Optimal. Leaf size=27 \[ 4 \sqrt [4]{x}+2 \sqrt {x}+4 \log \left (1-\sqrt [4]{x}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1607, 272, 45}
\begin {gather*} 2 \sqrt {x}+4 \sqrt [4]{x}+4 \log \left (1-\sqrt [4]{x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rule 1607
Rubi steps
\begin {align*} \int \frac {1}{-\sqrt [4]{x}+\sqrt {x}} \, dx &=\int \frac {1}{\left (-1+\sqrt [4]{x}\right ) \sqrt [4]{x}} \, dx\\ &=4 \text {Subst}\left (\int \frac {x^2}{-1+x} \, dx,x,\sqrt [4]{x}\right )\\ &=4 \text {Subst}\left (\int \left (1+\frac {1}{-1+x}+x\right ) \, dx,x,\sqrt [4]{x}\right )\\ &=4 \sqrt [4]{x}+2 \sqrt {x}+4 \log \left (1-\sqrt [4]{x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 25, normalized size = 0.93 \begin {gather*} 2 \left (2+\sqrt [4]{x}\right ) \sqrt [4]{x}+4 \log \left (-1+\sqrt [4]{x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 20, normalized size = 0.74
method | result | size |
derivativedivides | \(2 \sqrt {x}+4 x^{\frac {1}{4}}+4 \ln \left (-1+x^{\frac {1}{4}}\right )\) | \(20\) |
default | \(2 \sqrt {x}+4 x^{\frac {1}{4}}+4 \ln \left (-1+x^{\frac {1}{4}}\right )\) | \(20\) |
meijerg | \(\frac {2 x^{\frac {1}{4}} \left (3 x^{\frac {1}{4}}+6\right )}{3}+4 \ln \left (1-x^{\frac {1}{4}}\right )\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 19, normalized size = 0.70 \begin {gather*} 2 \, \sqrt {x} + 4 \, x^{\frac {1}{4}} + 4 \, \log \left (x^{\frac {1}{4}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 19, normalized size = 0.70 \begin {gather*} 2 \, \sqrt {x} + 4 \, x^{\frac {1}{4}} + 4 \, \log \left (x^{\frac {1}{4}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 22, normalized size = 0.81 \begin {gather*} 4 \sqrt [4]{x} + 2 \sqrt {x} + 4 \log {\left (\sqrt [4]{x} - 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.44, size = 20, normalized size = 0.74 \begin {gather*} 2 \, \sqrt {x} + 4 \, x^{\frac {1}{4}} + 4 \, \log \left ({\left | x^{\frac {1}{4}} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.10, size = 19, normalized size = 0.70 \begin {gather*} 4\,\ln \left (x^{1/4}-1\right )+2\,\sqrt {x}+4\,x^{1/4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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