3.1.70 \(\int \log (a^2+x^2) \, dx\) [70]

Optimal. Leaf size=23 \[ -2 x+2 a \tan ^{-1}\left (\frac {x}{a}\right )+x \log \left (a^2+x^2\right ) \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2498, 327, 209} \begin {gather*} x \log \left (a^2+x^2\right )+2 a \tan ^{-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*ArcTan[x/a] + x*Log[a^2 + x^2]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[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 \tan ^{-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 = 23, normalized size = 1.00 \begin {gather*} -2 x+2 a \tan ^{-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*ArcTan[x/a] + x*Log[a^2 + x^2]

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Mathics [C] Result contains complex when optimal does not.
time = 2.12, size = 32, normalized size = 1.39 \begin {gather*} -I a \left (\text {Log}\left [-I a+x\right ]-\text {Log}\left [I 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]

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

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Maple [A]
time = 0.02, size = 24, normalized size = 1.04

method	result	size
default	\(-2 x +2 a \arctan \left (\frac {x}{a}\right )+x \ln \left (a^{2}+x^{2}\right )\)	\(24\)
risch	\(-2 x +2 a \arctan \left (\frac {x}{a}\right )+x \ln \left (a^{2}+x^{2}\right )\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.35, size = 23, normalized size = 1.00 \begin {gather*} 2 \, a \arctan \left (\frac {x}{a}\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(log(a^2+x^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.32, size = 23, normalized size = 1.00 \begin {gather*} 2 \, a \arctan \left (\frac {x}{a}\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(log(a^2+x^2),x, algorithm="fricas")

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 0.08, size = 36, normalized size = 1.57 \begin {gather*} - 2 a \left (\frac {i \log {\left (- i a + x \right )}}{2} - \frac {i \log {\left (i 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*(I*log(-I*a + x)/2 - I*log(I*a + x)/2) + x*log(a**2 + x**2) - 2*x

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Giac [A]
time = 0.00, size = 31, normalized size = 1.35 \begin {gather*} x \ln \left (a^{2}+x^{2}\right )-2 \left (x-\frac {2 a^{2} \arctan \left (\frac {x}{a}\right )}{2 a}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.07, size = 23, normalized size = 1.00 \begin {gather*} x\,\ln \left (a^2+x^2\right )-2\,x+2\,a\,\mathrm {atan}\left (\frac {x}{a}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(a^2 + x^2),x)

[Out]

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

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