Optimal. Leaf size=115 \[ \frac {12 x^{13/12}}{13}-\frac {6 x^{5/6}}{5}+\frac {4 x^{3/4}}{3}+\frac {12 x^{7/12}}{7}-2 \sqrt {x}-3 \sqrt [3]{x}+4 \sqrt [4]{x}+12 \sqrt [12]{x}-8 \log \left (\sqrt [12]{x}+1\right )-2 \log \left (\sqrt [6]{x}-\sqrt [12]{x}+1\right )+4 \sqrt {3} \tan ^{-1}\left (\frac {1-2 \sqrt [12]{x}}{\sqrt {3}}\right ) \]
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Rubi [A] time = 0.15, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {1593, 1836, 1887, 1874, 31, 634, 618, 204, 628} \[ \frac {12 x^{13/12}}{13}-\frac {6 x^{5/6}}{5}+\frac {4 x^{3/4}}{3}+\frac {12 x^{7/12}}{7}-2 \sqrt {x}-3 \sqrt [3]{x}+4 \sqrt [4]{x}+12 \sqrt [12]{x}-8 \log \left (\sqrt [12]{x}+1\right )-2 \log \left (\sqrt [6]{x}-\sqrt [12]{x}+1\right )+4 \sqrt {3} \tan ^{-1}\left (\frac {1-2 \sqrt [12]{x}}{\sqrt {3}}\right ) \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 618
Rule 628
Rule 634
Rule 1593
Rule 1836
Rule 1874
Rule 1887
Rubi steps
\begin {align*} \int \frac {1+\sqrt [3]{x}}{1+\sqrt [4]{x}} \, dx &=12 \operatorname {Subst}\left (\int \frac {x^{11}+x^{15}}{1+x^3} \, dx,x,\sqrt [12]{x}\right )\\ &=12 \operatorname {Subst}\left (\int \frac {x^{11} \left (1+x^4\right )}{1+x^3} \, dx,x,\sqrt [12]{x}\right )\\ &=\frac {12 x^{13/12}}{13}+\frac {12}{13} \operatorname {Subst}\left (\int \frac {(13-13 x) x^{11}}{1+x^3} \, dx,x,\sqrt [12]{x}\right )\\ &=\frac {12 x^{13/12}}{13}+\frac {12}{13} \operatorname {Subst}\left (\int \left (13+13 x^2-13 x^3-13 x^5+13 x^6+13 x^8-13 x^9-\frac {13 \left (1+x^2\right )}{1+x^3}\right ) \, dx,x,\sqrt [12]{x}\right )\\ &=12 \sqrt [12]{x}+4 \sqrt [4]{x}-3 \sqrt [3]{x}-2 \sqrt {x}+\frac {12 x^{7/12}}{7}+\frac {4 x^{3/4}}{3}-\frac {6 x^{5/6}}{5}+\frac {12 x^{13/12}}{13}-12 \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^3} \, dx,x,\sqrt [12]{x}\right )\\ &=12 \sqrt [12]{x}+4 \sqrt [4]{x}-3 \sqrt [3]{x}-2 \sqrt {x}+\frac {12 x^{7/12}}{7}+\frac {4 x^{3/4}}{3}-\frac {6 x^{5/6}}{5}+\frac {12 x^{13/12}}{13}-4 \operatorname {Subst}\left (\int \frac {1+x}{1-x+x^2} \, dx,x,\sqrt [12]{x}\right )-8 \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt [12]{x}\right )\\ &=12 \sqrt [12]{x}+4 \sqrt [4]{x}-3 \sqrt [3]{x}-2 \sqrt {x}+\frac {12 x^{7/12}}{7}+\frac {4 x^{3/4}}{3}-\frac {6 x^{5/6}}{5}+\frac {12 x^{13/12}}{13}-8 \log \left (1+\sqrt [12]{x}\right )-2 \operatorname {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,\sqrt [12]{x}\right )-6 \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [12]{x}\right )\\ &=12 \sqrt [12]{x}+4 \sqrt [4]{x}-3 \sqrt [3]{x}-2 \sqrt {x}+\frac {12 x^{7/12}}{7}+\frac {4 x^{3/4}}{3}-\frac {6 x^{5/6}}{5}+\frac {12 x^{13/12}}{13}-8 \log \left (1+\sqrt [12]{x}\right )-2 \log \left (1-\sqrt [12]{x}+\sqrt [6]{x}\right )+12 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [12]{x}\right )\\ &=12 \sqrt [12]{x}+4 \sqrt [4]{x}-3 \sqrt [3]{x}-2 \sqrt {x}+\frac {12 x^{7/12}}{7}+\frac {4 x^{3/4}}{3}-\frac {6 x^{5/6}}{5}+\frac {12 x^{13/12}}{13}+4 \sqrt {3} \tan ^{-1}\left (\frac {1-2 \sqrt [12]{x}}{\sqrt {3}}\right )-8 \log \left (1+\sqrt [12]{x}\right )-2 \log \left (1-\sqrt [12]{x}+\sqrt [6]{x}\right )\\ \end {align*}
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Mathematica [A] time = 0.09, size = 123, normalized size = 1.07 \[ \frac {12 x^{13/12}}{13}-\frac {6 x^{5/6}}{5}+\frac {4 x^{3/4}}{3}+\frac {12 x^{7/12}}{7}-2 \sqrt {x}-3 \sqrt [3]{x}+4 \sqrt [4]{x}+12 \sqrt [12]{x}+4 \left (\sqrt [3]{-1}-1\right ) \log \left (\sqrt [3]{-1}-\sqrt [12]{x}\right )-4 \left (1+(-1)^{2/3}\right ) \log \left (-\sqrt [12]{x}-(-1)^{2/3}\right )-8 \log \left (\sqrt [12]{x}+1\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 80, normalized size = 0.70 \[ -4 \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} x^{\frac {1}{12}} - \frac {1}{3} \, \sqrt {3}\right ) + \frac {12}{13} \, {\left (x + 13\right )} x^{\frac {1}{12}} - \frac {6}{5} \, x^{\frac {5}{6}} + \frac {4}{3} \, x^{\frac {3}{4}} + \frac {12}{7} \, x^{\frac {7}{12}} - 2 \, \sqrt {x} - 3 \, x^{\frac {1}{3}} + 4 \, x^{\frac {1}{4}} - 2 \, \log \left (x^{\frac {1}{6}} - x^{\frac {1}{12}} + 1\right ) - 8 \, \log \left (x^{\frac {1}{12}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.43, size = 80, normalized size = 0.70 \[ -4 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{\frac {1}{12}} - 1\right )}\right ) + \frac {12}{13} \, x^{\frac {13}{12}} - \frac {6}{5} \, x^{\frac {5}{6}} + \frac {4}{3} \, x^{\frac {3}{4}} + \frac {12}{7} \, x^{\frac {7}{12}} - 2 \, \sqrt {x} - 3 \, x^{\frac {1}{3}} + 4 \, x^{\frac {1}{4}} + 12 \, x^{\frac {1}{12}} - 2 \, \log \left (x^{\frac {1}{6}} - x^{\frac {1}{12}} + 1\right ) - 8 \, \log \left (x^{\frac {1}{12}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 81, normalized size = 0.70 \[ \frac {12 x^{\frac {13}{12}}}{13}-4 \sqrt {3}\, \arctan \left (\frac {\left (2 x^{\frac {1}{12}}-1\right ) \sqrt {3}}{3}\right )-8 \ln \left (x^{\frac {1}{12}}+1\right )-2 \ln \left (x^{\frac {1}{6}}-x^{\frac {1}{12}}+1\right )-\frac {6 x^{\frac {5}{6}}}{5}+\frac {4 x^{\frac {3}{4}}}{3}+\frac {12 x^{\frac {7}{12}}}{7}-2 \sqrt {x}-3 x^{\frac {1}{3}}+4 x^{\frac {1}{4}}+12 x^{\frac {1}{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 80, normalized size = 0.70 \[ -4 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{\frac {1}{12}} - 1\right )}\right ) + \frac {12}{13} \, x^{\frac {13}{12}} - \frac {6}{5} \, x^{\frac {5}{6}} + \frac {4}{3} \, x^{\frac {3}{4}} + \frac {12}{7} \, x^{\frac {7}{12}} - 2 \, \sqrt {x} - 3 \, x^{\frac {1}{3}} + 4 \, x^{\frac {1}{4}} + 12 \, x^{\frac {1}{12}} - 2 \, \log \left (x^{\frac {1}{6}} - x^{\frac {1}{12}} + 1\right ) - 8 \, \log \left (x^{\frac {1}{12}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 130, normalized size = 1.13 \[ 4\,x^{1/4}+\ln \left (\left (-2+\sqrt {3}\,2{}\mathrm {i}\right )\,\left (54-36\,x^{1/12}+\sqrt {3}\,18{}\mathrm {i}\right )-144\,x^{1/12}+144\right )\,\left (-2+\sqrt {3}\,2{}\mathrm {i}\right )-\ln \left (\left (2+\sqrt {3}\,2{}\mathrm {i}\right )\,\left (36\,x^{1/12}-54+\sqrt {3}\,18{}\mathrm {i}\right )-144\,x^{1/12}+144\right )\,\left (2+\sqrt {3}\,2{}\mathrm {i}\right )-2\,\sqrt {x}-3\,x^{1/3}-8\,\ln \left (144\,x^{1/12}+144\right )+\frac {4\,x^{3/4}}{3}-\frac {6\,x^{5/6}}{5}+12\,x^{1/12}+\frac {12\,x^{7/12}}{7}+\frac {12\,x^{13/12}}{13} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 5.43, size = 221, normalized size = 1.92 \[ \frac {64 x^{\frac {13}{12}} \Gamma \left (\frac {16}{3}\right )}{13 \Gamma \left (\frac {19}{3}\right )} + \frac {64 x^{\frac {7}{12}} \Gamma \left (\frac {16}{3}\right )}{7 \Gamma \left (\frac {19}{3}\right )} + \frac {64 \sqrt [12]{x} \Gamma \left (\frac {16}{3}\right )}{\Gamma \left (\frac {19}{3}\right )} - \frac {32 x^{\frac {5}{6}} \Gamma \left (\frac {16}{3}\right )}{5 \Gamma \left (\frac {19}{3}\right )} + \frac {4 x^{\frac {3}{4}}}{3} + 4 \sqrt [4]{x} - \frac {16 \sqrt [3]{x} \Gamma \left (\frac {16}{3}\right )}{\Gamma \left (\frac {19}{3}\right )} - 2 \sqrt {x} - 4 \log {\left (\sqrt [4]{x} + 1 \right )} + \frac {64 e^{- \frac {i \pi }{3}} \log {\left (- \sqrt [12]{x} e^{\frac {i \pi }{3}} + 1 \right )} \Gamma \left (\frac {16}{3}\right )}{3 \Gamma \left (\frac {19}{3}\right )} - \frac {64 \log {\left (- \sqrt [12]{x} e^{i \pi } + 1 \right )} \Gamma \left (\frac {16}{3}\right )}{3 \Gamma \left (\frac {19}{3}\right )} + \frac {64 e^{\frac {i \pi }{3}} \log {\left (- \sqrt [12]{x} e^{\frac {5 i \pi }{3}} + 1 \right )} \Gamma \left (\frac {16}{3}\right )}{3 \Gamma \left (\frac {19}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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