3.3.78 \(\int \frac {\tanh ^{-1}(a x)^3}{x^2 (1-a^2 x^2)^2} \, dx\) [278]

Optimal. Leaf size=191 \[ -\frac {3 a}{8 \left (1-a^2 x^2\right )}+\frac {3 a^2 x \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )}+\frac {3}{8} a \tanh ^{-1}(a x)^2-\frac {3 a \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}+a \tanh ^{-1}(a x)^3-\frac {\tanh ^{-1}(a x)^3}{x}+\frac {a^2 x \tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {3}{8} a \tanh ^{-1}(a x)^4+3 a \tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-3 a \tanh ^{-1}(a x) \text {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )-\frac {3}{2} a \text {PolyLog}\left (3,-1+\frac {2}{1+a x}\right ) \]

[Out]

-3/8*a/(-a^2*x^2+1)+3/4*a^2*x*arctanh(a*x)/(-a^2*x^2+1)+3/8*a*arctanh(a*x)^2-3/4*a*arctanh(a*x)^2/(-a^2*x^2+1)
+a*arctanh(a*x)^3-arctanh(a*x)^3/x+1/2*a^2*x*arctanh(a*x)^3/(-a^2*x^2+1)+3/8*a*arctanh(a*x)^4+3*a*arctanh(a*x)
^2*ln(2-2/(a*x+1))-3*a*arctanh(a*x)*polylog(2,-1+2/(a*x+1))-3/2*a*polylog(3,-1+2/(a*x+1))

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Rubi [A]
time = 0.31, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6177, 6129, 6037, 6135, 6079, 6095, 6203, 6745, 6103, 6141, 267} \begin {gather*} -\frac {3 a}{8 \left (1-a^2 x^2\right )}+\frac {a^2 x \tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}-\frac {3 a \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}+\frac {3 a^2 x \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )}-\frac {3}{2} a \text {Li}_3\left (\frac {2}{a x+1}-1\right )-3 a \text {Li}_2\left (\frac {2}{a x+1}-1\right ) \tanh ^{-1}(a x)+\frac {3}{8} a \tanh ^{-1}(a x)^4+a \tanh ^{-1}(a x)^3-\frac {\tanh ^{-1}(a x)^3}{x}+\frac {3}{8} a \tanh ^{-1}(a x)^2+3 a \log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)^2 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*a)/(8*(1 - a^2*x^2)) + (3*a^2*x*ArcTanh[a*x])/(4*(1 - a^2*x^2)) + (3*a*ArcTanh[a*x]^2)/8 - (3*a*ArcTanh[a*
x]^2)/(4*(1 - a^2*x^2)) + a*ArcTanh[a*x]^3 - ArcTanh[a*x]^3/x + (a^2*x*ArcTanh[a*x]^3)/(2*(1 - a^2*x^2)) + (3*
a*ArcTanh[a*x]^4)/8 + 3*a*ArcTanh[a*x]^2*Log[2 - 2/(1 + a*x)] - 3*a*ArcTanh[a*x]*PolyLog[2, -1 + 2/(1 + a*x)]
- (3*a*PolyLog[3, -1 + 2/(1 + a*x)])/2

Rule 267

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

Rule 6037

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

Rule 6079

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_Symbol] :> Simp[(a + b*ArcTanh[c*x
])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - Dist[b*c*(p/d), Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[2 - 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 6103

