Optimal. Leaf size=59 \[ -\frac{\log \left ((a+b x)^2+c\right )}{2 \left (a^2+c\right )}-\frac{a \tan ^{-1}\left (\frac{a+b x}{\sqrt{c}}\right )}{\sqrt{c} \left (a^2+c\right )}+\frac{\log (x)}{a^2+c} \]
[Out]
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Rubi [A] time = 0.082148, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{\log \left ((a+b x)^2+c\right )}{2 \left (a^2+c\right )}-\frac{a \tan ^{-1}\left (\frac{a+b x}{\sqrt{c}}\right )}{\sqrt{c} \left (a^2+c\right )}+\frac{\log (x)}{a^2+c} \]
Antiderivative was successfully verified.
[In] Int[1/(x*(c + (a + b*x)^2)),x]
[Out]
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Rubi in Sympy [A] time = 10.9615, size = 53, normalized size = 0.9 \[ - \frac{a \operatorname{atan}{\left (\frac{a + b x}{\sqrt{c}} \right )}}{\sqrt{c} \left (a^{2} + c\right )} + \frac{\log{\left (- b x \right )}}{a^{2} + c} - \frac{\log{\left (c + \left (a + b x\right )^{2} \right )}}{2 \left (a^{2} + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x/(c+(b*x+a)**2),x)
[Out]
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Mathematica [A] time = 0.0572488, size = 48, normalized size = 0.81 \[ -\frac{\log \left ((a+b x)^2+c\right )+\frac{2 a \tan ^{-1}\left (\frac{a+b x}{\sqrt{c}}\right )}{\sqrt{c}}-2 \log (b x)}{2 \left (a^2+c\right )} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x*(c + (a + b*x)^2)),x]
[Out]
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Maple [A] time = 0.008, size = 72, normalized size = 1.2 \[ -{\frac{\ln \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2}+c \right ) }{2\,{a}^{2}+2\,c}}-{\frac{a}{{a}^{2}+c}\arctan \left ({\frac{2\,{b}^{2}x+2\,ab}{2\,b}{\frac{1}{\sqrt{c}}}} \right ){\frac{1}{\sqrt{c}}}}+{\frac{\ln \left ( x \right ) }{{a}^{2}+c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x/(c+(b*x+a)^2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(((b*x + a)^2 + c)*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.274189, size = 1, normalized size = 0.02 \[ \left [\frac{a \log \left (-\frac{2 \, b c x + 2 \, a c -{\left (b^{2} x^{2} + 2 \, a b x + a^{2} - c\right )} \sqrt{-c}}{b^{2} x^{2} + 2 \, a b x + a^{2} + c}\right ) - \sqrt{-c}{\left (\log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right ) - 2 \, \log \left (x\right )\right )}}{2 \,{\left (a^{2} + c\right )} \sqrt{-c}}, -\frac{2 \, a \arctan \left (\frac{b x + a}{\sqrt{c}}\right ) + \sqrt{c}{\left (\log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right ) - 2 \, \log \left (x\right )\right )}}{2 \,{\left (a^{2} + c\right )} \sqrt{c}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(((b*x + a)^2 + c)*x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.9779, size = 738, normalized size = 12.51 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x/(c+(b*x+a)**2),x)
[Out]
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GIAC/XCAS [A] time = 0.26141, size = 84, normalized size = 1.42 \[ -\frac{a \arctan \left (\frac{b x + a}{\sqrt{c}}\right )}{{\left (a^{2} + c\right )} \sqrt{c}} - \frac{{\rm ln}\left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right )}{2 \,{\left (a^{2} + c\right )}} + \frac{{\rm ln}\left ({\left | x \right |}\right )}{a^{2} + c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(((b*x + a)^2 + c)*x),x, algorithm="giac")
[Out]