3.742 \(\int \frac{1}{(a+b x) \left (a^2-b^2 x^2\right )} \, dx\)

Optimal. Leaf size=35 \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{2 a^2 b}-\frac{1}{2 a b (a+b x)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0819684, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{2 a^2 b}-\frac{1}{2 a b (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)*(a^2 - b^2*x^2)),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 19.124, size = 24, normalized size = 0.69 \[ - \frac{1}{2 a b \left (a + b x\right )} + \frac{\operatorname{atanh}{\left (\frac{b x}{a} \right )}}{2 a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)/(-b**2*x**2+a**2),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0219016, size = 47, normalized size = 1.34 \[ \frac{-(a+b x) \log (a-b x)+(a+b x) \log (a+b x)-2 a}{4 a^2 b (a+b x)} \]

Antiderivative was successfully verified.

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

_______________________________________________________________________________________

Maple [A]  time = 0.014, size = 47, normalized size = 1.3 \[ -{\frac{\ln \left ( bx-a \right ) }{4\,{a}^{2}b}}+{\frac{\ln \left ( bx+a \right ) }{4\,{a}^{2}b}}-{\frac{1}{2\,ab \left ( bx+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)/(-b^2*x^2+a^2),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.689612, size = 63, normalized size = 1.8 \[ -\frac{1}{2 \,{\left (a b^{2} x + a^{2} b\right )}} + \frac{\log \left (b x + a\right )}{4 \, a^{2} b} - \frac{\log \left (b x - a\right )}{4 \, a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((b^2*x^2 - a^2)*(b*x + a)),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)

_______________________________________________________________________________________

Fricas [A]  time = 0.21474, size = 66, normalized size = 1.89 \[ \frac{{\left (b x + a\right )} \log \left (b x + a\right ) -{\left (b x + a\right )} \log \left (b x - a\right ) - 2 \, a}{4 \,{\left (a^{2} b^{2} x + a^{3} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((b^2*x^2 - a^2)*(b*x + a)),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)

_______________________________________________________________________________________

Sympy [A]  time = 1.52908, size = 39, normalized size = 1.11 \[ - \frac{1}{2 a^{2} b + 2 a b^{2} x} - \frac{\frac{\log{\left (- \frac{a}{b} + x \right )}}{4} - \frac{\log{\left (\frac{a}{b} + x \right )}}{4}}{a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x+a)/(-b**2*x**2+a**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)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.216091, size = 65, normalized size = 1.86 \[ \frac{{\rm ln}\left ({\left | b x + a \right |}\right )}{4 \, a^{2} b} - \frac{{\rm ln}\left ({\left | b x - a \right |}\right )}{4 \, a^{2} b} - \frac{1}{2 \,{\left (b x + a\right )} a b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((b^2*x^2 - a^2)*(b*x + a)),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
)