3.20 \(\int x^3 \text{sech}^{-1}(\sqrt{x}) \, dx\)

Optimal. Leaf size=164 \[ \frac{1}{4} x^4 \text{sech}^{-1}\left (\sqrt{x}\right )+\frac{(1-x)^4}{28 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} \sqrt{x}}-\frac{3 (1-x)^3}{20 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} \sqrt{x}}+\frac{(1-x)^2}{4 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} \sqrt{x}}-\frac{1-x}{4 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} \sqrt{x}} \]

[Out]

-(1 - x)/(4*Sqrt[-1 + 1/Sqrt[x]]*Sqrt[1 + 1/Sqrt[x]]*Sqrt[x]) + (1 - x)^2/(4*Sqrt[-1 + 1/Sqrt[x]]*Sqrt[1 + 1/S
qrt[x]]*Sqrt[x]) - (3*(1 - x)^3)/(20*Sqrt[-1 + 1/Sqrt[x]]*Sqrt[1 + 1/Sqrt[x]]*Sqrt[x]) + (1 - x)^4/(28*Sqrt[-1
+ 1/Sqrt[x]]*Sqrt[1 + 1/Sqrt[x]]*Sqrt[x]) + (x^4*ArcSech[Sqrt[x]])/4

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Rubi [A]  time = 0.0304525, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {6345, 12, 43} \[ \frac{1}{4} x^4 \text{sech}^{-1}\left (\sqrt{x}\right )+\frac{(1-x)^4}{28 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} \sqrt{x}}-\frac{3 (1-x)^3}{20 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} \sqrt{x}}+\frac{(1-x)^2}{4 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} \sqrt{x}}-\frac{1-x}{4 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Int[x^3*ArcSech[Sqrt[x]],x]

[Out]

-(1 - x)/(4*Sqrt[-1 + 1/Sqrt[x]]*Sqrt[1 + 1/Sqrt[x]]*Sqrt[x]) + (1 - x)^2/(4*Sqrt[-1 + 1/Sqrt[x]]*Sqrt[1 + 1/S
qrt[x]]*Sqrt[x]) - (3*(1 - x)^3)/(20*Sqrt[-1 + 1/Sqrt[x]]*Sqrt[1 + 1/Sqrt[x]]*Sqrt[x]) + (1 - x)^4/(28*Sqrt[-1
+ 1/Sqrt[x]]*Sqrt[1 + 1/Sqrt[x]]*Sqrt[x]) + (x^4*ArcSech[Sqrt[x]])/4

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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x^3 \text{sech}^{-1}\left (\sqrt{x}\right ) \, dx &=\frac{1}{4} x^4 \text{sech}^{-1}\left (\sqrt{x}\right )+\frac{\sqrt{1-x} \int \frac{x^3}{2 \sqrt{1-x}} \, dx}{4 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}\\ &=\frac{1}{4} x^4 \text{sech}^{-1}\left (\sqrt{x}\right )+\frac{\sqrt{1-x} \int \frac{x^3}{\sqrt{1-x}} \, dx}{8 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}\\ &=\frac{1}{4} x^4 \text{sech}^{-1}\left (\sqrt{x}\right )+\frac{\sqrt{1-x} \int \left (\frac{1}{\sqrt{1-x}}-3 \sqrt{1-x}+3 (1-x)^{3/2}-(1-x)^{5/2}\right ) \, dx}{8 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}\\ &=-\frac{1-x}{4 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}+\frac{(1-x)^2}{4 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}-\frac{3 (1-x)^3}{20 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}+\frac{(1-x)^4}{28 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}+\frac{1}{4} x^4 \text{sech}^{-1}\left (\sqrt{x}\right )\\ \end{align*}

Mathematica [A]  time = 0.04074, size = 84, normalized size = 0.51 \[ \frac{1}{4} x^4 \text{sech}^{-1}\left (\sqrt{x}\right )-\frac{1}{140} \sqrt{\frac{1-\sqrt{x}}{\sqrt{x}+1}} \left (5 x^{7/2}+5 x^3+6 x^{5/2}+6 x^2+8 x^{3/2}+8 x+16 \sqrt{x}+16\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*ArcSech[Sqrt[x]],x]

[Out]

-(Sqrt[(1 - Sqrt[x])/(1 + Sqrt[x])]*(16 + 16*Sqrt[x] + 8*x + 8*x^(3/2) + 6*x^2 + 6*x^(5/2) + 5*x^3 + 5*x^(7/2)
))/140 + (x^4*ArcSech[Sqrt[x]])/4

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Maple [A]  time = 0.139, size = 54, normalized size = 0.3 \begin{align*}{\frac{{x}^{4}}{4}{\rm arcsech} \left (\sqrt{x}\right )}-{\frac{5\,{x}^{3}+6\,{x}^{2}+8\,x+16}{140}\sqrt{-{ \left ( -1+\sqrt{x} \right ){\frac{1}{\sqrt{x}}}}}\sqrt{x}\sqrt{{ \left ( 1+\sqrt{x} \right ){\frac{1}{\sqrt{x}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*arcsech(x^(1/2)),x)

[Out]

1/4*x^4*arcsech(x^(1/2))-1/140*(-(-1+x^(1/2))/x^(1/2))^(1/2)*x^(1/2)*((1+x^(1/2))/x^(1/2))^(1/2)*(5*x^3+6*x^2+
8*x+16)

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Maxima [A]  time = 0.982, size = 78, normalized size = 0.48 \begin{align*} \frac{1}{28} \, x^{\frac{7}{2}}{\left (\frac{1}{x} - 1\right )}^{\frac{7}{2}} - \frac{3}{20} \, x^{\frac{5}{2}}{\left (\frac{1}{x} - 1\right )}^{\frac{5}{2}} + \frac{1}{4} \, x^{4} \operatorname{arsech}\left (\sqrt{x}\right ) + \frac{1}{4} \, x^{\frac{3}{2}}{\left (\frac{1}{x} - 1\right )}^{\frac{3}{2}} - \frac{1}{4} \, \sqrt{x} \sqrt{\frac{1}{x} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arcsech(x^(1/2)),x, algorithm="maxima")

[Out]

1/28*x^(7/2)*(1/x - 1)^(7/2) - 3/20*x^(5/2)*(1/x - 1)^(5/2) + 1/4*x^4*arcsech(sqrt(x)) + 1/4*x^(3/2)*(1/x - 1)
^(3/2) - 1/4*sqrt(x)*sqrt(1/x - 1)

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Fricas [A]  time = 1.83839, size = 144, normalized size = 0.88 \begin{align*} \frac{1}{4} \, x^{4} \log \left (\frac{x \sqrt{-\frac{x - 1}{x}} + \sqrt{x}}{x}\right ) - \frac{1}{140} \,{\left (5 \, x^{3} + 6 \, x^{2} + 8 \, x + 16\right )} \sqrt{x} \sqrt{-\frac{x - 1}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arcsech(x^(1/2)),x, algorithm="fricas")

[Out]

1/4*x^4*log((x*sqrt(-(x - 1)/x) + sqrt(x))/x) - 1/140*(5*x^3 + 6*x^2 + 8*x + 16)*sqrt(x)*sqrt(-(x - 1)/x)

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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(x**3*asech(x**(1/2)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \operatorname{arsech}\left (\sqrt{x}\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arcsech(x^(1/2)),x, algorithm="giac")

[Out]

integrate(x^3*arcsech(sqrt(x)), x)