3.3.65 \(\int \frac {\tanh ^{-1}(a x)}{x^3 (1-a^2 x^2)^2} \, dx\) [265]

Optimal. Leaf size=123 \[ -\frac {a}{2 x}-\frac {a^3 x}{4 \left (1-a^2 x^2\right )}+\frac {1}{4} a^2 \tanh ^{-1}(a x)-\frac {\tanh ^{-1}(a x)}{2 x^2}+\frac {a^2 \tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}+a^2 \tanh ^{-1}(a x)^2+2 a^2 \tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-a^2 \text {PolyLog}\left (2,-1+\frac {2}{1+a x}\right ) \]

[Out]

-1/2*a/x-1/4*a^3*x/(-a^2*x^2+1)+1/4*a^2*arctanh(a*x)-1/2*arctanh(a*x)/x^2+1/2*a^2*arctanh(a*x)/(-a^2*x^2+1)+a^
2*arctanh(a*x)^2+2*a^2*arctanh(a*x)*ln(2-2/(a*x+1))-a^2*polylog(2,-1+2/(a*x+1))

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Rubi [A]
time = 0.27, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6177, 6129, 6037, 331, 212, 6135, 6079, 2497, 6141, 205} \begin {gather*} -a^2 \text {Li}_2\left (\frac {2}{a x+1}-1\right )+\frac {a^2 \tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}+a^2 \tanh ^{-1}(a x)^2+\frac {1}{4} a^2 \tanh ^{-1}(a x)+2 a^2 \log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)-\frac {a^3 x}{4 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)}{2 x^2}-\frac {a}{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcTanh[a*x]/(x^3*(1 - a^2*x^2)^2),x]

[Out]

-1/2*a/x - (a^3*x)/(4*(1 - a^2*x^2)) + (a^2*ArcTanh[a*x])/4 - ArcTanh[a*x]/(2*x^2) + (a^2*ArcTanh[a*x])/(2*(1
- a^2*x^2)) + a^2*ArcTanh[a*x]^2 + 2*a^2*ArcTanh[a*x]*Log[2 - 2/(1 + a*x)] - a^2*PolyLog[2, -1 + 2/(1 + a*x)]

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

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 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 2497

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/D[u, x])]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rule 6037

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcTanh[c*
x^n])^p/(m + 1)), x] - Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcTanh[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
]

Rule 6079

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_Symbol] :> Simp[(a + b*ArcTanh[c*x
])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - Dist[b*c*(p/d), Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[2 - 2/(1 + e*(x/
d))]/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0]

Rule 6129

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/d
, Int[(f*x)^m*(a + b*ArcTanh[c*x])^p, x], x] - Dist[e/(d*f^2), Int[(f*x)^(m + 2)*((a + b*ArcTanh[c*x])^p/(d +
e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]

Rule 6135

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[(a + b*ArcTanh[c
*x])^(p + 1)/(b*d*(p + 1)), x] + Dist[1/d, Int[(a + b*ArcTanh[c*x])^p/(x*(1 + c*x)), x], x] /; FreeQ[{a, b, c,
d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]

Rule 6141

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[(d + e*x^2)^
(q + 1)*((a + b*ArcTanh[c*x])^p/(2*e*(q + 1))), x] + Dist[b*(p/(2*c*(q + 1))), Int[(d + e*x^2)^q*(a + b*ArcTan
h[c*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0] && NeQ[q, -1]

Rule 6177

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Dist[1/d, Int
[x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^p, x], x] - Dist[e/d, Int[x^(m + 2)*(d + e*x^2)^q*(a + b*ArcTanh
[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegersQ[p, 2*q] && LtQ[q, -1] && ILtQ[
m, 0] && NeQ[p, -1]

Rubi steps

\begin {align*} \int \frac {\tanh ^{-1}(a x)}{x^3 \left (1-a^2 x^2\right )^2} \, dx &=a^2 \int \frac {\tanh ^{-1}(a x)}{x \left (1-a^2 x^2\right )^2} \, dx+\int \frac {\tanh ^{-1}(a x)}{x^3 \left (1-a^2 x^2\right )} \, dx\\ &=2 \left (a^2 \int \frac {\tanh ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx\right )+a^4 \int \frac {x \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx+\int \frac {\tanh ^{-1}(a x)}{x^3} \, dx\\ &=-\frac {\tanh ^{-1}(a x)}{2 x^2}+\frac {a^2 \tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {1}{2} a \int \frac {1}{x^2 \left (1-a^2 x^2\right )} \, dx+2 \left (\frac {1}{2} a^2 \tanh ^{-1}(a x)^2+a^2 \int \frac {\tanh ^{-1}(a x)}{x (1+a x)} \, dx\right )-\frac {1}{2} a^3 \int \frac {1}{\left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac {a}{2 x}-\frac {a^3 x}{4 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)}{2 x^2}+\frac {a^2 \tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4} a^3 \int \frac {1}{1-a^2 x^2} \, dx+\frac {1}{2} a^3 \int \frac {1}{1-a^2 x^2} \, dx+2 \left (\frac {1}{2} a^2 \tanh ^{-1}(a x)^2+a^2 \tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-a^3 \int \frac {\log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\right )\\ &=-\frac {a}{2 x}-\frac {a^3 x}{4 \left (1-a^2 x^2\right )}+\frac {1}{4} a^2 \tanh ^{-1}(a x)-\frac {\tanh ^{-1}(a x)}{2 x^2}+\frac {a^2 \tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}+2 \left (\frac {1}{2} a^2 \tanh ^{-1}(a x)^2+a^2 \tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-\frac {1}{2} a^2 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )\right )\\ \end {align*}

