Optimal. Leaf size=92 \[ -\frac{(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{x}-i a \tan ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-i a \tanh ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right ) \]
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Rubi [A] time = 0.0338023, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {5062, 94, 93, 212, 206, 203} \[ -\frac{(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{x}-i a \tan ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-i a \tanh ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right ) \]
Antiderivative was successfully verified.
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Rule 5062
Rule 94
Rule 93
Rule 212
Rule 206
Rule 203
Rubi steps
\begin{align*} \int \frac{e^{\frac{1}{2} i \tan ^{-1}(a x)}}{x^2} \, dx &=\int \frac{\sqrt [4]{1+i a x}}{x^2 \sqrt [4]{1-i a x}} \, dx\\ &=-\frac{(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{x}+\frac{1}{2} (i a) \int \frac{1}{x \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx\\ &=-\frac{(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{x}+(2 i a) \operatorname{Subst}\left (\int \frac{1}{-1+x^4} \, dx,x,\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )\\ &=-\frac{(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{x}-(i a) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-(i a) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )\\ &=-\frac{(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{x}-i a \tan ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-i a \tanh ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )\\ \end{align*}
Mathematica [C] time = 0.0149695, size = 71, normalized size = 0.77 \[ -\frac{i (1-i a x)^{3/4} \left (2 a x \text{Hypergeometric2F1}\left (\frac{3}{4},1,\frac{7}{4},\frac{a x+i}{-a x+i}\right )+3 a x-3 i\right )}{3 x (1+i a x)^{3/4}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.12, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}}\sqrt{{(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{\sqrt{\frac{i \, a x + 1}{\sqrt{a^{2} x^{2} + 1}}}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.63382, size = 363, normalized size = 3.95 \begin{align*} \frac{-i \, a x \log \left (\sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} + 1\right ) + a x \log \left (\sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} + i\right ) - a x \log \left (\sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} - i\right ) + i \, a x \log \left (\sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} - 1\right ) - 2 \,{\left (-i \, a x + 1\right )} \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}}{2 \, 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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{i \, a x + 1}{\sqrt{a^{2} x^{2} + 1}}}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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