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]

arccsc(a*x)+arctanh((1-1/a^2/x^2)^(1/2))-4*a*(1-1/a^2/x^2)^(1/2)/(a-1/x)

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Rubi [A]  time = 0.78, antiderivative size = 46, normalized size of antiderivative = 1.00, 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 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 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 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 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 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]]

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*}

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Mathematica [A]  time = 0.06, 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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fricas [B]  time = 0.59, size = 104, normalized size = 2.26 \[ -\frac {2 \, {\left (a x - 1\right )} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - {\left (a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + {\left (a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + 4 \, {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a x - 1} \]

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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giac [B]  time = 0.16, size = 91, normalized size = 1.98 \[ -a {\left (\frac {2 \, \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a} - \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a} + \frac {\log \left ({\left | \sqrt {\frac {a x - 1}{a x + 1}} - 1 \right |}\right )}{a} + \frac {4}{a \sqrt {\frac {a x - 1}{a x + 1}}}\right )} \]

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))))

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maple [B]  time = 0.06, size = 363, normalized size = 7.89 \[ \frac {\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{2} a^{2}+a^{2} x^{2} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+\ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{2} a^{3}+\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x^{2} a^{2}-2 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x a -2 a x \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )-2 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x \,a^{2}-2 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-2 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x a +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}+\arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) \sqrt {a^{2}}+a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )+\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((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))+ln((a^2*x+((a*x-1)*(a*x
+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x^2*a^3+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2)*x^2*a^2-2*(a^2*x^2-1)^(1/2)*(
a^2)^(1/2)*x*a-2*a*x*(a^2)^(1/2)*arctan(1/(a^2*x^2-1)^(1/2))-2*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/
(a^2)^(1/2))*x*a^2-2*((a*x-1)*(a*x+1))^(3/2)*(a^2)^(1/2)-2*((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2)*x*a+(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*ln((a^2*x+((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^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 [B]  time = 0.41, size = 90, normalized size = 1.96 \[ -a {\left (\frac {2 \, \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a} - \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a} + \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a} + \frac {4}{a \sqrt {\frac {a x - 1}{a x + 1}}}\right )} \]

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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mupad [B]  time = 0.04, size = 54, normalized size = 1.17 \[ 2\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )-2\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )-\frac {4}{\sqrt {\frac {a\,x-1}{a\,x+1}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*atanh(((a*x - 1)/(a*x + 1))^(1/2)) - 2*atan(((a*x - 1)/(a*x + 1))^(1/2)) - 4/((a*x - 1)/(a*x + 1))^(1/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\, dx \]

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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