3.2.91 \(\int \frac {1}{\sqrt {x^{3/2}-x^2}} \, dx\) [191]

Optimal. Leaf size=20 \[ 4 \arctan \left (\frac {x}{\sqrt {x^{3/2}-x^2}}\right ) \]

[Out]

4*arctan(x/(x^(3/2)-x^2)^(1/2))

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Rubi [A]
time = 0.00, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2033, 209} \begin {gather*} 4 \arctan \left (\frac {x}{\sqrt {x^{3/2}-x^2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[x^(3/2) - x^2],x]

[Out]

4*ArcTan[x/Sqrt[x^(3/2) - x^2]]

Rule 209

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

Rule 2033

Int[1/Sqrt[(a_.)*(x_)^2 + (b_.)*(x_)^(n_.)], x_Symbol] :> Dist[2/(2 - n), Subst[Int[1/(1 - a*x^2), x], x, x/Sq
rt[a*x^2 + b*x^n]], x] /; FreeQ[{a, b, n}, x] && NeQ[n, 2]

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=4 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt {x^{3/2}-x^2}}\right )\\ &=4 \arctan \left (\frac {x}{\sqrt {x^{3/2}-x^2}}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.06, size = 20, normalized size = 1.00 \begin {gather*} 4 \arctan \left (\frac {x}{\sqrt {x^{3/2}-x^2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[x^(3/2) - x^2],x]

[Out]

4*ArcTan[x/Sqrt[x^(3/2) - x^2]]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(45\) vs. \(2(16)=32\).
time = 0.08, size = 46, normalized size = 2.30

method	result	size
meijerg	\(4 \arcsin \left (x^{\frac {1}{4}}\right )\)	\(7\)
derivativedivides	\(\frac {2 \sqrt {x}\, \sqrt {-\sqrt {x}\, \left (\sqrt {x}-1\right )}\, \arcsin \left (2 \sqrt {x}-1\right )}{\sqrt {x^{\frac {3}{2}}-x^{2}}}\)	\(37\)
default	\(\frac {2 x \left (\sqrt {x}-1\right ) \ln \left (\sqrt {x}-\frac {1}{2}+\sqrt {x -\sqrt {x}}\right )}{\sqrt {x^{\frac {3}{2}}-x^{2}}\, \sqrt {\sqrt {x}\, \left (\sqrt {x}-1\right )}}\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(3/2)-x^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^(3/2)-x^2)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(-x^2 + x^(3/2)), x)

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Fricas [A]
time = 0.58, size = 18, normalized size = 0.90 \begin {gather*} -4 \, \arctan \left (\frac {\sqrt {-x^{2} + x^{\frac {3}{2}}}}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^(3/2)-x^2)^(1/2),x, algorithm="fricas")

[Out]

-4*arctan(sqrt(-x^2 + x^(3/2))/x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {x^{\frac {3}{2}} - x^{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x**(3/2)-x**2)**(1/2),x)

[Out]

Integral(1/sqrt(x**(3/2) - x**2), x)

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Giac [A]
time = 0.48, size = 15, normalized size = 0.75 \begin {gather*} {\left (\pi + 2 \, \arcsin \left (2 \, \sqrt {x} - 1\right )\right )} \mathrm {sgn}\left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^(3/2)-x^2)^(1/2),x, algorithm="giac")

[Out]

(pi + 2*arcsin(2*sqrt(x) - 1))*sgn(x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {1}{\sqrt {x^{3/2}-x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(3/2) - x^2)^(1/2),x)

[Out]

int(1/(x^(3/2) - x^2)^(1/2), x)

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Chatgpt [F] Failed to verify
time = 1.00, size = 14, normalized size = 0.70 \begin {gather*} \frac {4 \ln \left (\frac {3-2 \sqrt {x}}{\sqrt {x}}\right )}{3} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

int(1/(x^(3/2)-x^2)^(1/2),x)

[Out]

4/3*ln((3-2*x^(1/2))/x^(1/2))

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