3.128 \(\int \frac{x \tanh ^{-1}(a x)^3}{c+a c x} \, dx\)

Optimal. Leaf size=205 \[ \frac{3 \text{PolyLog}\left (3,1-\frac{2}{1-a x}\right )}{2 a^2 c}-\frac{3 \text{PolyLog}\left (4,1-\frac{2}{a x+1}\right )}{4 a^2 c}-\frac{3 \tanh ^{-1}(a x)^2 \text{PolyLog}\left (2,1-\frac{2}{a x+1}\right )}{2 a^2 c}-\frac{3 \tanh ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )}{a^2 c}-\frac{3 \tanh ^{-1}(a x) \text{PolyLog}\left (3,1-\frac{2}{a x+1}\right )}{2 a^2 c}+\frac{\tanh ^{-1}(a x)^3}{a^2 c}+\frac{\log \left (\frac{2}{a x+1}\right ) \tanh ^{-1}(a x)^3}{a^2 c}-\frac{3 \log \left (\frac{2}{1-a x}\right ) \tanh ^{-1}(a x)^2}{a^2 c}+\frac{x \tanh ^{-1}(a x)^3}{a c} \]

[Out]

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

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Rubi [A]  time = 0.374202, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.562, Rules used = {5930, 5910, 5984, 5918, 5948, 6058, 6610, 6056, 6060} \[ \frac{3 \text{PolyLog}\left (3,1-\frac{2}{1-a x}\right )}{2 a^2 c}-\frac{3 \text{PolyLog}\left (4,1-\frac{2}{a x+1}\right )}{4 a^2 c}-\frac{3 \tanh ^{-1}(a x)^2 \text{PolyLog}\left (2,1-\frac{2}{a x+1}\right )}{2 a^2 c}-\frac{3 \tanh ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )}{a^2 c}-\frac{3 \tanh ^{-1}(a x) \text{PolyLog}\left (3,1-\frac{2}{a x+1}\right )}{2 a^2 c}+\frac{\tanh ^{-1}(a x)^3}{a^2 c}+\frac{\log \left (\frac{2}{a x+1}\right ) \tanh ^{-1}(a x)^3}{a^2 c}-\frac{3 \log \left (\frac{2}{1-a x}\right ) \tanh ^{-1}(a x)^2}{a^2 c}+\frac{x \tanh ^{-1}(a x)^3}{a c} \]

Antiderivative was successfully verified.

[In]

Int[(x*ArcTanh[a*x]^3)/(c + a*c*x),x]

[Out]

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

Rule 5930

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

Rule 5910

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

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 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 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]

Rule 6056

Int[(Log[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[((a + b*ArcTa
nh[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 6060

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]

Rubi steps

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

Mathematica [A]  time = 0.274517, size = 126, normalized size = 0.61 \[ \frac{-\frac{3}{2} \left (\tanh ^{-1}(a x)-2\right ) \tanh ^{-1}(a x) \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(a x)}\right )-\frac{3}{2} \left (\tanh ^{-1}(a x)-1\right ) \text{PolyLog}\left (3,-e^{-2 \tanh ^{-1}(a x)}\right )-\frac{3}{4} \text{PolyLog}\left (4,-e^{-2 \tanh ^{-1}(a x)}\right )+a x \tanh ^{-1}(a x)^3-\tanh ^{-1}(a x)^3+\tanh ^{-1}(a x)^3 \log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )-3 \tanh ^{-1}(a x)^2 \log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )}{a^2 c} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(x*ArcTanh[a*x]^3)/(c + a*c*x),x]

[Out]

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

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Maple [C]  time = 0.449, size = 833, normalized size = 4.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*arctanh(a*x)^3/(a*c*x+c),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (a x - \log \left (a x + 1\right )\right )} \log \left (-a x + 1\right )^{3}}{8 \, a^{2} c} + \frac{1}{8} \, \int \frac{{\left (a^{2} x^{2} - a x\right )} \log \left (a x + 1\right )^{3} - 3 \,{\left (a^{2} x^{2} - a x\right )} \log \left (a x + 1\right )^{2} \log \left (-a x + 1\right ) + 3 \,{\left (a^{2} x^{2} + a x +{\left (a^{2} x^{2} - 2 \, a x - 1\right )} \log \left (a x + 1\right )\right )} \log \left (-a x + 1\right )^{2}}{a^{3} c x^{2} - a c}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x \operatorname{artanh}\left (a x\right )^{3}}{a c x + c}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x*arctanh(a*x)^3/(a*c*x + c), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*atanh(a*x)**3/(a*c*x+c),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \operatorname{artanh}\left (a x\right )^{3}}{a c x + c}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x*arctanh(a*x)^3/(a*c*x + c), x)