3.830 \(\int \frac{1+\sqrt [3]{x}}{1+\sqrt [4]{x}} \, dx\)

Optimal. Leaf size=115 \[ \frac{12 x^{13/12}}{13}-\frac{6 x^{5/6}}{5}+\frac{4 x^{3/4}}{3}+\frac{12 x^{7/12}}{7}-2 \sqrt{x}-3 \sqrt [3]{x}+4 \sqrt [4]{x}+12 \sqrt [12]{x}-8 \log \left (\sqrt [12]{x}+1\right )-2 \log \left (\sqrt [6]{x}-\sqrt [12]{x}+1\right )+4 \sqrt{3} \tan ^{-1}\left (\frac{1-2 \sqrt [12]{x}}{\sqrt{3}}\right ) \]

[Out]

12*x^(1/12) + 4*x^(1/4) - 3*x^(1/3) - 2*Sqrt[x] + (12*x^(7/12))/7 + (4*x^(3/4))/
3 - (6*x^(5/6))/5 + (12*x^(13/12))/13 + 4*Sqrt[3]*ArcTan[(1 - 2*x^(1/12))/Sqrt[3
]] - 8*Log[1 + x^(1/12)] - 2*Log[1 - x^(1/12) + x^(1/6)]

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Rubi [A]  time = 0.243356, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.529 \[ \frac{12 x^{13/12}}{13}-\frac{6 x^{5/6}}{5}+\frac{4 x^{3/4}}{3}+\frac{12 x^{7/12}}{7}-2 \sqrt{x}-3 \sqrt [3]{x}+4 \sqrt [4]{x}+12 \sqrt [12]{x}-8 \log \left (\sqrt [12]{x}+1\right )-2 \log \left (\sqrt [6]{x}-\sqrt [12]{x}+1\right )+4 \sqrt{3} \tan ^{-1}\left (\frac{1-2 \sqrt [12]{x}}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 + x^(1/3))/(1 + x^(1/4)),x]

[Out]

12*x^(1/12) + 4*x^(1/4) - 3*x^(1/3) - 2*Sqrt[x] + (12*x^(7/12))/7 + (4*x^(3/4))/
3 - (6*x^(5/6))/5 + (12*x^(13/12))/13 + 4*Sqrt[3]*ArcTan[(1 - 2*x^(1/12))/Sqrt[3
]] - 8*Log[1 + x^(1/12)] - 2*Log[1 - x^(1/12) + x^(1/6)]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{12 x^{\frac{13}{12}}}{13} + \frac{12 x^{\frac{7}{12}}}{7} + 12 \sqrt [12]{x} - \frac{6 x^{\frac{5}{6}}}{5} + \frac{4 x^{\frac{3}{4}}}{3} + 4 \sqrt [4]{x} - 3 \sqrt [3]{x} - 6 \log{\left (\sqrt [12]{x} + 1 \right )} - 2 \log{\left (\sqrt [4]{x} + 1 \right )} - 4 \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [12]{x}}{3} - \frac{1}{3}\right ) \right )} - 4 \int ^{\sqrt [4]{x}} x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+x**(1/3))/(1+x**(1/4)),x)

[Out]

12*x**(13/12)/13 + 12*x**(7/12)/7 + 12*x**(1/12) - 6*x**(5/6)/5 + 4*x**(3/4)/3 +
4*x**(1/4) - 3*x**(1/3) - 6*log(x**(1/12) + 1) - 2*log(x**(1/4) + 1) - 4*sqrt(3
)*atan(sqrt(3)*(2*x**(1/12)/3 - 1/3)) - 4*Integral(x, (x, x**(1/4)))

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Mathematica [A]  time = 0.0364198, size = 125, normalized size = 1.09 \[ \frac{12 x^{13/12}}{13}-\frac{6 x^{5/6}}{5}+\frac{4 x^{3/4}}{3}+\frac{12 x^{7/12}}{7}-2 \sqrt{x}-3 \sqrt [3]{x}+4 \sqrt [4]{x}+12 \sqrt [12]{x}-4 \log \left (\sqrt [12]{x}+1\right )+2 \log \left (\sqrt [6]{x}-\sqrt [12]{x}+1\right )-4 \log \left (\sqrt [4]{x}+1\right )-4 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [12]{x}-1}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + x^(1/3))/(1 + x^(1/4)),x]

