Optimal. Leaf size=132 \[ \frac{1}{4} a^2 \tan ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac{1}{4} a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac{(1-i a x)^{3/4} (1+i a x)^{5/4}}{2 x^2}-\frac{i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x} \]
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Rubi [A] time = 0.0426426, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {5062, 96, 94, 93, 212, 206, 203} \[ \frac{1}{4} a^2 \tan ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac{1}{4} a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac{(1-i a x)^{3/4} (1+i a x)^{5/4}}{2 x^2}-\frac{i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x} \]
Antiderivative was successfully verified.
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Rule 5062
Rule 96
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^3} \, dx &=\int \frac{\sqrt [4]{1+i a x}}{x^3 \sqrt [4]{1-i a x}} \, dx\\ &=-\frac{(1-i a x)^{3/4} (1+i a x)^{5/4}}{2 x^2}+\frac{1}{4} (i a) \int \frac{\sqrt [4]{1+i a x}}{x^2 \sqrt [4]{1-i a x}} \, dx\\ &=-\frac{i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x}-\frac{(1-i a x)^{3/4} (1+i a x)^{5/4}}{2 x^2}-\frac{1}{8} a^2 \int \frac{1}{x \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx\\ &=-\frac{i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x}-\frac{(1-i a x)^{3/4} (1+i a x)^{5/4}}{2 x^2}-\frac{1}{2} a^2 \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{i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x}-\frac{(1-i a x)^{3/4} (1+i a x)^{5/4}}{2 x^2}+\frac{1}{4} a^2 \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}{4} a^2 \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{i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x}-\frac{(1-i a x)^{3/4} (1+i a x)^{5/4}}{2 x^2}+\frac{1}{4} a^2 \tan ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac{1}{4} a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )\\ \end{align*}
Mathematica [C] time = 0.0181263, size = 81, normalized size = 0.61 \[ \frac{(1-i a x)^{3/4} \left (2 a^2 x^2 \text{Hypergeometric2F1}\left (\frac{3}{4},1,\frac{7}{4},\frac{a x+i}{-a x+i}\right )+9 a^2 x^2-15 i a x-6\right )}{12 x^2 (1+i a x)^{3/4}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.141, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}}\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^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80993, size = 401, normalized size = 3.04 \begin{align*} \frac{a^{2} x^{2} \log \left (\sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} + 1\right ) + i \, a^{2} x^{2} \log \left (\sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} + i\right ) - i \, a^{2} x^{2} \log \left (\sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} - i\right ) - a^{2} x^{2} \log \left (\sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} - 1\right ) -{\left (6 \, a^{2} x^{2} + 2 i \, a x + 4\right )} \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}}{8 \, x^{2}} \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^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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