\(\int x \text {arccosh}(\sqrt {x}) \, dx\) [233]

Optimal result

Integrand size = 8, antiderivative size = 86 \[ \int x \text {arccosh}\left (\sqrt {x}\right ) \, dx=-\frac {3}{16} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}-\frac {1}{8} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}-\frac {3 \text {arccosh}\left (\sqrt {x}\right )}{16}+\frac {1}{2} x^2 \text {arccosh}\left (\sqrt {x}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6017, 12, 329, 336, 54} \[ \int x \text {arccosh}\left (\sqrt {x}\right ) \, dx=\frac {1}{2} x^2 \text {arccosh}\left (\sqrt {x}\right )-\frac {3 \text {arccosh}\left (\sqrt {x}\right )}{16}-\frac {1}{8} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{3/2}-\frac {3}{16} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x} \]

[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 12

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

Rule 54

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 329

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[a1
*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 336

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 6017

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]*Sq
rt[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*} \text {integral}& = \frac {1}{2} x^2 \text {arccosh}\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 \text {arccosh}\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 \text {arccosh}\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 \text {arccosh}\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 \text {arccosh}\left (\sqrt {x}\right )-\frac {3}{16} \text {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 \text {arccosh}\left (\sqrt {x}\right )}{16}+\frac {1}{2} x^2 \text {arccosh}\left (\sqrt {x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.86 \[ \int x \text {arccosh}\left (\sqrt {x}\right ) \, dx=\frac {1}{16} \left (-\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x} (3+2 x)+8 x^2 \text {arccosh}\left (\sqrt {x}\right )-6 \text {arctanh}\left (\sqrt {\frac {-1+\sqrt {x}}{1+\sqrt {x}}}\right )\right ) \]

[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

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.76

method	result	size
derivativedivides	\(\frac {x^{2} \operatorname {arccosh}\left (\sqrt {x}\right )}{2}-\frac {\sqrt {-1+\sqrt {x}}\, \sqrt {1+\sqrt {x}}\, \left (2 x^{\frac {3}{2}} \sqrt {x -1}+3 \sqrt {x}\, \sqrt {x -1}+3 \ln \left (\sqrt {x}+\sqrt {x -1}\right )\right )}{16 \sqrt {x -1}}\)	\(65\)
default	\(\frac {x^{2} \operatorname {arccosh}\left (\sqrt {x}\right )}{2}-\frac {\sqrt {-1+\sqrt {x}}\, \sqrt {1+\sqrt {x}}\, \left (2 x^{\frac {3}{2}} \sqrt {x -1}+3 \sqrt {x}\, \sqrt {x -1}+3 \ln \left (\sqrt {x}+\sqrt {x -1}\right )\right )}{16 \sqrt {x -1}}\)	\(65\)
parts	\(\frac {x^{2} \operatorname {arccosh}\left (\sqrt {x}\right )}{2}-\frac {\sqrt {-1+\sqrt {x}}\, \sqrt {1+\sqrt {x}}\, \left (2 x^{\frac {3}{2}} \sqrt {x -1}+3 \sqrt {x}\, \sqrt {x -1}+3 \ln \left (\sqrt {x}+\sqrt {x -1}\right )\right )}{16 \sqrt {x -1}}\)	\(65\)

[In]

int(x*arccosh(x^(1/2)),x,method=_RETURNVERBOSE)

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.41 \[ \int x \text {arccosh}\left (\sqrt {x}\right ) \, dx=-\frac {1}{16} \, {\left (2 \, x + 3\right )} \sqrt {x - 1} \sqrt {x} + \frac {1}{16} \, {\left (8 \, x^{2} - 3\right )} \log \left (\sqrt {x - 1} + \sqrt {x}\right ) \]

[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))

Sympy [F]

\[ \int x \text {arccosh}\left (\sqrt {x}\right ) \, dx=\int x \operatorname {acosh}{\left (\sqrt {x} \right )}\, dx \]

[In]

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

[Out]

Integral(x*acosh(sqrt(x)), x)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.53 \[ \int x \text {arccosh}\left (\sqrt {x}\right ) \, dx=\frac {1}{2} \, x^{2} \operatorname {arcosh}\left (\sqrt {x}\right ) - \frac {1}{8} \, \sqrt {x - 1} x^{\frac {3}{2}} - \frac {3}{16} \, \sqrt {x - 1} \sqrt {x} - \frac {3}{16} \, \log \left (2 \, \sqrt {x - 1} + 2 \, \sqrt {x}\right ) \]

[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))

Giac [A] (verification not implemented)

none

Time = 0.50 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.64 \[ \int x \text {arccosh}\left (\sqrt {x}\right ) \, dx=\frac {1}{2} \, x^{2} \log \left (\sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} + \sqrt {x}\right ) - \frac {1}{16} \, {\left (2 \, x + 3\right )} \sqrt {x - 1} \sqrt {x} + \frac {3}{16} \, \log \left (-\sqrt {x - 1} + \sqrt {x}\right ) \]

[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(-sq
rt(x - 1) + sqrt(x))

Mupad [F(-1)]

Timed out. \[ \int x \text {arccosh}\left (\sqrt {x}\right ) \, dx=\int x\,\mathrm {acosh}\left (\sqrt {x}\right ) \,d x \]

[In]

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

[Out]

int(x*acosh(x^(1/2)), x)