3.3.11 \(\int \frac {(1-a^2 x^2)^2 \tanh ^{-1}(a x)^2}{x^4} \, dx\) [211]

Optimal. Leaf size=167 \[ -\frac {a^2}{3 x}+\frac {1}{3} a^3 \tanh ^{-1}(a x)-\frac {a \tanh ^{-1}(a x)}{3 x^2}-\frac {2}{3} a^3 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{3 x^3}+\frac {2 a^2 \tanh ^{-1}(a x)^2}{x}+a^4 x \tanh ^{-1}(a x)^2-2 a^3 \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )-\frac {10}{3} a^3 \tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-a^3 \text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )+\frac {5}{3} a^3 \text {PolyLog}\left (2,-1+\frac {2}{1+a x}\right ) \]

[Out]

-1/3*a^2/x+1/3*a^3*arctanh(a*x)-1/3*a*arctanh(a*x)/x^2-2/3*a^3*arctanh(a*x)^2-1/3*arctanh(a*x)^2/x^3+2*a^2*arc
tanh(a*x)^2/x+a^4*x*arctanh(a*x)^2-2*a^3*arctanh(a*x)*ln(2/(-a*x+1))-10/3*a^3*arctanh(a*x)*ln(2-2/(a*x+1))-a^3
*polylog(2,1-2/(-a*x+1))+5/3*a^3*polylog(2,-1+2/(a*x+1))

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Rubi [A]
time = 0.31, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 13, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {6159, 6021, 6131, 6055, 2449, 2352, 6037, 6129, 331, 212, 6135, 6079, 2497} \begin {gather*} a^4 x \tanh ^{-1}(a x)^2-a^3 \text {Li}_2\left (1-\frac {2}{1-a x}\right )+\frac {5}{3} a^3 \text {Li}_2\left (\frac {2}{a x+1}-1\right )-\frac {2}{3} a^3 \tanh ^{-1}(a x)^2+\frac {1}{3} a^3 \tanh ^{-1}(a x)-2 a^3 \log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)-\frac {10}{3} a^3 \log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)-\frac {a^2}{3 x}+\frac {2 a^2 \tanh ^{-1}(a x)^2}{x}-\frac {\tanh ^{-1}(a x)^2}{3 x^3}-\frac {a \tanh ^{-1}(a x)}{3 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 - a^2*x^2)^2*ArcTanh[a*x]^2)/x^4,x]

[Out]

-1/3*a^2/x + (a^3*ArcTanh[a*x])/3 - (a*ArcTanh[a*x])/(3*x^2) - (2*a^3*ArcTanh[a*x]^2)/3 - ArcTanh[a*x]^2/(3*x^
3) + (2*a^2*ArcTanh[a*x]^2)/x + a^4*x*ArcTanh[a*x]^2 - 2*a^3*ArcTanh[a*x]*Log[2/(1 - a*x)] - (10*a^3*ArcTanh[a
*x]*Log[2 - 2/(1 + a*x)])/3 - a^3*PolyLog[2, 1 - 2/(1 - a*x)] + (5*a^3*PolyLog[2, -1 + 2/(1 + a*x)])/3

Rule 212

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

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2497

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 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 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 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 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 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 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 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 6159

