3.2.37 \(\int x^3 \coth ^{-1}(\tanh (a+b x))^2 \, dx\) [137]
Optimal. Leaf size=42 \[ \frac {b^2 x^6}{60}-\frac {1}{10} b x^5 \coth ^{-1}(\tanh (a+b x))+\frac {1}{4} x^4 \coth ^{-1}(\tanh (a+b x))^2 \]
[Out]
1/60*b^2*x^6-1/10*b*x^5*arccoth(tanh(b*x+a))+1/4*x^4*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}{10} b x^5 \coth ^{-1}(\tanh (a+b x))+\frac {1}{4} x^4 \coth ^{-1}(\tanh (a+b x))^2+\frac {b^2 x^6}{60} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[x^3*ArcCoth[Tanh[a + b*x]]^2,x]
[Out]
(b^2*x^6)/60 - (b*x^5*ArcCoth[Tanh[a + b*x]])/10 + (x^4*ArcCoth[Tanh[a + b*x]]^2)/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 \coth ^{-1}(\tanh (a+b x))^2 \, dx &=\frac {1}{4} x^4 \coth ^{-1}(\tanh (a+b x))^2-\frac {1}{2} b \int x^4 \coth ^{-1}(\tanh (a+b x)) \, dx\\ &=-\frac {1}{10} b x^5 \coth ^{-1}(\tanh (a+b x))+\frac {1}{4} x^4 \coth ^{-1}(\tanh (a+b x))^2+\frac {1}{10} b^2 \int x^5 \, dx\\ &=\frac {b^2 x^6}{60}-\frac {1}{10} b x^5 \coth ^{-1}(\tanh (a+b x))+\frac {1}{4} x^4 \coth ^{-1}(\tanh (a+b x))^2\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 37, normalized size = 0.88 \begin {gather*} \frac {1}{60} x^4 \left (b^2 x^2-6 b x \coth ^{-1}(\tanh (a+b x))+15 \coth ^{-1}(\tanh (a+b x))^2\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[x^3*ArcCoth[Tanh[a + b*x]]^2,x]
[Out]
(x^4*(b^2*x^2 - 6*b*x*ArcCoth[Tanh[a + b*x]] + 15*ArcCoth[Tanh[a + b*x]]^2))/60
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 41.90, size = 3418, normalized size = 81.38
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\(\text {Expression too large to display}\) |
\(3418\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^3*arccoth(tanh(b*x+a))^2,x,method=_RETURNVERBOSE)
[Out]
1/4*x^4*ln(exp(b*x+a))^2-1/32*Pi^2*x^4*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))+1/32*Pi^2*x^4*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(b*x+a))^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+1/16*Pi^2*x^4*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))-1/16*Pi^2*x^4*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))^2-1/40*I*Pi*b
*x^5*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+1/32*Pi^2*x^4*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+1/40*I*Pi*b*x^5*csgn(I*exp(2*b*x+2
*a))^3+1/40*I*Pi*b*x^5*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3+1/60*b^2*x^6+1/32*Pi^2*x^4*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))^3-1/32*Pi^2*x^4*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))+1/32*Pi^2*x^4*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+1/40*I*Pi*b*x^5*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))+1/40*I*Pi*b*x^5*csgn(I*exp(b*x+a))
^2*csgn(I*exp(2*b*x+2*a))-1/20*I*Pi*b*x^5*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^2-1/40*I*Pi*b*x^5*csgn(I*e
xp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2+1/32*Pi^2*x^4*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))^2-1/16*Pi^2*x^4*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))^2+1/32*Pi^2*x^4*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a)/(e
xp(2*b*x+2*a)+1))^5-1/32*Pi^2*x^4*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+1/32*Pi
^2*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))^5-1/64*Pi^2*x^4*csgn(I*exp(b*x+a))^4*c
sgn(I*exp(2*b*x+2*a))^2+1/16*Pi^2*x^4*csgn(I*exp(b*x+a))^3*csgn(I*exp(2*b*x+2*a))^3-3/32*Pi^2*x^4*csgn(I*exp(b
*x+a))^2*csgn(I*exp(2*b*x+2*a))^4+1/16*Pi^2*x^4*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^5+1/32*Pi^2*x^4*csgn
(I*exp(2*b*x+2*a))^4*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2-1/64*Pi^2*x^4*csgn(I*exp(2*b*x+2*a))^2*csgn(I
*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^4-1/64*Pi^2*x^4*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))^4+1/16*Pi^2*x^4*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))^3-1/16*Pi^2*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
))^4-1/32*Pi^2*x^4*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3-1/6
4*Pi^2*x^4*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
/64*Pi^2*x^4*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^6-1/64*Pi^2*x^4*csgn(I*exp(2*b*x+2*a))^6-1/16*csgn(I/(e
