3.2.83 \(\int (1-a^2 x^2) \tanh ^{-1}(a x)^3 \, dx\) [183]

Optimal. Leaf size=157 \[ -x \tanh ^{-1}(a x)+\frac {\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{2 a}+\frac {2 \tanh ^{-1}(a x)^3}{3 a}+\frac {2}{3} x \tanh ^{-1}(a x)^3+\frac {1}{3} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3-\frac {2 \tanh ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a}-\frac {\log \left (1-a^2 x^2\right )}{2 a}-\frac {2 \tanh ^{-1}(a x) \text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a}+\frac {\text {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{a} \]

[Out]

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

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Rubi [A]
time = 0.14, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {6091, 6021, 6131, 6055, 6095, 6205, 6745, 266} \begin {gather*} -\frac {\log \left (1-a^2 x^2\right )}{2 a}+\frac {1}{3} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3+\frac {\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{2 a}+\frac {\text {Li}_3\left (1-\frac {2}{1-a x}\right )}{a}-\frac {2 \text {Li}_2\left (1-\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)}{a}+\frac {2}{3} x \tanh ^{-1}(a x)^3+\frac {2 \tanh ^{-1}(a x)^3}{3 a}-x \tanh ^{-1}(a x)-\frac {2 \log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)^2}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - a^2*x^2)*ArcTanh[a*x]^3,x]

[Out]

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

Rule 266

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

Rule 6021

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

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 6091

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

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

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Mathematica [A]
time = 0.21, size = 134, normalized size = 0.85 \begin {gather*} -\frac {6 a x \tanh ^{-1}(a x)-3 \tanh ^{-1}(a x)^2+3 a^2 x^2 \tanh ^{-1}(a x)^2+4 \tanh ^{-1}(a x)^3-6 a x \tanh ^{-1}(a x)^3+2 a^3 x^3 \tanh ^{-1}(a x)^3+12 \tanh ^{-1}(a x)^2 \log \left (1+e^{-2 \tanh ^{-1}(a x)}\right )+3 \log \left (1-a^2 x^2\right )-12 \tanh ^{-1}(a x) \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(a x)}\right )-6 \text {PolyLog}\left (3,-e^{-2 \tanh ^{-1}(a x)}\right )}{6 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*(6*a*x*ArcTanh[a*x] - 3*ArcTanh[a*x]^2 + 3*a^2*x^2*ArcTanh[a*x]^2 + 4*ArcTanh[a*x]^3 - 6*a*x*ArcTanh[a*x]
^3 + 2*a^3*x^3*ArcTanh[a*x]^3 + 12*ArcTanh[a*x]^2*Log[1 + E^(-2*ArcTanh[a*x])] + 3*Log[1 - a^2*x^2] - 12*ArcTa
nh[a*x]*PolyLog[2, -E^(-2*ArcTanh[a*x])] - 6*PolyLog[3, -E^(-2*ArcTanh[a*x])])/a

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 24.34, size = 749, normalized size = 4.77 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a*(-1/3*arctanh(a*x)^3*a^3*x^3+arctanh(a*x)^3*a*x-1/2*a^2*x^2*arctanh(a*x)^2+arctanh(a*x)^2*ln(a*x-1)+arctan
h(a*x)^2*ln(a*x+1)-2*arctanh(a*x)^2*ln((a*x+1)/(-a^2*x^2+1)^(1/2))-1/6*arctanh(a*x)*(3*I*Pi*csgn(I*(a*x+1)^2/(
a^2*x^2-1))^3*arctanh(a*x)+6*I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))^2*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))*arctanh(a
*x)-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*arctanh(a*
x)-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))*csgn(I/((a*x+
1)^2/(-a^2*x^2+1)+1))*arctanh(a*x)+3*I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))^2*a
rctanh(a*x)+3*I*Pi*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*I*Pi*csgn(I*(a*x+
1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^2*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))*arctanh(a*x)+6*I*Pi*csgn(I/(
(a*x+1)^2/(-a^2*x^2+1)+1))^3*arctanh(a*x)-6*I*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^2*arctanh(a*x)+6*I*Pi*arct
anh(a*x)+12*ln(2)*arctanh(a*x)-4*arctanh(a*x)^2-3*arctanh(a*x)+6*a*x+6)+ln((a*x+1)^2/(-a^2*x^2+1)+1)-2*arctanh
(a*x)*polylog(2,-(a*x+1)^2/(-a^2*x^2+1))+polylog(3,-(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((-a^2*x^2+1)*arctanh(a*x)^3,x, algorithm="maxima")

[Out]

1/48*(2*a^3*x^3 - 3*a^2*x^2 - 12*a*x - 6*(a^3*x^3 - 3*a*x - 2)*log(a*x + 1))*log(-a*x + 1)^2/a - 1/8*(log(-a*x
+ 1)^3 - 3*log(-a*x + 1)^2 + 6*log(-a*x + 1) - 6)*(a*x - 1)/a + 1/864*(4*(9*log(-a*x + 1)^3 - 9*log(-a*x + 1)
^2 + 6*log(-a*x + 1) - 2)*(a*x - 1)^3 + 27*(4*log(-a*x + 1)^3 - 6*log(-a*x + 1)^2 + 6*log(-a*x + 1) - 3)*(a*x
- 1)^2 + 108*(log(-a*x + 1)^3 - 3*log(-a*x + 1)^2 + 6*log(-a*x + 1) - 6)*(a*x - 1))/a + 1/8*integrate(-1/3*(3*
(a^3*x^3 - a^2*x^2 - a*x + 1)*log(a*x + 1)^3 + (2*a^3*x^3 - 3*a^2*x^2 - 9*(a^3*x^3 - a^2*x^2 - a*x + 1)*log(a*
x + 1)^2 - 12*a*x - 6*(a^3*x^3 - 3*a*x - 2)*log(a*x + 1))*log(-a*x + 1))/(a*x - 1), 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((-a^2*x^2+1)*arctanh(a*x)^3,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

[Out]

-int(atanh(a*x)^3*(a^2*x^2 - 1), x)

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