3.313 \(\int \frac {\tanh ^{-1}(a x)^2}{x^2 (1-a^2 x^2)^3} \, dx\)

Optimal. Leaf size=209 \[ \frac {31 a^2 x}{64 \left (1-a^2 x^2\right )}+\frac {a^2 x}{32 \left (1-a^2 x^2\right )^2}+\frac {7 a^2 x \tanh ^{-1}(a x)^2}{8 \left (1-a^2 x^2\right )}+\frac {a^2 x \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {7 a \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )}-\frac {a \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )^2}-a \text {Li}_2\left (\frac {2}{a x+1}-1\right )+\frac {5}{8} a \tanh ^{-1}(a x)^3+a \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{x}+\frac {31}{64} a \tanh ^{-1}(a x)+2 a \log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x) \]

[Out]

1/32*a^2*x/(-a^2*x^2+1)^2+31/64*a^2*x/(-a^2*x^2+1)+31/64*a*arctanh(a*x)-1/8*a*arctanh(a*x)/(-a^2*x^2+1)^2-7/8*
a*arctanh(a*x)/(-a^2*x^2+1)+a*arctanh(a*x)^2-arctanh(a*x)^2/x+1/4*a^2*x*arctanh(a*x)^2/(-a^2*x^2+1)^2+7/8*a^2*
x*arctanh(a*x)^2/(-a^2*x^2+1)+5/8*a*arctanh(a*x)^3+2*a*arctanh(a*x)*ln(2-2/(a*x+1))-a*polylog(2,-1+2/(a*x+1))

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Rubi [A]  time = 0.49, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 12, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.546, Rules used = {6030, 5982, 5916, 5988, 5932, 2447, 5948, 5956, 5994, 199, 206, 5964} \[ -a \text {PolyLog}\left (2,\frac {2}{a x+1}-1\right )+\frac {31 a^2 x}{64 \left (1-a^2 x^2\right )}+\frac {a^2 x}{32 \left (1-a^2 x^2\right )^2}+\frac {7 a^2 x \tanh ^{-1}(a x)^2}{8 \left (1-a^2 x^2\right )}+\frac {a^2 x \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {7 a \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )}-\frac {a \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )^2}+\frac {5}{8} a \tanh ^{-1}(a x)^3+a \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{x}+\frac {31}{64} a \tanh ^{-1}(a x)+2 a \log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*x)/(32*(1 - a^2*x^2)^2) + (31*a^2*x)/(64*(1 - a^2*x^2)) + (31*a*ArcTanh[a*x])/64 - (a*ArcTanh[a*x])/(8*(1
- a^2*x^2)^2) - (7*a*ArcTanh[a*x])/(8*(1 - a^2*x^2)) + a*ArcTanh[a*x]^2 - ArcTanh[a*x]^2/x + (a^2*x*ArcTanh[a
*x]^2)/(4*(1 - a^2*x^2)^2) + (7*a^2*x*ArcTanh[a*x]^2)/(8*(1 - a^2*x^2)) + (5*a*ArcTanh[a*x]^3)/8 + 2*a*ArcTanh
[a*x]*Log[2 - 2/(1 + a*x)] - a*PolyLog[2, -1 + 2/(1 + a*x)]

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2447

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[(Pq^m*(1 - u))/D[u, x]]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rule 5916

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

Rule 5932

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 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 5956

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 5964

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> -Simp[(b*p*(d + e*x^2)^(
q + 1)*(a + b*ArcTanh[c*x])^(p - 1))/(4*c*d*(q + 1)^2), x] + (Dist[(2*q + 3)/(2*d*(q + 1)), Int[(d + e*x^2)^(q
+ 1)*(a + b*ArcTanh[c*x])^p, x], x] + Dist[(b^2*p*(p - 1))/(4*(q + 1)^2), Int[(d + e*x^2)^q*(a + b*ArcTanh[c*
x])^(p - 2), x], x] - Simp[(x*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^p)/(2*d*(q + 1)), x]) /; FreeQ[{a, b, c
, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && GtQ[p, 1] && NeQ[q, -3/2]

Rule 5982

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 5988

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 5994

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 6030

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]