[Out]

12*x^(1/12) + 4*x^(1/4) - 3*x^(1/3) - 2*Sqrt[x] + (12*x^(7/12))/7 + (4*x^(3/4))/
3 - (6*x^(5/6))/5 + (12*x^(13/12))/13 - 4*Sqrt[3]*ArcTan[(-1 + 2*x^(1/12))/Sqrt[
3]] - 4*Log[1 + x^(1/12)] + 2*Log[1 - x^(1/12) + x^(1/6)] - 4*Log[1 + x^(1/4)]

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Maple [A]  time = 0.01, size = 81, normalized size = 0.7 \[{\frac{12}{13}{x}^{{\frac{13}{12}}}}-{\frac{6}{5}{x}^{{\frac{5}{6}}}}+{\frac{4}{3}{x}^{{\frac{3}{4}}}}+{\frac{12}{7}{x}^{{\frac{7}{12}}}}-2\,\sqrt{x}-3\,\sqrt [3]{x}+4\,\sqrt [4]{x}+12\,{x}^{1/12}-8\,\ln \left ( 1+{x}^{1/12} \right ) -2\,\ln \left ( 1-{x}^{1/12}+\sqrt [6]{x} \right ) -4\,\sqrt{3}\arctan \left ( 1/3\, \left ( 2\,{x}^{1/12}-1 \right ) \sqrt{3} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x^(1/3))/(1+x^(1/4)),x)

[Out]

12/13*x^(13/12)-6/5*x^(5/6)+4/3*x^(3/4)+12/7*x^(7/12)-2*x^(1/2)-3*x^(1/3)+4*x^(1
/4)+12*x^(1/12)-8*ln(1+x^(1/12))-2*ln(1-x^(1/12)+x^(1/6))-4*3^(1/2)*arctan(1/3*(
2*x^(1/12)-1)*3^(1/2))

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Maxima [A]  time = 0.787992, size = 108, normalized size = 0.94 \[ -4 \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{\frac{1}{12}} - 1\right )}\right ) + \frac{12}{13} \, x^{\frac{13}{12}} - \frac{6}{5} \, x^{\frac{5}{6}} + \frac{4}{3} \, x^{\frac{3}{4}} + \frac{12}{7} \, x^{\frac{7}{12}} - 2 \, \sqrt{x} - 3 \, x^{\frac{1}{3}} + 4 \, x^{\frac{1}{4}} + 12 \, x^{\frac{1}{12}} - 2 \, \log \left (x^{\frac{1}{6}} - x^{\frac{1}{12}} + 1\right ) - 8 \, \log \left (x^{\frac{1}{12}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^(1/3) + 1)/(x^(1/4) + 1),x, algorithm="maxima")

[Out]

-4*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^(1/12) - 1)) + 12/13*x^(13/12) - 6/5*x^(5/6)
+ 4/3*x^(3/4) + 12/7*x^(7/12) - 2*sqrt(x) - 3*x^(1/3) + 4*x^(1/4) + 12*x^(1/12)
- 2*log(x^(1/6) - x^(1/12) + 1) - 8*log(x^(1/12) + 1)

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Fricas [A]  time = 0.278469, size = 105, normalized size = 0.91 \[ -4 \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{\frac{1}{12}} - 1\right )}\right ) + \frac{12}{13} \,{\left (x + 13\right )} x^{\frac{1}{12}} - \frac{6}{5} \, x^{\frac{5}{6}} + \frac{4}{3} \, x^{\frac{3}{4}} + \frac{12}{7} \, x^{\frac{7}{12}} - 2 \, \sqrt{x} - 3 \, x^{\frac{1}{3}} + 4 \, x^{\frac{1}{4}} - 2 \, \log \left (x^{\frac{1}{6}} - x^{\frac{1}{12}} + 1\right ) - 8 \, \log \left (x^{\frac{1}{12}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^(1/3) + 1)/(x^(1/4) + 1),x, algorithm="fricas")

[Out]

-4*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^(1/12) - 1)) + 12/13*(x + 13)*x^(1/12) - 6/5*
x^(5/6) + 4/3*x^(3/4) + 12/7*x^(7/12) - 2*sqrt(x) - 3*x^(1/3) + 4*x^(1/4) - 2*lo
g(x^(1/6) - x^(1/12) + 1) - 8*log(x^(1/12) + 1)

