3.232 \(\int x^2 \cosh ^{-1}(\sqrt {x}) \, dx\)

Optimal. Leaf size=117 \[ -\frac {1}{18} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{5/2}-\frac {5}{72} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{3/2}+\frac {1}{3} x^3 \cosh ^{-1}\left (\sqrt {x}\right )-\frac {5}{48} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}-\frac {5}{48} \cosh ^{-1}\left (\sqrt {x}\right ) \]

[Out]

-5/48*arccosh(x^(1/2))+1/3*x^3*arccosh(x^(1/2))-5/72*x^(3/2)*(-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)-1/18*x^(5/2)
*(-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)-5/48*x^(1/2)*(-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5903, 12, 323, 330, 52} \[ -\frac {1}{18} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{5/2}-\frac {5}{72} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{3/2}+\frac {1}{3} x^3 \cosh ^{-1}\left (\sqrt {x}\right )-\frac {5}{48} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}-\frac {5}{48} \cosh ^{-1}\left (\sqrt {x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

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

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 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]

Rubi steps

\begin {align*} \int x^2 \cosh ^{-1}\left (\sqrt {x}\right ) \, dx &=\frac {1}{3} x^3 \cosh ^{-1}\left (\sqrt {x}\right )-\frac {1}{3} \int \frac {x^{5/2}}{2 \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx\\ &=\frac {1}{3} x^3 \cosh ^{-1}\left (\sqrt {x}\right )-\frac {1}{6} \int \frac {x^{5/2}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx\\ &=-\frac {1}{18} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2}+\frac {1}{3} x^3 \cosh ^{-1}\left (\sqrt {x}\right )-\frac {5}{36} \int \frac {x^{3/2}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx\\ &=-\frac {5}{72} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}-\frac {1}{18} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2}+\frac {1}{3} x^3 \cosh ^{-1}\left (\sqrt {x}\right )-\frac {5}{48} \int \frac {\sqrt {x}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx\\ &=-\frac {5}{48} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}-\frac {5}{72} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}-\frac {1}{18} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2}+\frac {1}{3} x^3 \cosh ^{-1}\left (\sqrt {x}\right )-\frac {5}{96} \int \frac {1}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}} \, dx\\ &=-\frac {5}{48} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}-\frac {5}{72} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}-\frac {1}{18} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2}+\frac {1}{3} x^3 \cosh ^{-1}\left (\sqrt {x}\right )-\frac {5}{48} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {5}{48} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}-\frac {5}{72} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}-\frac {1}{18} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2}-\frac {5}{48} \cosh ^{-1}\left (\sqrt {x}\right )+\frac {1}{3} x^3 \cosh ^{-1}\left (\sqrt {x}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 79, normalized size = 0.68 \[ \frac {1}{144} \left (48 x^3 \cosh ^{-1}\left (\sqrt {x}\right )-\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \left (8 x^2+10 x+15\right ) \sqrt {x}-30 \tanh ^{-1}\left (\sqrt {\frac {\sqrt {x}-1}{\sqrt {x}+1}}\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*Sqrt[x]*(15 + 10*x + 8*x^2)) + 48*x^3*ArcCosh[Sqrt[x]] - 30*ArcTanh[Sq
rt[(-1 + Sqrt[x])/(1 + Sqrt[x])]])/144

________________________________________________________________________________________

fricas [A]  time = 0.59, size = 40, normalized size = 0.34 \[ -\frac {1}{144} \, {\left (8 \, x^{2} + 10 \, x + 15\right )} \sqrt {x - 1} \sqrt {x} + \frac {1}{48} \, {\left (16 \, x^{3} - 5\right )} \log \left (\sqrt {x - 1} + \sqrt {x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/144*(8*x^2 + 10*x + 15)*sqrt(x - 1)*sqrt(x) + 1/48*(16*x^3 - 5)*log(sqrt(x - 1) + sqrt(x))

________________________________________________________________________________________

giac [A]  time = 1.21, size = 60, normalized size = 0.51 \[ \frac {1}{3} \, x^{3} \log \left (\sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} + \sqrt {x}\right ) - \frac {1}{144} \, {\left (2 \, {\left (4 \, x + 5\right )} x + 15\right )} \sqrt {x - 1} \sqrt {x} + \frac {5}{48} \, \log \left (-\sqrt {x - 1} + \sqrt {x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3*log(sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) + sqrt(x)) - 1/144*(2*(4*x + 5)*x + 15)*sqrt(x - 1)*sqrt(x) +
5/48*log(-sqrt(x - 1) + sqrt(x))

________________________________________________________________________________________

maple [A]  time = 0.03, size = 75, normalized size = 0.64 \[ \frac {x^{3} \mathrm {arccosh}\left (\sqrt {x}\right )}{3}-\frac {\sqrt {-1+\sqrt {x}}\, \sqrt {1+\sqrt {x}}\, \left (8 x^{\frac {5}{2}} \sqrt {-1+x}+10 x^{\frac {3}{2}} \sqrt {-1+x}+15 \sqrt {x}\, \sqrt {-1+x}+15 \ln \left (\sqrt {x}+\sqrt {-1+x}\right )\right )}{144 \sqrt {-1+x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.50, size = 56, normalized size = 0.48 \[ \frac {1}{3} \, x^{3} \operatorname {arcosh}\left (\sqrt {x}\right ) - \frac {1}{18} \, \sqrt {x - 1} x^{\frac {5}{2}} - \frac {5}{72} \, \sqrt {x - 1} x^{\frac {3}{2}} - \frac {5}{48} \, \sqrt {x - 1} \sqrt {x} - \frac {5}{48} \, \log \left (2 \, \sqrt {x - 1} + 2 \, \sqrt {x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3*arccosh(sqrt(x)) - 1/18*sqrt(x - 1)*x^(5/2) - 5/72*sqrt(x - 1)*x^(3/2) - 5/48*sqrt(x - 1)*sqrt(x) - 5/
48*log(2*sqrt(x - 1) + 2*sqrt(x))

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,\mathrm {acosh}\left (\sqrt {x}\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^2*acosh(x^(1/2)), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \operatorname {acosh}{\left (\sqrt {x} \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2*acosh(sqrt(x)), x)

________________________________________________________________________________________