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

Optimal. Leaf size=36 \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{2 a^2 b}+\frac{1}{2 a b (a-b x)} \]

[Out]

1/(2*a*b*(a - b*x)) + ArcTanh[(b*x)/a]/(2*a^2*b)

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Rubi [A]  time = 0.0669328, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{2 a^2 b}+\frac{1}{2 a b (a-b x)} \]

Antiderivative was successfully verified.

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

[Out]

1/(2*a*b*(a - b*x)) + ArcTanh[(b*x)/a]/(2*a^2*b)

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Rubi in Sympy [A]  time = 19.0701, size = 24, normalized size = 0.67 \[ \frac{1}{2 a b \left (a - b x\right )} + \frac{\operatorname{atanh}{\left (\frac{b x}{a} \right )}}{2 a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(2*a*b*(a - b*x)) + atanh(b*x/a)/(2*a**2*b)

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Mathematica [A]  time = 0.0189094, size = 50, normalized size = 1.39 \[ \frac{(b x-a) \log (a-b x)+(a-b x) \log (a+b x)+2 a}{4 a^2 b (a-b x)} \]

Antiderivative was successfully verified.

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

[Out]

(2*a + (-a + b*x)*Log[a - b*x] + (a - b*x)*Log[a + b*x])/(4*a^2*b*(a - b*x))

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Maple [A]  time = 0.013, size = 49, normalized size = 1.4 \[ -{\frac{\ln \left ( bx-a \right ) }{4\,{a}^{2}b}}-{\frac{1}{2\,ab \left ( bx-a \right ) }}+{\frac{\ln \left ( bx+a \right ) }{4\,{a}^{2}b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)/(-b^2*x^2+a^2)^2,x)

[Out]

-1/4/b/a^2*ln(b*x-a)-1/2/b/a/(b*x-a)+1/4/b/a^2*ln(b*x+a)

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Maxima [A]  time = 0.684686, size = 65, normalized size = 1.81 \[ -\frac{1}{2 \,{\left (a b^{2} x - a^{2} b\right )}} + \frac{\log \left (b x + a\right )}{4 \, a^{2} b} - \frac{\log \left (b x - a\right )}{4 \, a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/(a*b^2*x - a^2*b) + 1/4*log(b*x + a)/(a^2*b) - 1/4*log(b*x - a)/(a^2*b)

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Fricas [A]  time = 0.219603, size = 73, normalized size = 2.03 \[ \frac{{\left (b x - a\right )} \log \left (b x + a\right ) -{\left (b x - a\right )} \log \left (b x - a\right ) - 2 \, a}{4 \,{\left (a^{2} b^{2} x - a^{3} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*((b*x - a)*log(b*x + a) - (b*x - a)*log(b*x - a) - 2*a)/(a^2*b^2*x - a^3*b)

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Sympy [A]  time = 1.58567, size = 37, normalized size = 1.03 \[ - \frac{1}{- 2 a^{2} b + 2 a b^{2} x} + \frac{- \frac{\log{\left (- \frac{a}{b} + x \right )}}{4} + \frac{\log{\left (\frac{a}{b} + x \right )}}{4}}{a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(-2*a**2*b + 2*a*b**2*x) + (-log(-a/b + x)/4 + log(a/b + x)/4)/(a**2*b)

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GIAC/XCAS [A]  time = 0.216352, size = 68, normalized size = 1.89 \[ \frac{{\rm ln}\left ({\left | b x + a \right |}\right )}{4 \, a^{2} b} - \frac{{\rm ln}\left ({\left | b x - a \right |}\right )}{4 \, a^{2} b} - \frac{1}{2 \,{\left (b x - a\right )} a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*ln(abs(b*x + a))/(a^2*b) - 1/4*ln(abs(b*x - a))/(a^2*b) - 1/2/((b*x - a)*a*b
)