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Sympy [A]  time = 12.1051, size = 223, normalized size = 1.94 \[ \frac{64 x^{\frac{13}{12}} \Gamma \left (\frac{16}{3}\right )}{13 \Gamma \left (\frac{19}{3}\right )} + \frac{64 x^{\frac{7}{12}} \Gamma \left (\frac{16}{3}\right )}{7 \Gamma \left (\frac{19}{3}\right )} + \frac{64 \sqrt [12]{x} \Gamma \left (\frac{16}{3}\right )}{\Gamma \left (\frac{19}{3}\right )} - \frac{32 x^{\frac{5}{6}} \Gamma \left (\frac{16}{3}\right )}{5 \Gamma \left (\frac{19}{3}\right )} + \frac{4 x^{\frac{3}{4}}}{3} + 4 \sqrt [4]{x} - \frac{16 \sqrt [3]{x} \Gamma \left (\frac{16}{3}\right )}{\Gamma \left (\frac{19}{3}\right )} - 2 \sqrt{x} - 4 \log{\left (\sqrt [4]{x} + 1 \right )} + \frac{64 e^{\frac{5 i \pi }{3}} \log{\left (- \sqrt [12]{x} e^{\frac{i \pi }{3}} + 1 \right )} \Gamma \left (\frac{16}{3}\right )}{3 \Gamma \left (\frac{19}{3}\right )} - \frac{64 \log{\left (- \sqrt [12]{x} e^{i \pi } + 1 \right )} \Gamma \left (\frac{16}{3}\right )}{3 \Gamma \left (\frac{19}{3}\right )} + \frac{64 e^{\frac{i \pi }{3}} \log{\left (- \sqrt [12]{x} e^{\frac{5 i \pi }{3}} + 1 \right )} \Gamma \left (\frac{16}{3}\right )}{3 \Gamma \left (\frac{19}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+x**(1/3))/(1+x**(1/4)),x)

[Out]

64*x**(13/12)*gamma(16/3)/(13*gamma(19/3)) + 64*x**(7/12)*gamma(16/3)/(7*gamma(1
9/3)) + 64*x**(1/12)*gamma(16/3)/gamma(19/3) - 32*x**(5/6)*gamma(16/3)/(5*gamma(
19/3)) + 4*x**(3/4)/3 + 4*x**(1/4) - 16*x**(1/3)*gamma(16/3)/gamma(19/3) - 2*sqr
t(x) - 4*log(x**(1/4) + 1) + 64*exp(5*I*pi/3)*log(-x**(1/12)*exp_polar(I*pi/3) +
1)*gamma(16/3)/(3*gamma(19/3)) - 64*log(-x**(1/12)*exp_polar(I*pi) + 1)*gamma(1
6/3)/(3*gamma(19/3)) + 64*exp(I*pi/3)*log(-x**(1/12)*exp_polar(5*I*pi/3) + 1)*ga
mma(16/3)/(3*gamma(19/3))

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GIAC/XCAS [A]  time = 0.271114, size = 108, normalized size = 0.94 \[ -4 \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{\frac{1}{12}} - 1\right )}\right ) + \frac{12}{13} \, x^{\frac{13}{12}} - \frac{6}{5} \, x^{\frac{5}{6}} + \frac{4}{3} \, x^{\frac{3}{4}} + \frac{12}{7} \, x^{\frac{7}{12}} - 2 \, \sqrt{x} - 3 \, x^{\frac{1}{3}} + 4 \, x^{\frac{1}{4}} + 12 \, x^{\frac{1}{12}} - 2 \,{\rm ln}\left (x^{\frac{1}{6}} - x^{\frac{1}{12}} + 1\right ) - 8 \,{\rm ln}\left (x^{\frac{1}{12}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^(1/3) + 1)/(x^(1/4) + 1),x, algorithm="giac")

[Out]

-4*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^(1/12) - 1)) + 12/13*x^(13/12) - 6/5*x^(5/6)
+ 4/3*x^(3/4) + 12/7*x^(7/12) - 2*sqrt(x) - 3*x^(1/3) + 4*x^(1/4) + 12*x^(1/12)
- 2*ln(x^(1/6) - x^(1/12) + 1) - 8*ln(x^(1/12) + 1)