3.302 \(\int \frac{\left (1+x^7\right )^{2/3}}{x^8} \, dx\)

Optimal. Leaf size=70 \[ -\frac{\left (x^7+1\right )^{2/3}}{7 x^7}+\frac{1}{7} \log \left (1-\sqrt [3]{x^7+1}\right )+\frac{2 \tan ^{-1}\left (\frac{2 \sqrt [3]{x^7+1}+1}{\sqrt{3}}\right )}{7 \sqrt{3}}-\frac{\log (x)}{3} \]

[Out]

-(1 + x^7)^(2/3)/(7*x^7) + (2*ArcTan[(1 + 2*(1 + x^7)^(1/3))/Sqrt[3]])/(7*Sqrt[3
]) - Log[x]/3 + Log[1 - (1 + x^7)^(1/3)]/7

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Rubi [A]  time = 0.0784704, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462 \[ -\frac{\left (x^7+1\right )^{2/3}}{7 x^7}+\frac{1}{7} \log \left (1-\sqrt [3]{x^7+1}\right )+\frac{2 \tan ^{-1}\left (\frac{2 \sqrt [3]{x^7+1}+1}{\sqrt{3}}\right )}{7 \sqrt{3}}-\frac{\log (x)}{3} \]

Antiderivative was successfully verified.

[In]  Int[(1 + x^7)^(2/3)/x^8,x]

[Out]

-(1 + x^7)^(2/3)/(7*x^7) + (2*ArcTan[(1 + 2*(1 + x^7)^(1/3))/Sqrt[3]])/(7*Sqrt[3
]) - Log[x]/3 + Log[1 - (1 + x^7)^(1/3)]/7

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Rubi in Sympy [A]  time = 2.61101, size = 63, normalized size = 0.9 \[ - \frac{\log{\left (x^{7} \right )}}{21} + \frac{\log{\left (- \sqrt [3]{x^{7} + 1} + 1 \right )}}{7} + \frac{2 \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{x^{7} + 1}}{3} + \frac{1}{3}\right ) \right )}}{21} - \frac{\left (x^{7} + 1\right )^{\frac{2}{3}}}{7 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**7+1)**(2/3)/x**8,x)

[Out]

-log(x**7)/21 + log(-(x**7 + 1)**(1/3) + 1)/7 + 2*sqrt(3)*atan(sqrt(3)*(2*(x**7
+ 1)**(1/3)/3 + 1/3))/21 - (x**7 + 1)**(2/3)/(7*x**7)

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Mathematica [C]  time = 0.0223726, size = 54, normalized size = 0.77 \[ -\frac{2 \sqrt [3]{\frac{1}{x^7}+1} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};-\frac{1}{x^7}\right )}{7 \sqrt [3]{x^7+1}}-\frac{\left (x^7+1\right )^{2/3}}{7 x^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + x^7)^(2/3)/x^8,x]

[Out]

-(1 + x^7)^(2/3)/(7*x^7) - (2*(1 + x^(-7))^(1/3)*Hypergeometric2F1[1/3, 1/3, 4/3
, -x^(-7)])/(7*(1 + x^7)^(1/3))

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Maple [C]  time = 0.064, size = 76, normalized size = 1.1 \[ -{\frac{1}{7\,{x}^{7}} \left ({x}^{7}+1 \right ) ^{{\frac{2}{3}}}}+{\frac{\sqrt{3}\Gamma \left ({\frac{2}{3}} \right ) }{21\,\pi } \left ({\frac{2\,\pi \,\sqrt{3}}{3\,\Gamma \left ( 2/3 \right ) } \left ( -{\frac{\pi \,\sqrt{3}}{6}}-{\frac{3\,\ln \left ( 3 \right ) }{2}}+7\,\ln \left ( x \right ) \right ) }-{\frac{2\,\pi \,\sqrt{3}{x}^{7}}{9\,\Gamma \left ( 2/3 \right ) }{\mbox{$_3$F$_2$}(1,1,{\frac{4}{3}};\,2,2;\,-{x}^{7})}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^7+1)^(2/3)/x^8,x)

[Out]