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

Rule 6129

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

Rule 6135

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

Rule 6141

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

Rule 6177

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

Rule 6203

Int[(Log[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTan
h[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 \frac {\tanh ^{-1}(a x)^3}{x^2 \left (1-a^2 x^2\right )^2} \, dx &=a^2 \int \frac {\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^2} \, dx+\int \frac {\tanh ^{-1}(a x)^3}{x^2 \left (1-a^2 x^2\right )} \, dx\\ &=\frac {a^2 x \tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {1}{8} a \tanh ^{-1}(a x)^4+a^2 \int \frac {\tanh ^{-1}(a x)^3}{1-a^2 x^2} \, dx-\frac {1}{2} \left (3 a^3\right ) \int \frac {x \tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx+\int \frac {\tanh ^{-1}(a x)^3}{x^2} \, dx\\ &=-\frac {3 a \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^3}{x}+\frac {a^2 x \tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {3}{8} a \tanh ^{-1}(a x)^4+(3 a) \int \frac {\tanh ^{-1}(a x)^2}{x \left (1-a^2 x^2\right )} \, dx+\frac {1}{2} \left (3 a^2\right ) \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx\\ &=\frac {3 a^2 x \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )}+\frac {3}{8} a \tanh ^{-1}(a x)^2-\frac {3 a \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}+a \tanh ^{-1}(a x)^3-\frac {\tanh ^{-1}(a x)^3}{x}+\frac {a^2 x \tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {3}{8} a \tanh ^{-1}(a x)^4+(3 a) \int \frac {\tanh ^{-1}(a x)^2}{x (1+a x)} \, dx-\frac {1}{4} \left (3 a^3\right ) \int \frac {x}{\left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac {3 a}{8 \left (1-a^2 x^2\right )}+\frac {3 a^2 x \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )}+\frac {3}{8} a \tanh ^{-1}(a x)^2-\frac {3 a \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}+a \tanh ^{-1}(a x)^3-\frac {\tanh ^{-1}(a x)^3}{x}+\frac {a^2 x \tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {3}{8} a \tanh ^{-1}(a x)^4+3 a \tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-\left (6 a^2\right ) \int \frac {\tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac {3 a}{8 \left (1-a^2 x^2\right )}+\frac {3 a^2 x \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )}+\frac {3}{8} a \tanh ^{-1}(a x)^2-\frac {3 a \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}+a \tanh ^{-1}(a x)^3-\frac {\tanh ^{-1}(a x)^3}{x}+\frac {a^2 x \tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {3}{8} a \tanh ^{-1}(a x)^4+3 a \tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-3 a \tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+a x}\right )+\left (3 a^2\right ) \int \frac {\text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac {3 a}{8 \left (1-a^2 x^2\right )}+\frac {3 a^2 x \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )}+\frac {3}{8} a \tanh ^{-1}(a x)^2-\frac {3 a \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}+a \tanh ^{-1}(a x)^3-\frac {\tanh ^{-1}(a x)^3}{x}+\frac {a^2 x \tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {3}{8} a \tanh ^{-1}(a x)^4+3 a \tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-3 a \tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+a x}\right )-\frac {3}{2} a \text {Li}_3\left (-1+\frac {2}{1+a x}\right )\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.22, size = 144, normalized size = 0.75 \begin {gather*} \frac {1}{16} a \left (2 i \pi ^3-16 \tanh ^{-1}(a x)^3-\frac {16 \tanh ^{-1}(a x)^3}{a x}+6 \tanh ^{-1}(a x)^4-3 \cosh \left (2 \tanh ^{-1}(a x)\right )-6 \tanh ^{-1}(a x)^2 \cosh \left (2 \tanh ^{-1}(a x)\right )+48 \tanh ^{-1}(a x)^2 \log \left (1-e^{2 \tanh ^{-1}(a x)}\right )+48 \tanh ^{-1}(a x) \text {PolyLog}\left (2,e^{2 \tanh ^{-1}(a x)}\right )-24 \text {PolyLog}\left (3,e^{2 \tanh ^{-1}(a x)}\right )+6 \tanh ^{-1}(a x) \sinh \left (2 \tanh ^{-1}(a x)\right )+4 \tanh ^{-1}(a x)^3 \sinh \left (2 \tanh ^{-1}(a x)\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*((2*I)*Pi^3 - 16*ArcTanh[a*x]^3 - (16*ArcTanh[a*x]^3)/(a*x) + 6*ArcTanh[a*x]^4 - 3*Cosh[2*ArcTanh[a*x]] - 6
*ArcTanh[a*x]^2*Cosh[2*ArcTanh[a*x]] + 48*ArcTanh[a*x]^2*Log[1 - E^(2*ArcTanh[a*x])] + 48*ArcTanh[a*x]*PolyLog
[2, E^(2*ArcTanh[a*x])] - 24*PolyLog[3, E^(2*ArcTanh[a*x])] + 6*ArcTanh[a*x]*Sinh[2*ArcTanh[a*x]] + 4*ArcTanh[
a*x]^3*Sinh[2*ArcTanh[a*x]]))/16

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Maple [A]
time = 138.15, size = 271, normalized size = 1.42

method	result	size
derivativedivides	\(a \left (\frac {3 \arctanh \left (a x \right )^{4}}{8}-\frac {\left (a x +1\right ) \left (4 \arctanh \left (a x \right )^{3}-6 \arctanh \left (a x \right )^{2}+6 \arctanh \left (a x \right )-3\right )}{32 \left (a x -1\right )}+\frac {\left (4 \arctanh \left (a x \right )^{3}+6 \arctanh \left (a x \right )^{2}+6 \arctanh \left (a x \right )+3\right ) \left (a x -1\right )}{32 a x +32}+\frac {\arctanh \left (a x \right )^{3} \left (a x -1\right )}{a x}-2 \arctanh \left (a x \right )^{3}+3 \arctanh \left (a x \right )^{2} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \arctanh \left (a x \right ) \polylog \left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-6 \polylog \left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \arctanh \left (a x \right )^{2} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \arctanh \left (a x \right ) \polylog \left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-6 \polylog \left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )\)	\(271\)
default	\(a \left (\frac {3 \arctanh \left (a x \right )^{4}}{8}-\frac {\left (a x +1\right ) \left (4 \arctanh \left (a x \right )^{3}-6 \arctanh \left (a x \right )^{2}+6 \arctanh \left (a x \right )-3\right )}{32 \left (a x -1\right )}+\frac {\left (4 \arctanh \left (a x \right )^{3}+6 \arctanh \left (a x \right )^{2}+6 \arctanh \left (a x \right )+3\right ) \left (a x -1\right )}{32 a x +32}+\frac {\arctanh \left (a x \right )^{3} \left (a x -1\right )}{a x}-2 \arctanh \left (a x \right )^{3}+3 \arctanh \left (a x \right )^{2} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \arctanh \left (a x \right ) \polylog \left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-6 \polylog \left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \arctanh \left (a x \right )^{2} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \arctanh \left (a x \right ) \polylog \left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-6 \polylog \left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )\)	\(271\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*(3/8*arctanh(a*x)^4-1/32*(a*x+1)*(4*arctanh(a*x)^3-6*arctanh(a*x)^2+6*arctanh(a*x)-3)/(a*x-1)+1/32*(4*arctan
h(a*x)^3+6*arctanh(a*x)^2+6*arctanh(a*x)+3)*(a*x-1)/(a*x+1)+arctanh(a*x)^3/a/x*(a*x-1)-2*arctanh(a*x)^3+3*arct
anh(a*x)^2*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+6*arctanh(a*x)*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))-6*polylog(3,
-(a*x+1)/(-a^2*x^2+1)^(1/2))+3*arctanh(a*x)^2*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+6*arctanh(a*x)*polylog(2,(a*x+1
)/(-a^2*x^2+1)^(1/2))-6*polylog(3,(a*x+1)/(-a^2*x^2+1)^(1/2)))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

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

[Out]

integral(arctanh(a*x)^3/(a^4*x^6 - 2*a^2*x^4 + x^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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