3.136 \(\int \frac{\tanh ^{-1}(a x)^4}{c-a c x} \, dx\)

Optimal. Leaf size=131 \[ -\frac{3 \text{PolyLog}\left (5,1-\frac{2}{1-a x}\right )}{2 a c}+\frac{2 \tanh ^{-1}(a x)^3 \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )}{a c}-\frac{3 \tanh ^{-1}(a x)^2 \text{PolyLog}\left (3,1-\frac{2}{1-a x}\right )}{a c}+\frac{3 \tanh ^{-1}(a x) \text{PolyLog}\left (4,1-\frac{2}{1-a x}\right )}{a c}+\frac{\log \left (\frac{2}{1-a x}\right ) \tanh ^{-1}(a x)^4}{a c} \]

[Out]

(ArcTanh[a*x]^4*Log[2/(1 - a*x)])/(a*c) + (2*ArcTanh[a*x]^3*PolyLog[2, 1 - 2/(1 - a*x)])/(a*c) - (3*ArcTanh[a*
x]^2*PolyLog[3, 1 - 2/(1 - a*x)])/(a*c) + (3*ArcTanh[a*x]*PolyLog[4, 1 - 2/(1 - a*x)])/(a*c) - (3*PolyLog[5, 1
- 2/(1 - a*x)])/(2*a*c)

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Rubi [A]  time = 0.214482, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {5918, 5948, 6058, 6062, 6610} \[ -\frac{3 \text{PolyLog}\left (5,1-\frac{2}{1-a x}\right )}{2 a c}+\frac{2 \tanh ^{-1}(a x)^3 \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )}{a c}-\frac{3 \tanh ^{-1}(a x)^2 \text{PolyLog}\left (3,1-\frac{2}{1-a x}\right )}{a c}+\frac{3 \tanh ^{-1}(a x) \text{PolyLog}\left (4,1-\frac{2}{1-a x}\right )}{a c}+\frac{\log \left (\frac{2}{1-a x}\right ) \tanh ^{-1}(a x)^4}{a c} \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[a*x]^4/(c - a*c*x),x]

[Out]

(ArcTanh[a*x]^4*Log[2/(1 - a*x)])/(a*c) + (2*ArcTanh[a*x]^3*PolyLog[2, 1 - 2/(1 - a*x)])/(a*c) - (3*ArcTanh[a*
x]^2*PolyLog[3, 1 - 2/(1 - a*x)])/(a*c) + (3*ArcTanh[a*x]*PolyLog[4, 1 - 2/(1 - a*x)])/(a*c) - (3*PolyLog[5, 1
- 2/(1 - a*x)])/(2*a*c)

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 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{\tanh ^{-1}(a x)^4}{c-a c x} \, dx &=\frac{\tanh ^{-1}(a x)^4 \log \left (\frac{2}{1-a x}\right )}{a c}-\frac{4 \int \frac{\tanh ^{-1}(a x)^3 \log \left (\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx}{c}\\ &=\frac{\tanh ^{-1}(a x)^4 \log \left (\frac{2}{1-a x}\right )}{a c}+\frac{2 \tanh ^{-1}(a x)^3 \text{Li}_2\left (1-\frac{2}{1-a x}\right )}{a c}-\frac{6 \int \frac{\tanh ^{-1}(a x)^2 \text{Li}_2\left (1-\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx}{c}\\ &=\frac{\tanh ^{-1}(a x)^4 \log \left (\frac{2}{1-a x}\right )}{a c}+\frac{2 \tanh ^{-1}(a x)^3 \text{Li}_2\left (1-\frac{2}{1-a x}\right )}{a c}-\frac{3 \tanh ^{-1}(a x)^2 \text{Li}_3\left (1-\frac{2}{1-a x}\right )}{a c}+\frac{6 \int \frac{\tanh ^{-1}(a x) \text{Li}_3\left (1-\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx}{c}\\ &=\frac{\tanh ^{-1}(a x)^4 \log \left (\frac{2}{1-a x}\right )}{a c}+\frac{2 \tanh ^{-1}(a x)^3 \text{Li}_2\left (1-\frac{2}{1-a x}\right )}{a c}-\frac{3 \tanh ^{-1}(a x)^2 \text{Li}_3\left (1-\frac{2}{1-a x}\right )}{a c}+\frac{3 \tanh ^{-1}(a x) \text{Li}_4\left (1-\frac{2}{1-a x}\right )}{a c}-\frac{3 \int \frac{\text{Li}_4\left (1-\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx}{c}\\ &=\frac{\tanh ^{-1}(a x)^4 \log \left (\frac{2}{1-a x}\right )}{a c}+\frac{2 \tanh ^{-1}(a x)^3 \text{Li}_2\left (1-\frac{2}{1-a x}\right )}{a c}-\frac{3 \tanh ^{-1}(a x)^2 \text{Li}_3\left (1-\frac{2}{1-a x}\right )}{a c}+\frac{3 \tanh ^{-1}(a x) \text{Li}_4\left (1-\frac{2}{1-a x}\right )}{a c}-\frac{3 \text{Li}_5\left (1-\frac{2}{1-a x}\right )}{2 a c}\\ \end{align*}