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

Rubi steps

\begin {align*} \int \frac {\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{x^4} \, dx &=\int \left (a^4 \tanh ^{-1}(a x)^2+\frac {\tanh ^{-1}(a x)^2}{x^4}-\frac {2 a^2 \tanh ^{-1}(a x)^2}{x^2}\right ) \, dx\\ &=-\left (\left (2 a^2\right ) \int \frac {\tanh ^{-1}(a x)^2}{x^2} \, dx\right )+a^4 \int \tanh ^{-1}(a x)^2 \, dx+\int \frac {\tanh ^{-1}(a x)^2}{x^4} \, dx\\ &=-\frac {\tanh ^{-1}(a x)^2}{3 x^3}+\frac {2 a^2 \tanh ^{-1}(a x)^2}{x}+a^4 x \tanh ^{-1}(a x)^2+\frac {1}{3} (2 a) \int \frac {\tanh ^{-1}(a x)}{x^3 \left (1-a^2 x^2\right )} \, dx-\left (4 a^3\right ) \int \frac {\tanh ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx-\left (2 a^5\right ) \int \frac {x \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=-a^3 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{3 x^3}+\frac {2 a^2 \tanh ^{-1}(a x)^2}{x}+a^4 x \tanh ^{-1}(a x)^2+\frac {1}{3} (2 a) \int \frac {\tanh ^{-1}(a x)}{x^3} \, dx+\frac {1}{3} \left (2 a^3\right ) \int \frac {\tanh ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx-\left (4 a^3\right ) \int \frac {\tanh ^{-1}(a x)}{x (1+a x)} \, dx-\left (2 a^4\right ) \int \frac {\tanh ^{-1}(a x)}{1-a x} \, dx\\ &=-\frac {a \tanh ^{-1}(a x)}{3 x^2}-\frac {2}{3} a^3 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{3 x^3}+\frac {2 a^2 \tanh ^{-1}(a x)^2}{x}+a^4 x \tanh ^{-1}(a x)^2-2 a^3 \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )-4 a^3 \tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )+\frac {1}{3} a^2 \int \frac {1}{x^2 \left (1-a^2 x^2\right )} \, dx+\frac {1}{3} \left (2 a^3\right ) \int \frac {\tanh ^{-1}(a x)}{x (1+a x)} \, dx+\left (2 a^4\right ) \int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx+\left (4 a^4\right ) \int \frac {\log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac {a^2}{3 x}-\frac {a \tanh ^{-1}(a x)}{3 x^2}-\frac {2}{3} a^3 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{3 x^3}+\frac {2 a^2 \tanh ^{-1}(a x)^2}{x}+a^4 x \tanh ^{-1}(a x)^2-2 a^3 \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )-\frac {10}{3} a^3 \tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )+2 a^3 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )-\left (2 a^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-a x}\right )+\frac {1}{3} a^4 \int \frac {1}{1-a^2 x^2} \, dx-\frac {1}{3} \left (2 a^4\right ) \int \frac {\log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac {a^2}{3 x}+\frac {1}{3} a^3 \tanh ^{-1}(a x)-\frac {a \tanh ^{-1}(a x)}{3 x^2}-\frac {2}{3} a^3 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{3 x^3}+\frac {2 a^2 \tanh ^{-1}(a x)^2}{x}+a^4 x \tanh ^{-1}(a x)^2-2 a^3 \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )-\frac {10}{3} a^3 \tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-a^3 \text {Li}_2\left (1-\frac {2}{1-a x}\right )+\frac {5}{3} a^3 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 153, normalized size = 0.92 \begin {gather*} \frac {1}{3} \left (-\frac {a^2}{x}+a^3 \tanh ^{-1}(a x)-\frac {a \tanh ^{-1}(a x)}{x^2}-8 a^3 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{x^3}+\frac {6 a^2 \tanh ^{-1}(a x)^2}{x}+3 a^4 x \tanh ^{-1}(a x)^2-10 a^3 \tanh ^{-1}(a x) \log \left (1-e^{-2 \tanh ^{-1}(a x)}\right )-6 a^3 \tanh ^{-1}(a x) \log \left (1+e^{-2 \tanh ^{-1}(a x)}\right )+3 a^3 \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(a x)}\right )+5 a^3 \text {PolyLog}\left (2,e^{-2 \tanh ^{-1}(a x)}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 - a^2*x^2)^2*ArcTanh[a*x]^2)/x^4,x]

[Out]

(-(a^2/x) + a^3*ArcTanh[a*x] - (a*ArcTanh[a*x])/x^2 - 8*a^3*ArcTanh[a*x]^2 - ArcTanh[a*x]^2/x^3 + (6*a^2*ArcTa
nh[a*x]^2)/x + 3*a^4*x*ArcTanh[a*x]^2 - 10*a^3*ArcTanh[a*x]*Log[1 - E^(-2*ArcTanh[a*x])] - 6*a^3*ArcTanh[a*x]*
Log[1 + E^(-2*ArcTanh[a*x])] + 3*a^3*PolyLog[2, -E^(-2*ArcTanh[a*x])] + 5*a^3*PolyLog[2, E^(-2*ArcTanh[a*x])])
/3

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Maple [A]
time = 0.56, size = 208, normalized size = 1.25

method	result	size
derivativedivides	\(a^{3} \left (\arctanh \left (a x \right )^{2} a x -\frac {\arctanh \left (a x \right )^{2}}{3 a^{3} x^{3}}+\frac {2 \arctanh \left (a x \right )^{2}}{a x}+\frac {8 \arctanh \left (a x \right ) \ln \left (a x +1\right )}{3}+\frac {8 \arctanh \left (a x \right ) \ln \left (a x -1\right )}{3}-\frac {\arctanh \left (a x \right )}{3 a^{2} x^{2}}-\frac {10 \arctanh \left (a x \right ) \ln \left (a x \right )}{3}-\frac {\ln \left (a x -1\right )}{6}-\frac {1}{3 a x}+\frac {\ln \left (a x +1\right )}{6}+\frac {5 \dilog \left (a x +1\right )}{3}+\frac {5 \ln \left (a x \right ) \ln \left (a x +1\right )}{3}+\frac {5 \dilog \left (a x \right )}{3}-\frac {8 \dilog \left (\frac {a x}{2}+\frac {1}{2}\right )}{3}-\frac {4 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{3}+\frac {2 \ln \left (a x -1\right )^{2}}{3}-\frac {2 \ln \left (a x +1\right )^{2}}{3}+\frac {4 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{3}\right )\)	\(208\)
default	\(a^{3} \left (\arctanh \left (a x \right )^{2} a x -\frac {\arctanh \left (a x \right )^{2}}{3 a^{3} x^{3}}+\frac {2 \arctanh \left (a x \right )^{2}}{a x}+\frac {8 \arctanh \left (a x \right ) \ln \left (a x +1\right )}{3}+\frac {8 \arctanh \left (a x \right ) \ln \left (a x -1\right )}{3}-\frac {\arctanh \left (a x \right )}{3 a^{2} x^{2}}-\frac {10 \arctanh \left (a x \right ) \ln \left (a x \right )}{3}-\frac {\ln \left (a x -1\right )}{6}-\frac {1}{3 a x}+\frac {\ln \left (a x +1\right )}{6}+\frac {5 \dilog \left (a x +1\right )}{3}+\frac {5 \ln \left (a x \right ) \ln \left (a x +1\right )}{3}+\frac {5 \dilog \left (a x \right )}{3}-\frac {8 \dilog \left (\frac {a x}{2}+\frac {1}{2}\right )}{3}-\frac {4 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{3}+\frac {2 \ln \left (a x -1\right )^{2}}{3}-\frac {2 \ln \left (a x +1\right )^{2}}{3}+\frac {4 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{3}\right )\)	\(208\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a^2*x^2+1)^2*arctanh(a*x)^2/x^4,x,method=_RETURNVERBOSE)

