3.6.57 \(\int \frac {81+36 x^2+16 x^4}{729-64 x^6} \, dx\) [557]

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

[Out]

1/6*arctanh(2/3*x)

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Rubi [A]
time = 0.01, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1600, 212} \begin {gather*} \frac {1}{6} \tanh ^{-1}\left (\frac {2 x}{3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 1600

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rubi steps

\begin {align*} \int \frac {81+36 x^2+16 x^4}{729-64 x^6} \, dx &=\int \frac {1}{9-4 x^2} \, dx\\ &=\frac {1}{6} \tanh ^{-1}\left (\frac {2 x}{3}\right )\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(21\) vs. \(2(10)=20\).
time = 0.00, size = 21, normalized size = 2.10 \begin {gather*} -\frac {1}{12} \log (3-2 x)+\frac {1}{12} \log (3+2 x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/12*Log[3 - 2*x] + Log[3 + 2*x]/12

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(17\) vs. \(2(6)=12\).
time = 0.36, size = 18, normalized size = 1.80

method	result	size
default	\(\frac {\ln \left (2 x +3\right )}{12}-\frac {\ln \left (-3+2 x \right )}{12}\)	\(18\)
norman	\(\frac {\ln \left (2 x +3\right )}{12}-\frac {\ln \left (-3+2 x \right )}{12}\)	\(18\)
risch	\(\frac {\ln \left (2 x +3\right )}{12}-\frac {\ln \left (-3+2 x \right )}{12}\)	\(18\)
meijerg	\(-\frac {x \left (\ln \left (1-\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}\right )-\ln \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}\right )+\frac {\ln \left (1-\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{3-\left (x^{6}\right )^{\frac {1}{6}}}\right )-\frac {\ln \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{3+\left (x^{6}\right )^{\frac {1}{6}}}\right )\right )}{36 \left (x^{6}\right )^{\frac {1}{6}}}-\frac {x^{5} \left (\ln \left (1-\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}\right )-\ln \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}\right )+\frac {\ln \left (1-\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{3-\left (x^{6}\right )^{\frac {1}{6}}}\right )-\frac {\ln \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{3+\left (x^{6}\right )^{\frac {1}{6}}}\right )\right )}{36 \left (x^{6}\right )^{\frac {5}{6}}}+\frac {\arctanh \left (\frac {8 x^{3}}{27}\right )}{18}\)	\(256\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((16*x^4+36*x^2+81)/(-64*x^6+729),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 17 vs. \(2 (6) = 12\).
time = 0.28, size = 17, normalized size = 1.70 \begin {gather*} \frac {1}{12} \, \log \left (2 \, x + 3\right ) - \frac {1}{12} \, \log \left (2 \, x - 3\right ) \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 17 vs. \(2 (6) = 12\).
time = 0.35, size = 17, normalized size = 1.70 \begin {gather*} \frac {1}{12} \, \log \left (2 \, x + 3\right ) - \frac {1}{12} \, \log \left (2 \, x - 3\right ) \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 15 vs. \(2 (7) = 14\).
time = 0.03, size = 15, normalized size = 1.50 \begin {gather*} - \frac {\log {\left (x - \frac {3}{2} \right )}}{12} + \frac {\log {\left (x + \frac {3}{2} \right )}}{12} \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 15 vs. \(2 (6) = 12\).
time = 0.99, size = 15, normalized size = 1.50 \begin {gather*} \frac {1}{12} \, \log \left ({\left | x + \frac {3}{2} \right |}\right ) - \frac {1}{12} \, \log \left ({\left | x - \frac {3}{2} \right |}\right ) \end {gather*}

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

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Mupad [B]
time = 0.10, size = 6, normalized size = 0.60 \begin {gather*} \frac {\mathrm {atanh}\left (\frac {2\,x}{3}\right )}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atanh((2*x)/3)/6

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