3.1.3 \(\int x^3 \coth ^{-1}(a x) \, dx\) [3]

Optimal. Leaf size=41 \[ \frac {x}{4 a^3}+\frac {x^3}{12 a}+\frac {1}{4} x^4 \coth ^{-1}(a x)-\frac {\tanh ^{-1}(a x)}{4 a^4} \]

[Out]

1/4*x/a^3+1/12*x^3/a+1/4*x^4*arccoth(a*x)-1/4*arctanh(a*x)/a^4

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Rubi [A]
time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6038, 308, 212} \begin {gather*} -\frac {\tanh ^{-1}(a x)}{4 a^4}+\frac {x}{4 a^3}+\frac {1}{4} x^4 \coth ^{-1}(a x)+\frac {x^3}{12 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*ArcCoth[a*x],x]

[Out]

x/(4*a^3) + x^3/(12*a) + (x^4*ArcCoth[a*x])/4 - ArcTanh[a*x]/(4*a^4)

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 308

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 6038

Int[((a_.) + ArcCoth[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcCoth[c*
x^n])^p/(m + 1)), x] - Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcCoth[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))
), x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] && IntegerQ[m])) && NeQ[m, -1
]

Rubi steps

\begin {align*} \int x^3 \coth ^{-1}(a x) \, dx &=\frac {1}{4} x^4 \coth ^{-1}(a x)-\frac {1}{4} a \int \frac {x^4}{1-a^2 x^2} \, dx\\ &=\frac {1}{4} x^4 \coth ^{-1}(a x)-\frac {1}{4} a \int \left (-\frac {1}{a^4}-\frac {x^2}{a^2}+\frac {1}{a^4 \left (1-a^2 x^2\right )}\right ) \, dx\\ &=\frac {x}{4 a^3}+\frac {x^3}{12 a}+\frac {1}{4} x^4 \coth ^{-1}(a x)-\frac {\int \frac {1}{1-a^2 x^2} \, dx}{4 a^3}\\ &=\frac {x}{4 a^3}+\frac {x^3}{12 a}+\frac {1}{4} x^4 \coth ^{-1}(a x)-\frac {\tanh ^{-1}(a x)}{4 a^4}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 57, normalized size = 1.39 \begin {gather*} \frac {x}{4 a^3}+\frac {x^3}{12 a}+\frac {1}{4} x^4 \coth ^{-1}(a x)+\frac {\log (1-a x)}{8 a^4}-\frac {\log (1+a x)}{8 a^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*ArcCoth[a*x],x]

[Out]

x/(4*a^3) + x^3/(12*a) + (x^4*ArcCoth[a*x])/4 + Log[1 - a*x]/(8*a^4) - Log[1 + a*x]/(8*a^4)

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Maple [A]
time = 0.04, size = 46, normalized size = 1.12

method	result	size
derivativedivides	\(\frac {\frac {a^{4} x^{4} \mathrm {arccoth}\left (a x \right )}{4}+\frac {a^{3} x^{3}}{12}+\frac {a x}{4}+\frac {\ln \left (a x -1\right )}{8}-\frac {\ln \left (a x +1\right )}{8}}{a^{4}}\)	\(46\)
default	\(\frac {\frac {a^{4} x^{4} \mathrm {arccoth}\left (a x \right )}{4}+\frac {a^{3} x^{3}}{12}+\frac {a x}{4}+\frac {\ln \left (a x -1\right )}{8}-\frac {\ln \left (a x +1\right )}{8}}{a^{4}}\)	\(46\)
risch	\(\frac {x^{4} \ln \left (a x +1\right )}{8}-\frac {x^{4} \ln \left (a x -1\right )}{8}+\frac {x^{3}}{12 a}+\frac {x}{4 a^{3}}+\frac {\ln \left (-a x +1\right )}{8 a^{4}}-\frac {\ln \left (a x +1\right )}{8 a^{4}}\)	\(61\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*arccoth(a*x),x,method=_RETURNVERBOSE)

[Out]

1/a^4*(1/4*a^4*x^4*arccoth(a*x)+1/12*a^3*x^3+1/4*a*x+1/8*ln(a*x-1)-1/8*ln(a*x+1))

