3.233 \(\int x \cosh ^{-1}(\sqrt{x}) \, dx\)

Optimal. Leaf size=86 \[ -\frac{1}{8} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{3/2}+\frac{1}{2} x^2 \cosh ^{-1}\left (\sqrt{x}\right )-\frac{3}{16} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}-\frac{3}{16} \cosh ^{-1}\left (\sqrt{x}\right ) \]

[Out]

(-3*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*Sqrt[x])/16 - (Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(3/2))/8 - (3*A
rcCosh[Sqrt[x]])/16 + (x^2*ArcCosh[Sqrt[x]])/2

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Rubi [A]  time = 0.0503724, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {5903, 12, 323, 330, 52} \[ -\frac{1}{8} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{3/2}+\frac{1}{2} x^2 \cosh ^{-1}\left (\sqrt{x}\right )-\frac{3}{16} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}-\frac{3}{16} \cosh ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x*ArcCosh[Sqrt[x]],x]

[Out]

(-3*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*Sqrt[x])/16 - (Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(3/2))/8 - (3*A
rcCosh[Sqrt[x]])/16 + (x^2*ArcCosh[Sqrt[x]])/2

Rule 5903

Int[((a_.) + ArcCosh[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m + 1)*(a + b*ArcCos
h[u]))/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[SimplifyIntegrand[((c + d*x)^(m + 1)*D[u, x])/(Sqrt[-1 + u]*S
qrt[1 + u]), x], x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] &&  !Function
OfQ[(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 323

Int[((c_.)*(x_))^(m_)*((a1_) + (b1_.)*(x_)^(n_))^(p_)*((a2_) + (b2_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(2
*n - 1)*(c*x)^(m - 2*n + 1)*(a1 + b1*x^n)^(p + 1)*(a2 + b2*x^n)^(p + 1))/(b1*b2*(m + 2*n*p + 1)), x] - Dist[(a
1*a2*c^(2*n)*(m - 2*n + 1))/(b1*b2*(m + 2*n*p + 1)), Int[(c*x)^(m - 2*n)*(a1 + b1*x^n)^p*(a2 + b2*x^n)^p, x],
x] /; FreeQ[{a1, b1, a2, b2, c, p}, x] && EqQ[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && GtQ[m, 2*n - 1] && NeQ[m +
2*n*p + 1, 0] && IntBinomialQ[a1*a2, b1*b2, c, 2*n, m, p, x]

Rule 330

Int[((c_.)*(x_))^(m_)*((a1_) + (b1_.)*(x_)^(n_))^(p_)*((a2_) + (b2_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k =
Denominator[m]}, Dist[k/c, Subst[Int[x^(k*(m + 1) - 1)*(a1 + (b1*x^(k*n))/c^n)^p*(a2 + (b2*x^(k*n))/c^n)^p, x]
, x, (c*x)^(1/k)], x]] /; FreeQ[{a1, b1, a2, b2, c, p}, x] && EqQ[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && Fractio
nQ[m] && IntBinomialQ[a1*a2, b1*b2, c, 2*n, m, p, x]

Rule 52

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[(b*x)/a]/b, x] /; FreeQ[{a,
b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]

