3.243 \(\int \frac {x \tanh ^{-1}(a x)^3}{1-a^2 x^2} \, dx\)

Optimal. Leaf size=108 \[ \frac {3 \text {Li}_4\left (1-\frac {2}{1-a x}\right )}{4 a^2}+\frac {3 \text {Li}_2\left (1-\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)^2}{2 a^2}-\frac {3 \text {Li}_3\left (1-\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)}{2 a^2}-\frac {\tanh ^{-1}(a x)^4}{4 a^2}+\frac {\log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)^3}{a^2} \]

[Out]

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

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Rubi [A]  time = 0.21, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5984, 5918, 5948, 6058, 6062, 6610} \[ \frac {3 \text {PolyLog}\left (4,1-\frac {2}{1-a x}\right )}{4 a^2}+\frac {3 \tanh ^{-1}(a x)^2 \text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a^2}-\frac {3 \tanh ^{-1}(a x) \text {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{2 a^2}-\frac {\tanh ^{-1}(a x)^4}{4 a^2}+\frac {\log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)^3}{a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5918

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTanh[c*x])^p*
Log[2/(1 + (e*x)/d)])/e, x] + Dist[(b*c*p)/e, Int[((a + b*ArcTanh[c*x])^(p - 1)*Log[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 5948

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 5984

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

Rule 6058

Int[(Log[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[((a + b*ArcT
anh[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 6062

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

Rule 6610

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 {x \tanh ^{-1}(a x)^3}{1-a^2 x^2} \, dx &=-\frac {\tanh ^{-1}(a x)^4}{4 a^2}+\frac {\int \frac {\tanh ^{-1}(a x)^3}{1-a x} \, dx}{a}\\ &=-\frac {\tanh ^{-1}(a x)^4}{4 a^2}+\frac {\tanh ^{-1}(a x)^3 \log \left (\frac {2}{1-a x}\right )}{a^2}-\frac {3 \int \frac {\tanh ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a}\\ &=-\frac {\tanh ^{-1}(a x)^4}{4 a^2}+\frac {\tanh ^{-1}(a x)^3 \log \left (\frac {2}{1-a x}\right )}{a^2}+\frac {3 \tanh ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{2 a^2}-\frac {3 \int \frac {\tanh ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a}\\ &=-\frac {\tanh ^{-1}(a x)^4}{4 a^2}+\frac {\tanh ^{-1}(a x)^3 \log \left (\frac {2}{1-a x}\right )}{a^2}+\frac {3 \tanh ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{2 a^2}-\frac {3 \tanh ^{-1}(a x) \text {Li}_3\left (1-\frac {2}{1-a x}\right )}{2 a^2}+\frac {3 \int \frac {\text {Li}_3\left (1-\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{2 a}\\ &=-\frac {\tanh ^{-1}(a x)^4}{4 a^2}+\frac {\tanh ^{-1}(a x)^3 \log \left (\frac {2}{1-a x}\right )}{a^2}+\frac {3 \tanh ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{2 a^2}-\frac {3 \tanh ^{-1}(a x) \text {Li}_3\left (1-\frac {2}{1-a x}\right )}{2 a^2}+\frac {3 \text {Li}_4\left (1-\frac {2}{1-a x}\right )}{4 a^2}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 87, normalized size = 0.81 \[ -\frac {6 \tanh ^{-1}(a x)^2 \text {Li}_2\left (-e^{-2 \tanh ^{-1}(a x)}\right )+6 \tanh ^{-1}(a x) \text {Li}_3\left (-e^{-2 \tanh ^{-1}(a x)}\right )+3 \text {Li}_4\left (-e^{-2 \tanh ^{-1}(a x)}\right )-\tanh ^{-1}(a x)^4-4 \tanh ^{-1}(a x)^3 \log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )}{4 a^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/4*(-ArcTanh[a*x]^4 - 4*ArcTanh[a*x]^3*Log[1 + E^(-2*ArcTanh[a*x])] + 6*ArcTanh[a*x]^2*PolyLog[2, -E^(-2*Arc
Tanh[a*x])] + 6*ArcTanh[a*x]*PolyLog[3, -E^(-2*ArcTanh[a*x])] + 3*PolyLog[4, -E^(-2*ArcTanh[a*x])])/a^2

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fricas [F]  time = 1.02, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {x \operatorname {artanh}\left (a x\right )^{3}}{a^{2} x^{2} - 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {x \operatorname {artanh}\left (a x\right )^{3}}{a^{2} x^{2} - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.52, size = 776, normalized size = 7.19 \[ -\frac {\arctanh \left (a x \right )^{3} \ln \left (a x -1\right )}{2 a^{2}}-\frac {\arctanh \left (a x \right )^{3} \ln \left (a x +1\right )}{2 a^{2}}+\frac {\arctanh \left (a x \right )^{3} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2}}-\frac {\arctanh \left (a x \right )^{4}}{4 a^{2}}+\frac {3 \arctanh \left (a x \right )^{2} \polylog \left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2 a^{2}}-\frac {3 \arctanh \left (a x \right ) \polylog \left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2 a^{2}}+\frac {3 \polylog \left (4, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{4 a^{2}}+\frac {i \arctanh \left (a x \right )^{3} \pi }{2 a^{2}}+\frac {i \arctanh \left (a x \right )^{3} \mathrm {csgn}\left (\frac {i}{1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}}\right ) \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}\right )^{2} \pi }{4 a^{2}}+\frac {i \arctanh \left (a x \right )^{3} \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \mathrm {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )^{2} \pi }{4 a^{2}}+\frac {i \arctanh \left (a x \right )^{3} \mathrm {csgn}\left (\frac {i}{1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}}\right )^{3} \pi }{2 a^{2}}-\frac {i \arctanh \left (a x \right )^{3} \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \mathrm {csgn}\left (\frac {i}{1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}}\right ) \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}\right ) \pi }{4 a^{2}}+\frac {i \arctanh \left (a x \right )^{3} \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{2} \mathrm {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right ) \pi }{2 a^{2}}+\frac {i \arctanh \left (a x \right )^{3} \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}\right )^{3} \pi }{4 a^{2}}-\frac {i \arctanh \left (a x \right )^{3} \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}\right )^{2} \pi }{4 a^{2}}+\frac {i \arctanh \left (a x \right )^{3} \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{3} \pi }{4 a^{2}}-\frac {i \arctanh \left (a x \right )^{3} \mathrm {csgn}\left (\frac {i}{1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}}\right )^{2} \pi }{2 a^{2}}+\frac {\arctanh \left (a x \right )^{3} \ln \relax (2)}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {4 \, \log \left (a x + 1\right ) \log \left (-a x + 1\right )^{3} + \log \left (-a x + 1\right )^{4}}{64 \, a^{2}} - \frac {1}{8} \, \int \frac {2 \, a x \log \left (a x + 1\right )^{3} - 6 \, a x \log \left (a x + 1\right )^{2} \log \left (-a x + 1\right ) + 3 \, {\left (3 \, a x + 1\right )} \log \left (a x + 1\right ) \log \left (-a x + 1\right )^{2}}{2 \, {\left (a^{3} x^{2} - a\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(4*log(a*x + 1)*log(-a*x + 1)^3 + log(-a*x + 1)^4)/a^2 - 1/8*integrate(1/2*(2*a*x*log(a*x + 1)^3 - 6*a*x*
log(a*x + 1)^2*log(-a*x + 1) + 3*(3*a*x + 1)*log(a*x + 1)*log(-a*x + 1)^2)/(a^3*x^2 - a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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