3.3.36 \(\int \frac {\cosh ^{-1}(\sqrt {x})}{x^2} \, dx\) [236]

Optimal. Leaf size=40 \[ \frac {\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{\sqrt {x}}-\frac {\cosh ^{-1}\left (\sqrt {x}\right )}{x} \]

[Out]

-arccosh(x^(1/2))/x+(-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)/x^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6017, 12, 271} \begin {gather*} \frac {\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1}}{\sqrt {x}}-\frac {\cosh ^{-1}\left (\sqrt {x}\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/Sqrt[x] - ArcCosh[Sqrt[x]]/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 271

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

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*} \int \frac {\cosh ^{-1}\left (\sqrt {x}\right )}{x^2} \, dx &=-\frac {\cosh ^{-1}\left (\sqrt {x}\right )}{x}+\int \frac {1}{2 \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}} \, dx\\ &=-\frac {\cosh ^{-1}\left (\sqrt {x}\right )}{x}+\frac {1}{2} \int \frac {1}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}} \, dx\\ &=\frac {\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{\sqrt {x}}-\frac {\cosh ^{-1}\left (\sqrt {x}\right )}{x}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 40, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{\sqrt {x}}-\frac {\cosh ^{-1}\left (\sqrt {x}\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/Sqrt[x] - ArcCosh[Sqrt[x]]/x

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Maple [A]
time = 0.01, size = 29, normalized size = 0.72

method	result	size
derivativedivides	\(-\frac {\mathrm {arccosh}\left (\sqrt {x}\right )}{x}+\frac {\sqrt {-1+\sqrt {x}}\, \sqrt {1+\sqrt {x}}}{\sqrt {x}}\)	\(29\)
default	\(-\frac {\mathrm {arccosh}\left (\sqrt {x}\right )}{x}+\frac {\sqrt {-1+\sqrt {x}}\, \sqrt {1+\sqrt {x}}}{\sqrt {x}}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arccosh(x^(1/2))/x+(-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)/x^(1/2)

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Maxima [A]
time = 0.51, size = 19, normalized size = 0.48 \begin {gather*} \frac {\sqrt {x - 1}}{\sqrt {x}} - \frac {\operatorname {arcosh}\left (\sqrt {x}\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(x - 1)/sqrt(x) - arccosh(sqrt(x))/x

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Fricas [A]
time = 0.34, size = 26, normalized size = 0.65 \begin {gather*} \frac {\sqrt {x - 1} \sqrt {x} - \log \left (\sqrt {x - 1} + \sqrt {x}\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(sqrt(x - 1)*sqrt(x) - log(sqrt(x - 1) + sqrt(x)))/x

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {acosh}{\left (\sqrt {x} \right )}}{x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acosh(sqrt(x))/x**2, x)

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Giac [A]
time = 1.55, size = 45, normalized size = 1.12 \begin {gather*} -\frac {\log \left (\sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} + \sqrt {x}\right )}{x} + \frac {2}{{\left (\sqrt {x - 1} - \sqrt {x}\right )}^{2} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\mathrm {acosh}\left (\sqrt {x}\right )}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(acosh(x^(1/2))/x^2, x)

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