3.437 \(\int \frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x^7} \, dx\)

Optimal. Leaf size=243 \[ -\frac{1}{16} a^6 \text{PolyLog}\left (2,-\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right )+\frac{1}{16} a^6 \text{PolyLog}\left (2,\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right )+\frac{a^5 \sqrt{1-a^2 x^2}}{720 x}-\frac{11 a^3 \sqrt{1-a^2 x^2}}{360 x^3}-\frac{a \sqrt{1-a^2 x^2}}{30 x^5}+\frac{a^4 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{16 x^2}+\frac{a^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{24 x^4}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{6 x^6}+\frac{1}{8} a^6 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right ) \]

[Out]

-(a*Sqrt[1 - a^2*x^2])/(30*x^5) - (11*a^3*Sqrt[1 - a^2*x^2])/(360*x^3) + (a^5*Sqrt[1 - a^2*x^2])/(720*x) - (Sq
rt[1 - a^2*x^2]*ArcTanh[a*x])/(6*x^6) + (a^2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(24*x^4) + (a^4*Sqrt[1 - a^2*x^2]
*ArcTanh[a*x])/(16*x^2) + (a^6*ArcTanh[a*x]*ArcTanh[Sqrt[1 - a*x]/Sqrt[1 + a*x]])/8 - (a^6*PolyLog[2, -(Sqrt[1
- a*x]/Sqrt[1 + a*x])])/16 + (a^6*PolyLog[2, Sqrt[1 - a*x]/Sqrt[1 + a*x]])/16

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Rubi [A]  time = 0.412757, antiderivative size = 243, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {6010, 6026, 271, 264, 6018} \[ -\frac{1}{16} a^6 \text{PolyLog}\left (2,-\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right )+\frac{1}{16} a^6 \text{PolyLog}\left (2,\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right )+\frac{a^5 \sqrt{1-a^2 x^2}}{720 x}-\frac{11 a^3 \sqrt{1-a^2 x^2}}{360 x^3}-\frac{a \sqrt{1-a^2 x^2}}{30 x^5}+\frac{a^4 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{16 x^2}+\frac{a^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{24 x^4}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{6 x^6}+\frac{1}{8} a^6 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/x^7,x]

[Out]

-(a*Sqrt[1 - a^2*x^2])/(30*x^5) - (11*a^3*Sqrt[1 - a^2*x^2])/(360*x^3) + (a^5*Sqrt[1 - a^2*x^2])/(720*x) - (Sq
rt[1 - a^2*x^2]*ArcTanh[a*x])/(6*x^6) + (a^2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(24*x^4) + (a^4*Sqrt[1 - a^2*x^2]
*ArcTanh[a*x])/(16*x^2) + (a^6*ArcTanh[a*x]*ArcTanh[Sqrt[1 - a*x]/Sqrt[1 + a*x]])/8 - (a^6*PolyLog[2, -(Sqrt[1
- a*x]/Sqrt[1 + a*x])])/16 + (a^6*PolyLog[2, Sqrt[1 - a*x]/Sqrt[1 + a*x]])/16

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 6026

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((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])^p)/(d*f*(m + 1)), x] + (-Dist[(b*c*p)/(f*(m + 1)), Int[((
f*x)^(m + 1)*(a + b*ArcTanh[c*x])^(p - 1))/Sqrt[d + e*x^2], x], x] + Dist[(c^2*(m + 2))/(f^2*(m + 1)), 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] && LtQ[m, -1] && NeQ[m, -2]

