3.51 \(\int e^{-3 \coth ^{-1}(a x)} x^2 \, dx\)

Optimal. Leaf size=116 \[ \frac{1}{3} x^3 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a}+\frac{14 x \sqrt{1-\frac{1}{a^2 x^2}}}{3 a^2}+\frac{4 \sqrt{1-\frac{1}{a^2 x^2}}}{a^2 \left (a+\frac{1}{x}\right )}-\frac{11 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{2 a^3} \]

[Out]

(4*Sqrt[1 - 1/(a^2*x^2)])/(a^2*(a + x^(-1))) + (14*Sqrt[1 - 1/(a^2*x^2)]*x)/(3*a^2) - (3*Sqrt[1 - 1/(a^2*x^2)]
*x^2)/(2*a) + (Sqrt[1 - 1/(a^2*x^2)]*x^3)/3 - (11*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(2*a^3)

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Rubi [A]  time = 0.869178, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {6169, 6742, 271, 264, 266, 51, 63, 208, 651} \[ \frac{1}{3} x^3 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a}+\frac{14 x \sqrt{1-\frac{1}{a^2 x^2}}}{3 a^2}+\frac{4 \sqrt{1-\frac{1}{a^2 x^2}}}{a^2 \left (a+\frac{1}{x}\right )}-\frac{11 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{2 a^3} \]

Antiderivative was successfully verified.

[In]

Int[x^2/E^(3*ArcCoth[a*x]),x]

[Out]

(4*Sqrt[1 - 1/(a^2*x^2)])/(a^2*(a + x^(-1))) + (14*Sqrt[1 - 1/(a^2*x^2)]*x)/(3*a^2) - (3*Sqrt[1 - 1/(a^2*x^2)]
*x^2)/(2*a) + (Sqrt[1 - 1/(a^2*x^2)]*x^3)/3 - (11*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(2*a^3)

Rule 6169

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(x_)^(m_.), x_Symbol] :> -Subst[Int[(1 + x/a)^((n + 1)/2)/(x^(m + 2)*(1 - x/
a)^((n - 1)/2)*Sqrt[1 - x^2/a^2]), x], x, 1/x] /; FreeQ[a, x] && IntegerQ[(n - 1)/2] && IntegerQ[m]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

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 266

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

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rule 651

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^m*(a + c*x^2)^(p + 1))
/(2*c*d*(p + 1)), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + 2*p
+ 2, 0]

Rubi steps

\begin{align*} \int e^{-3 \coth ^{-1}(a x)} x^2 \, dx &=-\operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^2}{x^4 \left (1+\frac{x}{a}\right ) \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{1}{x^4 \sqrt{1-\frac{x^2}{a^2}}}-\frac{3}{a x^3 \sqrt{1-\frac{x^2}{a^2}}}+\frac{4}{a^2 x^2 \sqrt{1-\frac{x^2}{a^2}}}-\frac{4}{a^3 x \sqrt{1-\frac{x^2}{a^2}}}+\frac{4}{a^3 (a+x) \sqrt{1-\frac{x^2}{a^2}}}\right ) \, dx,x,\frac{1}{x}\right )\\ &=\frac{4 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a^3}-\frac{4 \operatorname{Subst}\left (\int \frac{1}{(a+x) \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a^3}-\frac{4 \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a^2}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a}-\operatorname{Subst}\left (\int \frac{1}{x^4 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{4 \sqrt{1-\frac{1}{a^2 x^2}}}{a^2 \left (a+\frac{1}{x}\right )}+\frac{4 \sqrt{1-\frac{1}{a^2 x^2}} x}{a^2}+\frac{1}{3} \sqrt{1-\frac{1}{a^2 x^2}} x^3+\frac{2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{a^3}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{3 a^2}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{2 a}\\ &=\frac{4 \sqrt{1-\frac{1}{a^2 x^2}}}{a^2 \left (a+\frac{1}{x}\right )}+\frac{14 \sqrt{1-\frac{1}{a^2 x^2}} x}{3 a^2}-\frac{3 \sqrt{1-\frac{1}{a^2 x^2}} x^2}{2 a}+\frac{1}{3} \sqrt{1-\frac{1}{a^2 x^2}} x^3+\frac{3 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{4 a^3}-\frac{4 \operatorname{Subst}\left (\int \frac{1}{a^2-a^2 x^2} \, dx,x,\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}\\ &=\frac{4 \sqrt{1-\frac{1}{a^2 x^2}}}{a^2 \left (a+\frac{1}{x}\right )}+\frac{14 \sqrt{1-\frac{1}{a^2 x^2}} x}{3 a^2}-\frac{3 \sqrt{1-\frac{1}{a^2 x^2}} x^2}{2 a}+\frac{1}{3} \sqrt{1-\frac{1}{a^2 x^2}} x^3-\frac{4 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a^3}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{a^2-a^2 x^2} \, dx,x,\sqrt{1-\frac{1}{a^2 x^2}}\right )}{2 a}\\ &=\frac{4 \sqrt{1-\frac{1}{a^2 x^2}}}{a^2 \left (a+\frac{1}{x}\right )}+\frac{14 \sqrt{1-\frac{1}{a^2 x^2}} x}{3 a^2}-\frac{3 \sqrt{1-\frac{1}{a^2 x^2}} x^2}{2 a}+\frac{1}{3} \sqrt{1-\frac{1}{a^2 x^2}} x^3-\frac{11 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{2 a^3}\\ \end{align*}

