3.364 \(\int \frac{x^5 \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx\)

Optimal. Leaf size=139 \[ -\frac{x^3 \sqrt{1-a^2 x^2}}{20 a^3}-\frac{5 x \sqrt{1-a^2 x^2}}{24 a^5}-\frac{x^4 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{5 a^2}-\frac{4 x^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{15 a^4}-\frac{8 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{15 a^6}+\frac{89 \sin ^{-1}(a x)}{120 a^6} \]

[Out]

(-5*x*Sqrt[1 - a^2*x^2])/(24*a^5) - (x^3*Sqrt[1 - a^2*x^2])/(20*a^3) + (89*ArcSin[a*x])/(120*a^6) - (8*Sqrt[1
- a^2*x^2]*ArcTanh[a*x])/(15*a^6) - (4*x^2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(15*a^4) - (x^4*Sqrt[1 - a^2*x^2]*A
rcTanh[a*x])/(5*a^2)

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Rubi [A]  time = 0.23484, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {6016, 321, 216, 5994} \[ -\frac{x^3 \sqrt{1-a^2 x^2}}{20 a^3}-\frac{5 x \sqrt{1-a^2 x^2}}{24 a^5}-\frac{x^4 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{5 a^2}-\frac{4 x^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{15 a^4}-\frac{8 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{15 a^6}+\frac{89 \sin ^{-1}(a x)}{120 a^6} \]

Antiderivative was successfully verified.

[In]

Int[(x^5*ArcTanh[a*x])/Sqrt[1 - a^2*x^2],x]

[Out]

(-5*x*Sqrt[1 - a^2*x^2])/(24*a^5) - (x^3*Sqrt[1 - a^2*x^2])/(20*a^3) + (89*ArcSin[a*x])/(120*a^6) - (8*Sqrt[1
- a^2*x^2]*ArcTanh[a*x])/(15*a^6) - (4*x^2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(15*a^4) - (x^4*Sqrt[1 - a^2*x^2]*A
rcTanh[a*x])/(5*a^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 \frac{x^5 \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx &=-\frac{x^4 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{5 a^2}+\frac{4 \int \frac{x^3 \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{5 a^2}+\frac{\int \frac{x^4}{\sqrt{1-a^2 x^2}} \, dx}{5 a}\\ &=-\frac{x^3 \sqrt{1-a^2 x^2}}{20 a^3}-\frac{4 x^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{15 a^4}-\frac{x^4 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{5 a^2}+\frac{8 \int \frac{x \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{15 a^4}+\frac{3 \int \frac{x^2}{\sqrt{1-a^2 x^2}} \, dx}{20 a^3}+\frac{4 \int \frac{x^2}{\sqrt{1-a^2 x^2}} \, dx}{15 a^3}\\ &=-\frac{5 x \sqrt{1-a^2 x^2}}{24 a^5}-\frac{x^3 \sqrt{1-a^2 x^2}}{20 a^3}-\frac{8 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{15 a^6}-\frac{4 x^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{15 a^4}-\frac{x^4 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{5 a^2}+\frac{3 \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{40 a^5}+\frac{2 \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{15 a^5}+\frac{8 \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{15 a^5}\\ &=-\frac{5 x \sqrt{1-a^2 x^2}}{24 a^5}-\frac{x^3 \sqrt{1-a^2 x^2}}{20 a^3}+\frac{89 \sin ^{-1}(a x)}{120 a^6}-\frac{8 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{15 a^6}-\frac{4 x^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{15 a^4}-\frac{x^4 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{5 a^2}\\ \end{align*}

Mathematica [A]  time = 0.0986469, size = 79, normalized size = 0.57 \[ -\frac{a x \sqrt{1-a^2 x^2} \left (6 a^2 x^2+25\right )+8 \sqrt{1-a^2 x^2} \left (3 a^4 x^4+4 a^2 x^2+8\right ) \tanh ^{-1}(a x)-89 \sin ^{-1}(a x)}{120 a^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^5*ArcTanh[a*x])/Sqrt[1 - a^2*x^2],x]

[Out]

-(a*x*Sqrt[1 - a^2*x^2]*(25 + 6*a^2*x^2) - 89*ArcSin[a*x] + 8*Sqrt[1 - a^2*x^2]*(8 + 4*a^2*x^2 + 3*a^4*x^4)*Ar
cTanh[a*x])/(120*a^6)

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Maple [C]  time = 0.282, size = 120, normalized size = 0.9 \begin{align*} -{\frac{24\,{a}^{4}{x}^{4}{\it Artanh} \left ( ax \right ) +6\,{x}^{3}{a}^{3}+32\,{a}^{2}{x}^{2}{\it Artanh} \left ( ax \right ) +25\,ax+64\,{\it Artanh} \left ( ax \right ) }{120\,{a}^{6}}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) }}+{\frac{{\frac{89\,i}{120}}}{{a}^{6}}\ln \left ({(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+i \right ) }-{\frac{{\frac{89\,i}{120}}}{{a}^{6}}\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^5*arctanh(a*x)/(-a^2*x^2+1)^(1/2),x)

