Optimal. Leaf size=35 \[ -\frac{\tan ^{-1}\left (\sqrt [4]{5} x\right )}{2 \sqrt [4]{5}}-\frac{\tanh ^{-1}\left (\sqrt [4]{5} x\right )}{2 \sqrt [4]{5}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0143085, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {212, 206, 203} \[ -\frac{\tan ^{-1}\left (\sqrt [4]{5} x\right )}{2 \sqrt [4]{5}}-\frac{\tanh ^{-1}\left (\sqrt [4]{5} x\right )}{2 \sqrt [4]{5}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 206
Rule 203
Rubi steps
\begin{align*} \int \frac{1}{-1+5 x^4} \, dx &=-\left (\frac{1}{2} \int \frac{1}{1-\sqrt{5} x^2} \, dx\right )-\frac{1}{2} \int \frac{1}{1+\sqrt{5} x^2} \, dx\\ &=-\frac{\tan ^{-1}\left (\sqrt [4]{5} x\right )}{2 \sqrt [4]{5}}-\frac{\tanh ^{-1}\left (\sqrt [4]{5} x\right )}{2 \sqrt [4]{5}}\\ \end{align*}
Mathematica [A] time = 0.0146024, size = 43, normalized size = 1.23 \[ -\frac{-\log \left (1-\sqrt [4]{5} x\right )+\log \left (\sqrt [4]{5} x+1\right )+2 \tan ^{-1}\left (\sqrt [4]{5} x\right )}{4 \sqrt [4]{5}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.003, size = 36, normalized size = 1. \begin{align*} -{\frac{\arctan \left ( \sqrt [4]{5}x \right ){5}^{{\frac{3}{4}}}}{10}}-{\frac{{5}^{{\frac{3}{4}}}}{20}\ln \left ({ \left ( x+{\frac{{5}^{{\frac{3}{4}}}}{5}} \right ) \left ( x-{\frac{{5}^{{\frac{3}{4}}}}{5}} \right ) ^{-1}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.43936, size = 55, normalized size = 1.57 \begin{align*} -\frac{1}{10} \cdot 5^{\frac{3}{4}} \arctan \left (5^{\frac{1}{4}} x\right ) + \frac{1}{20} \cdot 5^{\frac{3}{4}} \log \left (\frac{\sqrt{5} x - 5^{\frac{1}{4}}}{\sqrt{5} x + 5^{\frac{1}{4}}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.9271, size = 182, normalized size = 5.2 \begin{align*} \frac{1}{5} \cdot 5^{\frac{3}{4}} \arctan \left (\frac{1}{5} \cdot 5^{\frac{3}{4}} \sqrt{5 \, x^{2} + \sqrt{5}} - 5^{\frac{1}{4}} x\right ) - \frac{1}{20} \cdot 5^{\frac{3}{4}} \log \left (5 \, x + 5^{\frac{3}{4}}\right ) + \frac{1}{20} \cdot 5^{\frac{3}{4}} \log \left (5 \, x - 5^{\frac{3}{4}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.323315, size = 48, normalized size = 1.37 \begin{align*} \frac{5^{\frac{3}{4}} \log{\left (x - \frac{5^{\frac{3}{4}}}{5} \right )}}{20} - \frac{5^{\frac{3}{4}} \log{\left (x + \frac{5^{\frac{3}{4}}}{5} \right )}}{20} - \frac{5^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{5} x \right )}}{10} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.11051, size = 53, normalized size = 1.51 \begin{align*} -\frac{1}{10} \cdot 5^{\frac{3}{4}} \arctan \left (5 \, \left (\frac{1}{5}\right )^{\frac{3}{4}} x\right ) - \frac{1}{20} \cdot 5^{\frac{3}{4}} \log \left ({\left | x + \left (\frac{1}{5}\right )^{\frac{1}{4}} \right |}\right ) + \frac{1}{20} \cdot 5^{\frac{3}{4}} \log \left ({\left | x - \left (\frac{1}{5}\right )^{\frac{1}{4}} \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]