3.1.58 \(\int \frac {1}{x \sqrt {2+x-x^2}} \, dx\) [58]

Optimal. Leaf size=32 \[ -\frac {\tanh ^{-1}\left (\frac {4+x}{2 \sqrt {2} \sqrt {2+x-x^2}}\right )}{\sqrt {2}} \]

[Out]

-1/2*arctanh(1/4*(4+x)*2^(1/2)/(-x^2+x+2)^(1/2))*2^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {738, 212} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {x+4}{2 \sqrt {2} \sqrt {-x^2+x+2}}\right )}{\sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[2 + x - x^2]),x]

[Out]

-(ArcTanh[(4 + x)/(2*Sqrt[2]*Sqrt[2 + x - x^2])]/Sqrt[2])

Rule 212

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

Rule 738

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rubi steps

\begin {align*} \int \frac {1}{x \sqrt {2+x-x^2}} \, dx &=-\left (2 \text {Subst}\left (\int \frac {1}{8-x^2} \, dx,x,\frac {4+x}{\sqrt {2+x-x^2}}\right )\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {4+x}{2 \sqrt {2} \sqrt {2+x-x^2}}\right )}{\sqrt {2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.07, size = 34, normalized size = 1.06 \begin {gather*} i \sqrt {2} \tan ^{-1}\left (\frac {x+i \sqrt {2+x-x^2}}{\sqrt {2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[2 + x - x^2]),x]

[Out]

I*Sqrt[2]*ArcTan[(x + I*Sqrt[2 + x - x^2])/Sqrt[2]]

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Maple [A]
time = 0.11, size = 25, normalized size = 0.78

method	result	size
default	\(-\frac {\arctanh \left (\frac {\left (4+x \right ) \sqrt {2}}{4 \sqrt {-x^{2}+x +2}}\right ) \sqrt {2}}{2}\)	\(25\)
trager	\(\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}-2\right ) x +4 \sqrt {-x^{2}+x +2}-4 \RootOf \left (\textit {\_Z}^{2}-2\right )}{x}\right )}{2}\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*arctanh(1/4*(4+x)*2^(1/2)/(-x^2+x+2)^(1/2))*2^(1/2)

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Maxima [A]
time = 1.57, size = 33, normalized size = 1.03 \begin {gather*} -\frac {1}{2} \, \sqrt {2} \log \left (\frac {2 \, \sqrt {2} \sqrt {-x^{2} + x + 2}}{{\left | x \right |}} + \frac {4}{{\left | x \right |}} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(2)*log(2*sqrt(2)*sqrt(-x^2 + x + 2)/abs(x) + 4/abs(x) + 1)

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Fricas [A]
time = 0.58, size = 39, normalized size = 1.22 \begin {gather*} \frac {1}{4} \, \sqrt {2} \log \left (-\frac {4 \, \sqrt {2} \sqrt {-x^{2} + x + 2} {\left (x + 4\right )} + 7 \, x^{2} - 16 \, x - 32}{x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt(2)*log(-(4*sqrt(2)*sqrt(-x^2 + x + 2)*(x + 4) + 7*x^2 - 16*x - 32)/x^2)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \sqrt {- \left (x - 2\right ) \left (x + 1\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(x*sqrt(-(x - 2)*(x + 1))), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (24) = 48\).
time = 0.98, size = 71, normalized size = 2.22 \begin {gather*} -\frac {1}{2} \, \sqrt {2} \log \left (\frac {{\left | -4 \, \sqrt {2} + \frac {2 \, {\left (2 \, \sqrt {-x^{2} + x + 2} - 3\right )}}{2 \, x - 1} - 6 \right |}}{{\left | 4 \, \sqrt {2} + \frac {2 \, {\left (2 \, \sqrt {-x^{2} + x + 2} - 3\right )}}{2 \, x - 1} - 6 \right |}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(2)*log(abs(-4*sqrt(2) + 2*(2*sqrt(-x^2 + x + 2) - 3)/(2*x - 1) - 6)/abs(4*sqrt(2) + 2*(2*sqrt(-x^2 +
x + 2) - 3)/(2*x - 1) - 6))

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Mupad [B]
time = 0.34, size = 28, normalized size = 0.88 \begin {gather*} -\frac {\sqrt {2}\,\ln \left (\frac {x+2\,\sqrt {2}\,\sqrt {-x^2+x+2}+4}{x}\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(x - x^2 + 2)^(1/2)),x)

[Out]

-(2^(1/2)*log((x + 2*2^(1/2)*(x - x^2 + 2)^(1/2) + 4)/x))/2

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