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