[Out]

a^3*(arctanh(a*x)^2*a*x-1/3*arctanh(a*x)^2/a^3/x^3+2*arctanh(a*x)^2/a/x+8/3*arctanh(a*x)*ln(a*x+1)+8/3*arctanh
(a*x)*ln(a*x-1)-1/3*arctanh(a*x)/a^2/x^2-10/3*arctanh(a*x)*ln(a*x)-1/6*ln(a*x-1)-1/3/a/x+1/6*ln(a*x+1)+5/3*dil
og(a*x+1)+5/3*ln(a*x)*ln(a*x+1)+5/3*dilog(a*x)-8/3*dilog(1/2*a*x+1/2)-4/3*ln(a*x-1)*ln(1/2*a*x+1/2)+2/3*ln(a*x
-1)^2-2/3*ln(a*x+1)^2+4/3*(ln(a*x+1)-ln(1/2*a*x+1/2))*ln(-1/2*a*x+1/2))

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Maxima [A]
time = 0.27, size = 203, normalized size = 1.22 \begin {gather*} -\frac {1}{6} \, {\left (16 \, {\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 - 10 \, {\left (\log \left (a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-a x\right )\right )} a + 10 \, {\left (\log \left (-a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (a x\right )\right )} a - a \log \left (a x + 1\right ) + a \log \left (a x - 1\right ) + \frac {2 \, {\left (2 \, a x \log \left (a x + 1\right )^{2} - 4 \, a x \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 2 \, a x \log \left (a x - 1\right )^{2} + 1\right )}}{x}\right )} a^{2} + \frac {1}{3} \, {\left (8 \, a^{2} \log \left (a x + 1\right ) + 8 \, a^{2} \log \left (a x - 1\right ) - 10 \, a^{2} \log \left (x\right ) - \frac {1}{x^{2}}\right )} a \operatorname {artanh}\left (a x\right ) + \frac {1}{3} \, {\left (3 \, a^{4} x + \frac {6 \, a^{2} x^{2} - 1}{x^{3}}\right )} \operatorname {artanh}\left (a x\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*x^2+1)^2*arctanh(a*x)^2/x^4,x, algorithm="maxima")

[Out]

-1/6*(16*(log(a*x - 1)*log(1/2*a*x + 1/2) + dilog(-1/2*a*x + 1/2))*a - 10*(log(a*x + 1)*log(x) + dilog(-a*x))*
a + 10*(log(-a*x + 1)*log(x) + dilog(a*x))*a - a*log(a*x + 1) + a*log(a*x - 1) + 2*(2*a*x*log(a*x + 1)^2 - 4*a
*x*log(a*x + 1)*log(a*x - 1) - 2*a*x*log(a*x - 1)^2 + 1)/x)*a^2 + 1/3*(8*a^2*log(a*x + 1) + 8*a^2*log(a*x - 1)
- 10*a^2*log(x) - 1/x^2)*a*arctanh(a*x) + 1/3*(3*a^4*x + (6*a^2*x^2 - 1)/x^3)*arctanh(a*x)^2

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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)^2*arctanh(a*x)^2/x^4,x, algorithm="fricas")

[Out]

integral((a^4*x^4 - 2*a^2*x^2 + 1)*arctanh(a*x)^2/x^4, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a x - 1\right )^{2} \left (a x + 1\right )^{2} \operatorname {atanh}^{2}{\left (a x \right )}}{x^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a**2*x**2+1)**2*atanh(a*x)**2/x**4,x)

[Out]

Integral((a*x - 1)**2*(a*x + 1)**2*atanh(a*x)**2/x**4, 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)^2*arctanh(a*x)^2/x^4,x, algorithm="giac")

[Out]

integrate((a^2*x^2 - 1)^2*arctanh(a*x)^2/x^4, 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 )}^2\,{\left (a^2\,x^2-1\right )}^2}{x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((atanh(a*x)^2*(a^2*x^2 - 1)^2)/x^4,x)

[Out]

int((atanh(a*x)^2*(a^2*x^2 - 1)^2)/x^4, x)

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