3.2.68 \(\int \frac {(1-a^2 x^2) \tanh ^{-1}(a x)}{x^3} \, dx\) [168]

Optimal. Leaf size=56 \[ -\frac {a}{2 x}+\frac {1}{2} a^2 \tanh ^{-1}(a x)-\frac {\tanh ^{-1}(a x)}{2 x^2}+\frac {1}{2} a^2 \text {PolyLog}(2,-a x)-\frac {1}{2} a^2 \text {PolyLog}(2,a x) \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {6161, 6037, 331, 212, 6031} \begin {gather*} \frac {1}{2} a^2 \text {Li}_2(-a x)-\frac {1}{2} a^2 \text {Li}_2(a x)+\frac {1}{2} a^2 \tanh ^{-1}(a x)-\frac {\tanh ^{-1}(a x)}{2 x^2}-\frac {a}{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 6031

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (-Simp[(b/2)*PolyLog[2, (-c)*x]
, x] + Simp[(b/2)*PolyLog[2, c*x], x]) /; FreeQ[{a, b, c}, 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 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)}{x^3} \, dx &=-\left (a^2 \int \frac {\tanh ^{-1}(a x)}{x} \, dx\right )+\int \frac {\tanh ^{-1}(a x)}{x^3} \, dx\\ &=-\frac {\tanh ^{-1}(a x)}{2 x^2}+\frac {1}{2} a^2 \text {Li}_2(-a x)-\frac {1}{2} a^2 \text {Li}_2(a x)+\frac {1}{2} a \int \frac {1}{x^2 \left (1-a^2 x^2\right )} \, dx\\ &=-\frac {a}{2 x}-\frac {\tanh ^{-1}(a x)}{2 x^2}+\frac {1}{2} a^2 \text {Li}_2(-a x)-\frac {1}{2} a^2 \text {Li}_2(a x)+\frac {1}{2} a^3 \int \frac {1}{1-a^2 x^2} \, dx\\ &=-\frac {a}{2 x}+\frac {1}{2} a^2 \tanh ^{-1}(a x)-\frac {\tanh ^{-1}(a x)}{2 x^2}+\frac {1}{2} a^2 \text {Li}_2(-a x)-\frac {1}{2} a^2 \text {Li}_2(a x)\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 68, normalized size = 1.21 \begin {gather*} -\frac {a}{2 x}-\frac {\tanh ^{-1}(a x)}{2 x^2}-\frac {1}{4} a^2 \log (1-a x)+\frac {1}{4} a^2 \log (1+a x)-\frac {1}{2} a^2 (-\text {PolyLog}(2,-a x)+\text {PolyLog}(2,a x)) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.16, size = 78, normalized size = 1.39

method	result	size
derivativedivides	\(a^{2} \left (-\arctanh \left (a x \right ) \ln \left (a x \right )-\frac {\arctanh \left (a x \right )}{2 a^{2} x^{2}}-\frac {\ln \left (a x -1\right )}{4}+\frac {\ln \left (a x +1\right )}{4}-\frac {1}{2 a x}+\frac {\dilog \left (a x +1\right )}{2}+\frac {\ln \left (a x \right ) \ln \left (a x +1\right )}{2}+\frac {\dilog \left (a x \right )}{2}\right )\)	\(78\)
default	\(a^{2} \left (-\arctanh \left (a x \right ) \ln \left (a x \right )-\frac {\arctanh \left (a x \right )}{2 a^{2} x^{2}}-\frac {\ln \left (a x -1\right )}{4}+\frac {\ln \left (a x +1\right )}{4}-\frac {1}{2 a x}+\frac {\dilog \left (a x +1\right )}{2}+\frac {\ln \left (a x \right ) \ln \left (a x +1\right )}{2}+\frac {\dilog \left (a x \right )}{2}\right )\)	\(78\)
risch	\(-\frac {a}{2 x}+\frac {a^{2} \ln \left (-a x \right )}{4}-\frac {a^{2} \ln \left (-a x +1\right )}{4}+\frac {\ln \left (-a x +1\right )}{4 x^{2}}-\frac {a^{2} \dilog \left (-a x +1\right )}{2}-\frac {a^{2} \ln \left (a x \right )}{4}+\frac {a^{2} \ln \left (a x +1\right )}{4}-\frac {\ln \left (a x +1\right )}{4 x^{2}}+\frac {a^{2} \dilog \left (a x +1\right )}{2}\)	\(96\)
meijerg	\(\frac {i a^{2} \left (\frac {2 i}{x a}+\frac {2 i \left (-a x +1\right ) \left (a x +1\right ) \arctanh \left (a x \right )}{x^{2} a^{2}}\right )}{4}+\frac {i a^{2} \left (\frac {2 i a x \polylog \left (2, \sqrt {a^{2} x^{2}}\right )}{\sqrt {a^{2} x^{2}}}-\frac {2 i a x \polylog \left (2, -\sqrt {a^{2} x^{2}}\right )}{\sqrt {a^{2} x^{2}}}\right )}{4}\)	\(101\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^2*(-arctanh(a*x)*ln(a*x)-1/2*arctanh(a*x)/a^2/x^2-1/4*ln(a*x-1)+1/4*ln(a*x+1)-1/2/a/x+1/2*dilog(a*x+1)+1/2*l
n(a*x)*ln(a*x+1)+1/2*dilog(a*x))

