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

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

[Out]

-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]  time = 0.04, 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 = {6335, 30, 103, 12, 92, 208} \[ \frac {\sqrt {1-a x}}{2 a x^2 \sqrt {\frac {1}{a x+1}}}+\frac {1}{2 a x^2}+\frac {1}{2} a \sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {a x+1}\right )-\frac {e^{\text {sech}^{-1}(a x)}}{x} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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
+ a*x]*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/2

Rule 12

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

Rule 30

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

Rule 92

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] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

Rule 103

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] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

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 6335

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

Rubi steps

\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 ) \operatorname {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]  time = 0.06, size = 93, normalized size = 2.66 \[ \frac {1}{2} \left (-\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1)}{a x^2}-\frac {1}{a x^2}-a \log (x)+a \log \left (a x \sqrt {\frac {1-a x}{a x+1}}+\sqrt {\frac {1-a x}{a x+1}}+1\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [B]  time = 0.81, size = 128, normalized size = 3.66 \[ \frac {a^{2} x^{2} \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} + 1\right ) - a^{2} x^{2} \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 1\right ) - 2 \, a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 2}{4 \, a x^{2}} \]

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))/x^2,x, algorithm="fricas")

[Out]

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

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))/x^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.05, size = 91, normalized size = 2.60 \[ \frac {\sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}\, \left (a^{2} x^{2} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )-\sqrt {-a^{2} x^{2}+1}\right )}{2 x \sqrt {-a^{2} x^{2}+1}}-\frac {1}{2 a \,x^{2}} \]

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))/x^2,x)

[Out]

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

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

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))/x^2,x, algorithm="maxima")

[Out]

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

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

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))/x^2,x)

[Out]

(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
)^(1/2))/(2*x)

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

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))/x**2,x)

[Out]

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

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