3.709 \(\int \frac{1}{(1+x) \sqrt [6]{1+x^2}} \, dx\)

Optimal. Leaf size=203 \[ x F_1\left (\frac{1}{2};1,\frac{1}{6};\frac{3}{2};x^2,-x^2\right )+\frac{\log \left (\sqrt [3]{x^2+1}-\sqrt [6]{2} \sqrt [6]{x^2+1}+\sqrt [3]{2}\right )}{4 \sqrt [6]{2}}-\frac{\log \left (\sqrt [3]{x^2+1}+\sqrt [6]{2} \sqrt [6]{x^2+1}+\sqrt [3]{2}\right )}{4 \sqrt [6]{2}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{1-2^{5/6} \sqrt [6]{x^2+1}}{\sqrt{3}}\right )}{2 \sqrt [6]{2}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{2^{5/6} \sqrt [6]{x^2+1}+1}{\sqrt{3}}\right )}{2 \sqrt [6]{2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{x^2+1}}{\sqrt [6]{2}}\right )}{\sqrt [6]{2}} \]

[Out]

x*AppellF1[1/2, 1, 1/6, 3/2, x^2, -x^2] - (Sqrt[3]*ArcTan[(1 - 2^(5/6)*(1 + x^2)
^(1/6))/Sqrt[3]])/(2*2^(1/6)) + (Sqrt[3]*ArcTan[(1 + 2^(5/6)*(1 + x^2)^(1/6))/Sq
rt[3]])/(2*2^(1/6)) - ArcTanh[(1 + x^2)^(1/6)/2^(1/6)]/2^(1/6) + Log[2^(1/3) - 2
^(1/6)*(1 + x^2)^(1/6) + (1 + x^2)^(1/3)]/(4*2^(1/6)) - Log[2^(1/3) + 2^(1/6)*(1
+ x^2)^(1/6) + (1 + x^2)^(1/3)]/(4*2^(1/6))

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Rubi [A]  time = 0.870741, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667 \[ x F_1\left (\frac{1}{2};1,\frac{1}{6};\frac{3}{2};x^2,-x^2\right )+\frac{\log \left (\sqrt [3]{x^2+1}-\sqrt [6]{2} \sqrt [6]{x^2+1}+\sqrt [3]{2}\right )}{4 \sqrt [6]{2}}-\frac{\log \left (\sqrt [3]{x^2+1}+\sqrt [6]{2} \sqrt [6]{x^2+1}+\sqrt [3]{2}\right )}{4 \sqrt [6]{2}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{1-2^{5/6} \sqrt [6]{x^2+1}}{\sqrt{3}}\right )}{2 \sqrt [6]{2}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{2^{5/6} \sqrt [6]{x^2+1}+1}{\sqrt{3}}\right )}{2 \sqrt [6]{2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{x^2+1}}{\sqrt [6]{2}}\right )}{\sqrt [6]{2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((1 + x)*(1 + x^2)^(1/6)),x]

[Out]

x*AppellF1[1/2, 1, 1/6, 3/2, x^2, -x^2] - (Sqrt[3]*ArcTan[(1 - 2^(5/6)*(1 + x^2)
^(1/6))/Sqrt[3]])/(2*2^(1/6)) + (Sqrt[3]*ArcTan[(1 + 2^(5/6)*(1 + x^2)^(1/6))/Sq
rt[3]])/(2*2^(1/6)) - ArcTanh[(1 + x^2)^(1/6)/2^(1/6)]/2^(1/6) + Log[2^(1/3) - 2
^(1/6)*(1 + x^2)^(1/6) + (1 + x^2)^(1/3)]/(4*2^(1/6)) - Log[2^(1/3) + 2^(1/6)*(1
+ x^2)^(1/6) + (1 + x^2)^(1/3)]/(4*2^(1/6))

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Rubi in Sympy [A]  time = 10.6153, size = 51, normalized size = 0.25 \[ - \frac{3 \sqrt [6]{\frac{x - i}{x + 1}} \sqrt [6]{\frac{x + i}{x + 1}} \operatorname{appellf_{1}}{\left (\frac{1}{3},\frac{1}{6},\frac{1}{6},\frac{4}{3},\frac{1 - i}{x + 1},\frac{1 + i}{x + 1} \right )}}{\sqrt [6]{x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(1+x)/(x**2+1)**(1/6),x)

[Out]

-3*((x - I)/(x + 1))**(1/6)*((x + I)/(x + 1))**(1/6)*appellf1(1/3, 1/6, 1/6, 4/3
, (1 - I)/(x + 1), (1 + I)/(x + 1))/(x**2 + 1)**(1/6)

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Mathematica [C]  time = 0.2196, size = 154, normalized size = 0.76 \[ -\frac{(12+12 i) (x+1) F_1\left (\frac{1}{3};\frac{1}{6},\frac{1}{6};\frac{4}{3};\frac{1-i}{x+1},\frac{1+i}{x+1}\right )}{\sqrt [6]{x^2+1} \left ((4+4 i) (x+1) F_1\left (\frac{1}{3};\frac{1}{6},\frac{1}{6};\frac{4}{3};\frac{1-i}{x+1},\frac{1+i}{x+1}\right )+i F_1\left (\frac{4}{3};\frac{1}{6},\frac{7}{6};\frac{7}{3};\frac{1-i}{x+1},\frac{1+i}{x+1}\right )+F_1\left (\frac{4}{3};\frac{7}{6},\frac{1}{6};\frac{7}{3};\frac{1-i}{x+1},\frac{1+i}{x+1}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/((1 + x)*(1 + x^2)^(1/6)),x]

[Out]

((-12 - 12*I)*(1 + x)*AppellF1[1/3, 1/6, 1/6, 4/3, (1 - I)/(1 + x), (1 + I)/(1 +
x)])/((1 + x^2)^(1/6)*((4 + 4*I)*(1 + x)*AppellF1[1/3, 1/6, 1/6, 4/3, (1 - I)/(
1 + x), (1 + I)/(1 + x)] + I*AppellF1[4/3, 1/6, 7/6, 7/3, (1 - I)/(1 + x), (1 +
I)/(1 + x)] + AppellF1[4/3, 7/6, 1/6, 7/3, (1 - I)/(1 + x), (1 + I)/(1 + x)]))

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Maple [F]  time = 0.038, size = 0, normalized size = 0. \[ \int{\frac{1}{1+x}{\frac{1}{\sqrt [6]{{x}^{2}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1+x)/(x^2+1)^(1/6),x)

[Out]

int(1/(1+x)/(x^2+1)^(1/6),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{2} + 1\right )}^{\frac{1}{6}}{\left (x + 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 + 1)^(1/6)*(x + 1)),x, algorithm="maxima")

[Out]

integrate(1/((x^2 + 1)^(1/6)*(x + 1)), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 + 1)^(1/6)*(x + 1)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (x + 1\right ) \sqrt [6]{x^{2} + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(1+x)/(x**2+1)**(1/6),x)

[Out]

Integral(1/((x + 1)*(x**2 + 1)**(1/6)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{2} + 1\right )}^{\frac{1}{6}}{\left (x + 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 + 1)^(1/6)*(x + 1)),x, algorithm="giac")

[Out]

integrate(1/((x^2 + 1)^(1/6)*(x + 1)), x)