xp(2*b*x+2*a)+1))^4*Pi^2*x^4-1/8*csgn(I/(exp(2*b*x+2*a)+1))^3*Pi^2*x^4-1/16*Pi^2*x^4*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+1/16*Pi^2*x^4*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+1/16*Pi^2*x^4*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+1/8*Pi^2*x^4*csgn(I/(exp(2*b*x+2*a)+1))^3*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^2-1/16*Pi^2*x^4*csgn(I/(
exp(2*b*x+2*a)+1))^6+(-1/10*b*x^5-1/8*I*Pi*x^4*csgn(I*exp(2*b*x+2*a))^3-1/8*I*Pi*x^4*csgn(I*exp(2*b*x+2*a)/(ex
p(2*b*x+2*a)+1))^3+1/4*I*Pi*x^4*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^2+1/8*I*Pi*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))^2-1/8*I*Pi*x^4*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))-1/4
*I*Pi*x^4-1/8*I*Pi*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))+1/8*I*Pi*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))^2-1/4*I*Pi*x^4*csgn(I/(exp(
2*b*x+2*a)+1))^3+1/4*I*Pi*x^4*csgn(I/(exp(2*b*x+2*a)+1))^2)*ln(exp(b*x+a))+1/8*Pi^2*x^4*csgn(I/(exp(2*b*x+2*a)
+1))^5-1/20*I*Pi*b*x^5*csgn(I/(exp(2*b*x+2*a)+1))^2+1/20*I*Pi*b*x^5*csgn(I/(exp(2*b*x+2*a)+1))^3-1/16*Pi^2*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))-1/8*Pi^2*x^4*csgn
(I/(exp(2*b*x+2*a)+1))^2*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^2-1/16*Pi^2*x^4*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+1/16*Pi^2*x^4*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))-1/16*Pi^2*x^4*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))-1/16*Pi^2*x^4*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))^3-1/16*csgn(I*exp(2*b*x+2*a))^3*Pi^2*x^4+1/8*csgn(I/(exp(2*b*x+2*a)+1)
)^2*Pi^2*x^4+1/16*Pi^2*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*csgn(I/(exp(2*b*
x+2*a)+1))^3+1/20*I*Pi*b*x^5-1/16*Pi^2*x^4*csgn...
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Maxima [A]
time = 0.33, size = 36, normalized size = 0.86 \begin {gather*} \frac {1}{60} \, b^{2} x^{6} - \frac {1}{10} \, b x^{5} \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right ) + \frac {1}{4} \, x^{4} \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^3*arccoth(tanh(b*x+a))^2,x, algorithm="maxima")
[Out]
1/60*b^2*x^6 - 1/10*b*x^5*arccoth(tanh(b*x + a)) + 1/4*x^4*arccoth(tanh(b*x + a))^2
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Fricas [A]
time = 0.35, size = 30, normalized size = 0.71 \begin {gather*} \frac {1}{6} \, b^{2} x^{6} + \frac {2}{5} \, a b x^{5} - \frac {1}{16} \, {\left (\pi ^{2} - 4 \, a^{2}\right )} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3*arccoth(tanh(b*x+a))^2,x, algorithm="fricas")
[Out]
1/6*b^2*x^6 + 2/5*a*b*x^5 - 1/16*(pi^2 - 4*a^2)*x^4
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Sympy [A]
time = 0.26, size = 37, normalized size = 0.88 \begin {gather*} \frac {b^{2} x^{6}}{60} - \frac {b x^{5} \operatorname {acoth}{\left (\tanh {\left (a + b x \right )} \right )}}{10} + \frac {x^{4} \operatorname {acoth}^{2}{\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*acoth(tanh(b*x+a))**2,x)
[Out]
b**2*x**6/60 - b*x**5*acoth(tanh(a + b*x))/10 + x**4*acoth(tanh(a + b*x))**2/4
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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}{6} \, b^{2} x^{6} - \frac {1}{5} \, {\left (-i \, \pi b - 2 \, a b\right )} x^{5} - \frac {1}{16} \, {\left (\pi ^{2} - 4 i \, \pi a - 4 \, a^{2}\right )} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3*arccoth(tanh(b*x+a))^2,x, algorithm="giac")
[Out]
1/6*b^2*x^6 - 1/5*(-I*pi*b - 2*a*b)*x^5 - 1/16*(pi^2 - 4*I*pi*a - 4*a^2)*x^4
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Mupad [B]
time = 1.18, size = 36, normalized size = 0.86 \begin {gather*} \frac {b^2\,x^6}{60}-\frac {b\,x^5\,\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}{10}+\frac {x^4\,{\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^2}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^3*acoth(tanh(a + b*x))^2,x)
[Out]
(x^4*acoth(tanh(a + b*x))^2)/4 + (b^2*x^6)/60 - (b*x^5*acoth(tanh(a + b*x)))/10
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