Rubi steps

\begin {align*} \int \frac {\tanh ^{-1}(a x)^2}{x^2 \left (1-a^2 x^2\right )^3} \, dx &=a^2 \int \frac {\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^3} \, dx+\int \frac {\tanh ^{-1}(a x)^2}{x^2 \left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac {a \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )^2}+\frac {a^2 x \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )^2}+\frac {1}{8} a^2 \int \frac {1}{\left (1-a^2 x^2\right )^3} \, dx+\frac {1}{4} \left (3 a^2\right ) \int \frac {\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx+a^2 \int \frac {\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx+\int \frac {\tanh ^{-1}(a x)^2}{x^2 \left (1-a^2 x^2\right )} \, dx\\ &=\frac {a^2 x}{32 \left (1-a^2 x^2\right )^2}-\frac {a \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )^2}+\frac {a^2 x \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )^2}+\frac {7 a^2 x \tanh ^{-1}(a x)^2}{8 \left (1-a^2 x^2\right )}+\frac {7}{24} a \tanh ^{-1}(a x)^3+\frac {1}{32} \left (3 a^2\right ) \int \frac {1}{\left (1-a^2 x^2\right )^2} \, dx+a^2 \int \frac {\tanh ^{-1}(a x)^2}{1-a^2 x^2} \, dx-\frac {1}{4} \left (3 a^3\right ) \int \frac {x \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx-a^3 \int \frac {x \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx+\int \frac {\tanh ^{-1}(a x)^2}{x^2} \, dx\\ &=\frac {a^2 x}{32 \left (1-a^2 x^2\right )^2}+\frac {3 a^2 x}{64 \left (1-a^2 x^2\right )}-\frac {a \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )^2}-\frac {7 a \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^2}{x}+\frac {a^2 x \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )^2}+\frac {7 a^2 x \tanh ^{-1}(a x)^2}{8 \left (1-a^2 x^2\right )}+\frac {5}{8} a \tanh ^{-1}(a x)^3+(2 a) \int \frac {\tanh ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx+\frac {1}{64} \left (3 a^2\right ) \int \frac {1}{1-a^2 x^2} \, dx+\frac {1}{8} \left (3 a^2\right ) \int \frac {1}{\left (1-a^2 x^2\right )^2} \, dx+\frac {1}{2} a^2 \int \frac {1}{\left (1-a^2 x^2\right )^2} \, dx\\ &=\frac {a^2 x}{32 \left (1-a^2 x^2\right )^2}+\frac {31 a^2 x}{64 \left (1-a^2 x^2\right )}+\frac {3}{64} a \tanh ^{-1}(a x)-\frac {a \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )^2}-\frac {7 a \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )}+a \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{x}+\frac {a^2 x \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )^2}+\frac {7 a^2 x \tanh ^{-1}(a x)^2}{8 \left (1-a^2 x^2\right )}+\frac {5}{8} a \tanh ^{-1}(a x)^3+(2 a) \int \frac {\tanh ^{-1}(a x)}{x (1+a x)} \, dx+\frac {1}{16} \left (3 a^2\right ) \int \frac {1}{1-a^2 x^2} \, dx+\frac {1}{4} a^2 \int \frac {1}{1-a^2 x^2} \, dx\\ &=\frac {a^2 x}{32 \left (1-a^2 x^2\right )^2}+\frac {31 a^2 x}{64 \left (1-a^2 x^2\right )}+\frac {31}{64} a \tanh ^{-1}(a x)-\frac {a \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )^2}-\frac {7 a \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )}+a \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{x}+\frac {a^2 x \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )^2}+\frac {7 a^2 x \tanh ^{-1}(a x)^2}{8 \left (1-a^2 x^2\right )}+\frac {5}{8} a \tanh ^{-1}(a x)^3+2 a \tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-\left (2 a^2\right ) \int \frac {\log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=\frac {a^2 x}{32 \left (1-a^2 x^2\right )^2}+\frac {31 a^2 x}{64 \left (1-a^2 x^2\right )}+\frac {31}{64} a \tanh ^{-1}(a x)-\frac {a \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )^2}-\frac {7 a \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )}+a \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{x}+\frac {a^2 x \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )^2}+\frac {7 a^2 x \tanh ^{-1}(a x)^2}{8 \left (1-a^2 x^2\right )}+\frac {5}{8} a \tanh ^{-1}(a x)^3+2 a \tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-a \text {Li}_2\left (-1+\frac {2}{1+a x}\right )\\ \end {align*}

