Optimal. Leaf size=61 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1-3 x^2}}\right )}{3 \sqrt {6}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1-3 x^2}}\right )}{3 \sqrt {6}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {453}
\begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-3 x^2-1}}\right )}{3 \sqrt {6}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-3 x^2-1}}\right )}{3 \sqrt {6}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 453
Rubi steps
\begin {align*} \int \frac {x^2}{\left (-2-3 x^2\right ) \left (-1-3 x^2\right )^{3/4}} \, dx &=\frac {\tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1-3 x^2}}\right )}{3 \sqrt {6}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1-3 x^2}}\right )}{3 \sqrt {6}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 1.80, size = 54, normalized size = 0.89 \begin {gather*} -\frac {-\tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1-3 x^2}}\right )+\tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1-3 x^2}}\right )}{3 \sqrt {6}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 1.41, size = 139, normalized size = 2.28
method | result | size |
trager | \(-\frac {\RootOf \left (\textit {\_Z}^{2}+6\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+6\right ) \left (-3 x^{2}-1\right )^{\frac {3}{4}}-3 \sqrt {-3 x^{2}-1}\, x +\RootOf \left (\textit {\_Z}^{2}+6\right ) \left (-3 x^{2}-1\right )^{\frac {1}{4}}-3 x}{3 x^{2}+2}\right )}{18}+\frac {\RootOf \left (\textit {\_Z}^{2}-6\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-6\right ) \left (-3 x^{2}-1\right )^{\frac {3}{4}}-3 \sqrt {-3 x^{2}-1}\, x -\RootOf \left (\textit {\_Z}^{2}-6\right ) \left (-3 x^{2}-1\right )^{\frac {1}{4}}+3 x}{3 x^{2}+2}\right )}{18}\) | \(139\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] Result contains complex when optimal does not.
time = 0.61, size = 115, normalized size = 1.89 \begin {gather*} -\frac {1}{36} \, \sqrt {6} \log \left (\frac {\sqrt {6} x + 2 \, {\left (-3 \, x^{2} - 1\right )}^{\frac {1}{4}}}{2 \, x}\right ) + \frac {1}{36} \, \sqrt {6} \log \left (-\frac {\sqrt {6} x - 2 \, {\left (-3 \, x^{2} - 1\right )}^{\frac {1}{4}}}{2 \, x}\right ) - \frac {1}{36} i \, \sqrt {6} \log \left (\frac {i \, \sqrt {6} x + 2 \, {\left (-3 \, x^{2} - 1\right )}^{\frac {1}{4}}}{2 \, x}\right ) + \frac {1}{36} i \, \sqrt {6} \log \left (\frac {-i \, \sqrt {6} x + 2 \, {\left (-3 \, x^{2} - 1\right )}^{\frac {1}{4}}}{2 \, x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {x^{2}}{3 x^{2} \left (- 3 x^{2} - 1\right )^{\frac {3}{4}} + 2 \left (- 3 x^{2} - 1\right )^{\frac {3}{4}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} -\int \frac {x^2}{{\left (-3\,x^2-1\right )}^{3/4}\,\left (3\,x^2+2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________