-1/7*(x^7+1)^(2/3)/x^7+1/21/Pi*3^(1/2)*GAMMA(2/3)*(2/3*(-1/6*Pi*3^(1/2)-3/2*ln(3
)+7*ln(x))*Pi*3^(1/2)/GAMMA(2/3)-2/9*Pi*3^(1/2)/GAMMA(2/3)*x^7*hypergeom([1,1,4/
3],[2,2],-x^7))

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Maxima [A]  time = 1.6205, size = 89, normalized size = 1.27 \[ \frac{2}{21} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \,{\left (x^{7} + 1\right )}^{\frac{1}{3}} + 1\right )}\right ) - \frac{{\left (x^{7} + 1\right )}^{\frac{2}{3}}}{7 \, x^{7}} - \frac{1}{21} \, \log \left ({\left (x^{7} + 1\right )}^{\frac{2}{3}} +{\left (x^{7} + 1\right )}^{\frac{1}{3}} + 1\right ) + \frac{2}{21} \, \log \left ({\left (x^{7} + 1\right )}^{\frac{1}{3}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^7 + 1)^(2/3)/x^8,x, algorithm="maxima")

[Out]

2/21*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^7 + 1)^(1/3) + 1)) - 1/7*(x^7 + 1)^(2/3)/x
^7 - 1/21*log((x^7 + 1)^(2/3) + (x^7 + 1)^(1/3) + 1) + 2/21*log((x^7 + 1)^(1/3)
- 1)

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Fricas [A]  time = 0.221927, size = 117, normalized size = 1.67 \[ -\frac{\sqrt{3}{\left (\sqrt{3} x^{7} \log \left ({\left (x^{7} + 1\right )}^{\frac{2}{3}} +{\left (x^{7} + 1\right )}^{\frac{1}{3}} + 1\right ) - 2 \, \sqrt{3} x^{7} \log \left ({\left (x^{7} + 1\right )}^{\frac{1}{3}} - 1\right ) - 6 \, x^{7} \arctan \left (\frac{2}{3} \, \sqrt{3}{\left (x^{7} + 1\right )}^{\frac{1}{3}} + \frac{1}{3} \, \sqrt{3}\right ) + 3 \, \sqrt{3}{\left (x^{7} + 1\right )}^{\frac{2}{3}}\right )}}{63 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^7 + 1)^(2/3)/x^8,x, algorithm="fricas")

[Out]

-1/63*sqrt(3)*(sqrt(3)*x^7*log((x^7 + 1)^(2/3) + (x^7 + 1)^(1/3) + 1) - 2*sqrt(3
)*x^7*log((x^7 + 1)^(1/3) - 1) - 6*x^7*arctan(2/3*sqrt(3)*(x^7 + 1)^(1/3) + 1/3*
sqrt(3)) + 3*sqrt(3)*(x^7 + 1)^(2/3))/x^7

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Sympy [A]  time = 3.18736, size = 34, normalized size = 0.49 \[ - \frac{\Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{\frac{e^{i \pi }}{x^{7}}} \right )}}{7 x^{\frac{7}{3}} \Gamma \left (\frac{4}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**7+1)**(2/3)/x**8,x)

[Out]

-gamma(1/3)*hyper((-2/3, 1/3), (4/3,), exp_polar(I*pi)/x**7)/(7*x**(7/3)*gamma(4
/3))

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GIAC/XCAS [A]  time = 0.207936, size = 90, normalized size = 1.29 \[ \frac{2}{21} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \,{\left (x^{7} + 1\right )}^{\frac{1}{3}} + 1\right )}\right ) - \frac{{\left (x^{7} + 1\right )}^{\frac{2}{3}}}{7 \, x^{7}} - \frac{1}{21} \,{\rm ln}\left ({\left (x^{7} + 1\right )}^{\frac{2}{3}} +{\left (x^{7} + 1\right )}^{\frac{1}{3}} + 1\right ) + \frac{2}{21} \,{\rm ln}\left ({\left |{\left (x^{7} + 1\right )}^{\frac{1}{3}} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^7 + 1)^(2/3)/x^8,x, algorithm="giac")

[Out]

2/21*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^7 + 1)^(1/3) + 1)) - 1/7*(x^7 + 1)^(2/3)/x
^7 - 1/21*ln((x^7 + 1)^(2/3) + (x^7 + 1)^(1/3) + 1) + 2/21*ln(abs((x^7 + 1)^(1/3
) - 1))