3.2.38 \(\int x^2 \coth ^{-1}(\tanh (a+b x))^2 \, dx\) [138]
Optimal. Leaf size=42 \[ \frac {b^2 x^5}{30}-\frac {1}{6} b x^4 \coth ^{-1}(\tanh (a+b x))+\frac {1}{3} x^3 \coth ^{-1}(\tanh (a+b x))^2 \]
[Out]
1/30*b^2*x^5-1/6*b*x^4*arccoth(tanh(b*x+a))+1/3*x^3*arccoth(tanh(b*x+a))^2
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Rubi [A]
time = 0.02, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps
used = 3, 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}{6} b x^4 \coth ^{-1}(\tanh (a+b x))+\frac {1}{3} x^3 \coth ^{-1}(\tanh (a+b x))^2+\frac {b^2 x^5}{30} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[x^2*ArcCoth[Tanh[a + b*x]]^2,x]
[Out]
(b^2*x^5)/30 - (b*x^4*ArcCoth[Tanh[a + b*x]])/6 + (x^3*ArcCoth[Tanh[a + b*x]]^2)/3
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^2 \coth ^{-1}(\tanh (a+b x))^2 \, dx &=\frac {1}{3} x^3 \coth ^{-1}(\tanh (a+b x))^2-\frac {1}{3} (2 b) \int x^3 \coth ^{-1}(\tanh (a+b x)) \, dx\\ &=-\frac {1}{6} b x^4 \coth ^{-1}(\tanh (a+b x))+\frac {1}{3} x^3 \coth ^{-1}(\tanh (a+b x))^2+\frac {1}{6} b^2 \int x^4 \, dx\\ &=\frac {b^2 x^5}{30}-\frac {1}{6} b x^4 \coth ^{-1}(\tanh (a+b x))+\frac {1}{3} x^3 \coth ^{-1}(\tanh (a+b x))^2\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 37, normalized size = 0.88 \begin {gather*} \frac {1}{30} x^3 \left (b^2 x^2-5 b x \coth ^{-1}(\tanh (a+b x))+10 \coth ^{-1}(\tanh (a+b x))^2\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[x^2*ArcCoth[Tanh[a + b*x]]^2,x]
[Out]
(x^3*(b^2*x^2 - 5*b*x*ArcCoth[Tanh[a + b*x]] + 10*ArcCoth[Tanh[a + b*x]]^2))/30
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 34.22, size = 3418, normalized size = 81.38
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risch |
\(\text {Expression too large to display}\) |
\(3418\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^2*arccoth(tanh(b*x+a))^2,x,method=_RETURNVERBOSE)
[Out]
1/12*I*Pi*b*x^4-1/12*Pi^2*x^3*csgn(I/(exp(2*b*x+2*a)+1))^3*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2+1/12*Pi
^2*x^3*csgn(I/(exp(2*b*x+2*a)+1))^2*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3-1/12*Pi^2*x^3*csgn(I/(exp(2*b*
x+2*a)+1))^4*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))+1/12*I*Pi*b*x^4*csgn(I/(exp(2*b*
x+2*a)+1))^3+1/30*b^2*x^5+1/3*x^3*ln(exp(b*x+a))^2+1/24*I*Pi*b*x^4*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x
+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))-1/24*I*Pi*b*x^4*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2
*a)/(exp(2*b*x+2*a)+1))^2+1/24*I*Pi*b*x^4*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))-1/12*I*Pi*b*x^4*csgn(I*e
xp(b*x+a))*csgn(I*exp(2*b*x+2*a))^2-1/24*I*Pi*b*x^4*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*
a)+1))^2-1/24*Pi^2*x^3*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))^2*csgn(I*exp(2*b
*x+2*a)/(exp(2*b*x+2*a)+1))-1/24*Pi^2*x^3*csgn(I*exp(2*b*x+2*a))^3*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3
+1/24*Pi^2*x^3*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^5+1/24*Pi^2*x^3*csgn(I/(exp(2*
b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^5-1/12*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(b*x+a))^2*Pi^2
*x^3+1/6*csgn(I*exp(2*b*x+2*a))^2*csgn(I*exp(b*x+a))*Pi^2*x^3+1/6*Pi^2*x^3*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x
+2*a))^2*csgn(I/(exp(2*b*x+2*a)+1))^3+1/12*Pi^2*x^3*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*
a)+1))^2*csgn(I/(exp(2*b*x+2*a)+1))^3+1/6*Pi^2*x^3*csgn(I/(exp(2*b*x+2*a)+1))^5-1/6*csgn(I/(exp(2*b*x+2*a)+1))
^3*Pi^2*x^3-1/12*Pi^2*x^3*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3*csgn(I/(exp(2*b*x+2*a)+1))^3+1/12*Pi^2*x
