Optimal. Leaf size=429 \[ -\frac {11 \log \left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}-\frac {\sqrt {2} \sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}+1\right )}{128 \sqrt {2} a^3}+\frac {11 \log \left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}+\frac {\sqrt {2} \sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}+1\right )}{128 \sqrt {2} a^3}-\frac {11 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )}{64 \sqrt {2} a^3}+\frac {11 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}+1\right )}{64 \sqrt {2} a^3}+\frac {11 \tan ^{-1}\left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )}{64 a^3}+\frac {11 \tanh ^{-1}\left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )}{64 a^3}+\frac {37 x \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}}{96 a^2}+\frac {1}{3} x^3 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}+\frac {3 x^2 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}}{8 a} \]
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Rubi [A] time = 0.34, 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 = {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 12
Rule 93
Rule 99
Rule 151
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^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*}
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Mathematica [C] time = 5.34, 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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fricas [A] time = 0.49, size = 457, normalized size = 1.07 \[ \frac {132 \, \sqrt {2} a^{3} \frac {1}{a^{12}}^{\frac {1}{4}} \arctan \left (\sqrt {2} \sqrt {\sqrt {2} a^{9} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} \frac {1}{a^{12}}^{\frac {3}{4}} + a^{6} \sqrt {\frac {1}{a^{12}}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}} a^{3} \frac {1}{a^{12}}^{\frac {1}{4}} - \sqrt {2} a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} \frac {1}{a^{12}}^{\frac {1}{4}} - 1\right ) + 132 \, \sqrt {2} a^{3} \frac {1}{a^{12}}^{\frac {1}{4}} \arctan \left (\sqrt {2} \sqrt {-\sqrt {2} a^{9} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} \frac {1}{a^{12}}^{\frac {3}{4}} + a^{6} \sqrt {\frac {1}{a^{12}}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}} a^{3} \frac {1}{a^{12}}^{\frac {1}{4}} - \sqrt {2} a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} \frac {1}{a^{12}}^{\frac {1}{4}} + 1\right ) + 33 \, \sqrt {2} a^{3} \frac {1}{a^{12}}^{\frac {1}{4}} \log \left (\sqrt {2} a^{9} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} \frac {1}{a^{12}}^{\frac {3}{4}} + a^{6} \sqrt {\frac {1}{a^{12}}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right ) - 33 \, \sqrt {2} a^{3} \frac {1}{a^{12}}^{\frac {1}{4}} \log \left (-\sqrt {2} a^{9} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} \frac {1}{a^{12}}^{\frac {3}{4}} + a^{6} \sqrt {\frac {1}{a^{12}}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right ) + 8 \, {\left (32 \, a^{3} x^{3} + 68 \, a^{2} x^{2} + 73 \, a x + 37\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{8}} - 132 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}\right ) + 66 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + 1\right ) - 66 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} - 1\right )}{768 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 333, normalized size = 0.78 \[ -\frac {1}{768} \, a {\left (\frac {66 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}\right )}\right )}{a^{4}} + \frac {66 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}\right )}\right )}{a^{4}} - \frac {33 \, \sqrt {2} \log \left (\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{4}} + \frac {33 \, \sqrt {2} \log \left (-\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{4}} + \frac {132 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}\right )}{a^{4}} - \frac {66 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + 1\right )}{a^{4}} + \frac {66 \, \log \left ({\left | \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} - 1 \right |}\right )}{a^{4}} - \frac {16 \, {\left (\frac {10 \, {\left (a x - 1\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{8}}}{a x + 1} - \frac {33 \, {\left (a x - 1\right )}^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{8}}}{{\left (a x + 1\right )}^{2}} - 105 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{8}}\right )}}{a^{4} {\left (\frac {a x - 1}{a x + 1} - 1\right )}^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.15, 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 = 0.41, size = 341, normalized size = 0.79 \[ -\frac {1}{768} \, a {\left (\frac {16 \, {\left (33 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {23}{8}} - 10 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {15}{8}} + 105 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{8}}\right )}}{\frac {3 \, {\left (a x - 1\right )} a^{4}}{a x + 1} - \frac {3 \, {\left (a x - 1\right )}^{2} a^{4}}{{\left (a x + 1\right )}^{2}} + \frac {{\left (a x - 1\right )}^{3} a^{4}}{{\left (a x + 1\right )}^{3}} - a^{4}} + \frac {33 \, {\left (2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}\right )}\right ) + 2 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}\right )}\right ) - \sqrt {2} \log \left (\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) + \sqrt {2} \log \left (-\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )\right )}}{a^{4}} + \frac {132 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}\right )}{a^{4}} - \frac {66 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + 1\right )}{a^{4}} + \frac {66 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} - 1\right )}{a^{4}}\right )} \]
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 \[ \frac {\frac {35\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/8}}{16}-\frac {5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{15/8}}{24}+\frac {11\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{23/8}}{16}}{a^3+\frac {3\,a^3\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {a^3\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}-\frac {3\,a^3\,\left (a\,x-1\right )}{a\,x+1}}-\frac {\mathrm {atan}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\,1{}\mathrm {i}\right )\,11{}\mathrm {i}}{64\,a^3}-\frac {11\,\mathrm {atan}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\right )}{64\,a^3}+\frac {\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {11}{128}+\frac {11}{128}{}\mathrm {i}\right )}{a^3}+\frac {\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {11}{128}-\frac {11}{128}{}\mathrm {i}\right )}{a^3} \]
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 \[ \int \frac {x^{2}}{\sqrt [8]{\frac {a x - 1}{a x + 1}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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