3.70 \(\int \frac {e^{2 \text {sech}^{-1}(a x)}}{x} \, dx\)

Optimal. Leaf size=86 \[ \frac {2}{1-\sqrt {\frac {1-a x}{a x+1}}}-\frac {2}{\left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^2}-\log (a x+1)-2 \log \left (1-\sqrt {\frac {1-a x}{a x+1}}\right ) \]

[Out]

-ln(a*x+1)-2*ln(1-((-a*x+1)/(a*x+1))^(1/2))-2/(1-((-a*x+1)/(a*x+1))^(1/2))^2+2/(1-((-a*x+1)/(a*x+1))^(1/2))

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Rubi [A]  time = 0.45, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6337, 1629, 260} \[ \frac {2}{1-\sqrt {\frac {1-a x}{a x+1}}}-\frac {2}{\left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^2}-\log (a x+1)-2 \log \left (1-\sqrt {\frac {1-a x}{a x+1}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 260

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

Rule 1629

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*
Pq*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 6337

Int[E^(ArcSech[u_]*(n_.))*(x_)^(m_.), x_Symbol] :> Int[x^m*(1/u + Sqrt[(1 - u)/(1 + u)] + (1*Sqrt[(1 - u)/(1 +
u)])/u)^n, x] /; FreeQ[m, x] && IntegerQ[n]

Rubi steps

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

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Mathematica [A]  time = 0.06, size = 86, normalized size = 1.00 \[ -\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1)}{a^2 x^2}-\frac {1}{a^2 x^2}+\log \left (a x \sqrt {\frac {1-a x}{a x+1}}+\sqrt {\frac {1-a x}{a x+1}}+1\right )-2 \log (x) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [A]  time = 0.85, size = 138, normalized size = 1.60 \[ \frac {a^{2} x^{2} \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} + 1\right ) - a^{2} x^{2} \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 1\right ) - 2 \, a^{2} x^{2} \log \relax (x) - 2 \, a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 2}{2 \, a^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.06, size = 97, normalized size = 1.13 \[ -\ln \relax (x )-\frac {1}{x^{2} a^{2}}-\frac {\sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}\, \left (-a^{2} x^{2} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )+\sqrt {-a^{2} x^{2}+1}\right )}{a x \sqrt {-a^{2} x^{2}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2 \, {\left (\frac {1}{2} \, a^{2} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {1}{2} \, \sqrt {-a^{2} x^{2} + 1} a^{2} - \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{2 \, x^{2}}\right )}}{a^{2}} - \frac {1}{a^{2} x^{2}} - \int \frac {1}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 11.21, size = 323, normalized size = 3.76 \[ \ln \left (\frac {1}{x}\right )-4\,\mathrm {atanh}\left (\frac {\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}}{\sqrt {\frac {1}{a\,x}+1}-1}\right )+2\,\mathrm {acosh}\left (\frac {1}{a\,x}\right )+\frac {\frac {28\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^3}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^3}+\frac {28\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^5}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^5}+\frac {4\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^7}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^7}+\frac {4\,\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}{\sqrt {\frac {1}{a\,x}+1}-1}}{1+\frac {6\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4}-\frac {4\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^6}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^8}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^8}-\frac {4\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}}-\frac {1}{a^2\,x^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(1/x) - 4*atanh(((1/(a*x) - 1)^(1/2) - 1i)/((1/(a*x) + 1)^(1/2) - 1)) + 2*acosh(1/(a*x)) + ((28*((1/(a*x) -
1)^(1/2) - 1i)^3)/((1/(a*x) + 1)^(1/2) - 1)^3 + (28*((1/(a*x) - 1)^(1/2) - 1i)^5)/((1/(a*x) + 1)^(1/2) - 1)^5
+ (4*((1/(a*x) - 1)^(1/2) - 1i)^7)/((1/(a*x) + 1)^(1/2) - 1)^7 + (4*((1/(a*x) - 1)^(1/2) - 1i))/((1/(a*x) + 1
)^(1/2) - 1))/((6*((1/(a*x) - 1)^(1/2) - 1i)^4)/((1/(a*x) + 1)^(1/2) - 1)^4 - (4*((1/(a*x) - 1)^(1/2) - 1i)^2)
/((1/(a*x) + 1)^(1/2) - 1)^2 - (4*((1/(a*x) - 1)^(1/2) - 1i)^6)/((1/(a*x) + 1)^(1/2) - 1)^6 + ((1/(a*x) - 1)^(
1/2) - 1i)^8/((1/(a*x) + 1)^(1/2) - 1)^8 + 1) - 1/(a^2*x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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