Optimal. Leaf size=191 \[ -\frac{\left (3 x^2-1\right )^{3/4}}{4 x^2}-\frac{9 \log \left (\sqrt{3 x^2-1}-\sqrt{2} \sqrt [4]{3 x^2-1}+1\right )}{16 \sqrt{2}}+\frac{9 \log \left (\sqrt{3 x^2-1}+\sqrt{2} \sqrt [4]{3 x^2-1}+1\right )}{16 \sqrt{2}}+\frac{3}{4} \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )+\frac{9 \tan ^{-1}\left (1-\sqrt{2} \sqrt [4]{3 x^2-1}\right )}{8 \sqrt{2}}-\frac{9 \tan ^{-1}\left (\sqrt{2} \sqrt [4]{3 x^2-1}+1\right )}{8 \sqrt{2}}-\frac{3}{4} \tanh ^{-1}\left (\sqrt [4]{3 x^2-1}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.147334, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 13, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.542, Rules used = {446, 103, 156, 63, 297, 1162, 617, 204, 1165, 628, 298, 203, 206} \[ -\frac{\left (3 x^2-1\right )^{3/4}}{4 x^2}-\frac{9 \log \left (\sqrt{3 x^2-1}-\sqrt{2} \sqrt [4]{3 x^2-1}+1\right )}{16 \sqrt{2}}+\frac{9 \log \left (\sqrt{3 x^2-1}+\sqrt{2} \sqrt [4]{3 x^2-1}+1\right )}{16 \sqrt{2}}+\frac{3}{4} \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )+\frac{9 \tan ^{-1}\left (1-\sqrt{2} \sqrt [4]{3 x^2-1}\right )}{8 \sqrt{2}}-\frac{9 \tan ^{-1}\left (\sqrt{2} \sqrt [4]{3 x^2-1}+1\right )}{8 \sqrt{2}}-\frac{3}{4} \tanh ^{-1}\left (\sqrt [4]{3 x^2-1}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 446
Rule 103
Rule 156
Rule 63
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{x^3 \left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^2 (-2+3 x) \sqrt [4]{-1+3 x}} \, dx,x,x^2\right )\\ &=-\frac{\left (-1+3 x^2\right )^{3/4}}{4 x^2}-\frac{1}{4} \operatorname{Subst}\left (\int \frac{-\frac{9}{2}+\frac{9 x}{4}}{x (-2+3 x) \sqrt [4]{-1+3 x}} \, dx,x,x^2\right )\\ &=-\frac{\left (-1+3 x^2\right )^{3/4}}{4 x^2}-\frac{9}{16} \operatorname{Subst}\left (\int \frac{1}{x \sqrt [4]{-1+3 x}} \, dx,x,x^2\right )+\frac{9}{8} \operatorname{Subst}\left (\int \frac{1}{(-2+3 x) \sqrt [4]{-1+3 x}} \, dx,x,x^2\right )\\ &=-\frac{\left (-1+3 x^2\right )^{3/4}}{4 x^2}-\frac{3}{4} \operatorname{Subst}\left (\int \frac{x^2}{\frac{1}{3}+\frac{x^4}{3}} \, dx,x,\sqrt [4]{-1+3 x^2}\right )+\frac{3}{2} \operatorname{Subst}\left (\int \frac{x^2}{-1+x^4} \, dx,x,\sqrt [4]{-1+3 x^2}\right )\\ &=-\frac{\left (-1+3 x^2\right )^{3/4}}{4 x^2}+\frac{3}{8} \operatorname{Subst}\left (\int \frac{1-x^2}{\frac{1}{3}+\frac{x^4}{3}} \, dx,x,\sqrt [4]{-1+3 x^2}\right )-\frac{3}{8} \operatorname{Subst}\left (\int \frac{1+x^2}{\frac{1}{3}+\frac{x^4}{3}} \, dx,x,\sqrt [4]{-1+3 x^2}\right )-\frac{3}{4} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt [4]{-1+3 x^2}\right )+\frac{3}{4} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt [4]{-1+3 x^2}\right )\\ &=-\frac{\left (-1+3 x^2\right )^{3/4}}{4 x^2}+\frac{3}{4} \tan ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )-\frac{3}{4} \tanh ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )-\frac{9}{16} \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt [4]{-1+3 x^2}\right )-\frac{9}{16} \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\sqrt [4]{-1+3 x^2}\right )-\frac{9 \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\sqrt [4]{-1+3 x^2}\right )}{16 \sqrt{2}}-\frac{9 \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\sqrt [4]{-1+3 x^2}\right )}{16 \sqrt{2}}\\ &=-\frac{\left (-1+3 x^2\right )^{3/4}}{4 x^2}+\frac{3}{4} \tan ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )-\frac{3}{4} \tanh ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )-\frac{9 \log \left (1-\sqrt{2} \sqrt [4]{-1+3 x^2}+\sqrt{-1+3 x^2}\right )}{16 \sqrt{2}}+\frac{9 \log \left (1+\sqrt{2} \sqrt [4]{-1+3 x^2}+\sqrt{-1+3 x^2}\right )}{16 \sqrt{2}}-\frac{9 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt [4]{-1+3 x^2}\right )}{8 \sqrt{2}}+\frac{9 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt [4]{-1+3 x^2}\right )}{8 \sqrt{2}}\\ &=-\frac{\left (-1+3 x^2\right )^{3/4}}{4 x^2}+\frac{3}{4} \tan ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )+\frac{9 \tan ^{-1}\left (1-\sqrt{2} \sqrt [4]{-1+3 x^2}\right )}{8 \sqrt{2}}-\frac{9 \tan ^{-1}\left (1+\sqrt{2} \sqrt [4]{-1+3 x^2}\right )}{8 \sqrt{2}}-\frac{3}{4} \tanh ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )-\frac{9 \log \left (1-\sqrt{2} \sqrt [4]{-1+3 x^2}+\sqrt{-1+3 x^2}\right )}{16 \sqrt{2}}+\frac{9 \log \left (1+\sqrt{2} \sqrt [4]{-1+3 x^2}+\sqrt{-1+3 x^2}\right )}{16 \sqrt{2}}\\ \end{align*}
Mathematica [C] time = 0.0308546, size = 77, normalized size = 0.4 \[ \frac{1}{4} \left (-3 \left (3 x^2-1\right )^{3/4} \, _2F_1\left (\frac{3}{4},1;\frac{7}{4};1-3 x^2\right )-\frac{\left (3 x^2-1\right )^{3/4}}{x^2}+3 \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )-3 \tanh ^{-1}\left (\sqrt [4]{3 x^2-1}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.087, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3} \left ( 3\,{x}^{2}-2 \right ) }{\frac{1}{\sqrt [4]{3\,{x}^{2}-1}}}}\, 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} - 1\right )}^{\frac{1}{4}}{\left (3 \, x^{2} - 2\right )} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.36684, size = 720, normalized size = 3.77 \begin{align*} \frac{36 \, \sqrt{2} x^{2} \arctan \left (\sqrt{2} \sqrt{\sqrt{2}{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + \sqrt{3 \, x^{2} - 1} + 1} - \sqrt{2}{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} - 1\right ) + 36 \, \sqrt{2} x^{2} \arctan \left (\frac{1}{2} \, \sqrt{2} \sqrt{-4 \, \sqrt{2}{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + 4 \, \sqrt{3 \, x^{2} - 1} + 4} - \sqrt{2}{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + 1\right ) + 9 \, \sqrt{2} x^{2} \log \left (4 \, \sqrt{2}{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + 4 \, \sqrt{3 \, x^{2} - 1} + 4\right ) - 9 \, \sqrt{2} x^{2} \log \left (-4 \, \sqrt{2}{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + 4 \, \sqrt{3 \, x^{2} - 1} + 4\right ) + 24 \, x^{2} \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}}\right ) - 12 \, x^{2} \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + 1\right ) + 12 \, x^{2} \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} - 1\right ) - 8 \,{\left (3 \, x^{2} - 1\right )}^{\frac{3}{4}}}{32 \, x^{2}} \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}{x^{3} \left (3 x^{2} - 2\right ) \sqrt [4]{3 x^{2} - 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.23334, size = 228, normalized size = 1.19 \begin{align*} -\frac{9}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \,{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}}\right )}\right ) - \frac{9}{16} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \,{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}}\right )}\right ) + \frac{9}{32} \, \sqrt{2} \log \left (\sqrt{2}{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + \sqrt{3 \, x^{2} - 1} + 1\right ) - \frac{9}{32} \, \sqrt{2} \log \left (-\sqrt{2}{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + \sqrt{3 \, x^{2} - 1} + 1\right ) - \frac{{\left (3 \, x^{2} - 1\right )}^{\frac{3}{4}}}{4 \, x^{2}} + \frac{3}{4} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}}\right ) - \frac{3}{8} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + 1\right ) + \frac{3}{8} \, \log \left ({\left |{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]