3.27 \(\int \frac{\text{sech}^{-1}(\sqrt{x})}{x^4} \, dx\)

Optimal. Leaf size=172 \[ \frac{5 (1-x)}{48 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} x^{3/2}}+\frac{5 (1-x)}{72 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} x^{5/2}}+\frac{1-x}{18 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} x^{7/2}}-\frac{\text{sech}^{-1}\left (\sqrt{x}\right )}{3 x^3}+\frac{5 \sqrt{1-x} \tanh ^{-1}\left (\sqrt{1-x}\right )}{48 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} \sqrt{x}} \]

[Out]

(1 - x)/(18*Sqrt[-1 + 1/Sqrt[x]]*Sqrt[1 + 1/Sqrt[x]]*x^(7/2)) + (5*(1 - x))/(72*Sqrt[-1 + 1/Sqrt[x]]*Sqrt[1 +
1/Sqrt[x]]*x^(5/2)) + (5*(1 - x))/(48*Sqrt[-1 + 1/Sqrt[x]]*Sqrt[1 + 1/Sqrt[x]]*x^(3/2)) - ArcSech[Sqrt[x]]/(3*
x^3) + (5*Sqrt[1 - x]*ArcTanh[Sqrt[1 - x]])/(48*Sqrt[-1 + 1/Sqrt[x]]*Sqrt[1 + 1/Sqrt[x]]*Sqrt[x])

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Rubi [A]  time = 0.039765, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{5 (1-x)}{48 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} x^{3/2}}+\frac{5 (1-x)}{72 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} x^{5/2}}+\frac{1-x}{18 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} x^{7/2}}-\frac{\text{sech}^{-1}\left (\sqrt{x}\right )}{3 x^3}+\frac{5 \sqrt{1-x} \tanh ^{-1}\left (\sqrt{1-x}\right )}{48 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Int[ArcSech[Sqrt[x]]/x^4,x]

[Out]

(1 - x)/(18*Sqrt[-1 + 1/Sqrt[x]]*Sqrt[1 + 1/Sqrt[x]]*x^(7/2)) + (5*(1 - x))/(72*Sqrt[-1 + 1/Sqrt[x]]*Sqrt[1 +
1/Sqrt[x]]*x^(5/2)) + (5*(1 - x))/(48*Sqrt[-1 + 1/Sqrt[x]]*Sqrt[1 + 1/Sqrt[x]]*x^(3/2)) - ArcSech[Sqrt[x]]/(3*
x^3) + (5*Sqrt[1 - x]*ArcTanh[Sqrt[1 - x]])/(48*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^4} \, dx &=-\frac{\text{sech}^{-1}\left (\sqrt{x}\right )}{3 x^3}-\frac{\sqrt{1-x} \int \frac{1}{2 \sqrt{1-x} x^4} \, dx}{3 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}\\ &=-\frac{\text{sech}^{-1}\left (\sqrt{x}\right )}{3 x^3}-\frac{\sqrt{1-x} \int \frac{1}{\sqrt{1-x} x^4} \, dx}{6 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}\\ &=\frac{1-x}{18 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} x^{7/2}}-\frac{\text{sech}^{-1}\left (\sqrt{x}\right )}{3 x^3}-\frac{\left (5 \sqrt{1-x}\right ) \int \frac{1}{\sqrt{1-x} x^3} \, dx}{36 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}\\ &=\frac{1-x}{18 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} x^{7/2}}+\frac{5 (1-x)}{72 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} x^{5/2}}-\frac{\text{sech}^{-1}\left (\sqrt{x}\right )}{3 x^3}-\frac{\left (5 \sqrt{1-x}\right ) \int \frac{1}{\sqrt{1-x} x^2} \, dx}{48 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}\\ &=\frac{1-x}{18 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} x^{7/2}}+\frac{5 (1-x)}{72 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} x^{5/2}}+\frac{5 (1-x)}{48 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} x^{3/2}}-\frac{\text{sech}^{-1}\left (\sqrt{x}\right )}{3 x^3}-\frac{\left (5 \sqrt{1-x}\right ) \int \frac{1}{\sqrt{1-x} x} \, dx}{96 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}\\ &=\frac{1-x}{18 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} x^{7/2}}+\frac{5 (1-x)}{72 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} x^{5/2}}+\frac{5 (1-x)}{48 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} x^{3/2}}-\frac{\text{sech}^{-1}\left (\sqrt{x}\right )}{3 x^3}+\frac{\left (5 \sqrt{1-x}\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{1-x}\right )}{48 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}\\ &=\frac{1-x}{18 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} x^{7/2}}+\frac{5 (1-x)}{72 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} x^{5/2}}+\frac{5 (1-x)}{48 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} x^{3/2}}-\frac{\text{sech}^{-1}\left (\sqrt{x}\right )}{3 x^3}+\frac{5 \sqrt{1-x} \tanh ^{-1}\left (\sqrt{1-x}\right )}{48 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}\\ \end{align*}

