3.3.40 \(\int \frac {\tanh ^{-1}(a x)^2}{x^3 (1-a^2 x^2)} \, dx\) [240]

Optimal. Leaf size=138 \[ -\frac {a \tanh ^{-1}(a x)}{x}+\frac {1}{2} a^2 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{2 x^2}+\frac {1}{3} a^2 \tanh ^{-1}(a x)^3+a^2 \log (x)-\frac {1}{2} a^2 \log \left (1-a^2 x^2\right )+a^2 \tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-a^2 \tanh ^{-1}(a x) \text {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )-\frac {1}{2} a^2 \text {PolyLog}\left (3,-1+\frac {2}{1+a x}\right ) \]

[Out]

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

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Rubi [A]
time = 0.24, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6129, 6037, 272, 36, 29, 31, 6095, 6135, 6079, 6203, 6745} \begin {gather*} -\frac {1}{2} a^2 \text {Li}_3\left (\frac {2}{a x+1}-1\right )-a^2 \text {Li}_2\left (\frac {2}{a x+1}-1\right ) \tanh ^{-1}(a x)-\frac {1}{2} a^2 \log \left (1-a^2 x^2\right )+a^2 \log (x)+\frac {1}{3} a^2 \tanh ^{-1}(a x)^3+\frac {1}{2} a^2 \tanh ^{-1}(a x)^2+a^2 \log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{2 x^2}-\frac {a \tanh ^{-1}(a x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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 6095

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

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 6203

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

Rule 6745

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

Rubi steps

\begin {align*} \int \frac {\tanh ^{-1}(a x)^2}{x^3 \left (1-a^2 x^2\right )} \, dx &=a^2 \int \frac {\tanh ^{-1}(a x)^2}{x \left (1-a^2 x^2\right )} \, dx+\int \frac {\tanh ^{-1}(a x)^2}{x^3} \, dx\\ &=-\frac {\tanh ^{-1}(a x)^2}{2 x^2}+\frac {1}{3} a^2 \tanh ^{-1}(a x)^3+a \int \frac {\tanh ^{-1}(a x)}{x^2 \left (1-a^2 x^2\right )} \, dx+a^2 \int \frac {\tanh ^{-1}(a x)^2}{x (1+a x)} \, dx\\ &=-\frac {\tanh ^{-1}(a x)^2}{2 x^2}+\frac {1}{3} a^2 \tanh ^{-1}(a x)^3+a^2 \tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )+a \int \frac {\tanh ^{-1}(a x)}{x^2} \, dx+a^3 \int \frac {\tanh ^{-1}(a x)}{1-a^2 x^2} \, dx-\left (2 a^3\right ) \int \frac {\tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac {a \tanh ^{-1}(a x)}{x}+\frac {1}{2} a^2 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{2 x^2}+\frac {1}{3} a^2 \tanh ^{-1}(a x)^3+a^2 \tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-a^2 \tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+a x}\right )+a^2 \int \frac {1}{x \left (1-a^2 x^2\right )} \, dx+a^3 \int \frac {\text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac {a \tanh ^{-1}(a x)}{x}+\frac {1}{2} a^2 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{2 x^2}+\frac {1}{3} a^2 \tanh ^{-1}(a x)^3+a^2 \tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-a^2 \tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+a x}\right )-\frac {1}{2} a^2 \text {Li}_3\left (-1+\frac {2}{1+a x}\right )+\frac {1}{2} a^2 \text {Subst}\left (\int \frac {1}{x \left (1-a^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac {a \tanh ^{-1}(a x)}{x}+\frac {1}{2} a^2 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{2 x^2}+\frac {1}{3} a^2 \tanh ^{-1}(a x)^3+a^2 \tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-a^2 \tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+a x}\right )-\frac {1}{2} a^2 \text {Li}_3\left (-1+\frac {2}{1+a x}\right )+\frac {1}{2} a^2 \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\frac {1}{2} a^4 \text {Subst}\left (\int \frac {1}{1-a^2 x} \, dx,x,x^2\right )\\ &=-\frac {a \tanh ^{-1}(a x)}{x}+\frac {1}{2} a^2 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{2 x^2}+\frac {1}{3} a^2 \tanh ^{-1}(a x)^3+a^2 \log (x)-\frac {1}{2} a^2 \log \left (1-a^2 x^2\right )+a^2 \tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-a^2 \tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+a x}\right )-\frac {1}{2} a^2 \text {Li}_3\left (-1+\frac {2}{1+a x}\right )\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.24, size = 133, normalized size = 0.96 \begin {gather*} -a^2 \left (-\frac {i \pi ^3}{24}+\frac {\tanh ^{-1}(a x)}{a x}+\frac {\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{2 a^2 x^2}+\frac {1}{3} \tanh ^{-1}(a x)^3-\tanh ^{-1}(a x)^2 \log \left (1-e^{2 \tanh ^{-1}(a x)}\right )-\log \left (\frac {a x}{\sqrt {1-a^2 x^2}}\right )-\tanh ^{-1}(a x) \text {PolyLog}\left (2,e^{2 \tanh ^{-1}(a x)}\right )+\frac {1}{2} \text {PolyLog}\left (3,e^{2 \tanh ^{-1}(a x)}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^2*((-1/24*I)*Pi^3 + ArcTanh[a*x]/(a*x) + ((1 - a^2*x^2)*ArcTanh[a*x]^2)/(2*a^2*x^2) + ArcTanh[a*x]^3/3 - A
rcTanh[a*x]^2*Log[1 - E^(2*ArcTanh[a*x])] - Log[(a*x)/Sqrt[1 - a^2*x^2]] - ArcTanh[a*x]*PolyLog[2, E^(2*ArcTan
h[a*x])] + PolyLog[3, E^(2*ArcTanh[a*x])]/2))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 85.71, size = 1277, normalized size = 9.25

