Optimal. Leaf size=73 \[ -12 \sqrt [12]{x}+6 \sqrt [6]{x}-4 \sqrt [4]{x}+3 \sqrt [3]{x}-\frac {12 x^{5/12}}{5}+2 \sqrt {x}-\frac {12 x^{7/12}}{7}+\frac {3 x^{2/3}}{2}+12 \log \left (1+\sqrt [12]{x}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1607, 272, 45}
\begin {gather*} \frac {3 x^{2/3}}{2}-\frac {12 x^{7/12}}{7}-\frac {12 x^{5/12}}{5}+2 \sqrt {x}+3 \sqrt [3]{x}-4 \sqrt [4]{x}+6 \sqrt [6]{x}-12 \sqrt [12]{x}+12 \log \left (\sqrt [12]{x}+1\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 [3]{x}} \, dx &=\int \frac {1}{\left (1+\sqrt [12]{x}\right ) \sqrt [4]{x}} \, dx\\ &=12 \text {Subst}\left (\int \frac {x^8}{1+x} \, dx,x,\sqrt [12]{x}\right )\\ &=12 \text {Subst}\left (\int \left (-1+x-x^2+x^3-x^4+x^5-x^6+x^7+\frac {1}{1+x}\right ) \, dx,x,\sqrt [12]{x}\right )\\ &=-12 \sqrt [12]{x}+6 \sqrt [6]{x}-4 \sqrt [4]{x}+3 \sqrt [3]{x}-\frac {12 x^{5/12}}{5}+2 \sqrt {x}-\frac {12 x^{7/12}}{7}+\frac {3 x^{2/3}}{2}+12 \log \left (1+\sqrt [12]{x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 72, normalized size = 0.99 \begin {gather*} \frac {1}{70} \left (-840 \sqrt [12]{x}+420 \sqrt [6]{x}-280 \sqrt [4]{x}+210 \sqrt [3]{x}-168 x^{5/12}+140 \sqrt {x}-120 x^{7/12}+105 x^{2/3}\right )+12 \log \left (1+\sqrt [12]{x}\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. \(172\) vs.
\(2(49)=98\).
time = 0.21, size = 173, normalized size = 2.37
method | result | size |
derivativedivides | \(-12 x^{\frac {1}{12}}+6 x^{\frac {1}{6}}-4 x^{\frac {1}{4}}+3 x^{\frac {1}{3}}-\frac {12 x^{\frac {5}{12}}}{5}-\frac {12 x^{\frac {7}{12}}}{7}+\frac {3 x^{\frac {2}{3}}}{2}+12 \ln \left (1+x^{\frac {1}{12}}\right )+2 \sqrt {x}\) | \(50\) |
meijerg | \(-\frac {x^{\frac {1}{12}} \left (-315 x^{\frac {7}{12}}+360 \sqrt {x}-420 x^{\frac {5}{12}}+504 x^{\frac {1}{3}}-630 x^{\frac {1}{4}}+840 x^{\frac {1}{6}}-1260 x^{\frac {1}{12}}+2520\right )}{210}+12 \ln \left (1+x^{\frac {1}{12}}\right )\) | \(52\) |
default | \(-4 \ln \left (-1+x^{\frac {1}{12}}\right )+\frac {3 x^{\frac {2}{3}}}{2}-4 x^{\frac {1}{4}}+6 x^{\frac {1}{6}}+\ln \left (-1+x \right )-12 x^{\frac {1}{12}}-\frac {12 x^{\frac {5}{12}}}{5}-\frac {12 x^{\frac {7}{12}}}{7}-\ln \left (x^{\frac {2}{3}}+x^{\frac {1}{3}}+1\right )+2 \sqrt {x}+3 x^{\frac {1}{3}}+2 \ln \left (-1+x^{\frac {1}{6}}\right )-\ln \left (x^{\frac {1}{3}}+x^{\frac {1}{6}}+1\right )+\ln \left (-1+\sqrt {x}\right )-\ln \left (1+\sqrt {x}\right )+4 \ln \left (1+x^{\frac {1}{12}}\right )+2 \ln \left (x^{\frac {1}{6}}+x^{\frac {1}{12}}+1\right )-2 \ln \left (-1+x^{\frac {1}{4}}\right )+2 \ln \left (-1+x^{\frac {1}{3}}\right )-2 \ln \left (1+x^{\frac {1}{6}}\right )+2 \ln \left (1+x^{\frac {1}{4}}\right )+\ln \left (1-x^{\frac {1}{6}}+x^{\frac {1}{3}}\right )-2 \ln \left (1-x^{\frac {1}{12}}+x^{\frac {1}{6}}\right )\) | \(173\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 49, normalized size = 0.67 \begin {gather*} \frac {3}{2} \, x^{\frac {2}{3}} - \frac {12}{7} \, x^{\frac {7}{12}} + 2 \, \sqrt {x} - \frac {12}{5} \, x^{\frac {5}{12}} + 3 \, x^{\frac {1}{3}} - 4 \, x^{\frac {1}{4}} + 6 \, x^{\frac {1}{6}} - 12 \, x^{\frac {1}{12}} + 12 \, \log \left (x^{\frac {1}{12}} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 49, normalized size = 0.67 \begin {gather*} \frac {3}{2} \, x^{\frac {2}{3}} - \frac {12}{7} \, x^{\frac {7}{12}} + 2 \, \sqrt {x} - \frac {12}{5} \, x^{\frac {5}{12}} + 3 \, x^{\frac {1}{3}} - 4 \, x^{\frac {1}{4}} + 6 \, x^{\frac {1}{6}} - 12 \, x^{\frac {1}{12}} + 12 \, \log \left (x^{\frac {1}{12}} + 1\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 [4]{x} + \sqrt [3]{x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.36, size = 49, normalized size = 0.67 \begin {gather*} \frac {3}{2} \, x^{\frac {2}{3}} - \frac {12}{7} \, x^{\frac {7}{12}} + 2 \, \sqrt {x} - \frac {12}{5} \, x^{\frac {5}{12}} + 3 \, x^{\frac {1}{3}} - 4 \, x^{\frac {1}{4}} + 6 \, x^{\frac {1}{6}} - 12 \, x^{\frac {1}{12}} + 12 \, \log \left (x^{\frac {1}{12}} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 49, normalized size = 0.67 \begin {gather*} 12\,\ln \left (x^{1/12}+1\right )+2\,\sqrt {x}+3\,x^{1/3}-4\,x^{1/4}+\frac {3\,x^{2/3}}{2}+6\,x^{1/6}-12\,x^{1/12}-\frac {12\,x^{5/12}}{5}-\frac {12\,x^{7/12}}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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