3.1809 \(\int \frac{a+\frac{b}{x^2}}{x} \, dx\)

Optimal. Leaf size=13 \[ a \log (x)-\frac{b}{2 x^2} \]

[Out]

-b/(2*x^2) + a*Log[x]

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Rubi [A]  time = 0.0148258, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ a \log (x)-\frac{b}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)/x,x]

[Out]

-b/(2*x^2) + a*Log[x]

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Rubi in Sympy [A]  time = 2.78942, size = 10, normalized size = 0.77 \[ a \log{\left (x \right )} - \frac{b}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)/x,x)

[Out]

a*log(x) - b/(2*x**2)

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Mathematica [A]  time = 0.00431721, size = 13, normalized size = 1. \[ a \log (x)-\frac{b}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)/x,x]

[Out]

-b/(2*x^2) + a*Log[x]

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Maple [A]  time = 0.008, size = 12, normalized size = 0.9 \[ -{\frac{b}{2\,{x}^{2}}}+a\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)/x,x)

[Out]

-1/2*b/x^2+a*ln(x)

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Maxima [A]  time = 1.425, size = 19, normalized size = 1.46 \[ \frac{1}{2} \, a \log \left (x^{2}\right ) - \frac{b}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*a*log(x^2) - 1/2*b/x^2

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Fricas [A]  time = 0.220093, size = 23, normalized size = 1.77 \[ \frac{2 \, a x^{2} \log \left (x\right ) - b}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(2*a*x^2*log(x) - b)/x^2

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Sympy [A]  time = 1.0533, size = 10, normalized size = 0.77 \[ a \log{\left (x \right )} - \frac{b}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)/x,x)

[Out]

a*log(x) - b/(2*x**2)

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GIAC/XCAS [A]  time = 0.229645, size = 27, normalized size = 2.08 \[ \frac{1}{2} \, a{\rm ln}\left (x^{2}\right ) - \frac{a x^{2} + b}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*a*ln(x^2) - 1/2*(a*x^2 + b)/x^2