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Maxima [A]
time = 0.26, size = 52, normalized size = 1.27 \begin {gather*} \frac {1}{4} \, x^{4} \operatorname {arcoth}\left (a x\right ) + \frac {1}{24} \, a {\left (\frac {2 \, {\left (a^{2} x^{3} + 3 \, x\right )}}{a^{4}} - \frac {3 \, \log \left (a x + 1\right )}{a^{5}} + \frac {3 \, \log \left (a x - 1\right )}{a^{5}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arccoth(a*x),x, algorithm="maxima")

[Out]

1/4*x^4*arccoth(a*x) + 1/24*a*(2*(a^2*x^3 + 3*x)/a^4 - 3*log(a*x + 1)/a^5 + 3*log(a*x - 1)/a^5)

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Fricas [A]
time = 0.34, size = 43, normalized size = 1.05 \begin {gather*} \frac {2 \, a^{3} x^{3} + 6 \, a x + 3 \, {\left (a^{4} x^{4} - 1\right )} \log \left (\frac {a x + 1}{a x - 1}\right )}{24 \, a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arccoth(a*x),x, algorithm="fricas")

[Out]

1/24*(2*a^3*x^3 + 6*a*x + 3*(a^4*x^4 - 1)*log((a*x + 1)/(a*x - 1)))/a^4

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Sympy [C] Result contains complex when optimal does not.
time = 0.24, size = 41, normalized size = 1.00 \begin {gather*} \begin {cases} \frac {x^{4} \operatorname {acoth}{\left (a x \right )}}{4} + \frac {x^{3}}{12 a} + \frac {x}{4 a^{3}} - \frac {\operatorname {acoth}{\left (a x \right )}}{4 a^{4}} & \text {for}\: a \neq 0 \\\frac {i \pi x^{4}}{8} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*acoth(a*x),x)

[Out]

Piecewise((x**4*acoth(a*x)/4 + x**3/(12*a) + x/(4*a**3) - acoth(a*x)/(4*a**4), Ne(a, 0)), (I*pi*x**4/8, True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 195 vs. \(2 (33) = 66\).
time = 0.39, size = 195, normalized size = 4.76 \begin {gather*} \frac {1}{3} \, a {\left (\frac {\frac {3 \, {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} - \frac {3 \, {\left (a x + 1\right )}}{a x - 1} + 2}{a^{5} {\left (\frac {a x + 1}{a x - 1} - 1\right )}^{3}} + \frac {3 \, {\left (\frac {{\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}} + \frac {a x + 1}{a x - 1}\right )} \log \left (-\frac {\frac {\frac {{\left (a x + 1\right )} a}{a x - 1} - a}{a {\left (\frac {a x + 1}{a x - 1} + 1\right )}} + 1}{\frac {\frac {{\left (a x + 1\right )} a}{a x - 1} - a}{a {\left (\frac {a x + 1}{a x - 1} + 1\right )}} - 1}\right )}{a^{5} {\left (\frac {a x + 1}{a x - 1} - 1\right )}^{4}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arccoth(a*x),x, algorithm="giac")

[Out]

1/3*a*((3*(a*x + 1)^2/(a*x - 1)^2 - 3*(a*x + 1)/(a*x - 1) + 2)/(a^5*((a*x + 1)/(a*x - 1) - 1)^3) + 3*((a*x + 1
)^3/(a*x - 1)^3 + (a*x + 1)/(a*x - 1))*log(-(((a*x + 1)*a/(a*x - 1) - a)/(a*((a*x + 1)/(a*x - 1) + 1)) + 1)/((
(a*x + 1)*a/(a*x - 1) - a)/(a*((a*x + 1)/(a*x - 1) + 1)) - 1))/(a^5*((a*x + 1)/(a*x - 1) - 1)^4))

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Mupad [B]
time = 1.22, size = 33, normalized size = 0.80 \begin {gather*} \frac {\frac {a\,x}{4}-\frac {\mathrm {acoth}\left (a\,x\right )}{4}+\frac {a^3\,x^3}{12}}{a^4}+\frac {x^4\,\mathrm {acoth}\left (a\,x\right )}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*acoth(a*x),x)

[Out]

((a*x)/4 - acoth(a*x)/4 + (a^3*x^3)/12)/a^4 + (x^4*acoth(a*x))/4

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