Optimal. Leaf size=429 \[ \frac{37 x \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{\frac{1}{a x}+1}}{96 a^2}-\frac{11 \log \left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}-\frac{\sqrt{2} \sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}+1\right )}{128 \sqrt{2} a^3}+\frac{11 \log \left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}+\frac{\sqrt{2} \sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}+1\right )}{128 \sqrt{2} a^3}-\frac{11 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{64 \sqrt{2} a^3}+\frac{11 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}+1\right )}{64 \sqrt{2} a^3}+\frac{11 \tan ^{-1}\left (\frac{\sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{64 a^3}+\frac{11 \tanh ^{-1}\left (\frac{\sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{64 a^3}+\frac{1}{3} x^3 \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{\frac{1}{a x}+1}+\frac{3 x^2 \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{\frac{1}{a x}+1}}{8 a} \]
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Rubi [A] time = 0.343278, antiderivative size = 429, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 15, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.071, Rules used = {6171, 99, 151, 12, 93, 214, 212, 206, 203, 211, 1165, 628, 1162, 617, 204} \[ \frac{37 x \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{\frac{1}{a x}+1}}{96 a^2}-\frac{11 \log \left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}-\frac{\sqrt{2} \sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}+1\right )}{128 \sqrt{2} a^3}+\frac{11 \log \left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}+\frac{\sqrt{2} \sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}+1\right )}{128 \sqrt{2} a^3}-\frac{11 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{64 \sqrt{2} a^3}+\frac{11 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}+1\right )}{64 \sqrt{2} a^3}+\frac{11 \tan ^{-1}\left (\frac{\sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{64 a^3}+\frac{11 \tanh ^{-1}\left (\frac{\sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{64 a^3}+\frac{1}{3} x^3 \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{\frac{1}{a x}+1}+\frac{3 x^2 \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{\frac{1}{a x}+1}}{8 a} \]
Antiderivative was successfully verified.
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Rule 6171
Rule 99
Rule 151
Rule 12
Rule 93
Rule 214
Rule 212
Rule 206
Rule 203
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int e^{\frac{1}{4} \coth ^{-1}(a x)} x^2 \, dx &=-\operatorname{Subst}\left (\int \frac{\sqrt [8]{1+\frac{x}{a}}}{x^4 \sqrt [8]{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{3} \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x^3-\frac{1}{3} \operatorname{Subst}\left (\int \frac{\frac{9}{4 a}+\frac{2 x}{a^2}}{x^3 \sqrt [8]{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{7/8}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{3 \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x^2}{8 a}+\frac{1}{3} \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x^3+\frac{1}{6} \operatorname{Subst}\left (\int \frac{-\frac{37}{16 a^2}-\frac{9 x}{4 a^3}}{x^2 \sqrt [8]{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{7/8}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{37 \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x}{96 a^2}+\frac{3 \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x^2}{8 a}+\frac{1}{3} \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x^3-\frac{1}{6} \operatorname{Subst}\left (\int \frac{33}{64 a^3 x \sqrt [8]{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{7/8}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{37 \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x}{96 a^2}+\frac{3 \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x^2}{8 a}+\frac{1}{3} \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x^3-\frac{11 \operatorname{Subst}\left (\int \frac{1}{x \sqrt [8]{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{7/8}} \, dx,x,\frac{1}{x}\right )}{128 a^3}\\ &=\frac{37 \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x}{96 a^2}+\frac{3 \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x^2}{8 a}+\frac{1}{3} \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x^3-\frac{11 \operatorname{Subst}\left (\int \frac{1}{-1+x^8} \, dx,x,\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{16 a^3}\\ &=\frac{37 \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x}{96 a^2}+\frac{3 \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x^2}{8 a}+\frac{1}{3} \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x^3+\frac{11 \operatorname{Subst}\left (\int \frac{1}{1-x^4} \, dx,x,\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{32 a^3}+\frac{11 \operatorname{Subst}\left (\int \frac{1}{1+x^4} \, dx,x,\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{32 a^3}\\ &=\frac{37 \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x}{96 a^2}+\frac{3 \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x^2}{8 a}+\frac{1}{3} \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x^3+\frac{11 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{64 a^3}+\frac{11 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{64 a^3}+\frac{11 \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{64 a^3}+\frac{11 \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{64 a^3}\\ &=\frac{37 \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x}{96 a^2}+\frac{3 \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x^2}{8 a}+\frac{1}{3} \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x^3+\frac{11 \tan ^{-1}\left (\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{64 a^3}+\frac{11 \tanh ^{-1}\left (\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{64 a^3}+\frac{11 \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{128 a^3}+\frac{11 \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{128 a^3}-\frac{11 \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{128 \sqrt{2} a^3}-\frac{11 \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{128 \sqrt{2} a^3}\\ &=\frac{37 \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x}{96 a^2}+\frac{3 \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x^2}{8 a}+\frac{1}{3} \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x^3+\frac{11 \tan ^{-1}\left (\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{64 a^3}+\frac{11 \tanh ^{-1}\left (\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{64 a^3}-\frac{11 \log \left (1-\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}+\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{128 \sqrt{2} a^3}+\frac{11 \log \left (1+\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}+\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{128 \sqrt{2} a^3}+\frac{11 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{64 \sqrt{2} a^3}-\frac{11 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{64 \sqrt{2} a^3}\\ &=\frac{37 \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x}{96 a^2}+\frac{3 \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x^2}{8 a}+\frac{1}{3} \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x^3-\frac{11 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{64 \sqrt{2} a^3}+\frac{11 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{64 \sqrt{2} a^3}+\frac{11 \tan ^{-1}\left (\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{64 a^3}+\frac{11 \tanh ^{-1}\left (\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{64 a^3}-\frac{11 \log \left (1-\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}+\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{128 \sqrt{2} a^3}+\frac{11 \log \left (1+\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}+\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{128 \sqrt{2} a^3}\\ \end{align*}
Mathematica [C] time = 5.30153, size = 167, normalized size = 0.39 \[ \frac{-33 \text{RootSum}\left [\text{$\#$1}^4+1\& ,\frac{\coth ^{-1}(a x)-4 \log \left (e^{\frac{1}{4} \coth ^{-1}(a x)}-\text{$\#$1}\right )}{\text{$\#$1}^3}\& \right ]-4 \left (-\frac{840 e^{\frac{1}{4} \coth ^{-1}(a x)}}{e^{2 \coth ^{-1}(a x)}-1}-\frac{1600 e^{\frac{1}{4} \coth ^{-1}(a x)}}{\left (e^{2 \coth ^{-1}(a x)}-1\right )^2}-\frac{1024 e^{\frac{1}{4} \coth ^{-1}(a x)}}{\left (e^{2 \coth ^{-1}(a x)}-1\right )^3}+33 \log \left (1-e^{\frac{1}{4} \coth ^{-1}(a x)}\right )-33 \log \left (e^{\frac{1}{4} \coth ^{-1}(a x)}+1\right )-66 \tan ^{-1}\left (e^{\frac{1}{4} \coth ^{-1}(a x)}\right )\right )}{1536 a^3} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.168, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2}{\frac{1}{\sqrt [8]{{\frac{ax-1}{ax+1}}}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.55936, size = 460, normalized size = 1.07 \begin{align*} -\frac{1}{768} \, a{\left (\frac{16 \,{\left (33 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{23}{8}} - 10 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{15}{8}} + 105 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{8}}\right )}}{\frac{3 \,{\left (a x - 1\right )} a^{4}}{a x + 1} - \frac{3 \,{\left (a x - 1\right )}^{2} a^{4}}{{\left (a x + 1\right )}^{2}} + \frac{{\left (a x - 1\right )}^{3} a^{4}}{{\left (a x + 1\right )}^{3}} - a^{4}} + \frac{33 \,{\left (2 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}}\right )}\right ) + 2 \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}}\right )}\right ) - \sqrt{2} \log \left (\sqrt{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} + \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right ) + \sqrt{2} \log \left (-\sqrt{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} + \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right )\right )}}{a^{4}} + \frac{132 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}}\right )}{a^{4}} - \frac{66 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} + 1\right )}{a^{4}} + \frac{66 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} - 1\right )}{a^{4}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.91345, size = 1332, normalized size = 3.1 \begin{align*} \frac{132 \, \sqrt{2} a^{3} \frac{1}{a^{12}}^{\frac{1}{4}} \arctan \left (\sqrt{2} \sqrt{\sqrt{2} a^{9} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} \frac{1}{a^{12}}^{\frac{3}{4}} + a^{6} \sqrt{\frac{1}{a^{12}}} + \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}} a^{3} \frac{1}{a^{12}}^{\frac{1}{4}} - \sqrt{2} a^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} \frac{1}{a^{12}}^{\frac{1}{4}} - 1\right ) + 132 \, \sqrt{2} a^{3} \frac{1}{a^{12}}^{\frac{1}{4}} \arctan \left (\sqrt{2} \sqrt{-\sqrt{2} a^{9} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} \frac{1}{a^{12}}^{\frac{3}{4}} + a^{6} \sqrt{\frac{1}{a^{12}}} + \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}} a^{3} \frac{1}{a^{12}}^{\frac{1}{4}} - \sqrt{2} a^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} \frac{1}{a^{12}}^{\frac{1}{4}} + 1\right ) + 33 \, \sqrt{2} a^{3} \frac{1}{a^{12}}^{\frac{1}{4}} \log \left (\sqrt{2} a^{9} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} \frac{1}{a^{12}}^{\frac{3}{4}} + a^{6} \sqrt{\frac{1}{a^{12}}} + \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right ) - 33 \, \sqrt{2} a^{3} \frac{1}{a^{12}}^{\frac{1}{4}} \log \left (-\sqrt{2} a^{9} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} \frac{1}{a^{12}}^{\frac{3}{4}} + a^{6} \sqrt{\frac{1}{a^{12}}} + \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right ) + 8 \,{\left (32 \, a^{3} x^{3} + 68 \, a^{2} x^{2} + 73 \, a x + 37\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{8}} - 132 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}}\right ) + 66 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} + 1\right ) - 66 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} - 1\right )}{768 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22453, size = 450, normalized size = 1.05 \begin{align*} -\frac{1}{768} \, a{\left (\frac{66 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}}\right )}\right )}{a^{4}} + \frac{66 \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}}\right )}\right )}{a^{4}} - \frac{33 \, \sqrt{2} \log \left (\sqrt{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} + \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right )}{a^{4}} + \frac{33 \, \sqrt{2} \log \left (-\sqrt{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} + \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right )}{a^{4}} + \frac{132 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}}\right )}{a^{4}} - \frac{66 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} + 1\right )}{a^{4}} + \frac{66 \, \log \left ({\left | \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} - 1 \right |}\right )}{a^{4}} - \frac{16 \,{\left (\frac{10 \,{\left (a x - 1\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{8}}}{a x + 1} - \frac{33 \,{\left (a x - 1\right )}^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{8}}}{{\left (a x + 1\right )}^{2}} - 105 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{8}}\right )}}{a^{4}{\left (\frac{a x - 1}{a x + 1} - 1\right )}^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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