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

Optimal. Leaf size=12 \[ -\frac{\log (a-b x)}{b} \]

[Out]

-(Log[a - b*x]/b)

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Rubi [A]  time = 0.0170324, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{\log (a-b x)}{b} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)/(a^2 - b^2*x^2),x]

[Out]

-(Log[a - b*x]/b)

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Rubi in Sympy [A]  time = 4.29799, size = 8, normalized size = 0.67 \[ - \frac{\log{\left (a - b x \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(a - b*x)/b

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Mathematica [A]  time = 0.00165111, size = 12, normalized size = 1. \[ -\frac{\log (a-b x)}{b} \]

Antiderivative was successfully verified.

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

[Out]

-(Log[a - b*x]/b)

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Maple [A]  time = 0.002, size = 14, normalized size = 1.2 \[ -{\frac{\ln \left ( bx-a \right ) }{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)/(-b^2*x^2+a^2),x)

[Out]

-1/b*ln(b*x-a)

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Maxima [A]  time = 0.685594, size = 18, normalized size = 1.5 \[ -\frac{\log \left (b x - a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(b*x - a)/b

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Fricas [A]  time = 0.211073, size = 18, normalized size = 1.5 \[ -\frac{\log \left (b x - a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(b*x - a)/b

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Sympy [A]  time = 0.109073, size = 8, normalized size = 0.67 \[ - \frac{\log{\left (- a + b x \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)/(-b**2*x**2+a**2),x)

[Out]

-log(-a + b*x)/b

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GIAC/XCAS [A]  time = 0.215098, size = 19, normalized size = 1.58 \[ -\frac{{\rm ln}\left ({\left | b x - a \right |}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-ln(abs(b*x - a))/b