Optimal. Leaf size=20 \[ 4 \arctan \left (\frac {x}{\sqrt {x^{3/2}-x^2}}\right ) \]
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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.
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Rule 209
Rule 2033
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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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