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Mathematica [A]
time = 0.24, size = 83, normalized size = 0.67 \begin {gather*} \frac {1}{8} a^2 \left (-\frac {4}{a x}+8 \tanh ^{-1}(a x)^2+2 \tanh ^{-1}(a x) \left (2-\frac {2}{a^2 x^2}+\cosh \left (2 \tanh ^{-1}(a x)\right )+8 \log \left (1-e^{-2 \tanh ^{-1}(a x)}\right )\right )-8 \text {PolyLog}\left (2,e^{-2 \tanh ^{-1}(a x)}\right )-\sinh \left (2 \tanh ^{-1}(a x)\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcTanh[a*x]/(x^3*(1 - a^2*x^2)^2),x]

[Out]

(a^2*(-4/(a*x) + 8*ArcTanh[a*x]^2 + 2*ArcTanh[a*x]*(2 - 2/(a^2*x^2) + Cosh[2*ArcTanh[a*x]] + 8*Log[1 - E^(-2*A
rcTanh[a*x])]) - 8*PolyLog[2, E^(-2*ArcTanh[a*x])] - Sinh[2*ArcTanh[a*x]]))/8

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Maple [A]
time = 0.44, size = 213, normalized size = 1.73

method	result	size
derivativedivides	\(a^{2} \left (-\frac {\arctanh \left (a x \right )}{2 a^{2} x^{2}}+2 \arctanh \left (a x \right ) \ln \left (a x \right )+\frac {\arctanh \left (a x \right )}{4 a x +4}-\arctanh \left (a x \right ) \ln \left (a x +1\right )-\frac {\arctanh \left (a x \right )}{4 \left (a x -1\right )}-\arctanh \left (a x \right ) \ln \left (a x -1\right )-\dilog \left (a x \right )-\dilog \left (a x +1\right )-\ln \left (a x \right ) \ln \left (a x +1\right )-\frac {\ln \left (a x -1\right )^{2}}{4}+\dilog \left (\frac {a x}{2}+\frac {1}{2}\right )+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{2}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{2}+\frac {\ln \left (a x +1\right )^{2}}{4}-\frac {1}{2 a x}+\frac {1}{8 a x +8}+\frac {\ln \left (a x +1\right )}{8}+\frac {1}{8 a x -8}-\frac {\ln \left (a x -1\right )}{8}\right )\)	\(213\)
default	\(a^{2} \left (-\frac {\arctanh \left (a x \right )}{2 a^{2} x^{2}}+2 \arctanh \left (a x \right ) \ln \left (a x \right )+\frac {\arctanh \left (a x \right )}{4 a x +4}-\arctanh \left (a x \right ) \ln \left (a x +1\right )-\frac {\arctanh \left (a x \right )}{4 \left (a x -1\right )}-\arctanh \left (a x \right ) \ln \left (a x -1\right )-\dilog \left (a x \right )-\dilog \left (a x +1\right )-\ln \left (a x \right ) \ln \left (a x +1\right )-\frac {\ln \left (a x -1\right )^{2}}{4}+\dilog \left (\frac {a x}{2}+\frac {1}{2}\right )+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{2}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{2}+\frac {\ln \left (a x +1\right )^{2}}{4}-\frac {1}{2 a x}+\frac {1}{8 a x +8}+\frac {\ln \left (a x +1\right )}{8}+\frac {1}{8 a x -8}-\frac {\ln \left (a x -1\right )}{8}\right )\)	\(213\)
risch	\(-\frac {a}{2 x}+\frac {a^{2} \ln \left (a x -1\right )}{16}-\frac {a^{2} \ln \left (-a x -1\right )}{16}-\frac {a^{2}}{8 \left (-a x +1\right )}-\frac {a^{2} \ln \left (-a x +1\right )}{8 \left (-a x +1\right )}+\frac {a^{2} \ln \left (\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (-a x +1\right )}{2}+\frac {a^{2} \ln \left (-a x +1\right )}{-16 a x -16}+\frac {a^{2} \ln \left (a x +1\right )}{8 a x +8}+\frac {a^{2} \ln \left (-a x +1\right )^{2}}{4}-\frac {a^{2} \ln \left (a x +1\right )^{2}}{4}-\frac {a^{2} \ln \left (a x +1\right )}{16 \left (a x -1\right )}-\frac {a^{2} \ln \left (a x \right )}{4}+\frac {a^{2} \ln \left (a x +1\right )}{4}-\frac {\ln \left (a x +1\right )}{4 x^{2}}+\frac {a^{2} \ln \left (-a x \right )}{4}-\frac {a^{2} \ln \left (-a x +1\right )}{4}+\frac {\ln \left (-a x +1\right )}{4 x^{2}}-\frac {a^{2} \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (a x +1\right )}{2}+\frac {a^{2}}{8 a x +8}-\frac {a^{3} \ln \left (-a x +1\right ) x}{16 \left (-a x -1\right )}-\frac {a^{3} \ln \left (a x +1\right ) x}{16 \left (a x -1\right )}-\frac {a^{2} \dilog \left (-\frac {a x}{2}+\frac {1}{2}\right )}{2}+\frac {a^{2} \dilog \left (\frac {a x}{2}+\frac {1}{2}\right )}{2}-a^{2} \dilog \left (a x +1\right )+a^{2} \dilog \left (-a x +1\right )\)	\(347\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctanh(a*x)/x^3/(-a^2*x^2+1)^2,x,method=_RETURNVERBOSE)

