3.25 \(\int \frac{\text{sech}^{-1}(\sqrt{x})}{x^2} \, dx\)

Optimal. Leaf size=98 \[ \frac{1-x}{2 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} x^{3/2}}+\frac{\sqrt{1-x} \tanh ^{-1}\left (\sqrt{1-x}\right )}{2 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} \sqrt{x}}-\frac{\text{sech}^{-1}\left (\sqrt{x}\right )}{x} \]

[Out]

(1 - x)/(2*Sqrt[-1 + 1/Sqrt[x]]*Sqrt[1 + 1/Sqrt[x]]*x^(3/2)) - ArcSech[Sqrt[x]]/x + (Sqrt[1 - x]*ArcTanh[Sqrt[
1 - x]])/(2*Sqrt[-1 + 1/Sqrt[x]]*Sqrt[1 + 1/Sqrt[x]]*Sqrt[x])

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Rubi [A]  time = 0.0233795, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6345, 12, 51, 63, 206} \[ \frac{1-x}{2 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} x^{3/2}}+\frac{\sqrt{1-x} \tanh ^{-1}\left (\sqrt{1-x}\right )}{2 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} \sqrt{x}}-\frac{\text{sech}^{-1}\left (\sqrt{x}\right )}{x} \]

Antiderivative was successfully verified.

[In]

Int[ArcSech[Sqrt[x]]/x^2,x]

[Out]

(1 - x)/(2*Sqrt[-1 + 1/Sqrt[x]]*Sqrt[1 + 1/Sqrt[x]]*x^(3/2)) - ArcSech[Sqrt[x]]/x + (Sqrt[1 - x]*ArcTanh[Sqrt[
1 - x]])/(2*Sqrt[-1 + 1/Sqrt[x]]*Sqrt[1 + 1/Sqrt[x]]*Sqrt[x])

Rule 6345

Int[((a_.) + ArcSech[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m + 1)*(a + b*ArcSec
h[u]))/(d*(m + 1)), x] + Dist[(b*Sqrt[1 - u^2])/(d*(m + 1)*u*Sqrt[-1 + 1/u]*Sqrt[1 + 1/u]), Int[SimplifyIntegr
and[((c + d*x)^(m + 1)*D[u, x])/(u*Sqrt[1 - u^2]), x], x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && In
verseFunctionFreeQ[u, x] &&  !FunctionOfQ[(c + d*x)^(m + 1), u, x] &&  !FunctionOfExponentialQ[u, x]

Rule 12

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

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0771281, size = 111, normalized size = 1.13 \[ \frac{\sqrt{\frac{1-\sqrt{x}}{\sqrt{x}+1}} \left (\sqrt{x}+1\right )+x \log \left (\sqrt{x} \sqrt{\frac{1-\sqrt{x}}{\sqrt{x}+1}}+\sqrt{\frac{1-\sqrt{x}}{\sqrt{x}+1}}+1\right )-\frac{1}{2} x \log (x)-2 \text{sech}^{-1}\left (\sqrt{x}\right )}{2 x} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSech[Sqrt[x]]/x^2,x]

[Out]

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

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Maple [A]  time = 0.134, size = 64, normalized size = 0.7 \begin{align*} -{\frac{1}{x}{\rm arcsech} \left (\sqrt{x}\right )}+{\frac{1}{2}\sqrt{-{ \left ( -1+\sqrt{x} \right ){\frac{1}{\sqrt{x}}}}}\sqrt{{ \left ( 1+\sqrt{x} \right ){\frac{1}{\sqrt{x}}}}} \left ({\it Artanh} \left ({\frac{1}{\sqrt{1-x}}} \right ) x+\sqrt{1-x} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{1-x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsech(x^(1/2))/x^2,x)

[Out]

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

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Maxima [A]  time = 0.980277, size = 88, normalized size = 0.9 \begin{align*} -\frac{\sqrt{x} \sqrt{\frac{1}{x} - 1}}{2 \,{\left (x{\left (\frac{1}{x} - 1\right )} - 1\right )}} - \frac{\operatorname{arsech}\left (\sqrt{x}\right )}{x} + \frac{1}{4} \, \log \left (\sqrt{x} \sqrt{\frac{1}{x} - 1} + 1\right ) - \frac{1}{4} \, \log \left (\sqrt{x} \sqrt{\frac{1}{x} - 1} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsech(x^(1/2))/x^2,x, algorithm="maxima")

[Out]

-1/2*sqrt(x)*sqrt(1/x - 1)/(x*(1/x - 1) - 1) - arcsech(sqrt(x))/x + 1/4*log(sqrt(x)*sqrt(1/x - 1) + 1) - 1/4*l
og(sqrt(x)*sqrt(1/x - 1) - 1)

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Fricas [A]  time = 1.87779, size = 111, normalized size = 1.13 \begin{align*} \frac{{\left (x - 2\right )} \log \left (\frac{x \sqrt{-\frac{x - 1}{x}} + \sqrt{x}}{x}\right ) + \sqrt{x} \sqrt{-\frac{x - 1}{x}}}{2 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsech(x^(1/2))/x^2,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{asech}{\left (\sqrt{x} \right )}}{x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asech(x**(1/2))/x**2,x)

[Out]

Integral(asech(sqrt(x))/x**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arsech}\left (\sqrt{x}\right )}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsech(x^(1/2))/x^2,x, algorithm="giac")

[Out]

integrate(arcsech(sqrt(x))/x^2, x)