3.324 \(\int \frac{a+x}{a^2+x^2} \, dx\)

Optimal. Leaf size=19 \[ \frac{1}{2} \log \left (a^2+x^2\right )+\tan ^{-1}\left (\frac{x}{a}\right ) \]

[Out]

ArcTan[x/a] + Log[a^2 + x^2]/2

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Rubi [A]  time = 0.0170919, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{1}{2} \log \left (a^2+x^2\right )+\tan ^{-1}\left (\frac{x}{a}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(a + x)/(a^2 + x^2),x]

[Out]

ArcTan[x/a] + Log[a^2 + x^2]/2

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Rubi in Sympy [A]  time = 1.95558, size = 14, normalized size = 0.74 \[ \frac{\log{\left (a^{2} + x^{2} \right )}}{2} + \operatorname{atan}{\left (\frac{x}{a} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+x)/(a**2+x**2),x)

[Out]

log(a**2 + x**2)/2 + atan(x/a)

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Mathematica [A]  time = 0.00632638, size = 19, normalized size = 1. \[ \frac{1}{2} \log \left (a^2+x^2\right )+\tan ^{-1}\left (\frac{x}{a}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + x)/(a^2 + x^2),x]

[Out]

ArcTan[x/a] + Log[a^2 + x^2]/2

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Maple [A]  time = 0.004, size = 18, normalized size = 1. \[ \arctan \left ({\frac{x}{a}} \right ) +{\frac{\ln \left ({a}^{2}+{x}^{2} \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+x)/(a^2+x^2),x)

[Out]

arctan(x/a)+1/2*ln(a^2+x^2)

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Maxima [A]  time = 1.51017, size = 23, normalized size = 1.21 \[ \arctan \left (\frac{x}{a}\right ) + \frac{1}{2} \, \log \left (a^{2} + x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + x)/(a^2 + x^2),x, algorithm="maxima")

[Out]

arctan(x/a) + 1/2*log(a^2 + x^2)

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Fricas [A]  time = 0.201657, size = 23, normalized size = 1.21 \[ \arctan \left (\frac{x}{a}\right ) + \frac{1}{2} \, \log \left (a^{2} + x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + x)/(a^2 + x^2),x, algorithm="fricas")

[Out]

arctan(x/a) + 1/2*log(a^2 + x^2)

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Sympy [A]  time = 0.110728, size = 42, normalized size = 2.21 \[ \left (\frac{1}{2} - \frac{i}{2}\right ) \log{\left (- a + 2 a \left (\frac{1}{2} - \frac{i}{2}\right ) + x \right )} + \left (\frac{1}{2} + \frac{i}{2}\right ) \log{\left (- a + 2 a \left (\frac{1}{2} + \frac{i}{2}\right ) + x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+x)/(a**2+x**2),x)

[Out]

(1/2 - I/2)*log(-a + 2*a*(1/2 - I/2) + x) + (1/2 + I/2)*log(-a + 2*a*(1/2 + I/2)
+ x)

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GIAC/XCAS [A]  time = 0.21266, size = 23, normalized size = 1.21 \[ \arctan \left (\frac{x}{a}\right ) + \frac{1}{2} \,{\rm ln}\left (a^{2} + x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + x)/(a^2 + x^2),x, algorithm="giac")

[Out]

arctan(x/a) + 1/2*ln(a^2 + x^2)