∫ esech−1 (ax) _________________

Optimal. Leaf size=35 \[ -\frac {e^{\text {sech}^{-1}(a x)}}{2 x}+a \tanh ^{-1}\left (\sqrt {\frac {1-a x}{1+a x}}\right ) \]

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

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(99\) vs. \(2(35)=70\).
time = 0.03, antiderivative size = 99, normalized size of antiderivative = 2.83, number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6470, 30, 105, 12, 94, 214} \begin {gather*} \frac {\sqrt {1-a x}}{2 a x^2 \sqrt {\frac {1}{a x+1}}}+\frac {1}{2 a x^2}+\frac {1}{2} a \sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {a x+1}\right )-\frac {e^{\text {sech}^{-1}(a x)}}{x} \end {gather*}

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

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I

nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +

b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Dist[1/((m + 1)*(b*

c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +

c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] &

& (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])

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

Int[E^ArcSech[(a_.)*(x_)^(p_.)]*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(E^ArcSech[a*x^p]/(m + 1)), x] + (Dist

[p/(a*(m + 1)), Int[x^(m - p), x], x] + Dist[p*(Sqrt[1 + a*x^p]/(a*(m + 1)))*Sqrt[1/(1 + a*x^p)], Int[x^(m - p

)/(Sqrt[1 + a*x^p]*Sqrt[1 - a*x^p]), x], x]) /; FreeQ[{a, m, p}, x] && NeQ[m, -1]

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(93\) vs. \(2(35)=70\).
time = 0.04, size = 93, normalized size = 2.66 \begin {gather*} \frac {1}{2} \left (-\frac {1}{a x^2}-\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)}{a x^2}-a \log (x)+a \log \left (1+\sqrt {\frac {1-a x}{1+a x}}+a x \sqrt {\frac {1-a x}{1+a x}}\right )\right ) \end {gather*}

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

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method	result	size

default	\(\frac {\sqrt {\frac {a x +1}{a x}}\, \sqrt {-\frac {a x -1}{a x}}\, \left (a^{2} x^{2} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )-\sqrt {-a^{2} x^{2}+1}\right )}{2 x \sqrt {-a^{2} x^{2}+1}}-\frac {1}{2 a \,x^{2}}\)	\(91\)

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

1/2*((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^

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (56) = 112\).
time = 0.41, size = 128, normalized size = 3.66 \begin {gather*} \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 x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 2}{4 \, a x^{2}} \end {gather*}

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

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {1}{x^{3}}\, dx + \int \frac {a \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}}}{x^{2}}\, dx}{a} \end {gather*}

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

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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

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Mupad [B]
time = 1.84, size = 71, normalized size = 2.03 \begin {gather*} \frac {a\,\ln \left (\sqrt {\frac {1}{a\,x}-1}\,\sqrt {\frac {1}{a\,x}+1}+\frac {1}{a\,x}\right )}{2}-\frac {1}{2\,a\,x^2}-\frac {\sqrt {\frac {1}{a\,x}-1}\,\sqrt {\frac {1}{a\,x}+1}}{2\,x} \end {gather*}

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

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

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3.1.38 \(\int \frac {e^{\text {sech}^{-1}(a x)}}{x^2} \, dx\) [38]