Optimal. Leaf size=392 \[ -\frac {\log \left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}-\frac {\sqrt {2} \sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}+1\right )}{32 \sqrt {2} a^2}+\frac {\log \left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}+\frac {\sqrt {2} \sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}+1\right )}{32 \sqrt {2} a^2}-\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )}{16 \sqrt {2} a^2}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}+1\right )}{16 \sqrt {2} a^2}+\frac {\tan ^{-1}\left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )}{16 a^2}+\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )}{16 a^2}+\frac {1}{2} x^2 \left (1-\frac {1}{a x}\right )^{7/8} \left (\frac {1}{a x}+1\right )^{9/8}+\frac {x \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}}{8 a} \]
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Rubi [A] time = 0.25, antiderivative size = 392, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 14, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.167, Rules used = {6171, 96, 94, 93, 214, 212, 206, 203, 211, 1165, 628, 1162, 617, 204} \[ -\frac {\log \left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}-\frac {\sqrt {2} \sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}+1\right )}{32 \sqrt {2} a^2}+\frac {\log \left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}+\frac {\sqrt {2} \sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}+1\right )}{32 \sqrt {2} a^2}-\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )}{16 \sqrt {2} a^2}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}+1\right )}{16 \sqrt {2} a^2}+\frac {\tan ^{-1}\left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )}{16 a^2}+\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )}{16 a^2}+\frac {1}{2} x^2 \left (1-\frac {1}{a x}\right )^{7/8} \left (\frac {1}{a x}+1\right )^{9/8}+\frac {x \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}}{8 a} \]
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 96
Rule 203
Rule 204
Rule 206
Rule 211
Rule 212
Rule 214
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 6171
Rubi steps
\begin {align*} \int e^{\frac {1}{4} \coth ^{-1}(a x)} x \, dx &=-\operatorname {Subst}\left (\int \frac {\sqrt [8]{1+\frac {x}{a}}}{x^3 \sqrt [8]{1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{2} \left (1-\frac {1}{a x}\right )^{7/8} \left (1+\frac {1}{a x}\right )^{9/8} x^2-\frac {\operatorname {Subst}\left (\int \frac {\sqrt [8]{1+\frac {x}{a}}}{x^2 \sqrt [8]{1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{8 a}\\ &=\frac {\left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}} x}{8 a}+\frac {1}{2} \left (1-\frac {1}{a x}\right )^{7/8} \left (1+\frac {1}{a x}\right )^{9/8} x^2-\frac {\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 )}{32 a^2}\\ &=\frac {\left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}} x}{8 a}+\frac {1}{2} \left (1-\frac {1}{a x}\right )^{7/8} \left (1+\frac {1}{a x}\right )^{9/8} x^2-\frac {\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 )}{4 a^2}\\ &=\frac {\left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}} x}{8 a}+\frac {1}{2} \left (1-\frac {1}{a x}\right )^{7/8} \left (1+\frac {1}{a x}\right )^{9/8} x^2+\frac {\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 )}{8 a^2}+\frac {\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 )}{8 a^2}\\ &=\frac {\left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}} x}{8 a}+\frac {1}{2} \left (1-\frac {1}{a x}\right )^{7/8} \left (1+\frac {1}{a x}\right )^{9/8} x^2+\frac {\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 )}{16 a^2}+\frac {\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 )}{16 a^2}+\frac {\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 )}{16 a^2}+\frac {\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 )}{16 a^2}\\ &=\frac {\left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}} x}{8 a}+\frac {1}{2} \left (1-\frac {1}{a x}\right )^{7/8} \left (1+\frac {1}{a x}\right )^{9/8} x^2+\frac {\tan ^{-1}\left (\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )}{16 a^2}+\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )}{16 a^2}+\frac {\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 )}{32 a^2}+\frac {\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 )}{32 a^2}-\frac {\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 )}{32 \sqrt {2} a^2}-\frac {\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 )}{32 \sqrt {2} a^2}\\ &=\frac {\left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}} x}{8 a}+\frac {1}{2} \left (1-\frac {1}{a x}\right )^{7/8} \left (1+\frac {1}{a x}\right )^{9/8} x^2+\frac {\tan ^{-1}\left (\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )}{16 a^2}+\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )}{16 a^2}-\frac {\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 )}{32 \sqrt {2} a^2}+\frac {\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 )}{32 \sqrt {2} a^2}+\frac {\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 )}{16 \sqrt {2} a^2}-\frac {\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 )}{16 \sqrt {2} a^2}\\ &=\frac {\left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}} x}{8 a}+\frac {1}{2} \left (1-\frac {1}{a x}\right )^{7/8} \left (1+\frac {1}{a x}\right )^{9/8} x^2-\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )}{16 \sqrt {2} a^2}+\frac {\tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )}{16 \sqrt {2} a^2}+\frac {\tan ^{-1}\left (\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )}{16 a^2}+\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )}{16 a^2}-\frac {\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 )}{32 \sqrt {2} a^2}+\frac {\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 )}{32 \sqrt {2} a^2}\\ \end {align*}
