3.1.14 \(\int \sqrt {-8 x+x^2} \, dx\) [14]

Optimal. Leaf size=37 \[ -\frac {1}{2} (4-x) \sqrt {-8 x+x^2}-16 \tanh ^{-1}\left (\frac {x}{\sqrt {-8 x+x^2}}\right ) \]

[Out]

-16*arctanh(x/(x^2-8*x)^(1/2))-1/2*(4-x)*(x^2-8*x)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {626, 634, 212} \begin {gather*} -\frac {1}{2} \sqrt {x^2-8 x} (4-x)-16 \tanh ^{-1}\left (\frac {x}{\sqrt {x^2-8 x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[-8*x + x^2],x]

[Out]

-1/2*((4 - x)*Sqrt[-8*x + x^2]) - 16*ArcTanh[x/Sqrt[-8*x + x^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 626

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x)*((a + b*x + c*x^2)^p/(2*c*(2*p + 1
))), x] - Dist[p*((b^2 - 4*a*c)/(2*c*(2*p + 1))), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 634

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2
]], x] /; FreeQ[{b, c}, x]

Rubi steps

\begin {align*} \int \sqrt {-8 x+x^2} \, dx &=-\frac {1}{2} (4-x) \sqrt {-8 x+x^2}-8 \int \frac {1}{\sqrt {-8 x+x^2}} \, dx\\ &=-\frac {1}{2} (4-x) \sqrt {-8 x+x^2}-16 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {-8 x+x^2}}\right )\\ &=-\frac {1}{2} (4-x) \sqrt {-8 x+x^2}-16 \tanh ^{-1}\left (\frac {x}{\sqrt {-8 x+x^2}}\right )\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 42, normalized size = 1.14 \begin {gather*} \frac {1}{2} \sqrt {(-8+x) x} \left (-4+x-\frac {32 \tanh ^{-1}\left (\frac {1}{\sqrt {\frac {-8+x}{x}}}\right )}{\sqrt {-8+x} \sqrt {x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-8*x + x^2],x]

[Out]

(Sqrt[(-8 + x)*x]*(-4 + x - (32*ArcTanh[1/Sqrt[(-8 + x)/x]])/(Sqrt[-8 + x]*Sqrt[x])))/2

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Maple [A]
time = 0.41, size = 33, normalized size = 0.89

method	result	size
default	\(\frac {\left (2 x -8\right ) \sqrt {x^{2}-8 x}}{4}-8 \ln \left (x -4+\sqrt {x^{2}-8 x}\right )\)	\(33\)
risch	\(\frac {\left (x -4\right ) x \left (x -8\right )}{2 \sqrt {x \left (x -8\right )}}-8 \ln \left (x -4+\sqrt {x^{2}-8 x}\right )\)	\(33\)
trager	\(\left (\frac {x}{2}-2\right ) \sqrt {x^{2}-8 x}+8 \ln \left (4-x +\sqrt {x^{2}-8 x}\right )\)	\(34\)
meijerg	\(-\frac {32 i \sqrt {\mathrm {signum}\left (x -8\right )}\, \left (-\frac {i \sqrt {\pi }\, \sqrt {x}\, \sqrt {2}\, \left (-\frac {3 x}{4}+3\right ) \sqrt {-\frac {x}{8}+1}}{24}+\frac {i \sqrt {\pi }\, \arcsin \left (\frac {\sqrt {2}\, \sqrt {x}}{4}\right )}{2}\right )}{\sqrt {\pi }\, \sqrt {-\mathrm {signum}\left (x -8\right )}}\)	\(61\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2-8*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/4*(2*x-8)*(x^2-8*x)^(1/2)-8*ln(x-4+(x^2-8*x)^(1/2))

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Maxima [A]
time = 0.28, size = 43, normalized size = 1.16 \begin {gather*} \frac {1}{2} \, \sqrt {x^{2} - 8 \, x} x - 2 \, \sqrt {x^{2} - 8 \, x} - 8 \, \log \left (2 \, x + 2 \, \sqrt {x^{2} - 8 \, x} - 8\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-8*x)^(1/2),x, algorithm="maxima")

[Out]

1/2*sqrt(x^2 - 8*x)*x - 2*sqrt(x^2 - 8*x) - 8*log(2*x + 2*sqrt(x^2 - 8*x) - 8)

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Fricas [A]
time = 1.84, size = 32, normalized size = 0.86 \begin {gather*} \frac {1}{2} \, \sqrt {x^{2} - 8 \, x} {\left (x - 4\right )} + 8 \, \log \left (-x + \sqrt {x^{2} - 8 \, x} + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-8*x)^(1/2),x, algorithm="fricas")

[Out]

1/2*sqrt(x^2 - 8*x)*(x - 4) + 8*log(-x + sqrt(x^2 - 8*x) + 4)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {x^{2} - 8 x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2-8*x)**(1/2),x)

[Out]

Integral(sqrt(x**2 - 8*x), x)

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Giac [A]
time = 1.84, size = 33, normalized size = 0.89 \begin {gather*} \frac {1}{2} \, \sqrt {x^{2} - 8 \, x} {\left (x - 4\right )} + 8 \, \log \left ({\left | -x + \sqrt {x^{2} - 8 \, x} + 4 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-8*x)^(1/2),x, algorithm="giac")

[Out]

1/2*sqrt(x^2 - 8*x)*(x - 4) + 8*log(abs(-x + sqrt(x^2 - 8*x) + 4))

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Mupad [B]
time = 0.11, size = 29, normalized size = 0.78 \begin {gather*} \left (\frac {x}{2}-2\right )\,\sqrt {x^2-8\,x}-8\,\ln \left (x+\sqrt {x\,\left (x-8\right )}-4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 - 8*x)^(1/2),x)

[Out]

(x/2 - 2)*(x^2 - 8*x)^(1/2) - 8*log(x + (x*(x - 8))^(1/2) - 4)

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