Optimal. Leaf size=20 \[ -\frac{1}{8} \tanh ^{-1}\left (\frac{\sqrt{16-x^4}}{4}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01006, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {266, 63, 206} \[ -\frac{1}{8} \tanh ^{-1}\left (\frac{\sqrt{16-x^4}}{4}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{x \sqrt{16-x^4}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{\sqrt{16-x} x} \, dx,x,x^4\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{16-x^2} \, dx,x,\sqrt{16-x^4}\right )\right )\\ &=-\frac{1}{8} \tanh ^{-1}\left (\frac{\sqrt{16-x^4}}{4}\right )\\ \end{align*}
Mathematica [A] time = 0.002583, size = 20, normalized size = 1. \[ -\frac{1}{8} \tanh ^{-1}\left (\frac{\sqrt{16-x^4}}{4}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.008, size = 15, normalized size = 0.8 \begin{align*} -{\frac{1}{8}{\it Artanh} \left ( 4\,{\frac{1}{\sqrt{-{x}^{4}+16}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 0.964051, size = 39, normalized size = 1.95 \begin{align*} -\frac{1}{16} \, \log \left (\sqrt{-x^{4} + 16} + 4\right ) + \frac{1}{16} \, \log \left (\sqrt{-x^{4} + 16} - 4\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.49476, size = 86, normalized size = 4.3 \begin{align*} -\frac{1}{16} \, \log \left (\sqrt{-x^{4} + 16} + 4\right ) + \frac{1}{16} \, \log \left (\sqrt{-x^{4} + 16} - 4\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.30896, size = 24, normalized size = 1.2 \begin{align*} \begin{cases} - \frac{\operatorname{acosh}{\left (\frac{4}{x^{2}} \right )}}{8} & \text{for}\: \frac{16}{\left |{x^{4}}\right |} > 1 \\\frac{i \operatorname{asin}{\left (\frac{4}{x^{2}} \right )}}{8} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.23874, size = 42, normalized size = 2.1 \begin{align*} -\frac{1}{16} \, \log \left (\sqrt{-x^{4} + 16} + 4\right ) + \frac{1}{16} \, \log \left (-\sqrt{-x^{4} + 16} + 4\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]