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 = 0.132148, antiderivative size = 328, normalized size of antiderivative = 1., 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 5062
Rule 94
Rule 93
Rule 214
Rule 212
Rule 206
Rule 203
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
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*}
Mathematica [C] time = 0.0162501, size = 71, normalized size = 0.22 \[ -\frac{i (1-i a x)^{7/8} \left (2 a x \text{Hypergeometric2F1}\left (\frac{7}{8},1,\frac{15}{8},\frac{a x+i}{-a x+i}\right )+7 a x-7 i\right )}{7 x (1+i a x)^{7/8}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.063, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}}\sqrt [4]{{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\frac{i \, a x + 1}{\sqrt{a^{2} x^{2} + 1}}\right )^{\frac{1}{4}}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82177, size = 819, normalized size = 2.5 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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