Optimal. Leaf size=130 \[ \frac {4 x^{5/4}}{5}-\frac {6 x^{7/6}}{7}+\frac {12 x^{13/12}}{13}+\frac {12 x^{11/12}}{11}-\frac {6 x^{5/6}}{5}+\frac {4 x^{3/4}}{3}-\frac {3 x^{2/3}}{2}+\frac {12 x^{7/12}}{7}+\frac {12 x^{5/12}}{5}-x-2 \sqrt {x}-3 \sqrt [3]{x}+4 \sqrt [4]{x}-6 \sqrt [6]{x}+12 \sqrt [12]{x}-12 \log \left (\sqrt [12]{x}+1\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 130, 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 = {1593, 266, 43} \[ \frac {4 x^{5/4}}{5}-\frac {6 x^{7/6}}{7}+\frac {12 x^{13/12}}{13}+\frac {12 x^{11/12}}{11}-\frac {6 x^{5/6}}{5}+\frac {4 x^{3/4}}{3}-\frac {3 x^{2/3}}{2}+\frac {12 x^{7/12}}{7}+\frac {12 x^{5/12}}{5}-x-2 \sqrt {x}-3 \sqrt [3]{x}+4 \sqrt [4]{x}-6 \sqrt [6]{x}+12 \sqrt [12]{x}-12 \log \left (\sqrt [12]{x}+1\right ) \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 1593
Rubi steps
\begin {align*} \int \frac {1}{\frac {1}{\sqrt [3]{x}}+\frac {1}{\sqrt [4]{x}}} \, dx &=\int \frac {\sqrt [3]{x}}{1+\sqrt [12]{x}} \, dx\\ &=12 \operatorname {Subst}\left (\int \frac {x^{15}}{1+x} \, dx,x,\sqrt [12]{x}\right )\\ &=12 \operatorname {Subst}\left (\int \left (1+\frac {1}{-1-x}-x+x^2-x^3+x^4-x^5+x^6-x^7+x^8-x^9+x^{10}-x^{11}+x^{12}-x^{13}+x^{14}\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}+\frac {4 x^{3/4}}{3}-\frac {6 x^{5/6}}{5}+\frac {12 x^{11/12}}{11}-x+\frac {12 x^{13/12}}{13}-\frac {6 x^{7/6}}{7}+\frac {4 x^{5/4}}{5}-12 \log \left (1+\sqrt [12]{x}\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 130, normalized size = 1.00 \[ \frac {4 x^{5/4}}{5}-\frac {6 x^{7/6}}{7}+\frac {12 x^{13/12}}{13}+\frac {12 x^{11/12}}{11}-\frac {6 x^{5/6}}{5}+\frac {4 x^{3/4}}{3}-\frac {3 x^{2/3}}{2}+\frac {12 x^{7/12}}{7}+\frac {12 x^{5/12}}{5}-x-2 \sqrt {x}-3 \sqrt [3]{x}+4 \sqrt [4]{x}-6 \sqrt [6]{x}+12 \sqrt [12]{x}-12 \log \left (\sqrt [12]{x}+1\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 76, normalized size = 0.58 \[ \frac {4}{5} \, {\left (x + 5\right )} x^{\frac {1}{4}} - \frac {6}{7} \, {\left (x + 7\right )} x^{\frac {1}{6}} + \frac {12}{13} \, {\left (x + 13\right )} x^{\frac {1}{12}} - x + \frac {12}{11} \, x^{\frac {11}{12}} - \frac {6}{5} \, x^{\frac {5}{6}} + \frac {4}{3} \, x^{\frac {3}{4}} - \frac {3}{2} \, x^{\frac {2}{3}} + \frac {12}{7} \, x^{\frac {7}{12}} - 2 \, \sqrt {x} + \frac {12}{5} \, x^{\frac {5}{12}} - 3 \, x^{\frac {1}{3}} - 12 \, \log \left (x^{\frac {1}{12}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.03, size = 82, normalized size = 0.63 \[ \frac {4}{5} \, x^{\frac {5}{4}} - \frac {6}{7} \, x^{\frac {7}{6}} + \frac {12}{13} \, x^{\frac {13}{12}} - x + \frac {12}{11} \, x^{\frac {11}{12}} - \frac {6}{5} \, x^{\frac {5}{6}} + \frac {4}{3} \, x^{\frac {3}{4}} - \frac {3}{2} \, x^{\frac {2}{3}} + \frac {12}{7} \, x^{\frac {7}{12}} - 2 \, \sqrt {x} + \frac {12}{5} \, x^{\frac {5}{12}} - 3 \, x^{\frac {1}{3}} + 4 \, x^{\frac {1}{4}} - 6 \, x^{\frac {1}{6}} + 12 \, x^{\frac {1}{12}} - 12 \, \log \left (x^{\frac {1}{12}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 83, normalized size = 0.64 \[ \frac {4 x^{\frac {5}{4}}}{5}-\frac {6 x^{\frac {7}{6}}}{7}+\frac {12 x^{\frac {13}{12}}}{13}-x -12 \ln \left (x^{\frac {1}{12}}+1\right )+\frac {12 x^{\frac {11}{12}}}{11}-\frac {6 x^{\frac {5}{6}}}{5}+\frac {4 x^{\frac {3}{4}}}{3}-\frac {3 x^{\frac {2}{3}}}{2}+\frac {12 x^{\frac {7}{12}}}{7}-2 \sqrt {x}+\frac {12 x^{\frac {5}{12}}}{5}-3 x^{\frac {1}{3}}+4 x^{\frac {1}{4}}-6 x^{\frac {1}{6}}+12 x^{\frac {1}{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 82, normalized size = 0.63 \[ \frac {4}{5} \, x^{\frac {5}{4}} - \frac {6}{7} \, x^{\frac {7}{6}} + \frac {12}{13} \, x^{\frac {13}{12}} - x + \frac {12}{11} \, x^{\frac {11}{12}} - \frac {6}{5} \, x^{\frac {5}{6}} + \frac {4}{3} \, x^{\frac {3}{4}} - \frac {3}{2} \, x^{\frac {2}{3}} + \frac {12}{7} \, x^{\frac {7}{12}} - 2 \, \sqrt {x} + \frac {12}{5} \, x^{\frac {5}{12}} - 3 \, x^{\frac {1}{3}} + 4 \, x^{\frac {1}{4}} - 6 \, x^{\frac {1}{6}} + 12 \, x^{\frac {1}{12}} - 12 \, \log \left (x^{\frac {1}{12}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 82, normalized size = 0.63 \[ 4\,x^{1/4}-12\,\ln \left (x^{1/12}+1\right )-2\,\sqrt {x}-3\,x^{1/3}-x-\frac {3\,x^{2/3}}{2}-6\,x^{1/6}+\frac {4\,x^{3/4}}{3}+\frac {4\,x^{5/4}}{5}-\frac {6\,x^{5/6}}{5}+12\,x^{1/12}-\frac {6\,x^{7/6}}{7}+\frac {12\,x^{5/12}}{5}+\frac {12\,x^{7/12}}{7}+\frac {12\,x^{11/12}}{11}+\frac {12\,x^{13/12}}{13} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.17, size = 121, normalized size = 0.93 \[ \frac {12 x^{\frac {13}{12}}}{13} + \frac {12 x^{\frac {11}{12}}}{11} + \frac {12 x^{\frac {7}{12}}}{7} + \frac {12 x^{\frac {5}{12}}}{5} + 12 \sqrt [12]{x} - \frac {6 x^{\frac {7}{6}}}{7} - \frac {6 x^{\frac {5}{6}}}{5} - 6 \sqrt [6]{x} + \frac {4 x^{\frac {5}{4}}}{5} + \frac {4 x^{\frac {3}{4}}}{3} + 4 \sqrt [4]{x} - \frac {3 x^{\frac {2}{3}}}{2} - 3 \sqrt [3]{x} - 2 \sqrt {x} - x - 12 \log {\left (\sqrt [12]{x} + 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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