[Out]

a^2*(-1/2*arctanh(a*x)/a^2/x^2+2*arctanh(a*x)*ln(a*x)+1/4*arctanh(a*x)/(a*x+1)-arctanh(a*x)*ln(a*x+1)-1/4*arct
anh(a*x)/(a*x-1)-arctanh(a*x)*ln(a*x-1)-dilog(a*x)-dilog(a*x+1)-ln(a*x)*ln(a*x+1)-1/4*ln(a*x-1)^2+dilog(1/2*a*
x+1/2)+1/2*ln(a*x-1)*ln(1/2*a*x+1/2)-1/2*(ln(a*x+1)-ln(1/2*a*x+1/2))*ln(-1/2*a*x+1/2)+1/4*ln(a*x+1)^2-1/2/a/x+
1/8/(a*x+1)+1/8*ln(a*x+1)+1/8/(a*x-1)-1/8*ln(a*x-1))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (110) = 220\).
time = 0.27, size = 233, normalized size = 1.89 \begin {gather*} \frac {1}{8} \, {\left (8 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )} a - 8 \, {\left (\log \left (a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-a x\right )\right )} a + 8 \, {\left (\log \left (-a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (a x\right )\right )} a + a \log \left (a x + 1\right ) - a \log \left (a x - 1\right ) - \frac {2 \, {\left (a^{2} x^{2} - {\left (a^{3} x^{3} - a x\right )} \log \left (a x + 1\right )^{2} + 2 \, {\left (a^{3} x^{3} - a x\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) + {\left (a^{3} x^{3} - a x\right )} \log \left (a x - 1\right )^{2} - 2\right )}}{a^{2} x^{3} - x}\right )} a - \frac {1}{2} \, {\left (2 \, a^{2} \log \left (a^{2} x^{2} - 1\right ) - 2 \, a^{2} \log \left (x^{2}\right ) + \frac {2 \, a^{2} x^{2} - 1}{a^{2} x^{4} - x^{2}}\right )} \operatorname {artanh}\left (a x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x)/x^3/(-a^2*x^2+1)^2,x, algorithm="maxima")

[Out]

1/8*(8*(log(a*x - 1)*log(1/2*a*x + 1/2) + dilog(-1/2*a*x + 1/2))*a - 8*(log(a*x + 1)*log(x) + dilog(-a*x))*a +
8*(log(-a*x + 1)*log(x) + dilog(a*x))*a + a*log(a*x + 1) - a*log(a*x - 1) - 2*(a^2*x^2 - (a^3*x^3 - a*x)*log(
a*x + 1)^2 + 2*(a^3*x^3 - a*x)*log(a*x + 1)*log(a*x - 1) + (a^3*x^3 - a*x)*log(a*x - 1)^2 - 2)/(a^2*x^3 - x))*
a - 1/2*(2*a^2*log(a^2*x^2 - 1) - 2*a^2*log(x^2) + (2*a^2*x^2 - 1)/(a^2*x^4 - x^2))*arctanh(a*x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x)/x^3/(-a^2*x^2+1)^2,x, algorithm="fricas")

[Out]

integral(arctanh(a*x)/(a^4*x^7 - 2*a^2*x^5 + x^3), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {atanh}{\left (a x \right )}}{x^{3} \left (a x - 1\right )^{2} \left (a x + 1\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atanh(a*x)/x**3/(-a**2*x**2+1)**2,x)

[Out]

Integral(atanh(a*x)/(x**3*(a*x - 1)**2*(a*x + 1)**2), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x)/x^3/(-a^2*x^2+1)^2,x, algorithm="giac")

[Out]

integrate(arctanh(a*x)/((a^2*x^2 - 1)^2*x^3), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\mathrm {atanh}\left (a\,x\right )}{x^3\,{\left (a^2\,x^2-1\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atanh(a*x)/(x^3*(a^2*x^2 - 1)^2),x)

[Out]

int(atanh(a*x)/(x^3*(a^2*x^2 - 1)^2), x)

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