3.4.5 \(\int \frac {\tanh ^{-1}(a x)}{(1-a^2 x^2)^3} \, dx\) [305]

Optimal. Leaf size=94 \[ -\frac {1}{16 a \left (1-a^2 x^2\right )^2}-\frac {3}{16 a \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3 x \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )}+\frac {3 \tanh ^{-1}(a x)^2}{16 a} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {6107, 6103, 267} \begin {gather*} -\frac {3}{16 a \left (1-a^2 x^2\right )}-\frac {1}{16 a \left (1-a^2 x^2\right )^2}+\frac {3 x \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3 \tanh ^{-1}(a x)^2}{16 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcTanh[a*x]/(1 - a^2*x^2)^3,x]

[Out]

-1/16*1/(a*(1 - a^2*x^2)^2) - 3/(16*a*(1 - a^2*x^2)) + (x*ArcTanh[a*x])/(4*(1 - a^2*x^2)^2) + (3*x*ArcTanh[a*x
])/(8*(1 - a^2*x^2)) + (3*ArcTanh[a*x]^2)/(16*a)

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 6103

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 6107

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

Rubi steps

\begin {align*} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^3} \, dx &=-\frac {1}{16 a \left (1-a^2 x^2\right )^2}+\frac {x \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3}{4} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac {1}{16 a \left (1-a^2 x^2\right )^2}+\frac {x \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3 x \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )}+\frac {3 \tanh ^{-1}(a x)^2}{16 a}-\frac {1}{8} (3 a) \int \frac {x}{\left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac {1}{16 a \left (1-a^2 x^2\right )^2}-\frac {3}{16 a \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3 x \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )}+\frac {3 \tanh ^{-1}(a x)^2}{16 a}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 65, normalized size = 0.69 \begin {gather*} \frac {-4+3 a^2 x^2+\left (10 a x-6 a^3 x^3\right ) \tanh ^{-1}(a x)+3 \left (-1+a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{16 a \left (-1+a^2 x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4 + 3*a^2*x^2 + (10*a*x - 6*a^3*x^3)*ArcTanh[a*x] + 3*(-1 + a^2*x^2)^2*ArcTanh[a*x]^2)/(16*a*(-1 + a^2*x^2)^
2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(177\) vs. \(2(84)=168\).
time = 0.77, size = 178, normalized size = 1.89

