### 3.21 $$\int \frac{e^{3 \coth ^{-1}(a x)}}{x} \, dx$$

Optimal. Leaf size=46 $-\frac{4 a \sqrt{1-\frac{1}{a^2 x^2}}}{a-\frac{1}{x}}+\tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )+\csc ^{-1}(a x)$

[Out]

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

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Rubi [A]  time = 0.782756, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 12, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.583, Rules used = {6169, 6742, 216, 651, 266, 63, 208} $-\frac{4 a \sqrt{1-\frac{1}{a^2 x^2}}}{a-\frac{1}{x}}+\tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )+\csc ^{-1}(a x)$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.058624, size = 53, normalized size = 1.15 $-\frac{4 a x \sqrt{1-\frac{1}{a^2 x^2}}}{a x-1}+\log \left (x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )+\sin ^{-1}\left (\frac{1}{a x}\right )$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-4*a*Sqrt[1 - 1/(a^2*x^2)]*x)/(-1 + a*x) + ArcSin[1/(a*x)] + Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]

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Maple [B]  time = 0.173, size = 363, normalized size = 7.9 \begin{align*}{\frac{1}{ax+1} \left ( \ln \left ({ \left ({a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) } \right ){\frac{1}{\sqrt{{a}^{2}}}}} \right ){x}^{2}{a}^{3}+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{2}{a}^{2}+{a}^{2}{x}^{2}\sqrt{{a}^{2}}\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ) +\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{2}{a}^{2}-2\,\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}-2\,\sqrt{{a}^{2}}\sqrt{{a}^{2}{x}^{2}-1}xa-2\,ax\sqrt{{a}^{2}}\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ) -2\, \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}\sqrt{{a}^{2}}-2\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }xa+a\ln \left ({ \left ({a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) } \right ){\frac{1}{\sqrt{{a}^{2}}}}} \right ) +\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}+\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ) \sqrt{{a}^{2}}+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \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,x)

[Out]

(ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x^2*a^3+(a^2*x^2-1)^(1/2)*(a^2)^(1/2)*x^2*a^2+a^2
*x^2*(a^2)^(1/2)*arctan(1/(a^2*x^2-1)^(1/2))+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^2*a^2-2*ln((a^2*x+(a^2)^(1/
2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x*a^2-2*(a^2)^(1/2)*(a^2*x^2-1)^(1/2)*x*a-2*a*x*(a^2)^(1/2)*arctan(1/
(a^2*x^2-1)^(1/2))-2*((a*x-1)*(a*x+1))^(3/2)*(a^2)^(1/2)-2*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x*a+a*ln((a^2*x
+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))+(a^2*x^2-1)^(1/2)*(a^2)^(1/2)+arctan(1/(a^2*x^2-1)^(1/2))*(
a^2)^(1/2)+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(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 [B]  time = 1.4833, size = 122, normalized size = 2.65 \begin{align*} -a{\left (\frac{2 \, \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )}{a} - \frac{\log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a} + \frac{\log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a} + \frac{4}{a \sqrt{\frac{a x - 1}{a x + 1}}}\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,x, algorithm="maxima")

[Out]

-a*(2*arctan(sqrt((a*x - 1)/(a*x + 1)))/a - log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a + log(sqrt((a*x - 1)/(a*x + 1
)) - 1)/a + 4/(a*sqrt((a*x - 1)/(a*x + 1))))

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Fricas [B]  time = 1.86066, size = 262, normalized size = 5.7 \begin{align*} -\frac{2 \,{\left (a x - 1\right )} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) -{\left (a x - 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) +{\left (a x - 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) + 4 \,{\left (a x + 1\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{a x - 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*(a*x - 1)*arctan(sqrt((a*x - 1)/(a*x + 1))) - (a*x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) + (a*x - 1)*log
(sqrt((a*x - 1)/(a*x + 1)) - 1) + 4*(a*x + 1)*sqrt((a*x - 1)/(a*x + 1)))/(a*x - 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \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,x)

[Out]

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

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Giac [B]  time = 1.20681, size = 123, normalized size = 2.67 \begin{align*} -a{\left (\frac{2 \, \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )}{a} - \frac{\log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a} + \frac{\log \left ({\left | \sqrt{\frac{a x - 1}{a x + 1}} - 1 \right |}\right )}{a} + \frac{4}{a \sqrt{\frac{a x - 1}{a x + 1}}}\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,x, algorithm="giac")

[Out]

-a*(2*arctan(sqrt((a*x - 1)/(a*x + 1)))/a - log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a + log(abs(sqrt((a*x - 1)/(a*x
+ 1)) - 1))/a + 4/(a*sqrt((a*x - 1)/(a*x + 1))))