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1277\)
default	\(\text {Expression too large to display}\)	\(1277\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^2*(-1/2*arctanh(a*x)^2*ln(a*x-1)-1/2*arctanh(a*x)^2*ln(a*x+1)-1/2*arctanh(a*x)^2/a^2/x^2+arctanh(a*x)^2*ln(a
*x)+arctanh(a*x)^2*ln((a*x+1)/(-a^2*x^2+1)^(1/2))-arctanh(a*x)^2*ln((a*x+1)^2/(-a^2*x^2+1)-1)+arctanh(a*x)^2*l
n(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+2*arctanh(a*x)*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))-2*polylog(3,(a*x+1)/(-a^2
*x^2+1)^(1/2))+arctanh(a*x)^2*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+2*arctanh(a*x)*polylog(2,-(a*x+1)/(-a^2*x^2+1)^
(1/2))-2*polylog(3,-(a*x+1)/(-a^2*x^2+1)^(1/2))+1/12*arctanh(a*x)*(-6*I*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^2*P
i*arctanh(a*x)*a*x+6*I*csgn(I*((a*x+1)^2/(-a^2*x^2+1)-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^3*Pi*arctanh(a*x)*a*x+3*I
*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^2*Pi*arctanh(a*x)
*a*x-6*I*csgn(I*((a*x+1)^2/(-a^2*x^2+1)-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^2*csgn(I*((a*x+1)^2/(-a^2*x^2+1)-1))*Pi
*arctanh(a*x)*a*x-6*I*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))*csgn(I*((a*x+1)^2/(-a^2*x^2+1)-1)/((a*x+1)^2/(-a^2*x^
2+1)+1))^2*Pi*arctanh(a*x)*a*x+6*I*Pi*arctanh(a*x)*a*x+3*I*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3*Pi*arctanh(a*x)*a*x
+6*I*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))*csgn(I*(a*x+1)^2/(a^2*x^2-1))^2*Pi*arctanh(a*x)*a*x+3*I*csgn(I*(a*x+1)
/(-a^2*x^2+1)^(1/2))^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))*Pi*arctanh(a*x)*a*x-3*I*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csg
n(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^2*Pi*arctanh(a*x)*a*x+6*I*csgn(I/((a*x+1)^2/(-a^2*x^2+1)
+1))^3*Pi*arctanh(a*x)*a*x+6*I*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))*csgn(I*((a*x+1)^2/(-a^2*x^2+1)-1)/((a*x+1)^2
/(-a^2*x^2+1)+1))*csgn(I*((a*x+1)^2/(-a^2*x^2+1)-1))*Pi*arctanh(a*x)*a*x+3*I*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*
x+1)^2/(-a^2*x^2+1)+1))^3*Pi*arctanh(a*x)*a*x-3*I*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I/((a*x+1)^2/(-a^2*x^2+1)
+1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))*Pi*arctanh(a*x)*a*x+12*ln(2)*arctanh(a*x)*a*x-4*
arctanh(a*x)^2*a*x+6*a*x*arctanh(a*x)-12*a*x-12)/a/x+ln((a*x+1)/(-a^2*x^2+1)^(1/2)-1)+ln(1+(a*x+1)/(-a^2*x^2+1
)^(1/2)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(a^2*x^2*log(-a*x + 1)^3 + 3*(a^2*x^2*log(a*x + 1) + 1)*log(-a*x + 1)^2)/x^2 + 1/4*integrate(-(log(a*x +
1)^2 - (a^2*x^2 + a*x + (a^4*x^4 + a^3*x^3 + 2)*log(a*x + 1))*log(-a*x + 1))/(a^2*x^5 - x^3), 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)^2/x^3/(-a^2*x^2+1),x, algorithm="fricas")

[Out]

integral(-arctanh(a*x)^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}^{2}{\left (a x \right )}}{a^{2} x^{5} - x^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(atanh(a*x)**2/(a**2*x**5 - x**3), 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)^2/x^3/(-a^2*x^2+1),x, algorithm="giac")

[Out]

integrate(-arctanh(a*x)^2/((a^2*x^2 - 1)*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 )}^2}{x^3\,\left (a^2\,x^2-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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