### 3.18 $$\int e^{3 \coth ^{-1}(a x)} x^2 \, dx$$

Optimal. Leaf size=118 $\frac{1}{3} x^3 \sqrt{1-\frac{1}{a^2 x^2}}+\frac{3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a}+\frac{14 x \sqrt{1-\frac{1}{a^2 x^2}}}{3 a^2}-\frac{4 \sqrt{1-\frac{1}{a^2 x^2}}}{a^2 \left (a-\frac{1}{x}\right )}+\frac{11 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{2 a^3}$

[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 = 1.07141, antiderivative size = 118, 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, 651, 271, 264, 266, 51, 63, 208} $\frac{1}{3} x^3 \sqrt{1-\frac{1}{a^2 x^2}}+\frac{3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a}+\frac{14 x \sqrt{1-\frac{1}{a^2 x^2}}}{3 a^2}-\frac{4 \sqrt{1-\frac{1}{a^2 x^2}}}{a^2 \left (a-\frac{1}{x}\right )}+\frac{11 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{2 a^3}$

Antiderivative was successfully veriﬁed.

[In]

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

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]

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{4}{a^3 (a-x) \sqrt{1-\frac{x^2}{a^2}}}+\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}}}\right ) \, dx,x,\frac{1}{x}\right )\\ &=-\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 \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 = 0.0800755, size = 75, normalized size = 0.64 $\frac{\frac{a x \sqrt{1-\frac{1}{a^2 x^2}} \left (2 a^3 x^3+7 a^2 x^2+19 a x-52\right )}{a x-1}+33 \log \left (x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )}{6 a^3}$

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(3*ArcCoth[a*x])*x^2,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 = 0.166, size = 471, normalized size = 4. \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 ){\frac{1}{\sqrt{{a}^{2}}}}{\frac{1}{\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}} \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.00047, size = 246, normalized size = 2.08 \begin{align*} -\frac{1}{6} \, a{\left (\frac{2 \,{\left (\frac{75 \,{\left (a x - 1\right )}}{a x + 1} - \frac{88 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac{33 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - 12\right )}}{a^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} - 3 \, a^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} + 3 \, a^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - a^{4} \sqrt{\frac{a x - 1}{a x + 1}}} - \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}}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*a*(2*(75*(a*x - 1)/(a*x + 1) - 88*(a*x - 1)^2/(a*x + 1)^2 + 33*(a*x - 1)^3/(a*x + 1)^3 - 12)/(a^4*((a*x -
1)/(a*x + 1))^(7/2) - 3*a^4*((a*x - 1)/(a*x + 1))^(5/2) + 3*a^4*((a*x - 1)/(a*x + 1))^(3/2) - a^4*sqrt((a*x -
1)/(a*x + 1))) - 33*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^4 + 33*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/a^4)

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Fricas [A]  time = 1.98187, size = 267, normalized size = 2.26 \begin{align*} \frac{33 \,{\left (a x - 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 33 \,{\left (a x - 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) +{\left (2 \, a^{4} x^{4} + 9 \, a^{3} x^{3} + 26 \, a^{2} x^{2} - 33 \, a x - 52\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{6 \,{\left (a^{4} x - a^{3}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(33*(a*x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 33*(a*x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) - 1) + (2*a^
4*x^4 + 9*a^3*x^3 + 26*a^2*x^2 - 33*a*x - 52)*sqrt((a*x - 1)/(a*x + 1)))/(a^4*x - a^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.17711, size = 231, normalized size = 1.96 \begin{align*} \frac{1}{6} \, a{\left (\frac{33 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{4}} - \frac{33 \, \log \left ({\left | \sqrt{\frac{a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{4}} - \frac{24}{a^{4} \sqrt{\frac{a x - 1}{a x + 1}}} + \frac{2 \,{\left (\frac{52 \,{\left (a x - 1\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{a x + 1} - \frac{21 \,{\left (a x - 1\right )}^{2} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{2}} - 39 \, \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{a^{4}{\left (\frac{a x - 1}{a x + 1} - 1\right )}^{3}}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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