Optimal. Leaf size=38 \[ -\frac{\log \left (a+b x^6\right )}{6 a^2}+\frac{\log (x)}{a^2}+\frac{1}{6 a \left (a+b x^6\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0280392, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 44} \[ -\frac{\log \left (a+b x^6\right )}{6 a^2}+\frac{\log (x)}{a^2}+\frac{1}{6 a \left (a+b x^6\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x \left (a+b x^6\right )^2} \, dx &=\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{x (a+b x)^2} \, dx,x,x^6\right )\\ &=\frac{1}{6} \operatorname{Subst}\left (\int \left (\frac{1}{a^2 x}-\frac{b}{a (a+b x)^2}-\frac{b}{a^2 (a+b x)}\right ) \, dx,x,x^6\right )\\ &=\frac{1}{6 a \left (a+b x^6\right )}+\frac{\log (x)}{a^2}-\frac{\log \left (a+b x^6\right )}{6 a^2}\\ \end{align*}
Mathematica [A] time = 0.0130811, size = 33, normalized size = 0.87 \[ \frac{\frac{a}{a+b x^6}-\log \left (a+b x^6\right )+6 \log (x)}{6 a^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.01, size = 35, normalized size = 0.9 \begin{align*}{\frac{1}{6\,a \left ( b{x}^{6}+a \right ) }}+{\frac{\ln \left ( x \right ) }{{a}^{2}}}-{\frac{\ln \left ( b{x}^{6}+a \right ) }{6\,{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.973108, size = 50, normalized size = 1.32 \begin{align*} \frac{1}{6 \,{\left (a b x^{6} + a^{2}\right )}} - \frac{\log \left (b x^{6} + a\right )}{6 \, a^{2}} + \frac{\log \left (x^{6}\right )}{6 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.45866, size = 108, normalized size = 2.84 \begin{align*} -\frac{{\left (b x^{6} + a\right )} \log \left (b x^{6} + a\right ) - 6 \,{\left (b x^{6} + a\right )} \log \left (x\right ) - a}{6 \,{\left (a^{2} b x^{6} + a^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.4504, size = 34, normalized size = 0.89 \begin{align*} \frac{1}{6 a^{2} + 6 a b x^{6}} + \frac{\log{\left (x \right )}}{a^{2}} - \frac{\log{\left (\frac{a}{b} + x^{6} \right )}}{6 a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.17273, size = 63, normalized size = 1.66 \begin{align*} \frac{\log \left (x^{6}\right )}{6 \, a^{2}} - \frac{\log \left ({\left | b x^{6} + a \right |}\right )}{6 \, a^{2}} + \frac{b x^{6} + 2 \, a}{6 \,{\left (b x^{6} + a\right )} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]