Mathematica [A]  time = 0.128421, size = 140, normalized size = 0.81 \[ \frac{\sqrt{\frac{1-\sqrt{x}}{\sqrt{x}+1}} \left (15 x^{5/2}+15 x^2+10 x^{3/2}+10 x+8 \sqrt{x}+8\right )+15 x^3 \log \left (\sqrt{x} \sqrt{\frac{1-\sqrt{x}}{\sqrt{x}+1}}+\sqrt{\frac{1-\sqrt{x}}{\sqrt{x}+1}}+1\right )-\frac{15}{2} x^3 \log (x)-48 \text{sech}^{-1}\left (\sqrt{x}\right )}{144 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSech[Sqrt[x]]/x^4,x]

[Out]

(Sqrt[(1 - Sqrt[x])/(1 + Sqrt[x])]*(8 + 8*Sqrt[x] + 10*x + 10*x^(3/2) + 15*x^2 + 15*x^(5/2)) - 48*ArcSech[Sqrt
[x]] + 15*x^3*Log[1 + Sqrt[(1 - Sqrt[x])/(1 + Sqrt[x])] + Sqrt[(1 - Sqrt[x])/(1 + Sqrt[x])]*Sqrt[x]] - (15*x^3
*Log[x])/2)/(144*x^3)

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Maple [A]  time = 0.141, size = 91, normalized size = 0.5 \begin{align*} -{\frac{1}{3\,{x}^{3}}{\rm arcsech} \left (\sqrt{x}\right )}+{\frac{1}{144}\sqrt{-{ \left ( -1+\sqrt{x} \right ){\frac{1}{\sqrt{x}}}}}\sqrt{{ \left ( 1+\sqrt{x} \right ){\frac{1}{\sqrt{x}}}}} \left ( 15\,{\it Artanh} \left ({\frac{1}{\sqrt{1-x}}} \right ){x}^{3}+15\,\sqrt{1-x}{x}^{2}+10\,\sqrt{1-x}x+8\,\sqrt{1-x} \right ){x}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{1-x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsech(x^(1/2))/x^4,x)

[Out]

-1/3*arcsech(x^(1/2))/x^3+1/144*(-(-1+x^(1/2))/x^(1/2))^(1/2)/x^(5/2)*((1+x^(1/2))/x^(1/2))^(1/2)*(15*arctanh(
1/(1-x)^(1/2))*x^3+15*(1-x)^(1/2)*x^2+10*(1-x)^(1/2)*x+8*(1-x)^(1/2))/(1-x)^(1/2)

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Maxima [A]  time = 0.989615, size = 157, normalized size = 0.91 \begin{align*} -\frac{15 \, x^{\frac{5}{2}}{\left (\frac{1}{x} - 1\right )}^{\frac{5}{2}} - 40 \, x^{\frac{3}{2}}{\left (\frac{1}{x} - 1\right )}^{\frac{3}{2}} + 33 \, \sqrt{x} \sqrt{\frac{1}{x} - 1}}{144 \,{\left (x^{3}{\left (\frac{1}{x} - 1\right )}^{3} - 3 \, x^{2}{\left (\frac{1}{x} - 1\right )}^{2} + 3 \, x{\left (\frac{1}{x} - 1\right )} - 1\right )}} - \frac{\operatorname{arsech}\left (\sqrt{x}\right )}{3 \, x^{3}} + \frac{5}{96} \, \log \left (\sqrt{x} \sqrt{\frac{1}{x} - 1} + 1\right ) - \frac{5}{96} \, \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^4,x, algorithm="maxima")

[Out]

-1/144*(15*x^(5/2)*(1/x - 1)^(5/2) - 40*x^(3/2)*(1/x - 1)^(3/2) + 33*sqrt(x)*sqrt(1/x - 1))/(x^3*(1/x - 1)^3 -
3*x^2*(1/x - 1)^2 + 3*x*(1/x - 1) - 1) - 1/3*arcsech(sqrt(x))/x^3 + 5/96*log(sqrt(x)*sqrt(1/x - 1) + 1) - 5/9
6*log(sqrt(x)*sqrt(1/x - 1) - 1)

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Fricas [A]  time = 1.92981, size = 153, normalized size = 0.89 \begin{align*} \frac{{\left (15 \, x^{2} + 10 \, x + 8\right )} \sqrt{x} \sqrt{-\frac{x - 1}{x}} + 3 \,{\left (5 \, x^{3} - 16\right )} \log \left (\frac{x \sqrt{-\frac{x - 1}{x}} + \sqrt{x}}{x}\right )}{144 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsech(x^(1/2))/x^4,x, algorithm="fricas")

[Out]

1/144*((15*x^2 + 10*x + 8)*sqrt(x)*sqrt(-(x - 1)/x) + 3*(5*x^3 - 16)*log((x*sqrt(-(x - 1)/x) + sqrt(x))/x))/x^
3

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asech(x**(1/2))/x**4,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arsech}\left (\sqrt{x}\right )}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsech(x^(1/2))/x^4,x, algorithm="giac")

[Out]

integrate(arcsech(sqrt(x))/x^4, x)