3.4.6 \(\int \frac {\tanh ^{-1}(a x)}{x (1-a^2 x^2)^3} \, dx\) [306]

Optimal. Leaf size=129 \[ -\frac {a x}{16 \left (1-a^2 x^2\right )^2}-\frac {11 a x}{32 \left (1-a^2 x^2\right )}-\frac {11}{32} \tanh ^{-1}(a x)+\frac {\tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {\tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {1}{2} \tanh ^{-1}(a x)^2+\tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-\frac {1}{2} \text {PolyLog}\left (2,-1+\frac {2}{1+a x}\right ) \]

[Out]

-1/16*a*x/(-a^2*x^2+1)^2-11/32*a*x/(-a^2*x^2+1)-11/32*arctanh(a*x)+1/4*arctanh(a*x)/(-a^2*x^2+1)^2+1/2*arctanh
(a*x)/(-a^2*x^2+1)+1/2*arctanh(a*x)^2+arctanh(a*x)*ln(2-2/(a*x+1))-1/2*polylog(2,-1+2/(a*x+1))

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Rubi [A]
time = 0.19, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {6177, 6135, 6079, 2497, 6141, 205, 212} \begin {gather*} -\frac {11 a x}{32 \left (1-a^2 x^2\right )}-\frac {a x}{16 \left (1-a^2 x^2\right )^2}+\frac {\tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )^2}-\frac {1}{2} \text {Li}_2\left (\frac {2}{a x+1}-1\right )+\frac {1}{2} \tanh ^{-1}(a x)^2-\frac {11}{32} \tanh ^{-1}(a x)+\log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/16*(a*x)/(1 - a^2*x^2)^2 - (11*a*x)/(32*(1 - a^2*x^2)) - (11*ArcTanh[a*x])/32 + ArcTanh[a*x]/(4*(1 - a^2*x^
2)^2) + ArcTanh[a*x]/(2*(1 - a^2*x^2)) + ArcTanh[a*x]^2/2 + ArcTanh[a*x]*Log[2 - 2/(1 + a*x)] - PolyLog[2, -1
+ 2/(1 + a*x)]/2

Rule 205

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] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

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 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 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 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]

Rubi steps

\begin {align*} \int \frac {\tanh ^{-1}(a x)}{x \left (1-a^2 x^2\right )^3} \, dx &=a^2 \int \frac {x \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^3} \, dx+\int \frac {\tanh ^{-1}(a x)}{x \left (1-a^2 x^2\right )^2} \, dx\\ &=\frac {\tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )^2}-\frac {1}{4} a \int \frac {1}{\left (1-a^2 x^2\right )^3} \, dx+a^2 \int \frac {x \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx+\int \frac {\tanh ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx\\ &=-\frac {a x}{16 \left (1-a^2 x^2\right )^2}+\frac {\tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {\tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {1}{2} \tanh ^{-1}(a x)^2-\frac {1}{16} (3 a) \int \frac {1}{\left (1-a^2 x^2\right )^2} \, dx-\frac {1}{2} a \int \frac {1}{\left (1-a^2 x^2\right )^2} \, dx+\int \frac {\tanh ^{-1}(a x)}{x (1+a x)} \, dx\\ &=-\frac {a x}{16 \left (1-a^2 x^2\right )^2}-\frac {11 a x}{32 \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {\tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {1}{2} \tanh ^{-1}(a x)^2+\tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-\frac {1}{32} (3 a) \int \frac {1}{1-a^2 x^2} \, dx-\frac {1}{4} a \int \frac {1}{1-a^2 x^2} \, dx-a \int \frac {\log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac {a x}{16 \left (1-a^2 x^2\right )^2}-\frac {11 a x}{32 \left (1-a^2 x^2\right )}-\frac {11}{32} \tanh ^{-1}(a x)+\frac {\tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {\tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {1}{2} \tanh ^{-1}(a x)^2+\tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-\frac {1}{2} \text {Li}_2\left (-1+\frac {2}{1+a x}\right )\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 81, normalized size = 0.63 \begin {gather*} \frac {1}{128} \left (64 \tanh ^{-1}(a x)^2+4 \tanh ^{-1}(a x) \left (12 \cosh \left (2 \tanh ^{-1}(a x)\right )+\cosh \left (4 \tanh ^{-1}(a x)\right )+32 \log \left (1-e^{-2 \tanh ^{-1}(a x)}\right )\right )-64 \text {PolyLog}\left (2,e^{-2 \tanh ^{-1}(a x)}\right )-24 \sinh \left (2 \tanh ^{-1}(a x)\right )-\sinh \left (4 \tanh ^{-1}(a x)\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(64*ArcTanh[a*x]^2 + 4*ArcTanh[a*x]*(12*Cosh[2*ArcTanh[a*x]] + Cosh[4*ArcTanh[a*x]] + 32*Log[1 - E^(-2*ArcTanh
[a*x])]) - 64*PolyLog[2, E^(-2*ArcTanh[a*x])] - 24*Sinh[2*ArcTanh[a*x]] - Sinh[4*ArcTanh[a*x]])/128