Mathematica [A]  time = 0.108939, size = 112, normalized size = 0.85 \[ -\frac{2 \tanh ^{-1}(a x)^3 \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(a x)}\right )+3 \tanh ^{-1}(a x)^2 \text{PolyLog}\left (3,-e^{-2 \tanh ^{-1}(a x)}\right )+3 \tanh ^{-1}(a x) \text{PolyLog}\left (4,-e^{-2 \tanh ^{-1}(a x)}\right )+\frac{3}{2} \text{PolyLog}\left (5,-e^{-2 \tanh ^{-1}(a x)}\right )-\frac{2}{5} \tanh ^{-1}(a x)^5-\tanh ^{-1}(a x)^4 \log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )}{a c} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcTanh[a*x]^4/(c - a*c*x),x]

[Out]

-(((-2*ArcTanh[a*x]^5)/5 - ArcTanh[a*x]^4*Log[1 + E^(-2*ArcTanh[a*x])] + 2*ArcTanh[a*x]^3*PolyLog[2, -E^(-2*Ar
cTanh[a*x])] + 3*ArcTanh[a*x]^2*PolyLog[3, -E^(-2*ArcTanh[a*x])] + 3*ArcTanh[a*x]*PolyLog[4, -E^(-2*ArcTanh[a*
x])] + (3*PolyLog[5, -E^(-2*ArcTanh[a*x])])/2)/(a*c))

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Maple [C]  time = 0.201, size = 285, normalized size = 2.2 \begin{align*} -{\frac{ \left ({\it Artanh} \left ( ax \right ) \right ) ^{4}\ln \left ( ax-1 \right ) }{ac}}+{\frac{i \left ({\it Artanh} \left ( ax \right ) \right ) ^{4}\pi }{ac} \left ({\it csgn} \left ({i \left ({\frac{ \left ( ax+1 \right ) ^{2}}{-{a}^{2}{x}^{2}+1}}+1 \right ) ^{-1}} \right ) \right ) ^{3}}-{\frac{i \left ({\it Artanh} \left ( ax \right ) \right ) ^{4}\pi }{ac} \left ({\it csgn} \left ({i \left ({\frac{ \left ( ax+1 \right ) ^{2}}{-{a}^{2}{x}^{2}+1}}+1 \right ) ^{-1}} \right ) \right ) ^{2}}+{\frac{i \left ({\it Artanh} \left ( ax \right ) \right ) ^{4}\pi }{ac}}+{\frac{ \left ({\it Artanh} \left ( ax \right ) \right ) ^{4}\ln \left ( 2 \right ) }{ac}}+2\,{\frac{ \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}}{ac}{\it polylog} \left ( 2,-{\frac{ \left ( ax+1 \right ) ^{2}}{-{a}^{2}{x}^{2}+1}} \right ) }-3\,{\frac{ \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{ac}{\it polylog} \left ( 3,-{\frac{ \left ( ax+1 \right ) ^{2}}{-{a}^{2}{x}^{2}+1}} \right ) }+3\,{\frac{{\it Artanh} \left ( ax \right ) }{ac}{\it polylog} \left ( 4,-{\frac{ \left ( ax+1 \right ) ^{2}}{-{a}^{2}{x}^{2}+1}} \right ) }-{\frac{3}{2\,ac}{\it polylog} \left ( 5,-{\frac{ \left ( ax+1 \right ) ^{2}}{-{a}^{2}{x}^{2}+1}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctanh(a*x)^4/(-a*c*x+c),x)

[Out]

-1/a/c*arctanh(a*x)^4*ln(a*x-1)+I/a/c*arctanh(a*x)^4*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^3*Pi-I/a/c*arctanh(a*x
)^4*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^2*Pi+I/a/c*arctanh(a*x)^4*Pi+1/a/c*arctanh(a*x)^4*ln(2)+2/a/c*arctanh(a
*x)^3*polylog(2,-(a*x+1)^2/(-a^2*x^2+1))-3/a/c*arctanh(a*x)^2*polylog(3,-(a*x+1)^2/(-a^2*x^2+1))+3/a/c*arctanh
(a*x)*polylog(4,-(a*x+1)^2/(-a^2*x^2+1))-3/2/a/c*polylog(5,-(a*x+1)^2/(-a^2*x^2+1))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{\log \left (-a x + 1\right )^{5}}{80 \, a c} + \frac{1}{16} \, \int -\frac{\log \left (a x + 1\right )^{4} - 4 \, \log \left (a x + 1\right )^{3} \log \left (-a x + 1\right ) + 6 \, \log \left (a x + 1\right )^{2} \log \left (-a x + 1\right )^{2} - 4 \, \log \left (a x + 1\right ) \log \left (-a x + 1\right )^{3}}{a c x - c}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x)^4/(-a*c*x+c),x, algorithm="maxima")

[Out]

-1/80*log(-a*x + 1)^5/(a*c) + 1/16*integrate(-(log(a*x + 1)^4 - 4*log(a*x + 1)^3*log(-a*x + 1) + 6*log(a*x + 1
)^2*log(-a*x + 1)^2 - 4*log(a*x + 1)*log(-a*x + 1)^3)/(a*c*x - c), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\operatorname{artanh}\left (a x\right )^{4}}{a c x - c}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x)^4/(-a*c*x+c),x, algorithm="fricas")

[Out]

integral(-arctanh(a*x)^4/(a*c*x - c), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{\operatorname{atanh}^{4}{\left (a x \right )}}{a x - 1}\, dx}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atanh(a*x)**4/(-a*c*x+c),x)

[Out]

-Integral(atanh(a*x)**4/(a*x - 1), x)/c

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{\operatorname{artanh}\left (a x\right )^{4}}{a c x - c}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x)^4/(-a*c*x+c),x, algorithm="giac")

[Out]

integrate(-arctanh(a*x)^4/(a*c*x - c), x)