Optimal. Leaf size=36 \[ \frac{b \log \left (b x^n+2\right )}{4 n}-\frac{1}{4} b \log (x)-\frac{x^{-n}}{2 n} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0506997, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{b \log \left (b x^n+2\right )}{4 n}-\frac{1}{4} b \log (x)-\frac{x^{-n}}{2 n} \]
Antiderivative was successfully verified.
[In] Int[x^(-1 - n)/(2 + b*x^n),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 7.52773, size = 31, normalized size = 0.86 \[ - \frac{b \log{\left (x^{n} \right )}}{4 n} + \frac{b \log{\left (b x^{n} + 2 \right )}}{4 n} - \frac{x^{- n}}{2 n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(-1-n)/(2+b*x**n),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0186937, size = 31, normalized size = 0.86 \[ \frac{b \log \left (b+2 x^{-n}\right )}{4 n}-\frac{x^{-n}}{2 n} \]
Antiderivative was successfully verified.
[In] Integrate[x^(-1 - n)/(2 + b*x^n),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.026, size = 42, normalized size = 1.2 \[{\frac{1}{{{\rm e}^{n\ln \left ( x \right ) }}} \left ( -{\frac{b\ln \left ( x \right ){{\rm e}^{n\ln \left ( x \right ) }}}{4}}-{\frac{1}{2\,n}} \right ) }+{\frac{b\ln \left ( 2+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{4\,n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(-1-n)/(2+b*x^n),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.44282, size = 46, normalized size = 1.28 \[ -\frac{1}{4} \, b \log \left (x\right ) + \frac{b \log \left (\frac{b x^{n} + 2}{b}\right )}{4 \, n} - \frac{x^{-n}}{2 \, n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(-n - 1)/(b*x^n + 2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.229114, size = 46, normalized size = 1.28 \[ -\frac{b n x^{n} \log \left (x\right ) - b x^{n} \log \left (b x^{n} + 2\right ) + 2}{4 \, n x^{n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(-n - 1)/(b*x^n + 2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(-1-n)/(2+b*x**n),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{-n - 1}}{b x^{n} + 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(-n - 1)/(b*x^n + 2),x, algorithm="giac")
[Out]