (a+b x)) \, dx\\ &=\frac {1}{20} b^2 x^6 \tanh ^{-1}(\tanh (a+b x))-\frac {3}{20} b x^5 \tanh ^{-1}(\tanh (a+b x))^2+\frac {1}{4} x^4 \tanh ^{-1}(\tanh (a+b x))^3-\frac {1}{20} b^3 \int x^6 \, dx\\ &=-\frac {1}{140} b^3 x^7+\frac {1}{20} b^2 x^6 \tanh ^{-1}(\tanh (a+b x))-\frac {3}{20} b x^5 \tanh ^{-1}(\tanh (a+b x))^2+\frac {1}{4} x^4 \tanh ^{-1}(\tanh (a+b x))^3\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 54, normalized size = 0.89 \begin {gather*} -\frac {1}{140} x^4 \left (b^3 x^3-7 b^2 x^2 \tanh ^{-1}(\tanh (a+b x))+21 b x \tanh ^{-1}(\tanh (a+b x))^2-35 \tanh ^{-1}(\tanh (a+b x))^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*ArcTanh[Tanh[a + b*x]]^3,x]

[Out]

-1/140*(x^4*(b^3*x^3 - 7*b^2*x^2*ArcTanh[Tanh[a + b*x]] + 21*b*x*ArcTanh[Tanh[a + b*x]]^2 - 35*ArcTanh[Tanh[a
+ b*x]]^3))

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Maple [A]
time = 0.02, size = 56, normalized size = 0.92 \[\frac {x^{4} \arctanh \left (\tanh \left (b x +a \right )\right )^{3}}{4}-\frac {3 b \left (\frac {x^{5} \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}{5}-\frac {2 b \left (\frac {x^{6} \arctanh \left (\tanh \left (b x +a \right )\right )}{6}-\frac {b \,x^{7}}{42}\right )}{5}\right )}{4}\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*arctanh(tanh(b*x+a))^3,x)

[Out]

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

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Maxima [A]
time = 0.38, size = 54, normalized size = 0.89 \begin {gather*} -\frac {3}{20} \, b x^{5} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2} + \frac {1}{4} \, x^{4} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{3} - \frac {1}{140} \, {\left (b^{2} x^{7} - 7 \, b x^{6} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )\right )} b \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arctanh(tanh(b*x+a))^3,x, algorithm="maxima")

[Out]

-3/20*b*x^5*arctanh(tanh(b*x + a))^2 + 1/4*x^4*arctanh(tanh(b*x + a))^3 - 1/140*(b^2*x^7 - 7*b*x^6*arctanh(tan
h(b*x + a)))*b

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Fricas [A]
time = 0.32, size = 35, normalized size = 0.57 \begin {gather*} \frac {1}{7} \, b^{3} x^{7} + \frac {1}{2} \, a b^{2} x^{6} + \frac {3}{5} \, a^{2} b x^{5} + \frac {1}{4} \, a^{3} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arctanh(tanh(b*x+a))^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.57, size = 58, normalized size = 0.95 \begin {gather*} - \frac {b^{3} x^{7}}{140} + \frac {b^{2} x^{6} \operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}{20} - \frac {3 b x^{5} \operatorname {atanh}^{2}{\left (\tanh {\left (a + b x \right )} \right )}}{20} + \frac {x^{4} \operatorname {atanh}^{3}{\left (\tanh {\left (a + b x \right )} \right )}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*atanh(tanh(b*x+a))**3,x)

[Out]

-b**3*x**7/140 + b**2*x**6*atanh(tanh(a + b*x))/20 - 3*b*x**5*atanh(tanh(a + b*x))**2/20 + x**4*atanh(tanh(a +
b*x))**3/4

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Giac [A]
time = 0.38, size = 35, normalized size = 0.57 \begin {gather*} \frac {1}{7} \, b^{3} x^{7} + \frac {1}{2} \, a b^{2} x^{6} + \frac {3}{5} \, a^{2} b x^{5} + \frac {1}{4} \, a^{3} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arctanh(tanh(b*x+a))^3,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 1.05, size = 53, normalized size = 0.87 \begin {gather*} -\frac {b^3\,x^7}{140}+\frac {b^2\,x^6\,\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}{20}-\frac {3\,b\,x^5\,{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^2}{20}+\frac {x^4\,{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^3}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*atanh(tanh(a + b*x))^3,x)

[Out]

(x^4*atanh(tanh(a + b*x))^3)/4 - (b^3*x^7)/140 - (3*b*x^5*atanh(tanh(a + b*x))^2)/20 + (b^2*x^6*atanh(tanh(a +
b*x)))/20

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