3.3.22 \(\int \frac {1}{\sqrt {x} (-1+2 x)} \, dx\) [222]

Optimal. Leaf size=19 \[ -\sqrt {2} \tanh ^{-1}\left (\sqrt {2} \sqrt {x}\right ) \]

[Out]

-arctanh(2^(1/2)*x^(1/2))*2^(1/2)

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Rubi [A]
time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {65, 213} \begin {gather*} -\sqrt {2} \tanh ^{-1}\left (\sqrt {2} \sqrt {x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[x]*(-1 + 2*x)),x]

[Out]

-(Sqrt[2]*ArcTanh[Sqrt[2]*Sqrt[x]])

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {x} (-1+2 x)} \, dx &=2 \text {Subst}\left (\int \frac {1}{-1+2 x^2} \, dx,x,\sqrt {x}\right )\\ &=-\sqrt {2} \tanh ^{-1}\left (\sqrt {2} \sqrt {x}\right )\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 19, normalized size = 1.00 \begin {gather*} -\sqrt {2} \tanh ^{-1}\left (\sqrt {2} \sqrt {x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[x]*(-1 + 2*x)),x]

[Out]

-(Sqrt[2]*ArcTanh[Sqrt[2]*Sqrt[x]])

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Maple [A]
time = 0.07, size = 14, normalized size = 0.74

method	result	size
derivativedivides	\(-\arctanh \left (\sqrt {2}\, \sqrt {x}\right ) \sqrt {2}\)	\(14\)
default	\(-\arctanh \left (\sqrt {2}\, \sqrt {x}\right ) \sqrt {2}\)	\(14\)
meijerg	\(-\arctanh \left (\sqrt {2}\, \sqrt {x}\right ) \sqrt {2}\)	\(14\)
trager	\(-\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}-2\right ) x +4 \sqrt {x}+\RootOf \left (\textit {\_Z}^{2}-2\right )}{2 x -1}\right )}{2}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arctanh(2^(1/2)*x^(1/2))*2^(1/2)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 28 vs. \(2 (13) = 26\).
time = 1.60, size = 28, normalized size = 1.47 \begin {gather*} \frac {1}{2} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - 2 \, \sqrt {x}}{\sqrt {2} + 2 \, \sqrt {x}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 28 vs. \(2 (13) = 26\).
time = 0.62, size = 28, normalized size = 1.47 \begin {gather*} \frac {1}{2} \, \sqrt {2} \log \left (-\frac {2 \, \sqrt {2} \sqrt {x} - 2 \, x - 1}{2 \, x - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (17) = 34\).
time = 0.10, size = 39, normalized size = 2.05 \begin {gather*} \frac {\sqrt {2} \log {\left (\sqrt {x} - \frac {\sqrt {2}}{2} \right )}}{2} - \frac {\sqrt {2} \log {\left (\sqrt {x} + \frac {\sqrt {2}}{2} \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**(1/2)/(-1+2*x),x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (13) = 26\).
time = 0.48, size = 32, normalized size = 1.68 \begin {gather*} -\frac {1}{2} \, \sqrt {2} \log \left (\frac {1}{2} \, \sqrt {2} + \sqrt {x}\right ) + \frac {1}{2} \, \sqrt {2} \log \left ({\left | -\frac {1}{2} \, \sqrt {2} + \sqrt {x} \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.08, size = 13, normalized size = 0.68 \begin {gather*} -\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2\,x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(1/2)*(2*x - 1)),x)

[Out]

-2^(1/2)*atanh((2*x)^(1/2))

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