Optimal. Leaf size=429 \[ \frac {37 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}} x}{96 a^2}+\frac {3 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}} x^2}{8 a}+\frac {1}{3} \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}} x^3-\frac {11 \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )}{64 \sqrt {2} a^3}+\frac {11 \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )}{64 \sqrt {2} a^3}+\frac {11 \text {ArcTan}\left (\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )}{64 a^3}+\frac {11 \tanh ^{-1}\left (\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )}{64 a^3}-\frac {11 \log \left (1-\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{128 \sqrt {2} a^3}+\frac {11 \log \left (1+\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{128 \sqrt {2} a^3} \]
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Rubi [A]
time = 0.24, antiderivative size = 429, normalized size of antiderivative = 1.00, 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 = {6306, 101, 156, 12, 95, 220, 218, 212, 209, 217, 1179, 642, 1176, 631, 210}
\begin {gather*} -\frac {11 \text {ArcTan}\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 \text {ArcTan}\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 \text {ArcTan}\left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )}{64 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 \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 \tanh ^{-1}\left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )}{64 a^3}+\frac {37 x \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}}{96 a^2}+\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 95
Rule 101
Rule 156
Rule 209
Rule 210
Rule 212
Rule 217
Rule 218
Rule 220
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 6306
Rubi steps
\begin {align*} \int e^{\frac {1}{4} \coth ^{-1}(a x)} x^2 \, dx &=-\text {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} \text {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} \text {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} \text {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 \text {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 \text {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 \text {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 \text {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 \text {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 \text {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 \text {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 \text {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 \text {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 \text {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 \text {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 \text {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 \text {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 \text {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*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 6.11, size = 399, normalized size = 0.93 \begin {gather*} -\frac {e^{\frac {9}{4} \coth ^{-1}(a x)} \left (20905836325-62455078125 e^{-6 \coth ^{-1}(a x)}-51095314325 e^{-4 \coth ^{-1}(a x)}+25918688125 e^{-2 \coth ^{-1}(a x)}-1776332800 e^{2 \coth ^{-1}(a x)}-22053358800 \, _2F_1\left (\frac {1}{8},1;\frac {9}{8};e^{2 \coth ^{-1}(a x)}\right )+62455078125 e^{-6 \coth ^{-1}(a x)} \, _2F_1\left (\frac {1}{8},1;\frac {9}{8};e^{2 \coth ^{-1}(a x)}\right )+44155861200 e^{-4 \coth ^{-1}(a x)} \, _2F_1\left (\frac {1}{8},1;\frac {9}{8};e^{2 \coth ^{-1}(a x)}\right )-34498723050 e^{-2 \coth ^{-1}(a x)} \, _2F_1\left (\frac {1}{8},1;\frac {9}{8};e^{2 \coth ^{-1}(a x)}\right )+2913904125 e^{2 \coth ^{-1}(a x)} \, _2F_1\left (\frac {1}{8},1;\frac {9}{8};e^{2 \coth ^{-1}(a x)}\right )+12288 e^{2 \coth ^{-1}(a x)} \left (3145+5122 e^{2 \coth ^{-1}(a x)}+2105 e^{4 \coth ^{-1}(a x)}\right ) \, _4F_3\left (2,2,2,\frac {17}{8};1,1,\frac {41}{8};e^{2 \coth ^{-1}(a x)}\right )+196608 e^{2 \coth ^{-1}(a x)} \left (45+82 e^{2 \coth ^{-1}(a x)}+37 e^{4 \coth ^{-1}(a x)}\right ) \, _5F_4\left (2,2,2,2,\frac {17}{8};1,1,1,\frac {41}{8};e^{2 \coth ^{-1}(a x)}\right )+786432 e^{2 \coth ^{-1}(a x)} \, _6F_5\left (2,2,2,2,2,\frac {17}{8};1,1,1,1,\frac {41}{8};e^{2 \coth ^{-1}(a x)}\right )+1572864 e^{4 \coth ^{-1}(a x)} \, _6F_5\left (2,2,2,2,2,\frac {17}{8};1,1,1,1,\frac {41}{8};e^{2 \coth ^{-1}(a x)}\right )+786432 e^{6 \coth ^{-1}(a x)} \, _6F_5\left (2,2,2,2,2,\frac {17}{8};1,1,1,1,\frac {41}{8};e^{2 \coth ^{-1}(a x)}\right )\right )}{64627200 a^3} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {x^{2}}{\left (\frac {a x -1}{a x +1}\right )^{\frac {1}{8}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 341, normalized size = 0.79 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 457, normalized size = 1.07 \begin {gather*} \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 {gather*}
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 \begin {gather*} \int \frac {x^{2}}{\sqrt [8]{\frac {a x - 1}{a x + 1}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 308, normalized size = 0.72 \begin {gather*} -\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 (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + 1\right )}{a^{4}} + \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 )}}{a^{4} {\left (\frac {a x - 1}{a x + 1} - 1\right )}^{3}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.31, size = 227, normalized size = 0.53 \begin {gather*} \frac {\frac {35\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/8}}{16}-\frac {5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{15/8}}{24}+\frac {11\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{23/8}}{16}}{a^3+\frac {3\,a^3\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {a^3\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}-\frac {3\,a^3\,\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 )\,11{}\mathrm {i}}{64\,a^3}-\frac {11\,\mathrm {atan}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\right )}{64\,a^3}+\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 {11}{128}+\frac {11}{128}{}\mathrm {i}\right )}{a^3}+\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 {11}{128}-\frac {11}{128}{}\mathrm {i}\right )}{a^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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