### 3.15 $$\int x^2 \text{csch}^{-1}(\sqrt{x}) \, dx$$

Optimal. Leaf size=89 $\frac{1}{3} x^3 \text{csch}^{-1}\left (\sqrt{x}\right )+\frac{(-x-1)^{5/2} \sqrt{x}}{15 \sqrt{-x}}+\frac{2 (-x-1)^{3/2} \sqrt{x}}{9 \sqrt{-x}}+\frac{\sqrt{-x-1} \sqrt{x}}{3 \sqrt{-x}}$

[Out]

(Sqrt[-1 - x]*Sqrt[x])/(3*Sqrt[-x]) + (2*(-1 - x)^(3/2)*Sqrt[x])/(9*Sqrt[-x]) + ((-1 - x)^(5/2)*Sqrt[x])/(15*S
qrt[-x]) + (x^3*ArcCsch[Sqrt[x]])/3

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Rubi [A]  time = 0.0253573, antiderivative size = 89, 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 = {6346, 12, 43} $\frac{1}{3} x^3 \text{csch}^{-1}\left (\sqrt{x}\right )+\frac{(-x-1)^{5/2} \sqrt{x}}{15 \sqrt{-x}}+\frac{2 (-x-1)^{3/2} \sqrt{x}}{9 \sqrt{-x}}+\frac{\sqrt{-x-1} \sqrt{x}}{3 \sqrt{-x}}$

Antiderivative was successfully veriﬁed.

[In]

Int[x^2*ArcCsch[Sqrt[x]],x]

[Out]

(Sqrt[-1 - x]*Sqrt[x])/(3*Sqrt[-x]) + (2*(-1 - x)^(3/2)*Sqrt[x])/(9*Sqrt[-x]) + ((-1 - x)^(5/2)*Sqrt[x])/(15*S
qrt[-x]) + (x^3*ArcCsch[Sqrt[x]])/3

Rule 6346

Int[((a_.) + ArcCsch[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m + 1)*(a + b*ArcCsc
h[u]))/(d*(m + 1)), x] - Dist[(b*u)/(d*(m + 1)*Sqrt[-u^2]), Int[SimplifyIntegrand[((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] && InverseFunctionFreeQ[u, x] &&  !F
unctionOfQ[(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^2 \text{csch}^{-1}\left (\sqrt{x}\right ) \, dx &=\frac{1}{3} x^3 \text{csch}^{-1}\left (\sqrt{x}\right )-\frac{\sqrt{x} \int \frac{x^2}{2 \sqrt{-1-x}} \, dx}{3 \sqrt{-x}}\\ &=\frac{1}{3} x^3 \text{csch}^{-1}\left (\sqrt{x}\right )-\frac{\sqrt{x} \int \frac{x^2}{\sqrt{-1-x}} \, dx}{6 \sqrt{-x}}\\ &=\frac{1}{3} x^3 \text{csch}^{-1}\left (\sqrt{x}\right )-\frac{\sqrt{x} \int \left (\frac{1}{\sqrt{-1-x}}+2 \sqrt{-1-x}+(-1-x)^{3/2}\right ) \, dx}{6 \sqrt{-x}}\\ &=\frac{\sqrt{-1-x} \sqrt{x}}{3 \sqrt{-x}}+\frac{2 (-1-x)^{3/2} \sqrt{x}}{9 \sqrt{-x}}+\frac{(-1-x)^{5/2} \sqrt{x}}{15 \sqrt{-x}}+\frac{1}{3} x^3 \text{csch}^{-1}\left (\sqrt{x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0302423, size = 42, normalized size = 0.47 $\frac{1}{45} \sqrt{\frac{1}{x}+1} \left (3 x^2-4 x+8\right ) \sqrt{x}+\frac{1}{3} x^3 \text{csch}^{-1}\left (\sqrt{x}\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x^2*ArcCsch[Sqrt[x]],x]

[Out]

(Sqrt[1 + x^(-1)]*Sqrt[x]*(8 - 4*x + 3*x^2))/45 + (x^3*ArcCsch[Sqrt[x]])/3

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Maple [A]  time = 0.118, size = 38, normalized size = 0.4 \begin{align*}{\frac{{x}^{3}}{3}{\rm arccsch} \left (\sqrt{x}\right )}+{\frac{ \left ( 1+x \right ) \left ( 3\,{x}^{2}-4\,x+8 \right ) }{45}{\frac{1}{\sqrt{{\frac{1+x}{x}}}}}{\frac{1}{\sqrt{x}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*arccsch(x^(1/2)),x)

[Out]

1/3*x^3*arccsch(x^(1/2))+1/45*(1+x)*(3*x^2-4*x+8)/((1+x)/x)^(1/2)/x^(1/2)

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Maxima [A]  time = 0.99561, size = 62, normalized size = 0.7 \begin{align*} \frac{1}{15} \, x^{\frac{5}{2}}{\left (\frac{1}{x} + 1\right )}^{\frac{5}{2}} + \frac{1}{3} \, x^{3} \operatorname{arcsch}\left (\sqrt{x}\right ) - \frac{2}{9} \, x^{\frac{3}{2}}{\left (\frac{1}{x} + 1\right )}^{\frac{3}{2}} + \frac{1}{3} \, \sqrt{x} \sqrt{\frac{1}{x} + 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*x^(5/2)*(1/x + 1)^(5/2) + 1/3*x^3*arccsch(sqrt(x)) - 2/9*x^(3/2)*(1/x + 1)^(3/2) + 1/3*sqrt(x)*sqrt(1/x +
1)

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Fricas [A]  time = 2.63231, size = 128, normalized size = 1.44 \begin{align*} \frac{1}{3} \, x^{3} \log \left (\frac{x \sqrt{\frac{x + 1}{x}} + \sqrt{x}}{x}\right ) + \frac{1}{45} \,{\left (3 \, x^{2} - 4 \, x + 8\right )} \sqrt{x} \sqrt{\frac{x + 1}{x}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3*log((x*sqrt((x + 1)/x) + sqrt(x))/x) + 1/45*(3*x^2 - 4*x + 8)*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**2*acsch(x**(1/2)),x)

[Out]

Timed out

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2*arccsch(sqrt(x)), x)