Optimal. Leaf size=328 \[ -\frac {(1-i a x)^{7/8} \sqrt [8]{1+i a x}}{x}+\frac {i a \log \left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}-\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}+1\right )}{4 \sqrt {2}}-\frac {i a \log \left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}+\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}+1\right )}{4 \sqrt {2}}-\frac {1}{2} i a \tan ^{-1}\left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )+\frac {i a \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )}{2 \sqrt {2}}-\frac {i a \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )}{2 \sqrt {2}}-\frac {1}{2} i a \tanh ^{-1}\left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right ) \]
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Rubi [A] time = 0.13, antiderivative size = 328, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.812, Rules used = {5062, 94, 93, 214, 212, 206, 203, 211, 1165, 628, 1162, 617, 204} \[ -\frac {(1-i a x)^{7/8} \sqrt [8]{1+i a x}}{x}+\frac {i a \log \left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}-\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}+1\right )}{4 \sqrt {2}}-\frac {i a \log \left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}+\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}+1\right )}{4 \sqrt {2}}-\frac {1}{2} i a \tan ^{-1}\left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )+\frac {i a \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )}{2 \sqrt {2}}-\frac {i a \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )}{2 \sqrt {2}}-\frac {1}{2} i a \tanh ^{-1}\left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right ) \]
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 203
Rule 204
Rule 206
Rule 211
Rule 212
Rule 214
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 5062
Rubi steps
\begin {align*} \int \frac {e^{\frac {1}{4} i \tan ^{-1}(a x)}}{x^2} \, dx &=\int \frac {\sqrt [8]{1+i a x}}{x^2 \sqrt [8]{1-i a x}} \, dx\\ &=-\frac {(1-i a x)^{7/8} \sqrt [8]{1+i a x}}{x}+\frac {1}{4} (i a) \int \frac {1}{x \sqrt [8]{1-i a x} (1+i a x)^{7/8}} \, dx\\ &=-\frac {(1-i a x)^{7/8} \sqrt [8]{1+i a x}}{x}+(2 i a) \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 )\\ &=-\frac {(1-i a x)^{7/8} \sqrt [8]{1+i a x}}{x}-(i a) \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 )-(i a) \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 )\\ &=-\frac {(1-i a x)^{7/8} \sqrt [8]{1+i a x}}{x}-\frac {1}{2} (i a) \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 )-\frac {1}{2} (i a) \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 )-\frac {1}{2} (i a) \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 )-\frac {1}{2} (i a) \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 )\\ &=-\frac {(1-i a x)^{7/8} \sqrt [8]{1+i a x}}{x}-\frac {1}{2} i a \tan ^{-1}\left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-\frac {1}{2} i a \tanh ^{-1}\left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-\frac {1}{4} (i a) \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 )-\frac {1}{4} (i a) \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 )+\frac {(i a) \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 )}{4 \sqrt {2}}+\frac {(i a) \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 )}{4 \sqrt {2}}\\ &=-\frac {(1-i a x)^{7/8} \sqrt [8]{1+i a x}}{x}-\frac {1}{2} i a \tan ^{-1}\left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-\frac {1}{2} i a \tanh ^{-1}\left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )+\frac {i a \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 )}{4 \sqrt {2}}-\frac {i a \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 )}{4 \sqrt {2}}-\frac {(i a) \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 )}{2 \sqrt {2}}+\frac {(i a) \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 )}{2 \sqrt {2}}\\ &=-\frac {(1-i a x)^{7/8} \sqrt [8]{1+i a x}}{x}-\frac {1}{2} i a \tan ^{-1}\left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )+\frac {i a \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )}{2 \sqrt {2}}-\frac {i a \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )}{2 \sqrt {2}}-\frac {1}{2} i a \tanh ^{-1}\left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )+\frac {i a \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 )}{4 \sqrt {2}}-\frac {i a \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 )}{4 \sqrt {2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 71, normalized size = 0.22 \[ -\frac {i (1-i a x)^{7/8} \left (2 a x \, _2F_1\left (\frac {7}{8},1;\frac {15}{8};\frac {a x+i}{i-a x}\right )+7 a x-7 i\right )}{7 x (1+i a x)^{7/8}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.54, size = 345, normalized size = 1.05 \[ \frac {-i \, a x \log \left (\left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}} + 1\right ) + a x \log \left (\left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}} + i\right ) - a x \log \left (\left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}} - i\right ) + i \, a x \log \left (\left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}} - 1\right ) + \sqrt {i \, a^{2}} x \log \left (\frac {a \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}} + i \, \sqrt {i \, a^{2}}}{a}\right ) - \sqrt {i \, a^{2}} x \log \left (\frac {a \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}} - i \, \sqrt {i \, a^{2}}}{a}\right ) + \sqrt {-i \, a^{2}} x \log \left (\frac {a \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}} + i \, \sqrt {-i \, a^{2}}}{a}\right ) - \sqrt {-i \, a^{2}} x \log \left (\frac {a \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}} - i \, \sqrt {-i \, a^{2}}}{a}\right ) - 4 \, {\left (-i \, a x + 1\right )} \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}}{4 \, x} \]
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 = 0.15, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )^{\frac {1}{4}}}{x^{2}}\, dx \]
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 \[ \int \frac {\left (\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}\right )^{\frac {1}{4}}}{x^{2}}\,{d x} \]
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 \[ \int \frac {{\left (\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}\right )}^{1/4}}{x^2} \,d x \]
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 \[ \int \frac {\sqrt [4]{\frac {i \left (a x - i\right )}{\sqrt {a^{2} x^{2} + 1}}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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