3.3.43 \(\int \frac {x \tanh ^{-1}(a x)^3}{1-a^2 x^2} \, dx\) [243]

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

[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 = 0.15, 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 = {6131, 6055, 6095, 6205, 6209, 6745} \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*ArcTanh[a*x]^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 6055

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 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 6131

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 6205

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 6209

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 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 {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 = 0.05, size = 87, normalized size = 0.81 \begin {gather*} -\frac {-\tanh ^{-1}(a x)^4-4 \tanh ^{-1}(a x)^3 \log \left (1+e^{-2 \tanh ^{-1}(a x)}\right )+6 \tanh ^{-1}(a x)^2 \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(a x)}\right )+6 \tanh ^{-1}(a x) \text {PolyLog}\left (3,-e^{-2 \tanh ^{-1}(a x)}\right )+3 \text {PolyLog}\left (4,-e^{-2 \tanh ^{-1}(a x)}\right )}{4 a^2} \end {gather*}

Antiderivative was successfully verified.

[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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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 46.66, size = 670, normalized size = 6.20

method	result	size
derivativedivides	\(\frac {-\frac {\arctanh \left (a x \right )^{3} \ln \left (a x -1\right )}{2}-\frac {\arctanh \left (a x \right )^{3} \ln \left (a x +1\right )}{2}+\arctanh \left (a x \right )^{3} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\frac {\arctanh \left (a x \right )^{4}}{4}+\frac {\left (i \pi \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{3}+2 i \pi \,\mathrm {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right ) \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{2}-i \pi \,\mathrm {csgn}\left (\frac {i}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right ) \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 (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )}\right )-i \pi \,\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 (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )}\right )^{2}+i \pi \mathrm {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )^{2} \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )+2 i \pi \mathrm {csgn}\left (\frac {i}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right )^{3}+i \pi \,\mathrm {csgn}\left (\frac {i}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right ) \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )}\right )^{2}+i \pi \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )}\right )^{3}-2 i \pi \mathrm {csgn}\left (\frac {i}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right )^{2}+2 i \pi +4 \ln \left (2\right )\right ) \arctanh \left (a x \right )^{3}}{4}+\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}-\frac {3 \arctanh \left (a x \right ) \polylog \left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2}+\frac {3 \polylog \left (4, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{4}}{a^{2}}\)	\(670\)
default	\(\frac {-\frac {\arctanh \left (a x \right )^{3} \ln \left (a x -1\right )}{2}-\frac {\arctanh \left (a x \right )^{3} \ln \left (a x +1\right )}{2}+\arctanh \left (a x \right )^{3} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\frac {\arctanh \left (a x \right )^{4}}{4}+\frac {\left (i \pi \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{3}+2 i \pi \,\mathrm {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right ) \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{2}-i \pi \,\mathrm {csgn}\left (\frac {i}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right ) \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 (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )}\right )-i \pi \,\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 (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )}\right )^{2}+i \pi \mathrm {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )^{2} \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )+2 i \pi \mathrm {csgn}\left (\frac {i}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right )^{3}+i \pi \,\mathrm {csgn}\left (\frac {i}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right ) \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )}\right )^{2}+i \pi \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )}\right )^{3}-2 i \pi \mathrm {csgn}\left (\frac {i}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right )^{2}+2 i \pi +4 \ln \left (2\right )\right ) \arctanh \left (a x \right )^{3}}{4}+\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}-\frac {3 \arctanh \left (a x \right ) \polylog \left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2}+\frac {3 \polylog \left (4, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{4}}{a^{2}}\)	\(670\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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

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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