^3*csgn(I*exp(2*b*x+2*a))^3*csgn(I/(exp(2*b*x+2*a)+1))^2+1/12*Pi^2*x^3*csgn(I/(exp(2*b*x+2*a)+1))^4*csgn(I*exp
(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2+1/12*Pi^2*x^3*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*
a)+1))^2+1/6*csgn(I/(exp(2*b*x+2*a)+1))^2*Pi^2*x^3-1/12*I*Pi*b*x^4*csgn(I/(exp(2*b*x+2*a)+1))^2-1/12*csgn(I*ex
p(2*b*x+2*a))^3*Pi^2*x^3-1/24*Pi^2*x^3*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a))^4*csgn(I*exp(2*b*x+2*
a)/(exp(2*b*x+2*a)+1))-1/12*Pi^2*x^3*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))*csgn(I/(exp(2*b*x+2*a)+1))^3-
1/12*Pi^2*x^3*csgn(I/(exp(2*b*x+2*a)+1))^6+(-1/6*b*x^4+1/6*I*Pi*x^3*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*
a)/(exp(2*b*x+2*a)+1))^2-1/6*I*Pi*x^3*csgn(I*exp(2*b*x+2*a))^3+1/6*I*Pi*x^3*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*
exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2-1/6*I*Pi*x^3*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))+1/3*I*Pi*x^3*csg
n(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^2-1/6*I*Pi*x^3*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3-1/6*I*Pi*x^3
*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))+1/3*I*Pi*x^3*csgn
(I/(exp(2*b*x+2*a)+1))^2-1/3*I*Pi*x^3-1/3*I*Pi*x^3*csgn(I/(exp(2*b*x+2*a)+1))^3)*ln(exp(b*x+a))+1/24*Pi^2*x^3*
csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a))^3*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2+1/24*Pi^2*x^3*
csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a))^2*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3-1/12*Pi^2*x^3*
csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))-1/48*Pi^2*x^3*csgn
(I*exp(2*b*x+2*a))^6+1/12*Pi^2*x^3*csgn(I/(exp(2*b*x+2*a)+1))^2*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))-1/
6*Pi^2*x^3*csgn(I/(exp(2*b*x+2*a)+1))^2*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^2-1/12*Pi^2*x^3*csgn(I/(exp(
2*b*x+2*a)+1))^2*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2-1/12*Pi^2*x^3-1/12*csgn(I/
(exp(2*b*x+2*a)+1))^4*Pi^2*x^3+1/12*Pi^2*x^3*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^
2-1/12*Pi^2*x^3*csgn(I*exp(2*b*x+2*a))^3*csgn(I/(exp(2*b*x+2*a)+1))^3+1/12*Pi^2*x^3*csgn(I*exp(b*x+a))*csgn(I*
exp(2*b*x+2*a))^2*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3+1/24*Pi^2*x^3*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*
b*x+2*a))^2*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2+1/24*I*Pi*b*x^4*csgn(I*exp(2*b*x+2*a))^3-1/12*Pi^2*x^3
*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3+1/24*Pi^2*x^3*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(b*x+a))^2*csg
n(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2+1/12*Pi^2*x^3*csgn(I/(exp(2*b*x+2*a)+1))*csgn(
I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^3*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))-1/12*Pi^2*x^3*csgn(I/(exp(2*b
*x+2*a)+1))*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^2*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2-1/48*Pi^2*
x^3*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^6+1/12*Pi^2*x^3*csgn(I/(exp(2*b*x+2*a)+1))^3*csgn(I*exp(2*b*x+2*
a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))-1/48*Pi^2*x^3*csgn(I/(exp(2*b*x+2*a)+1))^2*csgn(I*exp(2*b*x+2*a)
)^2*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2+1/24*Pi^2*x^3*csgn(I/(exp(2*b*x+2*a)+1))^2*csgn(I*exp(2*b*x+2*
a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3-1/12*Pi^2*x^3*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a)
)*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^4+1/12*Pi^2*x^3*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^5+1/24*P
i^2*x^3*csgn(I*exp(2*b*x+2*a))^4*csgn(I*exp(2*b...
