Optimal. Leaf size=35 \[ \frac{c \log \left (a+c x^4\right )}{4 a^2}-\frac{c \log (x)}{a^2}-\frac{1}{4 a x^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0231348, antiderivative size = 35, 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} \[ \frac{c \log \left (a+c x^4\right )}{4 a^2}-\frac{c \log (x)}{a^2}-\frac{1}{4 a x^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^5 \left (a+c x^4\right )} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{x^2 (a+c x)} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (\frac{1}{a x^2}-\frac{c}{a^2 x}+\frac{c^2}{a^2 (a+c x)}\right ) \, dx,x,x^4\right )\\ &=-\frac{1}{4 a x^4}-\frac{c \log (x)}{a^2}+\frac{c \log \left (a+c x^4\right )}{4 a^2}\\ \end{align*}
Mathematica [A] time = 0.0066834, size = 35, normalized size = 1. \[ \frac{c \log \left (a+c x^4\right )}{4 a^2}-\frac{c \log (x)}{a^2}-\frac{1}{4 a x^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.005, size = 32, normalized size = 0.9 \begin{align*} -{\frac{1}{4\,a{x}^{4}}}-{\frac{c\ln \left ( x \right ) }{{a}^{2}}}+{\frac{c\ln \left ( c{x}^{4}+a \right ) }{4\,{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.969551, size = 45, normalized size = 1.29 \begin{align*} \frac{c \log \left (c x^{4} + a\right )}{4 \, a^{2}} - \frac{c \log \left (x^{4}\right )}{4 \, a^{2}} - \frac{1}{4 \, a x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.70416, size = 80, normalized size = 2.29 \begin{align*} \frac{c x^{4} \log \left (c x^{4} + a\right ) - 4 \, c x^{4} \log \left (x\right ) - a}{4 \, a^{2} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.744159, size = 31, normalized size = 0.89 \begin{align*} - \frac{1}{4 a x^{4}} - \frac{c \log{\left (x \right )}}{a^{2}} + \frac{c \log{\left (\frac{a}{c} + x^{4} \right )}}{4 a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.62182, size = 58, normalized size = 1.66 \begin{align*} -\frac{c \log \left (x^{4}\right )}{4 \, a^{2}} + \frac{c \log \left ({\left | c x^{4} + a \right |}\right )}{4 \, a^{2}} + \frac{c x^{4} - a}{4 \, a^{2} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]