Rule 271

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

Rule 264

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

Rule 6018

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

Rubi steps

\begin{align*} \int \frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x^7} \, dx &=-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{5 x^6}-\frac{1}{5} \int \frac{\tanh ^{-1}(a x)}{x^7 \sqrt{1-a^2 x^2}} \, dx+\frac{1}{5} a \int \frac{1}{x^6 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{a \sqrt{1-a^2 x^2}}{25 x^5}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{6 x^6}-\frac{1}{30} a \int \frac{1}{x^6 \sqrt{1-a^2 x^2}} \, dx-\frac{1}{6} a^2 \int \frac{\tanh ^{-1}(a x)}{x^5 \sqrt{1-a^2 x^2}} \, dx+\frac{1}{25} \left (4 a^3\right ) \int \frac{1}{x^4 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{a \sqrt{1-a^2 x^2}}{30 x^5}-\frac{4 a^3 \sqrt{1-a^2 x^2}}{75 x^3}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{6 x^6}+\frac{a^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{24 x^4}-\frac{1}{75} \left (2 a^3\right ) \int \frac{1}{x^4 \sqrt{1-a^2 x^2}} \, dx-\frac{1}{24} a^3 \int \frac{1}{x^4 \sqrt{1-a^2 x^2}} \, dx-\frac{1}{8} a^4 \int \frac{\tanh ^{-1}(a x)}{x^3 \sqrt{1-a^2 x^2}} \, dx+\frac{1}{75} \left (8 a^5\right ) \int \frac{1}{x^2 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{a \sqrt{1-a^2 x^2}}{30 x^5}-\frac{11 a^3 \sqrt{1-a^2 x^2}}{360 x^3}-\frac{8 a^5 \sqrt{1-a^2 x^2}}{75 x}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{6 x^6}+\frac{a^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{24 x^4}+\frac{a^4 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{16 x^2}-\frac{1}{225} \left (4 a^5\right ) \int \frac{1}{x^2 \sqrt{1-a^2 x^2}} \, dx-\frac{1}{36} a^5 \int \frac{1}{x^2 \sqrt{1-a^2 x^2}} \, dx-\frac{1}{16} a^5 \int \frac{1}{x^2 \sqrt{1-a^2 x^2}} \, dx-\frac{1}{16} a^6 \int \frac{\tanh ^{-1}(a x)}{x \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{a \sqrt{1-a^2 x^2}}{30 x^5}-\frac{11 a^3 \sqrt{1-a^2 x^2}}{360 x^3}+\frac{a^5 \sqrt{1-a^2 x^2}}{720 x}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{6 x^6}+\frac{a^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{24 x^4}+\frac{a^4 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{16 x^2}+\frac{1}{8} a^6 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )-\frac{1}{16} a^6 \text{Li}_2\left (-\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )+\frac{1}{16} a^6 \text{Li}_2\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )\\ \end{align*}

Mathematica [A]  time = 3.37897, size = 307, normalized size = 1.26 \[ \frac{a^6 \left (-360 \text{PolyLog}\left (2,-e^{-\tanh ^{-1}(a x)}\right )+360 \text{PolyLog}\left (2,e^{-\tanh ^{-1}(a x)}\right )-\frac{416 \left (1-a^2 x^2\right )^{3/2} \sinh ^4\left (\frac{1}{2} \tanh ^{-1}(a x)\right )}{a^3 x^3}-\frac{3 a x \text{csch}^6\left (\frac{1}{2} \tanh ^{-1}(a x)\right )}{\sqrt{1-a^2 x^2}}-\frac{26 a x \text{csch}^4\left (\frac{1}{2} \tanh ^{-1}(a x)\right )}{\sqrt{1-a^2 x^2}}+76 \tanh \left (\frac{1}{2} \tanh ^{-1}(a x)\right )-360 \tanh ^{-1}(a x) \log \left (1-e^{-\tanh ^{-1}(a x)}\right )+360 \tanh ^{-1}(a x) \log \left (e^{-\tanh ^{-1}(a x)}+1\right )-76 \coth \left (\frac{1}{2} \tanh ^{-1}(a x)\right )-15 \tanh ^{-1}(a x) \text{csch}^6\left (\frac{1}{2} \tanh ^{-1}(a x)\right )-90 \tanh ^{-1}(a x) \text{csch}^4\left (\frac{1}{2} \tanh ^{-1}(a x)\right )-90 \tanh ^{-1}(a x) \text{csch}^2\left (\frac{1}{2} \tanh ^{-1}(a x)\right )-15 \tanh ^{-1}(a x) \text{sech}^6\left (\frac{1}{2} \tanh ^{-1}(a x)\right )+90 \tanh ^{-1}(a x) \text{sech}^4\left (\frac{1}{2} \tanh ^{-1}(a x)\right )-90 \tanh ^{-1}(a x) \text{sech}^2\left (\frac{1}{2} \tanh ^{-1}(a x)\right )+6 \tanh \left (\frac{1}{2} \tanh ^{-1}(a x)\right ) \text{sech}^4\left (\frac{1}{2} \tanh ^{-1}(a x)\right )\right )}{5760} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/x^7,x]