Rubi steps

\begin{align*} \int x \cosh ^{-1}\left (\sqrt{x}\right ) \, dx &=\frac{1}{2} x^2 \cosh ^{-1}\left (\sqrt{x}\right )-\frac{1}{2} \int \frac{x^{3/2}}{2 \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}} \, dx\\ &=\frac{1}{2} x^2 \cosh ^{-1}\left (\sqrt{x}\right )-\frac{1}{4} \int \frac{x^{3/2}}{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}} \, dx\\ &=-\frac{1}{8} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{3/2}+\frac{1}{2} x^2 \cosh ^{-1}\left (\sqrt{x}\right )-\frac{3}{16} \int \frac{\sqrt{x}}{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}} \, dx\\ &=-\frac{3}{16} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} \sqrt{x}-\frac{1}{8} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{3/2}+\frac{1}{2} x^2 \cosh ^{-1}\left (\sqrt{x}\right )-\frac{3}{32} \int \frac{1}{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} \sqrt{x}} \, dx\\ &=-\frac{3}{16} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} \sqrt{x}-\frac{1}{8} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{3/2}+\frac{1}{2} x^2 \cosh ^{-1}\left (\sqrt{x}\right )-\frac{3}{16} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,\sqrt{x}\right )\\ &=-\frac{3}{16} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} \sqrt{x}-\frac{1}{8} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{3/2}-\frac{3}{16} \cosh ^{-1}\left (\sqrt{x}\right )+\frac{1}{2} x^2 \cosh ^{-1}\left (\sqrt{x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0383801, size = 74, normalized size = 0.86 \[ \frac{1}{16} \left (8 x^2 \cosh ^{-1}\left (\sqrt{x}\right )-\sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} (2 x+3) \sqrt{x}-6 \tanh ^{-1}\left (\sqrt{\frac{\sqrt{x}-1}{\sqrt{x}+1}}\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x*ArcCosh[Sqrt[x]],x]

[Out]

(-(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*Sqrt[x]*(3 + 2*x)) + 8*x^2*ArcCosh[Sqrt[x]] - 6*ArcTanh[Sqrt[(-1 + Sqr
t[x])/(1 + Sqrt[x])]])/16

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Maple [A]  time = 0.005, size = 65, normalized size = 0.8 \begin{align*}{\frac{{x}^{2}}{2}{\rm arccosh} \left (\sqrt{x}\right )}-{\frac{1}{16}\sqrt{-1+\sqrt{x}}\sqrt{1+\sqrt{x}} \left ( 2\,{x}^{3/2}\sqrt{-1+x}+3\,\sqrt{x}\sqrt{-1+x}+3\,\ln \left ( \sqrt{x}+\sqrt{-1+x} \right ) \right ){\frac{1}{\sqrt{-1+x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2*arccosh(x^(1/2))-1/16*(-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)*(2*x^(3/2)*(-1+x)^(1/2)+3*x^(1/2)*(-1+x)^(1
/2)+3*ln(x^(1/2)+(-1+x)^(1/2)))/(-1+x)^(1/2)

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Maxima [A]  time = 1.02128, size = 62, normalized size = 0.72 \begin{align*} \frac{1}{2} \, x^{2} \operatorname{arcosh}\left (\sqrt{x}\right ) - \frac{1}{8} \, \sqrt{x - 1} x^{\frac{3}{2}} - \frac{3}{16} \, \sqrt{x - 1} \sqrt{x} - \frac{3}{16} \, \log \left (2 \, \sqrt{x - 1} + 2 \, \sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2*arccosh(sqrt(x)) - 1/8*sqrt(x - 1)*x^(3/2) - 3/16*sqrt(x - 1)*sqrt(x) - 3/16*log(2*sqrt(x - 1) + 2*sqr
t(x))

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Fricas [A]  time = 1.95737, size = 112, normalized size = 1.3 \begin{align*} -\frac{1}{16} \,{\left (2 \, x + 3\right )} \sqrt{x - 1} \sqrt{x} + \frac{1}{16} \,{\left (8 \, x^{2} - 3\right )} \log \left (\sqrt{x - 1} + \sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(2*x + 3)*sqrt(x - 1)*sqrt(x) + 1/16*(8*x^2 - 3)*log(sqrt(x - 1) + sqrt(x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*acosh(sqrt(x)), x)

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Giac [A]  time = 1.16708, size = 76, normalized size = 0.88 \begin{align*} \frac{1}{2} \, x^{2} \log \left (\sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} + \sqrt{x}\right ) - \frac{1}{16} \,{\left (2 \, x + 3\right )} \sqrt{x - 1} \sqrt{x} + \frac{3}{16} \, \log \left ({\left | \sqrt{x - 1} - \sqrt{x} \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2*log(sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) + sqrt(x)) - 1/16*(2*x + 3)*sqrt(x - 1)*sqrt(x) + 3/16*log(abs
(sqrt(x - 1) - sqrt(x)))