[Out]

-1/120/a^6*(-(a*x-1)*(a*x+1))^(1/2)*(24*a^4*x^4*arctanh(a*x)+6*x^3*a^3+32*a^2*x^2*arctanh(a*x)+25*a*x+64*arcta
nh(a*x))+89/120*I*ln((a*x+1)/(-a^2*x^2+1)^(1/2)+I)/a^6-89/120*I*ln((a*x+1)/(-a^2*x^2+1)^(1/2)-I)/a^6

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Maxima [A]  time = 1.45784, size = 269, normalized size = 1.94 \begin{align*} -\frac{1}{120} \, a{\left (\frac{3 \,{\left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1} x^{3}}{a^{2}} + \frac{3 \, \sqrt{-a^{2} x^{2} + 1} x}{a^{4}} - \frac{3 \, \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{\sqrt{a^{2}} a^{4}}\right )}}{a^{2}} + \frac{16 \,{\left (\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^{4}} - \frac{64 \, \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{\sqrt{a^{2}} a^{6}}\right )} - \frac{1}{15} \,{\left (\frac{3 \, \sqrt{-a^{2} x^{2} + 1} x^{4}}{a^{2}} + \frac{4 \, \sqrt{-a^{2} x^{2} + 1} x^{2}}{a^{4}} + \frac{8 \, \sqrt{-a^{2} x^{2} + 1}}{a^{6}}\right )} \operatorname{artanh}\left (a x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*arctanh(a*x)/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

-1/120*a*(3*(2*sqrt(-a^2*x^2 + 1)*x^3/a^2 + 3*sqrt(-a^2*x^2 + 1)*x/a^4 - 3*arcsin(a^2*x/sqrt(a^2))/(sqrt(a^2)*
a^4))/a^2 + 16*(sqrt(-a^2*x^2 + 1)*x/a^2 - arcsin(a^2*x/sqrt(a^2))/(sqrt(a^2)*a^2))/a^4 - 64*arcsin(a^2*x/sqrt
(a^2))/(sqrt(a^2)*a^6)) - 1/15*(3*sqrt(-a^2*x^2 + 1)*x^4/a^2 + 4*sqrt(-a^2*x^2 + 1)*x^2/a^4 + 8*sqrt(-a^2*x^2
+ 1)/a^6)*arctanh(a*x)

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Fricas [A]  time = 2.0824, size = 212, normalized size = 1.53 \begin{align*} -\frac{{\left (6 \, a^{3} x^{3} + 25 \, a x + 4 \,{\left (3 \, a^{4} x^{4} + 4 \, a^{2} x^{2} + 8\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )\right )} \sqrt{-a^{2} x^{2} + 1} + 178 \, \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right )}{120 \, a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*arctanh(a*x)/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

-1/120*((6*a^3*x^3 + 25*a*x + 4*(3*a^4*x^4 + 4*a^2*x^2 + 8)*log(-(a*x + 1)/(a*x - 1)))*sqrt(-a^2*x^2 + 1) + 17
8*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)))/a^6

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5} \operatorname{atanh}{\left (a x \right )}}{\sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*atanh(a*x)/(-a**2*x**2+1)**(1/2),x)

[Out]

Integral(x**5*atanh(a*x)/sqrt(-(a*x - 1)*(a*x + 1)), x)

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Giac [A]  time = 1.25854, size = 161, normalized size = 1.16 \begin{align*} -\frac{1}{120} \, \sqrt{-a^{2} x^{2} + 1} x{\left (\frac{6 \, x^{2}}{a^{3}} + \frac{25}{a^{5}}\right )} - \frac{{\left (3 \,{\left (a^{2} x^{2} - 1\right )}^{2} \sqrt{-a^{2} x^{2} + 1} - 10 \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} + 15 \, \sqrt{-a^{2} x^{2} + 1}\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )}{30 \, a^{6}} + \frac{89 \, \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{120 \, a^{5}{\left | a \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*arctanh(a*x)/(-a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

-1/120*sqrt(-a^2*x^2 + 1)*x*(6*x^2/a^3 + 25/a^5) - 1/30*(3*(a^2*x^2 - 1)^2*sqrt(-a^2*x^2 + 1) - 10*(-a^2*x^2 +
1)^(3/2) + 15*sqrt(-a^2*x^2 + 1))*log(-(a*x + 1)/(a*x - 1))/a^6 + 89/120*arcsin(a*x)*sgn(a)/(a^5*abs(a))