3.1053 \(\int \frac{1}{(a+b x) (a c-b c x)} \, dx\)

Optimal. Leaf size=17 \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{a b c} \]

[Out]

ArcTanh[(b*x)/a]/(a*b*c)

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Rubi [A]  time = 0.0281547, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{a b c} \]

Antiderivative was successfully verified.

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

[Out]

ArcTanh[(b*x)/a]/(a*b*c)

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Rubi in Sympy [A]  time = 10.9698, size = 10, normalized size = 0.59 \[ \frac{\operatorname{atanh}{\left (\frac{b x}{a} \right )}}{a b c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atanh(b*x/a)/(a*b*c)

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Mathematica [A]  time = 0.0104122, size = 17, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{a b c} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)*(a*c - b*c*x)),x]

[Out]

ArcTanh[(b*x)/a]/(a*b*c)

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Maple [B]  time = 0.009, size = 38, normalized size = 2.2 \[{\frac{\ln \left ( bx+a \right ) }{2\,bca}}-{\frac{\ln \left ( bx-a \right ) }{2\,bca}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)/(-b*c*x+a*c),x)

[Out]

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

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Maxima [A]  time = 1.33874, size = 50, normalized size = 2.94 \[ \frac{\log \left (b x + a\right )}{2 \, a b c} - \frac{\log \left (b x - a\right )}{2 \, a b c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((b*c*x - a*c)*(b*x + a)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.204205, size = 38, normalized size = 2.24 \[ \frac{\log \left (b x + a\right ) - \log \left (b x - a\right )}{2 \, a b c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((b*c*x - a*c)*(b*x + a)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.41556, size = 22, normalized size = 1.29 \[ - \frac{\frac{\log{\left (- \frac{a}{b} + x \right )}}{2} - \frac{\log{\left (\frac{a}{b} + x \right )}}{2}}{a b c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(log(-a/b + x)/2 - log(a/b + x)/2)/(a*b*c)

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GIAC/XCAS [A]  time = 0.205112, size = 53, normalized size = 3.12 \[ \frac{{\rm ln}\left ({\left | b x + a \right |}\right )}{2 \, a b c} - \frac{{\rm ln}\left ({\left | b x - a \right |}\right )}{2 \, a b c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((b*c*x - a*c)*(b*x + a)),x, algorithm="giac")

[Out]

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