method	result	size
derivativedivides	\(\frac {\frac {\arctanh \left (a x \right )}{16 \left (a x -1\right )^{2}}-\frac {3 \arctanh \left (a x \right )}{16 \left (a x -1\right )}-\frac {3 \arctanh \left (a x \right ) \ln \left (a x -1\right )}{16}-\frac {\arctanh \left (a x \right )}{16 \left (a x +1\right )^{2}}-\frac {3 \arctanh \left (a x \right )}{16 \left (a x +1\right )}+\frac {3 \arctanh \left (a x \right ) \ln \left (a x +1\right )}{16}+\frac {3 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{32}-\frac {3 \ln \left (a x -1\right )^{2}}{64}+\frac {3 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{32}-\frac {3 \ln \left (a x +1\right )^{2}}{64}-\frac {1}{64 \left (a x -1\right )^{2}}+\frac {7}{64 \left (a x -1\right )}-\frac {1}{64 \left (a x +1\right )^{2}}-\frac {7}{64 \left (a x +1\right )}}{a}\)	\(178\)
default	\(\frac {\frac {\arctanh \left (a x \right )}{16 \left (a x -1\right )^{2}}-\frac {3 \arctanh \left (a x \right )}{16 \left (a x -1\right )}-\frac {3 \arctanh \left (a x \right ) \ln \left (a x -1\right )}{16}-\frac {\arctanh \left (a x \right )}{16 \left (a x +1\right )^{2}}-\frac {3 \arctanh \left (a x \right )}{16 \left (a x +1\right )}+\frac {3 \arctanh \left (a x \right ) \ln \left (a x +1\right )}{16}+\frac {3 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{32}-\frac {3 \ln \left (a x -1\right )^{2}}{64}+\frac {3 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{32}-\frac {3 \ln \left (a x +1\right )^{2}}{64}-\frac {1}{64 \left (a x -1\right )^{2}}+\frac {7}{64 \left (a x -1\right )}-\frac {1}{64 \left (a x +1\right )^{2}}-\frac {7}{64 \left (a x +1\right )}}{a}\)	\(178\)
risch	\(\frac {3 \ln \left (a x +1\right )^{2}}{64 a}-\frac {\left (3 x^{4} \ln \left (-a x +1\right ) a^{4}+6 a^{3} x^{3}-6 x^{2} \ln \left (-a x +1\right ) a^{2}-10 a x +3 \ln \left (-a x +1\right )\right ) \ln \left (a x +1\right )}{32 \left (a^{2} x^{2}-1\right )^{2} a}+\frac {3 a^{4} x^{4} \ln \left (-a x +1\right )^{2}+12 a^{3} x^{3} \ln \left (-a x +1\right )-6 a^{2} x^{2} \ln \left (-a x +1\right )^{2}+12 a^{2} x^{2}-20 a x \ln \left (-a x +1\right )+3 \ln \left (-a x +1\right )^{2}-16}{64 a \left (a x +1\right ) \left (a x -1\right ) \left (a^{2} x^{2}-1\right )}\)	\(200\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a*(1/16*arctanh(a*x)/(a*x-1)^2-3/16*arctanh(a*x)/(a*x-1)-3/16*arctanh(a*x)*ln(a*x-1)-1/16*arctanh(a*x)/(a*x+
1)^2-3/16*arctanh(a*x)/(a*x+1)+3/16*arctanh(a*x)*ln(a*x+1)+3/32*ln(a*x-1)*ln(1/2*a*x+1/2)-3/64*ln(a*x-1)^2+3/3
2*(ln(a*x+1)-ln(1/2*a*x+1/2))*ln(-1/2*a*x+1/2)-3/64*ln(a*x+1)^2-1/64/(a*x-1)^2+7/64/(a*x-1)-1/64/(a*x+1)^2-7/6
4/(a*x+1))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 182 vs. \(2 (80) = 160\).
time = 0.26, size = 182, normalized size = 1.94 \begin {gather*} -\frac {1}{16} \, {\left (\frac {2 \, {\left (3 \, a^{2} x^{3} - 5 \, x\right )}}{a^{4} x^{4} - 2 \, a^{2} x^{2} + 1} - \frac {3 \, \log \left (a x + 1\right )}{a} + \frac {3 \, \log \left (a x - 1\right )}{a}\right )} \operatorname {artanh}\left (a x\right ) + \frac {{\left (12 \, a^{2} x^{2} - 3 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{2} + 6 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 3 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2} - 16\right )} a}{64 \, {\left (a^{6} x^{4} - 2 \, a^{4} x^{2} + a^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(2*(3*a^2*x^3 - 5*x)/(a^4*x^4 - 2*a^2*x^2 + 1) - 3*log(a*x + 1)/a + 3*log(a*x - 1)/a)*arctanh(a*x) + 1/6
4*(12*a^2*x^2 - 3*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)^2 + 6*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)*log(a*x
- 1) - 3*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1)^2 - 16)*a/(a^6*x^4 - 2*a^4*x^2 + a^2)

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Fricas [A]
time = 0.34, size = 97, normalized size = 1.03 \begin {gather*} \frac {12 \, a^{2} x^{2} + 3 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - 4 \, {\left (3 \, a^{3} x^{3} - 5 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) - 16}{64 \, {\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(12*a^2*x^2 + 3*(a^4*x^4 - 2*a^2*x^2 + 1)*log(-(a*x + 1)/(a*x - 1))^2 - 4*(3*a^3*x^3 - 5*a*x)*log(-(a*x +
1)/(a*x - 1)) - 16)/(a^5*x^4 - 2*a^3*x^2 + a)

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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^{6} - 3 a^{4} x^{4} + 3 a^{2} x^{2} - 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [B]
time = 1.37, size = 154, normalized size = 1.64 \begin {gather*} \frac {\frac {3\,a\,x^2}{2}-\frac {2}{a}}{8\,a^4\,x^4-16\,a^2\,x^2+8}-\ln \left (1-a\,x\right )\,\left (\frac {3\,\ln \left (a\,x+1\right )}{32\,a}+\frac {\frac {5\,x}{8}-\frac {3\,a^2\,x^3}{8}}{2\,a^4\,x^4-4\,a^2\,x^2+2}\right )+\frac {3\,{\ln \left (a\,x+1\right )}^2}{64\,a}+\frac {3\,{\ln \left (1-a\,x\right )}^2}{64\,a}+\frac {\ln \left (a\,x+1\right )\,\left (\frac {5\,x}{16\,a}-\frac {3\,a\,x^3}{16}\right )}{\frac {1}{a}-2\,a\,x^2+a^3\,x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-atanh(a*x)/(a^2*x^2 - 1)^3,x)

[Out]

((3*a*x^2)/2 - 2/a)/(8*a^4*x^4 - 16*a^2*x^2 + 8) - log(1 - a*x)*((3*log(a*x + 1))/(32*a) + ((5*x)/8 - (3*a^2*x
^3)/8)/(2*a^4*x^4 - 4*a^2*x^2 + 2)) + (3*log(a*x + 1)^2)/(64*a) + (3*log(1 - a*x)^2)/(64*a) + (log(a*x + 1)*((
5*x)/(16*a) - (3*a*x^3)/16))/(1/a - 2*a*x^2 + a^3*x^4)

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