3.427 \(\int x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x) \, dx\)

Optimal. Leaf size=136 \[ \frac{x^3 \sqrt{1-a^2 x^2}}{20 a}+\frac{x \sqrt{1-a^2 x^2}}{24 a^3}+\frac{1}{5} x^4 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-\frac{x^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{15 a^2}-\frac{2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{15 a^4}+\frac{11 \sin ^{-1}(a x)}{120 a^4} \]

[Out]

(x*Sqrt[1 - a^2*x^2])/(24*a^3) + (x^3*Sqrt[1 - a^2*x^2])/(20*a) + (11*ArcSin[a*x])/(120*a^4) - (2*Sqrt[1 - a^2
*x^2]*ArcTanh[a*x])/(15*a^4) - (x^2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(15*a^2) + (x^4*Sqrt[1 - a^2*x^2]*ArcTanh[
a*x])/5

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Rubi [A]  time = 0.203294, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {6010, 6016, 321, 216, 5994} \[ \frac{x^3 \sqrt{1-a^2 x^2}}{20 a}+\frac{x \sqrt{1-a^2 x^2}}{24 a^3}+\frac{1}{5} x^4 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-\frac{x^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{15 a^2}-\frac{2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{15 a^4}+\frac{11 \sin ^{-1}(a x)}{120 a^4} \]

Antiderivative was successfully verified.

[In]

Int[x^3*Sqrt[1 - a^2*x^2]*ArcTanh[a*x],x]

[Out]

(x*Sqrt[1 - a^2*x^2])/(24*a^3) + (x^3*Sqrt[1 - a^2*x^2])/(20*a) + (11*ArcSin[a*x])/(120*a^4) - (2*Sqrt[1 - a^2
*x^2]*ArcTanh[a*x])/(15*a^4) - (x^2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(15*a^2) + (x^4*Sqrt[1 - a^2*x^2]*ArcTanh[
a*x])/5

Rule 6010

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

Rule 6016

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

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 5994

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]

Rubi steps

\begin{align*} \int x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x) \, dx &=\frac{1}{5} x^4 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)+\frac{1}{5} \int \frac{x^3 \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx-\frac{1}{5} a \int \frac{x^4}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{x^3 \sqrt{1-a^2 x^2}}{20 a}-\frac{x^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{15 a^2}+\frac{1}{5} x^4 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)+\frac{2 \int \frac{x \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{15 a^2}+\frac{\int \frac{x^2}{\sqrt{1-a^2 x^2}} \, dx}{15 a}-\frac{3 \int \frac{x^2}{\sqrt{1-a^2 x^2}} \, dx}{20 a}\\ &=\frac{x \sqrt{1-a^2 x^2}}{24 a^3}+\frac{x^3 \sqrt{1-a^2 x^2}}{20 a}-\frac{2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{15 a^4}-\frac{x^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{15 a^2}+\frac{1}{5} x^4 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)+\frac{\int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{30 a^3}-\frac{3 \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{40 a^3}+\frac{2 \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{15 a^3}\\ &=\frac{x \sqrt{1-a^2 x^2}}{24 a^3}+\frac{x^3 \sqrt{1-a^2 x^2}}{20 a}+\frac{11 \sin ^{-1}(a x)}{120 a^4}-\frac{2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{15 a^4}-\frac{x^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{15 a^2}+\frac{1}{5} x^4 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)\\ \end{align*}

Mathematica [A]  time = 0.0649956, size = 79, normalized size = 0.58 \[ \frac{a x \sqrt{1-a^2 x^2} \left (6 a^2 x^2+5\right )+8 \sqrt{1-a^2 x^2} \left (3 a^4 x^4-a^2 x^2-2\right ) \tanh ^{-1}(a x)+11 \sin ^{-1}(a x)}{120 a^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*Sqrt[1 - a^2*x^2]*ArcTanh[a*x],x]

[Out]

(a*x*Sqrt[1 - a^2*x^2]*(5 + 6*a^2*x^2) + 11*ArcSin[a*x] + 8*Sqrt[1 - a^2*x^2]*(-2 - a^2*x^2 + 3*a^4*x^4)*ArcTa
nh[a*x])/(120*a^4)