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Mathematica [A]  time = 0.59, size = 127, normalized size = 0.61 \[ -a \left (\tanh ^{-1}(a x)^2 \left (\frac {a x}{a^2 x^2-1}+\frac {1}{a x}-\frac {1}{32} \sinh \left (4 \tanh ^{-1}(a x)\right )-1\right )+\text {Li}_2\left (e^{-2 \tanh ^{-1}(a x)}\right )-\frac {5}{8} \tanh ^{-1}(a x)^3-\frac {1}{4} \sinh \left (2 \tanh ^{-1}(a x)\right )-\frac {1}{256} \sinh \left (4 \tanh ^{-1}(a x)\right )+\frac {1}{64} \tanh ^{-1}(a x) \left (-128 \log \left (1-e^{-2 \tanh ^{-1}(a x)}\right )+32 \cosh \left (2 \tanh ^{-1}(a x)\right )+\cosh \left (4 \tanh ^{-1}(a x)\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*((-5*ArcTanh[a*x]^3)/8 + (ArcTanh[a*x]*(32*Cosh[2*ArcTanh[a*x]] + Cosh[4*ArcTanh[a*x]] - 128*Log[1 - E^(-2
*ArcTanh[a*x])]))/64 + PolyLog[2, E^(-2*ArcTanh[a*x])] - Sinh[2*ArcTanh[a*x]]/4 + ArcTanh[a*x]^2*(-1 + 1/(a*x)
+ (a*x)/(-1 + a^2*x^2) - Sinh[4*ArcTanh[a*x]]/32) - Sinh[4*ArcTanh[a*x]]/256))