Mathematica [A]  time = 0.0761989, size = 75, normalized size = 0.65 \[ \frac{\frac{a x \sqrt{1-\frac{1}{a^2 x^2}} \left (2 a^3 x^3-7 a^2 x^2+19 a x+52\right )}{a x+1}-33 \log \left (x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )}{6 a^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^2/E^(3*ArcCoth[a*x]),x]

[Out]

((a*Sqrt[1 - 1/(a^2*x^2)]*x*(52 + 19*a*x - 7*a^2*x^2 + 2*a^3*x^3))/(1 + a*x) - 33*Log[(1 + Sqrt[1 - 1/(a^2*x^2
)])*x])/(6*a^3)

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Maple [B]  time = 0.132, size = 471, normalized size = 4.1 \begin{align*} -{\frac{1}{6\,{a}^{3} \left ( ax-1 \right ) } \left ( 9\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{3}{a}^{3}-2\,\sqrt{{a}^{2}} \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}{x}^{2}{a}^{2}+18\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{2}{a}^{2}-9\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ){x}^{2}{a}^{3}-4\,\sqrt{{a}^{2}} \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}xa-42\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{2}{a}^{2}+42\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ){x}^{2}{a}^{3}+9\,\sqrt{{a}^{2}}\sqrt{{a}^{2}{x}^{2}-1}xa-18\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ) x{a}^{2}+10\, \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}\sqrt{{a}^{2}}-84\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }xa+84\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ) x{a}^{2}-9\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ) a-42\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }+42\,a\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ) \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt{{a}^{2}}}}{\frac{1}{\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(9*(a^2*x^2-1)^(1/2)*(a^2)^(1/2)*x^3*a^3-2*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x^2*a^2+18*(a^2*x^2-1)^(1/
2)*(a^2)^(1/2)*x^2*a^2-9*ln((a^2*x+(a^2*x^2-1)^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x^2*a^3-4*(a^2)^(1/2)*((a*x-1)*
(a*x+1))^(3/2)*x*a-42*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^2*a^2+42*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(
1/2))/(a^2)^(1/2))*x^2*a^3+9*(a^2)^(1/2)*(a^2*x^2-1)^(1/2)*x*a-18*ln((a^2*x+(a^2*x^2-1)^(1/2)*(a^2)^(1/2))/(a^
2)^(1/2))*x*a^2+10*((a*x-1)*(a*x+1))^(3/2)*(a^2)^(1/2)-84*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x*a+84*ln((a^2*x
+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x*a^2-9*ln((a^2*x+(a^2*x^2-1)^(1/2)*(a^2)^(1/2))/(a^2)^(1/2
))*a-42*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)+42*a*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2)))/
a^3*((a*x-1)/(a*x+1))^(3/2)/(a^2)^(1/2)/((a*x-1)*(a*x+1))^(1/2)/(a*x-1)

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Maxima [A]  time = 1.01547, size = 251, normalized size = 2.16 \begin{align*} -\frac{1}{6} \, a{\left (\frac{2 \,{\left (39 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} - 52 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} + 21 \, \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{\frac{3 \,{\left (a x - 1\right )} a^{4}}{a x + 1} - \frac{3 \,{\left (a x - 1\right )}^{2} a^{4}}{{\left (a x + 1\right )}^{2}} + \frac{{\left (a x - 1\right )}^{3} a^{4}}{{\left (a x + 1\right )}^{3}} - a^{4}} + \frac{33 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{4}} - \frac{33 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{4}} - \frac{24 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{4}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*a*(2*(39*((a*x - 1)/(a*x + 1))^(5/2) - 52*((a*x - 1)/(a*x + 1))^(3/2) + 21*sqrt((a*x - 1)/(a*x + 1)))/(3*
(a*x - 1)*a^4/(a*x + 1) - 3*(a*x - 1)^2*a^4/(a*x + 1)^2 + (a*x - 1)^3*a^4/(a*x + 1)^3 - a^4) + 33*log(sqrt((a*
x - 1)/(a*x + 1)) + 1)/a^4 - 33*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/a^4 - 24*sqrt((a*x - 1)/(a*x + 1))/a^4)

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Fricas [A]  time = 1.69842, size = 209, normalized size = 1.8 \begin{align*} \frac{{\left (2 \, a^{3} x^{3} - 7 \, a^{2} x^{2} + 19 \, a x + 52\right )} \sqrt{\frac{a x - 1}{a x + 1}} - 33 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) + 33 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{6 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*((2*a^3*x^3 - 7*a^2*x^2 + 19*a*x + 52)*sqrt((a*x - 1)/(a*x + 1)) - 33*log(sqrt((a*x - 1)/(a*x + 1)) + 1) +
33*log(sqrt((a*x - 1)/(a*x + 1)) - 1))/a^3

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{undef} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

undef