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Maxima [A]
time = 0.33, size = 36, normalized size = 0.86 \begin {gather*} \frac {1}{30} \, b^{2} x^{5} - \frac {1}{6} \, b x^{4} \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right ) + \frac {1}{3} \, x^{3} \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*arccoth(tanh(b*x+a))^2,x, algorithm="maxima")
[Out]
1/30*b^2*x^5 - 1/6*b*x^4*arccoth(tanh(b*x + a)) + 1/3*x^3*arccoth(tanh(b*x + a))^2
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Fricas [A]
time = 0.34, size = 30, normalized size = 0.71 \begin {gather*} \frac {1}{5} \, b^{2} x^{5} + \frac {1}{2} \, a b x^{4} - \frac {1}{12} \, {\left (\pi ^{2} - 4 \, a^{2}\right )} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*arccoth(tanh(b*x+a))^2,x, algorithm="fricas")
[Out]
1/5*b^2*x^5 + 1/2*a*b*x^4 - 1/12*(pi^2 - 4*a^2)*x^3
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Sympy [A]
time = 0.17, size = 37, normalized size = 0.88 \begin {gather*} \frac {b^{2} x^{5}}{30} - \frac {b x^{4} \operatorname {acoth}{\left (\tanh {\left (a + b x \right )} \right )}}{6} + \frac {x^{3} \operatorname {acoth}^{2}{\left (\tanh {\left (a + b x \right )} \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**2*acoth(tanh(b*x+a))**2,x)
[Out]
b**2*x**5/30 - b*x**4*acoth(tanh(a + b*x))/6 + x**3*acoth(tanh(a + b*x))**2/3
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Giac [C] Result contains complex when optimal does not.
time = 0.41, size = 41, normalized size = 0.98 \begin {gather*} \frac {1}{5} \, b^{2} x^{5} - \frac {1}{4} \, {\left (-i \, \pi b - 2 \, a b\right )} x^{4} - \frac {1}{12} \, {\left (\pi ^{2} - 4 i \, \pi a - 4 \, a^{2}\right )} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*arccoth(tanh(b*x+a))^2,x, algorithm="giac")
[Out]
1/5*b^2*x^5 - 1/4*(-I*pi*b - 2*a*b)*x^4 - 1/12*(pi^2 - 4*I*pi*a - 4*a^2)*x^3
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Mupad [B]
time = 1.18, size = 36, normalized size = 0.86 \begin {gather*} \frac {b^2\,x^5}{30}-\frac {b\,x^4\,\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}{6}+\frac {x^3\,{\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^2}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^2*acoth(tanh(a + b*x))^2,x)
[Out]
(x^3*acoth(tanh(a + b*x))^2)/3 + (b^2*x^5)/30 - (b*x^4*acoth(tanh(a + b*x)))/6
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