3.55 \(\int x^3 \tanh ^{-1}(\tanh (a+b x))^3 \, dx\)

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

[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 = 0.04, 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 = {2168, 30} \[ \frac {1}{20} b^2 x^6 \tanh ^{-1}(\tanh (a+b x))-\frac {3}{20} b x^5 \tanh ^{-1}(\tanh (a+b x))^2+\frac {1}{4} x^4 \tanh ^{-1}(\tanh (a+b x))^3-\frac {1}{140} b^3 x^7 \]

Antiderivative was successfully verified.

[In]

Int[x^3*ArcTanh[Tanh[a + b*x]]^3,x]

[Out]

-(b^3*x^7)/140 + (b^2*x^6*ArcTanh[Tanh[a + b*x]])/20 - (3*b*x^5*ArcTanh[Tanh[a + b*x]]^2)/20 + (x^4*ArcTanh[Ta
nh[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 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^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 = 0.03, size = 54, normalized size = 0.89 \[ -\frac {1}{140} x^4 \left (-7 b^2 x^2 \tanh ^{-1}(\tanh (a+b x))+21 b x \tanh ^{-1}(\tanh (a+b x))^2-35 \tanh ^{-1}(\tanh (a+b x))^3+b^3 x^3\right ) \]

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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fricas [A]  time = 1.02, size = 35, normalized size = 0.57 \[ \frac {1}{7} \, b^{3} x^{7} + \frac {1}{2} \, a b^{2} x^{6} + \frac {3}{5} \, a^{2} b x^{5} + \frac {1}{4} \, a^{3} x^{4} \]

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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giac [A]  time = 0.21, size = 35, normalized size = 0.57 \[ \frac {1}{7} \, b^{3} x^{7} + \frac {1}{2} \, a b^{2} x^{6} + \frac {3}{5} \, a^{2} b x^{5} + \frac {1}{4} \, a^{3} x^{4} \]

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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maple [A]  time = 0.14, size = 56, normalized size = 0.92 \[ \frac {x^{4} \arctanh \left (\tanh \left (b x +a \right )\right )^{3}}{4}-\frac {3 b \left (\frac {x^{5} \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}{5}-\frac {2 b \left (\frac {x^{6} \arctanh \left (\tanh \left (b x +a \right )\right )}{6}-\frac {x^{7} b}{42}\right )}{5}\right )}{4} \]

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*
x^7*b))

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maxima [A]  time = 0.52, size = 54, normalized size = 0.89 \[ -\frac {3}{20} \, b x^{5} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2} + \frac {1}{4} \, x^{4} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{3} - \frac {1}{140} \, {\left (b^{2} x^{7} - 7 \, b x^{6} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )\right )} b \]

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

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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sympy [A]  time = 4.09, size = 80, normalized size = 1.31 \[ \begin {cases} \frac {x^{3} \operatorname {atanh}^{4}{\left (\tanh {\left (a + b x \right )} \right )}}{4 b} - \frac {3 x^{2} \operatorname {atanh}^{5}{\left (\tanh {\left (a + b x \right )} \right )}}{20 b^{2}} + \frac {x \operatorname {atanh}^{6}{\left (\tanh {\left (a + b x \right )} \right )}}{20 b^{3}} - \frac {\operatorname {atanh}^{7}{\left (\tanh {\left (a + b x \right )} \right )}}{140 b^{4}} & \text {for}\: b \neq 0 \\\frac {x^{4} \operatorname {atanh}^{3}{\left (\tanh {\relax (a )} \right )}}{4} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x**3*atanh(tanh(a + b*x))**4/(4*b) - 3*x**2*atanh(tanh(a + b*x))**5/(20*b**2) + x*atanh(tanh(a + b*
x))**6/(20*b**3) - atanh(tanh(a + b*x))**7/(140*b**4), Ne(b, 0)), (x**4*atanh(tanh(a))**3/4, True))

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