Optimal. Leaf size=859 \[ -2 \tan ^{-1}\left (\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )+\sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{\sqrt {2+\sqrt {2}}}\right )+\sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{\sqrt {2-\sqrt {2}}}\right )-\sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )-\sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )+\sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )-\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}+1\right )-2 \tanh ^{-1}\left (\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )-\frac {1}{2} \sqrt {2-\sqrt {2}} \log \left (\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1\right )+\frac {1}{2} \sqrt {2-\sqrt {2}} \log \left (\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1\right )-\frac {1}{2} \sqrt {2+\sqrt {2}} \log \left (\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1\right )+\frac {1}{2} \sqrt {2+\sqrt {2}} \log \left (\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1\right )+\frac {\log \left (\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}-\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}+1\right )}{\sqrt {2}}-\frac {\log \left (\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}+\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}+1\right )}{\sqrt {2}} \]
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Rubi [A] time = 0.55, antiderivative size = 859, normalized size of antiderivative = 1.00, number of steps used = 39, number of rules used = 20, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.250, Rules used = {5062, 105, 63, 331, 299, 1122, 1169, 634, 618, 204, 628, 93, 214, 212, 206, 203, 211, 1165, 1162, 617} \[ -2 \tan ^{-1}\left (\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )+\sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{\sqrt {2+\sqrt {2}}}\right )+\sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{\sqrt {2-\sqrt {2}}}\right )-\sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )-\sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )+\sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )-\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}+1\right )-2 \tanh ^{-1}\left (\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )-\frac {1}{2} \sqrt {2-\sqrt {2}} \log \left (\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1\right )+\frac {1}{2} \sqrt {2-\sqrt {2}} \log \left (\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1\right )-\frac {1}{2} \sqrt {2+\sqrt {2}} \log \left (\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1\right )+\frac {1}{2} \sqrt {2+\sqrt {2}} \log \left (\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1\right )+\frac {\log \left (\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}-\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}+1\right )}{\sqrt {2}}-\frac {\log \left (\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}+\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}+1\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 93
Rule 105
Rule 203
Rule 204
Rule 206
Rule 211
Rule 212
Rule 214
Rule 299
Rule 331
Rule 617
Rule 618
Rule 628
Rule 634
Rule 1122
Rule 1162
Rule 1165
Rule 1169
Rule 5062
Rubi steps
\begin {align*} \int \frac {e^{\frac {1}{4} i \tan ^{-1}(a x)}}{x} \, dx &=\int \frac {\sqrt [8]{1+i a x}}{x \sqrt [8]{1-i a x}} \, dx\\ &=(i a) \int \frac {1}{\sqrt [8]{1-i a x} (1+i a x)^{7/8}} \, dx+\int \frac {1}{x \sqrt [8]{1-i a x} (1+i a x)^{7/8}} \, dx\\ &=-\left (8 \operatorname {Subst}\left (\int \frac {x^6}{\left (2-x^8\right )^{7/8}} \, dx,x,\sqrt [8]{1-i a x}\right )\right )+8 \operatorname {Subst}\left (\int \frac {1}{-1+x^8} \, dx,x,\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )\\ &=-\left (4 \operatorname {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )\right )-4 \operatorname {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-8 \operatorname {Subst}\left (\int \frac {x^6}{1+x^8} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )\\ &=-\left (2 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )\right )-2 \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-2 \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-2 \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-\left (2 \sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {x^4}{1-\sqrt {2} x^2+x^4} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )+\left (2 \sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {x^4}{1+\sqrt {2} x^2+x^4} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )\\ &=-2 \tan ^{-1}\left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-2 \tanh ^{-1}\left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )}{\sqrt {2}}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )}{\sqrt {2}}+\left (2 \sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1-\sqrt {2} x^2}{1-\sqrt {2} x^2+x^4} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )-\left (2 \sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1+\sqrt {2} x^2}{1+\sqrt {2} x^2+x^4} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )-\operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-\operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )\\ &=-2 \tan ^{-1}\left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-2 \tanh ^{-1}\left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )+\frac {\log \left (1-\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}+\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )}{\sqrt {2}}-\frac {\log \left (1+\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}+\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )}{\sqrt {2}}-\sqrt {2} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )+\sqrt {2} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )+\sqrt {2-\sqrt {2}} \operatorname {Subst}\left (\int \frac {\sqrt {2+\sqrt {2}}-\left (1+\sqrt {2}\right ) x}{1-\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )+\sqrt {2-\sqrt {2}} \operatorname {Subst}\left (\int \frac {\sqrt {2+\sqrt {2}}+\left (1+\sqrt {2}\right ) x}{1+\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )-\sqrt {2+\sqrt {2}} \operatorname {Subst}\left (\int \frac {\sqrt {2-\sqrt {2}}-\left (1-\sqrt {2}\right ) x}{1-\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )-\sqrt {2+\sqrt {2}} \operatorname {Subst}\left (\int \frac {\sqrt {2-\sqrt {2}}+\left (1-\sqrt {2}\right ) x}{1+\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )\\ &=-2 \tan ^{-1}\left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )+\sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-\sqrt {2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-2 \tanh ^{-1}\left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )+\frac {\log \left (1-\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}+\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )}{\sqrt {2}}-\frac {\log \left (1+\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}+\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )}{\sqrt {2}}-\frac {1}{2} \sqrt {2-\sqrt {2}} \operatorname {Subst}\left (\int \frac {-\sqrt {2-\sqrt {2}}+2 x}{1-\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )+\frac {1}{2} \sqrt {2-\sqrt {2}} \operatorname {Subst}\left (\int \frac {\sqrt {2-\sqrt {2}}+2 x}{1+\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )+\frac {1}{2} \left (-2+\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )+\frac {1}{2} \left (-2+\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )-\frac {1}{2} \sqrt {2+\sqrt {2}} \operatorname {Subst}\left (\int \frac {-\sqrt {2+\sqrt {2}}+2 x}{1-\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )+\frac {1}{2} \sqrt {2+\sqrt {2}} \operatorname {Subst}\left (\int \frac {\sqrt {2+\sqrt {2}}+2 x}{1+\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )-\frac {1}{2} \left (2+\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )-\frac {1}{2} \left (2+\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )\\ &=-2 \tan ^{-1}\left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )+\sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-\sqrt {2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-2 \tanh ^{-1}\left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-\frac {1}{2} \sqrt {2-\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )+\frac {1}{2} \sqrt {2-\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )-\frac {1}{2} \sqrt {2+\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )+\frac {1}{2} \sqrt {2+\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )+\frac {\log \left (1-\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}+\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )}{\sqrt {2}}-\frac {\log \left (1+\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}+\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )}{\sqrt {2}}+\left (2-\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{-2+\sqrt {2}-x^2} \, dx,x,-\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )+\left (2-\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{-2+\sqrt {2}-x^2} \, dx,x,\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )+\left (2+\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{-2-\sqrt {2}-x^2} \, dx,x,-\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )+\left (2+\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{-2-\sqrt {2}-x^2} \, dx,x,\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )\\ &=-2 \tan ^{-1}\left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )+\sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2+\sqrt {2}}}\right )+\sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2-\sqrt {2}}}\right )-\sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2+\sqrt {2}}}\right )-\sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2-\sqrt {2}}}\right )+\sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-\sqrt {2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-2 \tanh ^{-1}\left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-\frac {1}{2} \sqrt {2-\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )+\frac {1}{2} \sqrt {2-\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )-\frac {1}{2} \sqrt {2+\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )+\frac {1}{2} \sqrt {2+\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )+\frac {\log \left (1-\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}+\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )}{\sqrt {2}}-\frac {\log \left (1+\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}+\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 97, normalized size = 0.11 \[ -\frac {4 (1-i a x)^{7/8} \left (\sqrt [8]{2} (1+i a x)^{7/8} \, _2F_1\left (\frac {7}{8},\frac {7}{8};\frac {15}{8};\frac {1}{2} (1-i a x)\right )+2 \, _2F_1\left (\frac {7}{8},1;\frac {15}{8};\frac {a x+i}{i-a x}\right )\right )}{7 (1+i a x)^{7/8}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.63, size = 509, normalized size = 0.59 \[ -\frac {1}{2} \, \sqrt {4 i} \log \left (\frac {1}{2} \, \sqrt {4 i} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) + \frac {1}{2} \, \sqrt {4 i} \log \left (-\frac {1}{2} \, \sqrt {4 i} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) - \frac {1}{2} \, \sqrt {-4 i} \log \left (\frac {1}{2} \, \sqrt {-4 i} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) + \frac {1}{2} \, \sqrt {-4 i} \log \left (-\frac {1}{2} \, \sqrt {-4 i} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) + i^{\frac {1}{4}} \log \left (i^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) + i \, i^{\frac {1}{4}} \log \left (i \, i^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) - i \, i^{\frac {1}{4}} \log \left (-i \, i^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) - i^{\frac {1}{4}} \log \left (-i^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) + \left (-i\right )^{\frac {1}{4}} \log \left (\left (-i\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) + i \, \left (-i\right )^{\frac {1}{4}} \log \left (i \, \left (-i\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) - i \, \left (-i\right )^{\frac {1}{4}} \log \left (-i \, \left (-i\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) - \left (-i\right )^{\frac {1}{4}} \log \left (-\left (-i\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) - \log \left (\left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}} + 1\right ) - i \, \log \left (\left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}} + i\right ) + i \, \log \left (\left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}} - i\right ) + \log \left (\left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.16, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )^{\frac {1}{4}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}\right )^{\frac {1}{4}}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}\right )}^{1/4}}{x} \,d x \]
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 \[ \int \frac {\sqrt [4]{\frac {i \left (a x - i\right )}{\sqrt {a^{2} x^{2} + 1}}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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