[Out]

(a^6*(-76*Coth[ArcTanh[a*x]/2] - 90*ArcTanh[a*x]*Csch[ArcTanh[a*x]/2]^2 - (26*a*x*Csch[ArcTanh[a*x]/2]^4)/Sqrt
[1 - a^2*x^2] - 90*ArcTanh[a*x]*Csch[ArcTanh[a*x]/2]^4 - (3*a*x*Csch[ArcTanh[a*x]/2]^6)/Sqrt[1 - a^2*x^2] - 15
*ArcTanh[a*x]*Csch[ArcTanh[a*x]/2]^6 - 360*ArcTanh[a*x]*Log[1 - E^(-ArcTanh[a*x])] + 360*ArcTanh[a*x]*Log[1 +
E^(-ArcTanh[a*x])] - 360*PolyLog[2, -E^(-ArcTanh[a*x])] + 360*PolyLog[2, E^(-ArcTanh[a*x])] - 90*ArcTanh[a*x]*
Sech[ArcTanh[a*x]/2]^2 + 90*ArcTanh[a*x]*Sech[ArcTanh[a*x]/2]^4 - 15*ArcTanh[a*x]*Sech[ArcTanh[a*x]/2]^6 - (41
6*(1 - a^2*x^2)^(3/2)*Sinh[ArcTanh[a*x]/2]^4)/(a^3*x^3) + 76*Tanh[ArcTanh[a*x]/2] + 6*Sech[ArcTanh[a*x]/2]^4*T
anh[ArcTanh[a*x]/2]))/5760

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Maple [A]  time = 0.286, size = 183, normalized size = 0.8 \begin{align*}{\frac{{x}^{5}{a}^{5}+45\,{a}^{4}{x}^{4}{\it Artanh} \left ( ax \right ) -22\,{x}^{3}{a}^{3}+30\,{a}^{2}{x}^{2}{\it Artanh} \left ( ax \right ) -24\,ax-120\,{\it Artanh} \left ( ax \right ) }{720\,{x}^{6}}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) }}+{\frac{{a}^{6}{\it Artanh} \left ( ax \right ) }{16}\ln \left ( 1+{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }+{\frac{{a}^{6}}{16}{\it polylog} \left ( 2,-{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }-{\frac{{a}^{6}{\it Artanh} \left ( ax \right ) }{16}\ln \left ( 1-{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }-{\frac{{a}^{6}}{16}{\it polylog} \left ( 2,{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctanh(a*x)*(-a^2*x^2+1)^(1/2)/x^7,x)

[Out]

1/720*(-(a*x-1)*(a*x+1))^(1/2)*(x^5*a^5+45*a^4*x^4*arctanh(a*x)-22*x^3*a^3+30*a^2*x^2*arctanh(a*x)-24*a*x-120*
arctanh(a*x))/x^6+1/16*a^6*arctanh(a*x)*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+1/16*a^6*polylog(2,-(a*x+1)/(-a^2*x^2
+1)^(1/2))-1/16*a^6*arctanh(a*x)*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))-1/16*a^6*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2
))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-a^{2} x^{2} + 1} \operatorname{artanh}\left (a x\right )}{x^{7}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x)*(-a^2*x^2+1)^(1/2)/x^7,x, algorithm="maxima")

[Out]

integrate(sqrt(-a^2*x^2 + 1)*arctanh(a*x)/x^7, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-a^{2} x^{2} + 1} \operatorname{artanh}\left (a x\right )}{x^{7}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x)*(-a^2*x^2+1)^(1/2)/x^7,x, algorithm="fricas")

[Out]

integral(sqrt(-a^2*x^2 + 1)*arctanh(a*x)/x^7, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname{atanh}{\left (a x \right )}}{x^{7}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atanh(a*x)*(-a**2*x**2+1)**(1/2)/x**7,x)

[Out]

Integral(sqrt(-(a*x - 1)*(a*x + 1))*atanh(a*x)/x**7, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-a^{2} x^{2} + 1} \operatorname{artanh}\left (a x\right )}{x^{7}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x)*(-a^2*x^2+1)^(1/2)/x^7,x, algorithm="giac")

[Out]

integrate(sqrt(-a^2*x^2 + 1)*arctanh(a*x)/x^7, x)