3.1.74 \(\int \log (-a^2+x^2) \, dx\) [74]

Optimal. Leaf size=25 \[ -2 x+2 a \tanh ^{-1}\left (\frac {x}{a}\right )+x \log \left (-a^2+x^2\right ) \]

[Out]

-2*x+2*a*arctanh(x/a)+x*ln(-a^2+x^2)

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Rubi [A]
time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2498, 327, 213} \begin {gather*} x \log \left (x^2-a^2\right )+2 a \tanh ^{-1}\left (\frac {x}{a}\right )-2 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[-a^2 + x^2],x]

[Out]

-2*x + 2*a*ArcTanh[x/a] + x*Log[-a^2 + x^2]

Rule 213

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

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 2498

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]

Rubi steps

\begin {align*} \int \log \left (-a^2+x^2\right ) \, dx &=x \log \left (-a^2+x^2\right )-2 \int \frac {x^2}{-a^2+x^2} \, dx\\ &=-2 x+x \log \left (-a^2+x^2\right )-\left (2 a^2\right ) \int \frac {1}{-a^2+x^2} \, dx\\ &=-2 x+2 a \tanh ^{-1}\left (\frac {x}{a}\right )+x \log \left (-a^2+x^2\right )\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 25, normalized size = 1.00 \begin {gather*} -2 x+2 a \tanh ^{-1}\left (\frac {x}{a}\right )+x \log \left (-a^2+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Log[-a^2 + x^2],x]

[Out]

-2*x + 2*a*ArcTanh[x/a] + x*Log[-a^2 + x^2]

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Mathics [A]
time = 2.05, size = 32, normalized size = 1.28 \begin {gather*} -a \left (\text {Log}\left [-a+x\right ]-\text {Log}\left [a+x\right ]\right )-2 x+x \text {Log}\left [-a^2+x^2\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[Log[x^2-a^2],x]')

[Out]

-a (Log[-a + x] - Log[a + x]) - 2 x + x Log[-a ^ 2 + x ^ 2]

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Maple [A]
time = 0.01, size = 32, normalized size = 1.28

method	result	size
default	\(x \ln \left (-a^{2}+x^{2}\right )-2 x +a \ln \left (a +x \right )-a \ln \left (a -x \right )\)	\(32\)
risch	\(x \ln \left (-a^{2}+x^{2}\right )-2 x +a \ln \left (a +x \right )-a \ln \left (-a +x \right )\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(-a^2+x^2),x,method=_RETURNVERBOSE)

[Out]

x*ln(-a^2+x^2)-2*x+a*ln(a+x)-a*ln(a-x)

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Maxima [A]
time = 0.28, size = 31, normalized size = 1.24 \begin {gather*} x \log \left (-a^{2} + x^{2}\right ) + a \log \left (a + x\right ) - a \log \left (-a + x\right ) - 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(-a^2+x^2),x, algorithm="maxima")

[Out]

x*log(-a^2 + x^2) + a*log(a + x) - a*log(-a + x) - 2*x

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Fricas [A]
time = 0.40, size = 31, normalized size = 1.24 \begin {gather*} x \log \left (-a^{2} + x^{2}\right ) + a \log \left (a + x\right ) - a \log \left (-a + x\right ) - 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(-a^2+x^2),x, algorithm="fricas")

[Out]

x*log(-a^2 + x^2) + a*log(a + x) - a*log(-a + x) - 2*x

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Sympy [A]
time = 0.08, size = 29, normalized size = 1.16 \begin {gather*} - 2 a \left (\frac {\log {\left (- a + x \right )}}{2} - \frac {\log {\left (a + x \right )}}{2}\right ) + x \log {\left (- a^{2} + x^{2} \right )} - 2 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(-a**2+x**2),x)

[Out]

-2*a*(log(-a + x)/2 - log(a + x)/2) + x*log(-a**2 + x**2) - 2*x

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Giac [A]
time = 0.00, size = 37, normalized size = 1.48 \begin {gather*} x \ln \left (-a^{2}+x^{2}\right )+2 \left (-\frac {1}{2} a \ln \left |x-a\right |+\frac {1}{2} a \ln \left |x+a\right |-x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(-a^2+x^2),x)

[Out]

x*log(-a^2 + x^2) + a*log(abs(a + x)) - a*log(abs(-a + x)) - 2*x

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Mupad [B]
time = 0.07, size = 25, normalized size = 1.00 \begin {gather*} x\,\ln \left (x^2-a^2\right )-2\,x+2\,a\,\mathrm {atanh}\left (\frac {x}{a}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(x^2 - a^2),x)

[Out]

x*log(x^2 - a^2) - 2*x + 2*a*atanh(x/a)

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