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Maxima [A]
time = 0.27, size = 81, normalized size = 1.45 \begin {gather*} \frac {1}{4} \, {\left (2 \, {\left (\log \left (a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-a x\right )\right )} a - 2 \, {\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}{x}\right )} a - \frac {1}{2} \, {\left (a^{2} \log \left (x^{2}\right ) + \frac {1}{x^{2}}\right )} \operatorname {artanh}\left (a x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*(log(a*x + 1)*log(x) + dilog(-a*x))*a - 2*(log(-a*x + 1)*log(x) + dilog(a*x))*a + a*log(a*x + 1) - a*lo
g(a*x - 1) - 2/x)*a - 1/2*(a^2*log(x^2) + 1/x^2)*arctanh(a*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)/x^3,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(-atanh(a*x)/x**3, x) - Integral(a**2*atanh(a*x)/x, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 330 vs. \(2 (44) = 88\).
time = 1.31, size = 330, normalized size = 5.89 \begin {gather*} a^{2} {\left (\frac {\log \left (\frac {{\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}}\right )}{a} - \frac {\log \left ({\left | \frac {{\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} - 1 \right |}\right )}{a} + \frac {\frac {{\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} - 2}{a {\left (\frac {{\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} - 1\right )}} - \frac {2 \, \log \left (-\frac {\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{a - \frac {a {\left (\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} + 1\right )}}{\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} - 1}} - 1}{\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{a - \frac {a {\left (\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} + 1\right )}}{\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} - 1}} + 1}\right )}{a {\left (\frac {{\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} - 1\right )}^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^2*(log((a*x + 1)^2/(a*x - 1)^2)/a - log(abs((a*x + 1)^2/(a*x - 1)^2 - 1))/a + ((a*x + 1)^2/(a*x - 1)^2 - 2)/
(a*((a*x + 1)^2/(a*x - 1)^2 - 1)) - 2*log(-(a*((a*x + 1)/(a*x - 1) + 1)/(a - a*(a*((a*x + 1)/(a*x - 1) + 1)/((
a*x + 1)*a/(a*x - 1) - a) + 1)/(a*((a*x + 1)/(a*x - 1) + 1)/((a*x + 1)*a/(a*x - 1) - a) - 1)) - 1)/(a*((a*x +
1)/(a*x - 1) + 1)/(a - a*(a*((a*x + 1)/(a*x - 1) + 1)/((a*x + 1)*a/(a*x - 1) - a) + 1)/(a*((a*x + 1)/(a*x - 1)
+ 1)/((a*x + 1)*a/(a*x - 1) - a) - 1)) + 1))/(a*((a*x + 1)^2/(a*x - 1)^2 - 1)^2))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} -\int \frac {\mathrm {atanh}\left (a\,x\right )\,\left (a^2\,x^2-1\right )}{x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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