3.7.91 \(\int -\frac {18 x}{-1+9 x^2} \, dx\)

Optimal. Leaf size=16 \[ \frac {1}{4}+\log \left (\frac {2}{-1+9 x^2}\right ) \]

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Rubi [A]  time = 0.00, antiderivative size = 10, normalized size of antiderivative = 0.62, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {12, 260} \begin {gather*} -\log \left (1-9 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-18*x)/(-1 + 9*x^2),x]

[Out]

-Log[1 - 9*x^2]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\left (18 \int \frac {x}{-1+9 x^2} \, dx\right )\\ &=-\log \left (1-9 x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 10, normalized size = 0.62 \begin {gather*} -\log \left (1-9 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-18*x)/(-1 + 9*x^2),x]

[Out]

-Log[1 - 9*x^2]

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fricas [A]  time = 0.76, size = 10, normalized size = 0.62 \begin {gather*} -\log \left (9 \, x^{2} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-18*x/(9*x^2-1),x, algorithm="fricas")

[Out]

-log(9*x^2 - 1)

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giac [A]  time = 0.31, size = 11, normalized size = 0.69 \begin {gather*} -\log \left ({\left | 9 \, x^{2} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-18*x/(9*x^2-1),x, algorithm="giac")

[Out]

-log(abs(9*x^2 - 1))

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maple [A]  time = 0.17, size = 11, normalized size = 0.69

method	result	size
derivativedivides	\(-\ln \left (9 x^{2}-1\right )\)	\(11\)
default	\(-\ln \left (9 x^{2}-1\right )\)	\(11\)
meijerg	\(-\ln \left (-9 x^{2}+1\right )\)	\(11\)
risch	\(-\ln \left (9 x^{2}-1\right )\)	\(11\)
norman	\(-\ln \left (3 x -1\right )-\ln \left (3 x +1\right )\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-18*x/(9*x^2-1),x,method=_RETURNVERBOSE)

[Out]

-ln(9*x^2-1)

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maxima [A]  time = 0.46, size = 10, normalized size = 0.62 \begin {gather*} -\log \left (9 \, x^{2} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-18*x/(9*x^2-1),x, algorithm="maxima")

[Out]

-log(9*x^2 - 1)

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mupad [B]  time = 0.06, size = 8, normalized size = 0.50 \begin {gather*} -\ln \left (x^2-\frac {1}{9}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(18*x)/(9*x^2 - 1),x)

[Out]

-log(x^2 - 1/9)

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sympy [A]  time = 0.07, size = 8, normalized size = 0.50 \begin {gather*} - \log {\left (9 x^{2} - 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-18*x/(9*x**2-1),x)

[Out]

-log(9*x**2 - 1)

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