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fricas [F]  time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\operatorname {artanh}\left (a x\right )^{2}}{a^{6} x^{8} - 3 \, a^{4} x^{6} + 3 \, a^{2} x^{4} - x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-arctanh(a*x)^2/(a^6*x^8 - 3*a^4*x^6 + 3*a^2*x^4 - x^2), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {\operatorname {artanh}\left (a x\right )^{2}}{{\left (a^{2} x^{2} - 1\right )}^{3} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.95, size = 4797, normalized size = 22.95 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*arctanh(a*x)^2+1/4*a*arctanh(a*x)/(a*x-1)-1/4*a*arctanh(a*x)/(a*x+1)+1/512*a/(a*x-1)^2-15/32*I*a*Pi*csgn(I/
(1+(a*x+1)^2/(-a^2*x^2+1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+
1)))*arctanh(a*x)*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))-arctanh(a*x)^2/x+15/16*I*a*Pi*arctanh(a*x)^2+15/16*I*a*Pi*c
sgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))^2*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))+15/16*I*a*Pi*csgn(I/(1+(a*x+1)^2/(-
a^2*x^2+1)))^2*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))+15/32*I*a*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3*polylog(2,-(
a*x+1)/(-a^2*x^2+1)^(1/2))+15/32*I*a*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))+
15/32*I*a*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^3*dilog((a*x+1)/(-a^2*x^2+1)^(1/2))-15/1
6*I*a*Pi*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))^3*dilog((a*x+1)/(-a^2*x^2+1)^(1/2))-15/16*I*a*Pi*arctanh(a*x)*ln(1
-(a*x+1)/(-a^2*x^2+1)^(1/2))-15/16*I*a*Pi*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))^2*dilog(1+(a*x+1)/(-a^2*x^2+1)^(1
/2))+15/16*I*a*Pi*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))^2*dilog((a*x+1)/(-a^2*x^2+1)^(1/2))+15/32*I*a*Pi*csgn(I*(
a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^3*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))+15/32*I*a*Pi*csgn(I*
(a*x+1)^2/(a^2*x^2-1))^3*dilog((a*x+1)/(-a^2*x^2+1)^(1/2))-15/32*I*a*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3*dilog(
1+(a*x+1)/(-a^2*x^2+1)^(1/2))-1/8*a^2*x/(a*x-1)+1/8/(a*x+1)*a^2*x-7/16*a*arctanh(a*x)^2/(a*x-1)-7/16*a*arctanh
(a*x)^2/(a*x+1)+5/8*a*arctanh(a*x)^3-15/32*I*a*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^3*a
rctanh(a*x)^2-1/512*a/(a*x+1)^2+15/32*I*a*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*
x+1)^2/(-a^2*x^2+1)))^2*dilog(1+(a*x+1)/(-a^2*x^2+1)^(1/2))-15/32*I*a*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*
(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^2*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))-15/16*I*a*Pi*csgn(I/
(1+(a*x+1)^2/(-a^2*x^2+1)))^3*arctanh(a*x)*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+15/16*I*a*Pi*csgn(I/(1+(a*x+1)^2/(
-a^2*x^2+1)))^2*arctanh(a*x)*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+15/16*I*a*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*dilog((a*x+1)/(-a^2*x^2+1)^(1/2))-15/16*I*a*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*dilog(1+(a*x+1)/(-a^2*x^2+1)^(1/2))-15/32*I*a*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))*dilog(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+15/16*I*a*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*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))+15/32*I*a*Pi*csgn(I*(a*x
+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^3*arctanh(a*x)*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+15/32*I*a*Pi*csg
n(I/(1+(a*x+1)^2/(-a^2*x^2+1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^2*dilog((a*x+1)/(-a^2
*x^2+1)^(1/2))+15/32*I*a*Pi*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2
*x^2+1)))^2*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))+15/32*I*a*Pi*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))*csgn(I*(a*x+
1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^2*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))+15/16*I*a*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*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))-15/32*I*a*Pi*cs
gn(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)^2+15/32*I*a*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))*dilog((a*x+1)/(-a^2*x^2+1)^(1/2))-15/32*I*a*Pi*csgn(I/(1+(a*
x+1)^2/(-a^2*x^2+1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^2*dilog(1+(a*x+1)/(-a^2*x^2+1)^
(1/2))-1/8*a/(a*x-1)-1/8*a/(a*x+1)+a*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))-a*dilog((a*x+1)/(-a^2*x^2+1)^(1/2)
)+a*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))+a*dilog(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+15/32*I*a*Pi*csgn(I*(a*x+1)^2/
(a^2*x^2-1))^3*arctanh(a*x)*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+15/32*I*a*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))*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))+15/32*I*a*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))*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))-15/16*I*a*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)^2-15/32*I*a*Pi*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))
*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^2*arctanh(a*x)^2+15/32*I*a*Pi*csgn(I*(a*x+1)^2/(a^2*
x^2-1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^2*arctanh(a*x)^2-15/32*I*a*Pi*csgn(I*(a*x+1)^
2/(a^2*x^2-1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^2*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2
))-15/32*I*a*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^2*dilog
((a*x+1)/(-a^2*x^2+1)^(1/2))-15/32*I*a*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3*arctanh(a*x)^2-15/16*I*a*Pi*csgn(I/(
1+(a*x+1)^2/(-a^2*x^2+1)))^3*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))-15/16*I*a*Pi*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+
1)))^3*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))-15/16*I*a*Pi*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))^2*arctanh(a*x)^2
+15/16*I*a*Pi*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))^3*dilog(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+15/16*I*a*Pi*csgn(I/(1+
(a*x+1)^2/(-a^2*x^2+1)))^3*arctanh(a*x)^2-15/32*I*a*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1))
)^3*dilog(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+15/32*I*a*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^
3*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))+1/512/(a*x-1)^2*a^3*x^2-15/8*a*arctanh(a*x)^2*ln((a*x+1)/(-a^2*x^2+1)^
(1/2))+2*a*arctanh(a*x)*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+a*arctanh(a*x)*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+1/4*a
rctanh(a*x)/(a*x-1)*a^2*x+1/4*arctanh(a*x)/(a*x+1)*a^2*x+15/16*I*a*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)*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+15/32*I*a*Pi*csgn(I/(1+(a*x+1)^2/(-a^
2*x^2+1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^2*arctanh(a*x)*ln(1-(a*x+1)/(-a^2*x^2+1)^(
1/2))+15/32*I*a*Pi*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)^2/(a^2*x^2-
1)/(1+(a*x+1)^2/(-a^2*x^2+1)))*dilog(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+15/32*I*a*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)*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))-15/32*I*a*Pi*csgn(I*(a*x+1)
^2/(a^2*x^2-1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^2*arctanh(a*x)*ln(1-(a*x+1)/(-a^2*x^2
+1)^(1/2))-15/32*I*a*Pi*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)^2/(a^2
*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))-15/32*I*a*Pi*csgn(I/(1+(a*x+1)^2/(-
a^2*x^2+1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))*dilog((a*x
+1)/(-a^2*x^2+1)^(1/2))+15/32*I*a*Pi*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(
a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))*arctanh(a*x)^2-15/32*I*a*Pi*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1))
)*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))*polylog(2,(a*x+1)/(-a
^2*x^2+1)^(1/2))-15/16*I*a*Pi*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))-15/16*I*a*Pi*polylog(2,-(a*x+1)/(-a^2*x^2+
1)^(1/2))+15/16*I*a*Pi*dilog(1+(a*x+1)/(-a^2*x^2+1)^(1/2))-15/16*I*a*Pi*dilog((a*x+1)/(-a^2*x^2+1)^(1/2))-1/12
8*arctanh(a*x)/(a*x+1)^2*a^3*x^2+1/64*arctanh(a*x)/(a*x+1)^2*a^2*x-1/128*arctanh(a*x)/(a*x-1)^2*a^3*x^2-1/64*a
rctanh(a*x)/(a*x-1)^2*a^2*x+1/256/(a*x-1)^2*a^2*x-1/512/(a*x+1)^2*a^3*x^2+1/256/(a*x+1)^2*a^2*x+1/16*a*arctanh
(a*x)^2/(a*x-1)^2-1/16*a*arctanh(a*x)^2/(a*x+1)^2-15/16*a*arctanh(a*x)^2*ln(a*x-1)+15/16*a*arctanh(a*x)^2*ln(a
*x+1)-1/128*a*arctanh(a*x)/(a*x-1)^2-1/128*a*arctanh(a*x)/(a*x+1)^2

