3.545 \(\int \frac{81+36 x^2+16 x^4}{729-64 x^6} \, dx\)

Optimal. Leaf size=10 \[ \frac{1}{6} \tanh ^{-1}\left (\frac{2 x}{3}\right ) \]

[Out]

ArcTanh[(2*x)/3]/6

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Rubi [A]  time = 0.0118707, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{1}{6} \tanh ^{-1}\left (\frac{2 x}{3}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(81 + 36*x^2 + 16*x^4)/(729 - 64*x^6),x]

[Out]

ArcTanh[(2*x)/3]/6

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Rubi in Sympy [A]  time = 3.89002, size = 7, normalized size = 0.7 \[ \frac{\operatorname{atanh}{\left (\frac{2 x}{3} \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((16*x**4+36*x**2+81)/(-64*x**6+729),x)

[Out]

atanh(2*x/3)/6

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Mathematica [B]  time = 0.00429929, size = 21, normalized size = 2.1 \[ \frac{1}{12} \log (2 x+3)-\frac{1}{12} \log (3-2 x) \]

Antiderivative was successfully verified.

[In]  Integrate[(81 + 36*x^2 + 16*x^4)/(729 - 64*x^6),x]

[Out]

-Log[3 - 2*x]/12 + Log[3 + 2*x]/12

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Maple [B]  time = 0.008, size = 18, normalized size = 1.8 \[{\frac{\ln \left ( 2\,x+3 \right ) }{12}}-{\frac{\ln \left ( -3+2\,x \right ) }{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((16*x^4+36*x^2+81)/(-64*x^6+729),x)

[Out]

1/12*ln(2*x+3)-1/12*ln(-3+2*x)

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Maxima [A]  time = 1.36482, size = 23, normalized size = 2.3 \[ \frac{1}{12} \, \log \left (2 \, x + 3\right ) - \frac{1}{12} \, \log \left (2 \, x - 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(16*x^4 + 36*x^2 + 81)/(64*x^6 - 729),x, algorithm="maxima")

[Out]

1/12*log(2*x + 3) - 1/12*log(2*x - 3)

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Fricas [A]  time = 0.205763, size = 23, normalized size = 2.3 \[ \frac{1}{12} \, \log \left (2 \, x + 3\right ) - \frac{1}{12} \, \log \left (2 \, x - 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(16*x^4 + 36*x^2 + 81)/(64*x^6 - 729),x, algorithm="fricas")

[Out]

1/12*log(2*x + 3) - 1/12*log(2*x - 3)

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Sympy [A]  time = 0.099868, size = 15, normalized size = 1.5 \[ - \frac{\log{\left (x - \frac{3}{2} \right )}}{12} + \frac{\log{\left (x + \frac{3}{2} \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((16*x**4+36*x**2+81)/(-64*x**6+729),x)

[Out]

-log(x - 3/2)/12 + log(x + 3/2)/12

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GIAC/XCAS [A]  time = 0.218568, size = 20, normalized size = 2. \[ \frac{1}{12} \,{\rm ln}\left ({\left | x + \frac{3}{2} \right |}\right ) - \frac{1}{12} \,{\rm ln}\left ({\left | x - \frac{3}{2} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(16*x^4 + 36*x^2 + 81)/(64*x^6 - 729),x, algorithm="giac")

[Out]

1/12*ln(abs(x + 3/2)) - 1/12*ln(abs(x - 3/2))