3.2.80 \(\int \frac {(1-a^2 x^2) \tanh ^{-1}(a x)^2}{x^4} \, dx\) [180]

Optimal. Leaf size=116 \[ -\frac {a^2}{3 x}+\frac {1}{3} a^3 \tanh ^{-1}(a x)-\frac {a \tanh ^{-1}(a x)}{3 x^2}-\frac {2}{3} a^3 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{3 x^3}+\frac {a^2 \tanh ^{-1}(a x)^2}{x}-\frac {4}{3} a^3 \tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )+\frac {2}{3} a^3 \text {PolyLog}\left (2,-1+\frac {2}{1+a x}\right ) \]

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

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Rubi [A]
time = 0.23, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6161, 6037, 6129, 331, 212, 6135, 6079, 2497} \begin {gather*} \frac {2}{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)-\frac {4}{3} a^3 \log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)-\frac {a^2}{3 x}+\frac {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)*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) + (a^2*ArcTanh[a*x]^2)/x - (4*a^3*ArcTanh[a*x]*Log[2 - 2/(1 + a*x)])/3 + (2*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 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 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 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 6161

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

Rubi steps

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

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Mathematica [A]
time = 0.21, size = 93, normalized size = 0.80 \begin {gather*} \frac {-a^2 x^2-(-1+a x)^2 (1+2 a x) \tanh ^{-1}(a x)^2+\tanh ^{-1}(a x) \left (-a x+a^3 x^3-4 a^3 x^3 \log \left (1-e^{-2 \tanh ^{-1}(a x)}\right )\right )+2 a^3 x^3 \text {PolyLog}\left (2,e^{-2 \tanh ^{-1}(a x)}\right )}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.15, size = 198, normalized size = 1.71

method	result	size
derivativedivides	\(a^{3} \left (-\frac {\arctanh \left (a x \right )^{2}}{3 a^{3} x^{3}}+\frac {\arctanh \left (a x \right )^{2}}{a x}+\frac {2 \arctanh \left (a x \right ) \ln \left (a x -1\right )}{3}+\frac {2 \arctanh \left (a x \right ) \ln \left (a x +1\right )}{3}-\frac {\arctanh \left (a x \right )}{3 a^{2} x^{2}}-\frac {4 \arctanh \left (a x \right ) \ln \left (a x \right )}{3}-\frac {\ln \left (a x -1\right )}{6}-\frac {1}{3 a x}+\frac {\ln \left (a x +1\right )}{6}+\frac {2 \dilog \left (a x \right )}{3}+\frac {2 \dilog \left (a x +1\right )}{3}+\frac {2 \ln \left (a x \right ) \ln \left (a x +1\right )}{3}-\frac {2 \dilog \left (\frac {a x}{2}+\frac {1}{2}\right )}{3}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{3}+\frac {\ln \left (a x -1\right )^{2}}{6}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{3}-\frac {\ln \left (a x +1\right )^{2}}{6}\right )\)	\(198\)
default	\(a^{3} \left (-\frac {\arctanh \left (a x \right )^{2}}{3 a^{3} x^{3}}+\frac {\arctanh \left (a x \right )^{2}}{a x}+\frac {2 \arctanh \left (a x \right ) \ln \left (a x -1\right )}{3}+\frac {2 \arctanh \left (a x \right ) \ln \left (a x +1\right )}{3}-\frac {\arctanh \left (a x \right )}{3 a^{2} x^{2}}-\frac {4 \arctanh \left (a x \right ) \ln \left (a x \right )}{3}-\frac {\ln \left (a x -1\right )}{6}-\frac {1}{3 a x}+\frac {\ln \left (a x +1\right )}{6}+\frac {2 \dilog \left (a x \right )}{3}+\frac {2 \dilog \left (a x +1\right )}{3}+\frac {2 \ln \left (a x \right ) \ln \left (a x +1\right )}{3}-\frac {2 \dilog \left (\frac {a x}{2}+\frac {1}{2}\right )}{3}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{3}+\frac {\ln \left (a x -1\right )^{2}}{6}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{3}-\frac {\ln \left (a x +1\right )^{2}}{6}\right )\)	\(198\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^3*(-1/3*arctanh(a*x)^2/a^3/x^3+arctanh(a*x)^2/a/x+2/3*arctanh(a*x)*ln(a*x-1)+2/3*arctanh(a*x)*ln(a*x+1)-1/3*
arctanh(a*x)/a^2/x^2-4/3*arctanh(a*x)*ln(a*x)-1/6*ln(a*x-1)-1/3/a/x+1/6*ln(a*x+1)+2/3*dilog(a*x)+2/3*dilog(a*x
+1)+2/3*ln(a*x)*ln(a*x+1)-2/3*dilog(1/2*a*x+1/2)-1/3*ln(a*x-1)*ln(1/2*a*x+1/2)+1/6*ln(a*x-1)^2+1/3*(ln(a*x+1)-
ln(1/2*a*x+1/2))*ln(-1/2*a*x+1/2)-1/6*ln(a*x+1)^2)

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Maxima [A]
time = 0.26, size = 188, normalized size = 1.62 \begin {gather*} -\frac {1}{6} \, {\left (4 \, {\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 - 4 \, {\left (\log \left (a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-a x\right )\right )} a + 4 \, {\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 {a x \log \left (a x + 1\right )^{2} - 2 \, a x \log \left (a x + 1\right ) \log \left (a x - 1\right ) - a x \log \left (a x - 1\right )^{2} + 2}{x}\right )} a^{2} + \frac {1}{3} \, {\left (2 \, a^{2} \log \left (a^{2} x^{2} - 1\right ) - 2 \, a^{2} \log \left (x^{2}\right ) - \frac {1}{x^{2}}\right )} a \operatorname {artanh}\left (a x\right ) + \frac {{\left (3 \, a^{2} x^{2} - 1\right )} \operatorname {artanh}\left (a x\right )^{2}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)^2/x^4,x, algorithm="fricas")

[Out]

integral(-(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 \left (- \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{x^{4}}\right )\, dx - \int \frac {a^{2} \operatorname {atanh}^{2}{\left (a x \right )}}{x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(-(a^2*x^2 - 1)*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 )}{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))/x^4,x)

[Out]

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

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