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maxima [B]  time = 0.35, size = 534, normalized size = 2.56 \[ -\frac {1}{128} \, a^{2} {\left (\frac {2 \, {\left (31 \, a^{3} x^{3} - 5 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{3} + 5 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{3} - {\left (16 \, a^{4} x^{4} - 32 \, a^{2} x^{2} - 15 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right ) + 16\right )} \log \left (a x + 1\right )^{2} + 16 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2} - 33 \, a x - {\left (15 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2} - 32 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )\right )} \log \left (a x + 1\right )\right )}}{a^{5} x^{4} - 2 \, a^{3} x^{2} + a} - \frac {128 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a} + \frac {128 \, {\left (\log \left (a x + 1\right ) \log \relax (x) + {\rm Li}_2\left (-a x\right )\right )}}{a} - \frac {128 \, {\left (\log \left (-a x + 1\right ) \log \relax (x) + {\rm Li}_2\left (a x\right )\right )}}{a} - \frac {31 \, \log \left (a x + 1\right )}{a} + \frac {31 \, \log \left (a x - 1\right )}{a}\right )} + \frac {1}{32} \, a {\left (\frac {28 \, a^{2} x^{2} - 15 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{2} + 30 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 15 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2} - 32}{a^{4} x^{4} - 2 \, a^{2} x^{2} + 1} - 32 \, \log \left (a x + 1\right ) - 32 \, \log \left (a x - 1\right ) + 64 \, \log \relax (x)\right )} \operatorname {artanh}\left (a x\right ) + \frac {1}{16} \, {\left (15 \, a \log \left (a x + 1\right ) - 15 \, a \log \left (a x - 1\right ) - \frac {2 \, {\left (15 \, a^{4} x^{4} - 25 \, a^{2} x^{2} + 8\right )}}{a^{4} x^{5} - 2 \, a^{2} x^{3} + x}\right )} \operatorname {artanh}\left (a x\right )^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/128*a^2*(2*(31*a^3*x^3 - 5*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)^3 + 5*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x -
1)^3 - (16*a^4*x^4 - 32*a^2*x^2 - 15*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1) + 16)*log(a*x + 1)^2 + 16*(a^4*x^
4 - 2*a^2*x^2 + 1)*log(a*x - 1)^2 - 33*a*x - (15*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1)^2 - 32*(a^4*x^4 - 2*a^
2*x^2 + 1)*log(a*x - 1))*log(a*x + 1))/(a^5*x^4 - 2*a^3*x^2 + a) - 128*(log(a*x - 1)*log(1/2*a*x + 1/2) + dilo
g(-1/2*a*x + 1/2))/a + 128*(log(a*x + 1)*log(x) + dilog(-a*x))/a - 128*(log(-a*x + 1)*log(x) + dilog(a*x))/a -
31*log(a*x + 1)/a + 31*log(a*x - 1)/a) + 1/32*a*((28*a^2*x^2 - 15*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)^2 +
30*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)*log(a*x - 1) - 15*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1)^2 - 32)/(a^
4*x^4 - 2*a^2*x^2 + 1) - 32*log(a*x + 1) - 32*log(a*x - 1) + 64*log(x))*arctanh(a*x) + 1/16*(15*a*log(a*x + 1)
- 15*a*log(a*x - 1) - 2*(15*a^4*x^4 - 25*a^2*x^2 + 8)/(a^4*x^5 - 2*a^2*x^3 + x))*arctanh(a*x)^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{a^{6} x^{8} - 3 a^{4} x^{6} + 3 a^{2} x^{4} - x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(atanh(a*x)**2/(a**6*x**8 - 3*a**4*x**6 + 3*a**2*x**4 - x**2), x)

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