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Maple [C]  time = 0.259, size = 120, normalized size = 0.9 \begin{align*}{\frac{24\,{a}^{4}{x}^{4}{\it Artanh} \left ( ax \right ) +6\,{x}^{3}{a}^{3}-8\,{a}^{2}{x}^{2}{\it Artanh} \left ( ax \right ) +5\,ax-16\,{\it Artanh} \left ( ax \right ) }{120\,{a}^{4}}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) }}+{\frac{{\frac{11\,i}{120}}}{{a}^{4}}\ln \left ({(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+i \right ) }-{\frac{{\frac{11\,i}{120}}}{{a}^{4}}\ln \left ({(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-i \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120/a^4*(-(a*x-1)*(a*x+1))^(1/2)*(24*a^4*x^4*arctanh(a*x)+6*x^3*a^3-8*a^2*x^2*arctanh(a*x)+5*a*x-16*arctanh(
a*x))+11/120*I*ln((a*x+1)/(-a^2*x^2+1)^(1/2)+I)/a^4-11/120*I*ln((a*x+1)/(-a^2*x^2+1)^(1/2)-I)/a^4

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Maxima [A]  time = 1.47476, size = 201, normalized size = 1.48 \begin{align*} -\frac{1}{120} \, a{\left (\frac{3 \,{\left (\frac{2 \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} x}{a^{2}} - \frac{\sqrt{-a^{2} x^{2} + 1} x}{a^{2}} - \frac{\arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{\sqrt{a^{2}} a^{2}}\right )}}{a^{2}} - \frac{8 \,{\left (\sqrt{-a^{2} x^{2} + 1} x + \frac{\arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{\sqrt{a^{2}}}\right )}}{a^{4}}\right )} - \frac{1}{15} \,{\left (\frac{3 \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} x^{2}}{a^{2}} + \frac{2 \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{a^{4}}\right )} \operatorname{artanh}\left (a x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/120*a*(3*(2*(-a^2*x^2 + 1)^(3/2)*x/a^2 - sqrt(-a^2*x^2 + 1)*x/a^2 - arcsin(a^2*x/sqrt(a^2))/(sqrt(a^2)*a^2)
)/a^2 - 8*(sqrt(-a^2*x^2 + 1)*x + arcsin(a^2*x/sqrt(a^2))/sqrt(a^2))/a^4) - 1/15*(3*(-a^2*x^2 + 1)^(3/2)*x^2/a
^2 + 2*(-a^2*x^2 + 1)^(3/2)/a^4)*arctanh(a*x)

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Fricas [A]  time = 2.24214, size = 205, normalized size = 1.51 \begin{align*} \frac{{\left (6 \, a^{3} x^{3} + 5 \, a x + 4 \,{\left (3 \, a^{4} x^{4} - a^{2} x^{2} - 2\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )\right )} \sqrt{-a^{2} x^{2} + 1} - 22 \, \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right )}{120 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*((6*a^3*x^3 + 5*a*x + 4*(3*a^4*x^4 - a^2*x^2 - 2)*log(-(a*x + 1)/(a*x - 1)))*sqrt(-a^2*x^2 + 1) - 22*arc
tan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)))/a^4

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \sqrt{- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname{atanh}{\left (a x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3*sqrt(-(a*x - 1)*(a*x + 1))*atanh(a*x), x)

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Giac [A]  time = 1.18705, size = 139, normalized size = 1.02 \begin{align*} \frac{{\left (6 \, a^{2} x^{2} + 5\right )} \sqrt{-a^{2} x^{2} + 1} x + \frac{11 \, \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{{\left | a \right |}}}{120 \, a^{3}} + \frac{{\left (3 \,{\left (a^{2} x^{2} - 1\right )}^{2} \sqrt{-a^{2} x^{2} + 1} - 5 \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )}{30 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*((6*a^2*x^2 + 5)*sqrt(-a^2*x^2 + 1)*x + 11*arcsin(a*x)*sgn(a)/abs(a))/a^3 + 1/30*(3*(a^2*x^2 - 1)^2*sqrt
(-a^2*x^2 + 1) - 5*(-a^2*x^2 + 1)^(3/2))*log(-(a*x + 1)/(a*x - 1))/a^4