3.165 \(\int (1-a^2 x^2) \tanh ^{-1}(a x) \, dx\)

Optimal. Leaf size=64 \[ \frac {1-a^2 x^2}{6 a}+\frac {\log \left (1-a^2 x^2\right )}{3 a}+\frac {1}{3} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)+\frac {2}{3} x \tanh ^{-1}(a x) \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5942, 5910, 260} \[ \frac {1-a^2 x^2}{6 a}+\frac {\log \left (1-a^2 x^2\right )}{3 a}+\frac {1}{3} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)+\frac {2}{3} x \tanh ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[(1 - a^2*x^2)*ArcTanh[a*x],x]

[Out]

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

Rule 260

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

Rule 5910

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTanh[c*x])^p, x] - Dist[b*c*p, In
t[(x*(a + b*ArcTanh[c*x])^(p - 1))/(1 - c^2*x^2), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 0]

Rule 5942

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

Rubi steps

\begin {align*} \int \left (1-a^2 x^2\right ) \tanh ^{-1}(a x) \, dx &=\frac {1-a^2 x^2}{6 a}+\frac {1}{3} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)+\frac {2}{3} \int \tanh ^{-1}(a x) \, dx\\ &=\frac {1-a^2 x^2}{6 a}+\frac {2}{3} x \tanh ^{-1}(a x)+\frac {1}{3} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)-\frac {1}{3} (2 a) \int \frac {x}{1-a^2 x^2} \, dx\\ &=\frac {1-a^2 x^2}{6 a}+\frac {2}{3} x \tanh ^{-1}(a x)+\frac {1}{3} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)+\frac {\log \left (1-a^2 x^2\right )}{3 a}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 47, normalized size = 0.73 \[ -\frac {1}{3} a^2 x^3 \tanh ^{-1}(a x)+\frac {\log \left (1-a^2 x^2\right )}{3 a}-\frac {a x^2}{6}+x \tanh ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.48, size = 53, normalized size = 0.83 \[ -\frac {a^{2} x^{2} + {\left (a^{3} x^{3} - 3 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) - 2 \, \log \left (a^{2} x^{2} - 1\right )}{6 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.18, size = 203, normalized size = 3.17 \[ \frac {2}{3} \, a {\left (\frac {\log \left (\frac {{\left | -a x - 1 \right |}}{{\left | a x - 1 \right |}}\right )}{a^{2}} - \frac {\log \left ({\left | -\frac {a x + 1}{a x - 1} + 1 \right |}\right )}{a^{2}} - \frac {{\left (\frac {3 \, {\left (a x + 1\right )}}{a x - 1} - 1\right )} \log \left (-\frac {\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} + 1}{\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} - 1}\right )}{a^{2} {\left (\frac {a x + 1}{a x - 1} - 1\right )}^{3}} - \frac {a x + 1}{{\left (a x - 1\right )} a^{2} {\left (\frac {a x + 1}{a x - 1} - 1\right )}^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*a*(log(abs(-a*x - 1)/abs(a*x - 1))/a^2 - log(abs(-(a*x + 1)/(a*x - 1) + 1))/a^2 - (3*(a*x + 1)/(a*x - 1) -
1)*log(-(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))/(a^2*((a*x + 1)/(a*x - 1) - 1)^3) - (a*x + 1)/((a*x - 1)*a^2*((a*x + 1)/(a*x - 1) - 1)
^2))

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maple [A]  time = 0.03, size = 48, normalized size = 0.75 \[ -\frac {a^{2} \arctanh \left (a x \right ) x^{3}}{3}+x \arctanh \left (a x \right )-\frac {a \,x^{2}}{6}+\frac {\ln \left (a x -1\right )}{3 a}+\frac {\ln \left (a x +1\right )}{3 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.31, size = 47, normalized size = 0.73 \[ -\frac {1}{6} \, {\left (x^{2} - \frac {2 \, \log \left (a x + 1\right )}{a^{2}} - \frac {2 \, \log \left (a x - 1\right )}{a^{2}}\right )} a - \frac {1}{3} \, {\left (a^{2} x^{3} - 3 \, x\right )} \operatorname {artanh}\left (a x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.84, size = 40, normalized size = 0.62 \[ x\,\mathrm {atanh}\left (a\,x\right )-\frac {a\,x^2}{6}+\frac {\ln \left (a^2\,x^2-1\right )}{3\,a}-\frac {a^2\,x^3\,\mathrm {atanh}\left (a\,x\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-atanh(a*x)*(a^2*x^2 - 1),x)

[Out]

x*atanh(a*x) - (a*x^2)/6 + log(a^2*x^2 - 1)/(3*a) - (a^2*x^3*atanh(a*x))/3

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sympy [A]  time = 0.68, size = 49, normalized size = 0.77 \[ \begin {cases} - \frac {a^{2} x^{3} \operatorname {atanh}{\left (a x \right )}}{3} - \frac {a x^{2}}{6} + x \operatorname {atanh}{\left (a x \right )} + \frac {2 \log {\left (x - \frac {1}{a} \right )}}{3 a} + \frac {2 \operatorname {atanh}{\left (a x \right )}}{3 a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a**2*x**3*atanh(a*x)/3 - a*x**2/6 + x*atanh(a*x) + 2*log(x - 1/a)/(3*a) + 2*atanh(a*x)/(3*a), Ne(a
, 0)), (0, True))

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