Optimal. Leaf size=120 \[ \frac{\tan ^{-1}\left (\frac{2-\sqrt{2} \sqrt{2-3 x^2}}{\sqrt [4]{2} \sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{2\ 2^{3/4} \sqrt{3}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{2-3 x^2}+2}{\sqrt [4]{2} \sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{2\ 2^{3/4} \sqrt{3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0158787, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {397} \[ \frac{\tan ^{-1}\left (\frac{2-\sqrt{2} \sqrt{2-3 x^2}}{\sqrt [4]{2} \sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{2\ 2^{3/4} \sqrt{3}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{2-3 x^2}+2}{\sqrt [4]{2} \sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{2\ 2^{3/4} \sqrt{3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 397
Rubi steps
\begin{align*} \int \frac{1}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx &=\frac{\tan ^{-1}\left (\frac{2-\sqrt{2} \sqrt{2-3 x^2}}{\sqrt [4]{2} \sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{2\ 2^{3/4} \sqrt{3}}+\frac{\tanh ^{-1}\left (\frac{2+\sqrt{2} \sqrt{2-3 x^2}}{\sqrt [4]{2} \sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{2\ 2^{3/4} \sqrt{3}}\\ \end{align*}
Mathematica [C] time = 0.120949, size = 135, normalized size = 1.12 \[ -\frac{4 x F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right )}{\sqrt [4]{2-3 x^2} \left (3 x^2-4\right ) \left (x^2 \left (2 F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right )+F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right )\right )+4 F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right )\right )} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.051, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{-3\,{x}^{2}+4}{\frac{1}{\sqrt [4]{-3\,{x}^{2}+2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{1}{{\left (3 \, x^{2} - 4\right )}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 28.2156, size = 1553, normalized size = 12.94 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{1}{3 x^{2} \sqrt [4]{2 - 3 x^{2}} - 4 \sqrt [4]{2 - 3 x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{1}{{\left (3 \, x^{2} - 4\right )}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]