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(233\) vs. \(2(115)=230\).
time = 0.92, size = 234, normalized size = 1.81

method	result	size
derivativedivides	\(\frac {\arctanh \left (a x \right )}{16 \left (a x -1\right )^{2}}-\frac {5 \arctanh \left (a x \right )}{16 \left (a x -1\right )}-\frac {\arctanh \left (a x \right ) \ln \left (a x -1\right )}{2}+\arctanh \left (a x \right ) \ln \left (a x \right )+\frac {\arctanh \left (a x \right )}{16 \left (a x +1\right )^{2}}+\frac {5 \arctanh \left (a x \right )}{16 \left (a x +1\right )}-\frac {\arctanh \left (a x \right ) \ln \left (a x +1\right )}{2}-\frac {\dilog \left (a x \right )}{2}-\frac {\dilog \left (a x +1\right )}{2}-\frac {\ln \left (a x \right ) \ln \left (a x +1\right )}{2}+\frac {\dilog \left (\frac {a x}{2}+\frac {1}{2}\right )}{2}+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{4}-\frac {\ln \left (a x -1\right )^{2}}{8}+\frac {\ln \left (a x +1\right )^{2}}{8}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{4}-\frac {1}{64 \left (a x -1\right )^{2}}+\frac {11}{64 \left (a x -1\right )}+\frac {11 \ln \left (a x -1\right )}{64}+\frac {1}{64 \left (a x +1\right )^{2}}+\frac {11}{64 \left (a x +1\right )}-\frac {11 \ln \left (a x +1\right )}{64}\)	\(234\)
default	\(\frac {\arctanh \left (a x \right )}{16 \left (a x -1\right )^{2}}-\frac {5 \arctanh \left (a x \right )}{16 \left (a x -1\right )}-\frac {\arctanh \left (a x \right ) \ln \left (a x -1\right )}{2}+\arctanh \left (a x \right ) \ln \left (a x \right )+\frac {\arctanh \left (a x \right )}{16 \left (a x +1\right )^{2}}+\frac {5 \arctanh \left (a x \right )}{16 \left (a x +1\right )}-\frac {\arctanh \left (a x \right ) \ln \left (a x +1\right )}{2}-\frac {\dilog \left (a x \right )}{2}-\frac {\dilog \left (a x +1\right )}{2}-\frac {\ln \left (a x \right ) \ln \left (a x +1\right )}{2}+\frac {\dilog \left (\frac {a x}{2}+\frac {1}{2}\right )}{2}+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{4}-\frac {\ln \left (a x -1\right )^{2}}{8}+\frac {\ln \left (a x +1\right )^{2}}{8}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{4}-\frac {1}{64 \left (a x -1\right )^{2}}+\frac {11}{64 \left (a x -1\right )}+\frac {11 \ln \left (a x -1\right )}{64}+\frac {1}{64 \left (a x +1\right )^{2}}+\frac {11}{64 \left (a x +1\right )}-\frac {11 \ln \left (a x +1\right )}{64}\)	\(234\)
risch	\(\frac {11 \ln \left (a x -1\right )}{128}-\frac {5 \ln \left (a x +1\right ) \left (a x +1\right )}{64 \left (a x -1\right )}+\frac {1}{64 a x -64}-\frac {\ln \left (a x +1\right ) \left (a x +1\right ) \left (a x -3\right )}{128 \left (a x -1\right )^{2}}-\frac {\ln \left (a x +1\right )^{2}}{8}+\frac {\ln \left (a x +1\right )}{32 \left (a x +1\right )^{2}}+\frac {1}{64 \left (a x +1\right )^{2}}-\frac {\dilog \left (a x +1\right )}{2}+\frac {5 \ln \left (a x +1\right )}{32 \left (a x +1\right )}+\frac {5}{32 \left (a x +1\right )}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{4}+\frac {\dilog \left (\frac {a x}{2}+\frac {1}{2}\right )}{4}-\frac {11 \ln \left (-a x -1\right )}{128}+\frac {5 \ln \left (-a x +1\right ) \left (-a x +1\right )}{64 \left (-a x -1\right )}-\frac {1}{64 \left (-a x -1\right )}+\frac {\ln \left (-a x +1\right ) \left (-a x +1\right ) \left (-a x -3\right )}{128 \left (-a x -1\right )^{2}}+\frac {\ln \left (-a x +1\right )^{2}}{8}-\frac {\ln \left (-a x +1\right )}{32 \left (-a x +1\right )^{2}}-\frac {1}{64 \left (-a x +1\right )^{2}}+\frac {\dilog \left (-a x +1\right )}{2}-\frac {5 \ln \left (-a x +1\right )}{32 \left (-a x +1\right )}-\frac {5}{32 \left (-a x +1\right )}+\frac {\left (\ln \left (-a x +1\right )-\ln \left (-\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{4}-\frac {\dilog \left (-\frac {a x}{2}+\frac {1}{2}\right )}{4}\)	\(344\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*arctanh(a*x)/(a*x-1)^2-5/16*arctanh(a*x)/(a*x-1)-1/2*arctanh(a*x)*ln(a*x-1)+arctanh(a*x)*ln(a*x)+1/16*arc
tanh(a*x)/(a*x+1)^2+5/16*arctanh(a*x)/(a*x+1)-1/2*arctanh(a*x)*ln(a*x+1)-1/2*dilog(a*x)-1/2*dilog(a*x+1)-1/2*l
n(a*x)*ln(a*x+1)+1/2*dilog(1/2*a*x+1/2)+1/4*ln(a*x-1)*ln(1/2*a*x+1/2)-1/8*ln(a*x-1)^2+1/8*ln(a*x+1)^2-1/4*(ln(
a*x+1)-ln(1/2*a*x+1/2))*ln(-1/2*a*x+1/2)-1/64/(a*x-1)^2+11/64/(a*x-1)+11/64*ln(a*x-1)+1/64/(a*x+1)^2+11/64/(a*
x+1)-11/64*ln(a*x+1)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 268 vs. \(2 (110) = 220\).
time = 0.27, size = 268, normalized size = 2.08 \begin {gather*} \frac {1}{64} \, a {\left (\frac {2 \, {\left (11 \, a^{3} x^{3} + 4 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{2} - 8 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 4 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2} - 13 \, a x\right )}}{a^{5} x^{4} - 2 \, a^{3} x^{2} + a} + \frac {32 \, {\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 {32 \, {\left (\log \left (a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-a x\right )\right )}}{a} + \frac {32 \, {\left (\log \left (-a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (a x\right )\right )}}{a} - \frac {11 \, \log \left (a x + 1\right )}{a} + \frac {11 \, \log \left (a x - 1\right )}{a}\right )} - \frac {1}{4} \, {\left (\frac {2 \, a^{2} x^{2} - 3}{a^{4} x^{4} - 2 \, a^{2} x^{2} + 1} + 2 \, \log \left (a^{2} x^{2} - 1\right ) - 2 \, \log \left (x^{2}\right )\right )} \operatorname {artanh}\left (a x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*a*(2*(11*a^3*x^3 + 4*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)^2 - 8*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)*
log(a*x - 1) - 4*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1)^2 - 13*a*x)/(a^5*x^4 - 2*a^3*x^2 + a) + 32*(log(a*x -
1)*log(1/2*a*x + 1/2) + dilog(-1/2*a*x + 1/2))/a - 32*(log(a*x + 1)*log(x) + dilog(-a*x))/a + 32*(log(-a*x + 1
)*log(x) + dilog(a*x))/a - 11*log(a*x + 1)/a + 11*log(a*x - 1)/a) - 1/4*((2*a^2*x^2 - 3)/(a^4*x^4 - 2*a^2*x^2
+ 1) + 2*log(a^2*x^2 - 1) - 2*log(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(arctanh(a*x)/x/(-a^2*x^2+1)^3,x, algorithm="fricas")

[Out]

integral(-arctanh(a*x)/(a^6*x^7 - 3*a^4*x^5 + 3*a^2*x^3 - x), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {\operatorname {atanh}{\left (a x \right )}}{a^{6} x^{7} - 3 a^{4} x^{5} + 3 a^{2} x^{3} - x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(atanh(a*x)/(a**6*x**7 - 3*a**4*x**5 + 3*a**2*x**3 - x), 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)/x/(-a^2*x^2+1)^3,x, algorithm="giac")

[Out]

integrate(-arctanh(a*x)/((a^2*x^2 - 1)^3*x), 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 )}{x\,{\left (a^2\,x^2-1\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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