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Mathematica [A] time = 0.80, size = 319, normalized size = 0.81 \[ \frac {\frac {6}{e^{\frac {1}{4} \coth ^{-1}(a x)}-1}+\frac {6}{e^{\frac {1}{4} \coth ^{-1}(a x)}+1}-\frac {12 e^{\frac {1}{4} \coth ^{-1}(a x)}}{e^{\frac {1}{2} \coth ^{-1}(a x)}+1}-\frac {40 e^{\frac {1}{4} \coth ^{-1}(a x)}}{e^{\coth ^{-1}(a x)}+1}+\frac {2}{\left (e^{\frac {1}{4} \coth ^{-1}(a x)}-1\right )^2}-\frac {2}{\left (e^{\frac {1}{4} \coth ^{-1}(a x)}+1\right )^2}+\frac {8 e^{\frac {1}{4} \coth ^{-1}(a x)}}{\left (e^{\frac {1}{2} \coth ^{-1}(a x)}+1\right )^2}+\frac {32 e^{\frac {1}{4} \coth ^{-1}(a x)}}{\left (e^{\coth ^{-1}(a x)}+1\right )^2}-\sqrt {2} \log \left (-\sqrt {2} e^{\frac {1}{4} \coth ^{-1}(a x)}+e^{\frac {1}{2} \coth ^{-1}(a x)}+1\right )+\sqrt {2} \log \left (\sqrt {2} e^{\frac {1}{4} \coth ^{-1}(a x)}+e^{\frac {1}{2} \coth ^{-1}(a x)}+1\right )+4 \tan ^{-1}\left (e^{\frac {1}{4} \coth ^{-1}(a x)}\right )-2 \sqrt {2} \tan ^{-1}\left (1-\sqrt {2} e^{\frac {1}{4} \coth ^{-1}(a x)}\right )+2 \sqrt {2} \tan ^{-1}\left (\sqrt {2} e^{\frac {1}{4} \coth ^{-1}(a x)}+1\right )+4 \tanh ^{-1}\left (e^{\frac {1}{4} \coth ^{-1}(a x)}\right )}{64 a^2} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.50, size = 448, normalized size = 1.14 \[ \frac {4 \, \sqrt {2} a^{2} \frac {1}{a^{8}}^{\frac {1}{4}} \arctan \left (\sqrt {2} \sqrt {\sqrt {2} a^{6} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} \frac {1}{a^{8}}^{\frac {3}{4}} + a^{4} \sqrt {\frac {1}{a^{8}}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}} a^{2} \frac {1}{a^{8}}^{\frac {1}{4}} - \sqrt {2} a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} \frac {1}{a^{8}}^{\frac {1}{4}} - 1\right ) + 4 \, \sqrt {2} a^{2} \frac {1}{a^{8}}^{\frac {1}{4}} \arctan \left (\sqrt {2} \sqrt {-\sqrt {2} a^{6} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} \frac {1}{a^{8}}^{\frac {3}{4}} + a^{4} \sqrt {\frac {1}{a^{8}}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}} a^{2} \frac {1}{a^{8}}^{\frac {1}{4}} - \sqrt {2} a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} \frac {1}{a^{8}}^{\frac {1}{4}} + 1\right ) + \sqrt {2} a^{2} \frac {1}{a^{8}}^{\frac {1}{4}} \log \left (\sqrt {2} a^{6} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} \frac {1}{a^{8}}^{\frac {3}{4}} + a^{4} \sqrt {\frac {1}{a^{8}}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right ) - \sqrt {2} a^{2} \frac {1}{a^{8}}^{\frac {1}{4}} \log \left (-\sqrt {2} a^{6} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} \frac {1}{a^{8}}^{\frac {3}{4}} + a^{4} \sqrt {\frac {1}{a^{8}}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right ) + 8 \, {\left (4 \, a^{2} x^{2} + 9 \, a x + 5\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{8}} - 4 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}\right ) + 2 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + 1\right ) - 2 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} - 1\right )}{64 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 300, normalized size = 0.77 \[ -\frac {1}{64} \, a {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}\right )}\right )}{a^{3}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}\right )}\right )}{a^{3}} - \frac {\sqrt {2} \log \left (\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{3}} + \frac {\sqrt {2} \log \left (-\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{3}} + \frac {4 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}\right )}{a^{3}} - \frac {2 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + 1\right )}{a^{3}} + \frac {2 \, \log \left ({\left | \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} - 1 \right |}\right )}{a^{3}} + \frac {16 \, {\left (\frac {{\left (a x - 1\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{8}}}{a x + 1} - 9 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{8}}\right )}}{a^{3} {\left (\frac {a x - 1}{a x + 1} - 1\right )}^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (\frac {a x -1}{a x +1}\right )^{\frac {1}{8}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 304, normalized size = 0.78 \[ \frac {1}{64} \, a {\left (\frac {16 \, {\left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {15}{8}} - 9 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{8}}\right )}}{\frac {2 \, {\left (a x - 1\right )} a^{3}}{a x + 1} - \frac {{\left (a x - 1\right )}^{2} a^{3}}{{\left (a x + 1\right )}^{2}} - a^{3}} - \frac {2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}\right )}\right ) + 2 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}\right )}\right ) - \sqrt {2} \log \left (\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) + \sqrt {2} \log \left (-\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{3}} - \frac {4 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}\right )}{a^{3}} + \frac {2 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + 1\right )}{a^{3}} - \frac {2 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} - 1\right )}{a^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.28, size = 190, normalized size = 0.48 \[ \frac {\frac {9\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/8}}{4}-\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{15/8}}{4}}{a^2+\frac {a^2\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {2\,a^2\,\left (a\,x-1\right )}{a\,x+1}}-\frac {\mathrm {atan}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{16\,a^2}-\frac {\mathrm {atan}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\right )}{16\,a^2}+\frac {\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {1}{32}+\frac {1}{32}{}\mathrm {i}\right )}{a^2}+\frac {\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {1}{32}-\frac {1}{32}{}\mathrm {i}\right )}{a^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sqrt [8]{\frac {a x - 1}{a x + 1}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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