3.2.47 \(\int x^4 \coth ^{-1}(\tanh (a+b x))^3 \, dx\) [147]

Optimal. Leaf size=61 \[ -\frac {1}{280} b^3 x^8+\frac {1}{35} b^2 x^7 \coth ^{-1}(\tanh (a+b x))-\frac {1}{10} b x^6 \coth ^{-1}(\tanh (a+b x))^2+\frac {1}{5} x^5 \coth ^{-1}(\tanh (a+b x))^3 \]

[Out]

-1/280*b^3*x^8+1/35*b^2*x^7*arccoth(tanh(b*x+a))-1/10*b*x^6*arccoth(tanh(b*x+a))^2+1/5*x^5*arccoth(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}{35} b^2 x^7 \coth ^{-1}(\tanh (a+b x))-\frac {1}{10} b x^6 \coth ^{-1}(\tanh (a+b x))^2+\frac {1}{5} x^5 \coth ^{-1}(\tanh (a+b x))^3-\frac {1}{280} b^3 x^8 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4*ArcCoth[Tanh[a + b*x]]^3,x]

[Out]

-1/280*(b^3*x^8) + (b^2*x^7*ArcCoth[Tanh[a + b*x]])/35 - (b*x^6*ArcCoth[Tanh[a + b*x]]^2)/10 + (x^5*ArcCoth[Ta
nh[a + b*x]]^3)/5

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^4 \coth ^{-1}(\tanh (a+b x))^3 \, dx &=\frac {1}{5} x^5 \coth ^{-1}(\tanh (a+b x))^3-\frac {1}{5} (3 b) \int x^5 \coth ^{-1}(\tanh (a+b x))^2 \, dx\\ &=-\frac {1}{10} b x^6 \coth ^{-1}(\tanh (a+b x))^2+\frac {1}{5} x^5 \coth ^{-1}(\tanh (a+b x))^3+\frac {1}{5} b^2 \int x^6 \coth ^{-1}(\tanh (a+b x)) \, dx\\ &=\frac {1}{35} b^2 x^7 \coth ^{-1}(\tanh (a+b x))-\frac {1}{10} b x^6 \coth ^{-1}(\tanh (a+b x))^2+\frac {1}{5} x^5 \coth ^{-1}(\tanh (a+b x))^3-\frac {1}{35} b^3 \int x^7 \, dx\\ &=-\frac {1}{280} b^3 x^8+\frac {1}{35} b^2 x^7 \coth ^{-1}(\tanh (a+b x))-\frac {1}{10} b x^6 \coth ^{-1}(\tanh (a+b x))^2+\frac {1}{5} x^5 \coth ^{-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}{280} x^5 \left (b^3 x^3-8 b^2 x^2 \coth ^{-1}(\tanh (a+b x))+28 b x \coth ^{-1}(\tanh (a+b x))^2-56 \coth ^{-1}(\tanh (a+b x))^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4*ArcCoth[Tanh[a + b*x]]^3,x]

[Out]

-1/280*(x^5*(b^3*x^3 - 8*b^2*x^2*ArcCoth[Tanh[a + b*x]] + 28*b*x*ArcCoth[Tanh[a + b*x]]^2 - 56*ArcCoth[Tanh[a
+ b*x]]^3))

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int x^{4} \mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )^{3}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*arccoth(tanh(b*x+a))^3,x)

[Out]

int(x^4*arccoth(tanh(b*x+a))^3,x)

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Maxima [A]
time = 0.37, size = 54, normalized size = 0.89 \begin {gather*} -\frac {1}{10} \, b x^{6} \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{2} + \frac {1}{5} \, x^{5} \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{3} - \frac {1}{280} \, {\left (b^{2} x^{8} - 8 \, b x^{7} \operatorname {arcoth}\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^4*arccoth(tanh(b*x+a))^3,x, algorithm="maxima")

[Out]

-1/10*b*x^6*arccoth(tanh(b*x + a))^2 + 1/5*x^5*arccoth(tanh(b*x + a))^3 - 1/280*(b^2*x^8 - 8*b*x^7*arccoth(tan
h(b*x + a)))*b

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Fricas [A]
time = 0.47, size = 52, normalized size = 0.85 \begin {gather*} \frac {1}{8} \, b^{3} x^{8} + \frac {3}{7} \, a b^{2} x^{7} - \frac {1}{8} \, {\left (\pi ^{2} b - 4 \, a^{2} b\right )} x^{6} - \frac {1}{20} \, {\left (3 \, \pi ^{2} a - 4 \, a^{3}\right )} x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arccoth(tanh(b*x+a))^3,x, algorithm="fricas")

[Out]

1/8*b^3*x^8 + 3/7*a*b^2*x^7 - 1/8*(pi^2*b - 4*a^2*b)*x^6 - 1/20*(3*pi^2*a - 4*a^3)*x^5

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Sympy [A]
time = 0.62, size = 56, normalized size = 0.92 \begin {gather*} - \frac {b^{3} x^{8}}{280} + \frac {b^{2} x^{7} \operatorname {acoth}{\left (\tanh {\left (a + b x \right )} \right )}}{35} - \frac {b x^{6} \operatorname {acoth}^{2}{\left (\tanh {\left (a + b x \right )} \right )}}{10} + \frac {x^{5} \operatorname {acoth}^{3}{\left (\tanh {\left (a + b x \right )} \right )}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*acoth(tanh(b*x+a))**3,x)

[Out]

-b**3*x**8/280 + b**2*x**7*acoth(tanh(a + b*x))/35 - b*x**6*acoth(tanh(a + b*x))**2/10 + x**5*acoth(tanh(a + b
*x))**3/5

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Giac [C] Result contains complex when optimal does not.
time = 0.42, size = 77, normalized size = 1.26 \begin {gather*} \frac {1}{8} \, b^{3} x^{8} - \frac {3}{14} \, {\left (-i \, \pi b^{2} - 2 \, a b^{2}\right )} x^{7} - \frac {1}{8} \, {\left (\pi ^{2} b - 4 i \, \pi a b - 4 \, a^{2} b\right )} x^{6} - \frac {1}{40} \, {\left (i \, \pi ^{3} + 6 \, \pi ^{2} a - 12 i \, \pi a^{2} - 8 \, a^{3}\right )} x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arccoth(tanh(b*x+a))^3,x, algorithm="giac")

[Out]

1/8*b^3*x^8 - 3/14*(-I*pi*b^2 - 2*a*b^2)*x^7 - 1/8*(pi^2*b - 4*I*pi*a*b - 4*a^2*b)*x^6 - 1/40*(I*pi^3 + 6*pi^2
*a - 12*I*pi*a^2 - 8*a^3)*x^5

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Mupad [B]
time = 1.25, size = 53, normalized size = 0.87 \begin {gather*} -\frac {b^3\,x^8}{280}+\frac {b^2\,x^7\,\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}{35}-\frac {b\,x^6\,{\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^2}{10}+\frac {x^5\,{\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^3}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*acoth(tanh(a + b*x))^3,x)

[Out]

(x^5*acoth(tanh(a + b*x))^3)/5 - (b^3*x^8)/280 - (b*x^6*acoth(tanh(a + b*x))^2)/10 + (b^2*x^7*acoth(tanh(a + b
*x)))/35

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