3.1542 \(\int \frac{x^5}{9-x^{12}} \, dx\)

Optimal. Leaf size=12 \[ \frac{1}{18} \tanh ^{-1}\left (\frac{x^6}{3}\right ) \]

[Out]

ArcTanh[x^6/3]/18

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Rubi [A]  time = 0.0191459, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{1}{18} \tanh ^{-1}\left (\frac{x^6}{3}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^5/(9 - x^12),x]

[Out]

ArcTanh[x^6/3]/18

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Rubi in Sympy [A]  time = 3.02734, size = 7, normalized size = 0.58 \[ \frac{\operatorname{atanh}{\left (\frac{x^{6}}{3} \right )}}{18} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(-x**12+9),x)

[Out]

atanh(x**6/3)/18

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Mathematica [A]  time = 0.00614431, size = 23, normalized size = 1.92 \[ \frac{1}{36} \log \left (x^6+3\right )-\frac{1}{36} \log \left (3-x^6\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(9 - x^12),x]

[Out]

-Log[3 - x^6]/36 + Log[3 + x^6]/36

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Maple [B]  time = 0.009, size = 18, normalized size = 1.5 \[{\frac{\ln \left ({x}^{6}+3 \right ) }{36}}-{\frac{\ln \left ({x}^{6}-3 \right ) }{36}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(-x^12+9),x)

[Out]

1/36*ln(x^6+3)-1/36*ln(x^6-3)

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Maxima [A]  time = 1.41764, size = 23, normalized size = 1.92 \[ \frac{1}{36} \, \log \left (x^{6} + 3\right ) - \frac{1}{36} \, \log \left (x^{6} - 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^5/(x^12 - 9),x, algorithm="maxima")

[Out]

1/36*log(x^6 + 3) - 1/36*log(x^6 - 3)

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Fricas [A]  time = 0.223566, size = 23, normalized size = 1.92 \[ \frac{1}{36} \, \log \left (x^{6} + 3\right ) - \frac{1}{36} \, \log \left (x^{6} - 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^5/(x^12 - 9),x, algorithm="fricas")

[Out]

1/36*log(x^6 + 3) - 1/36*log(x^6 - 3)

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Sympy [A]  time = 0.332021, size = 15, normalized size = 1.25 \[ - \frac{\log{\left (x^{6} - 3 \right )}}{36} + \frac{\log{\left (x^{6} + 3 \right )}}{36} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(-x**12+9),x)

[Out]

-log(x**6 - 3)/36 + log(x**6 + 3)/36

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GIAC/XCAS [A]  time = 0.228385, size = 24, normalized size = 2. \[ \frac{1}{36} \,{\rm ln}\left (x^{6} + 3\right ) - \frac{1}{36} \,{\rm ln}\left ({\left | x^{6} - 3 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^5/(x^12 - 9),x, algorithm="giac")

[Out]

1/36*ln(x^6 + 3) - 1/36*ln(abs(x^6 - 3))