Optimal. Leaf size=121 \[ -\frac{(1-i a x)^{5/4}}{x \sqrt [4]{1+i a x}}-\frac{10 i a \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-5 i a \tan ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+5 i a \tanh ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right ) \]
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Rubi [A] time = 0.0392875, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {5062, 94, 93, 298, 203, 206} \[ -\frac{(1-i a x)^{5/4}}{x \sqrt [4]{1+i a x}}-\frac{10 i a \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-5 i a \tan ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+5 i a \tanh ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right ) \]
Antiderivative was successfully verified.
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Rule 5062
Rule 94
Rule 93
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{-\frac{5}{2} i \tan ^{-1}(a x)}}{x^2} \, dx &=\int \frac{(1-i a x)^{5/4}}{x^2 (1+i a x)^{5/4}} \, dx\\ &=-\frac{(1-i a x)^{5/4}}{x \sqrt [4]{1+i a x}}-\frac{1}{2} (5 i a) \int \frac{\sqrt [4]{1-i a x}}{x (1+i a x)^{5/4}} \, dx\\ &=-\frac{10 i a \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-\frac{(1-i a x)^{5/4}}{x \sqrt [4]{1+i a x}}-\frac{1}{2} (5 i a) \int \frac{1}{x (1-i a x)^{3/4} \sqrt [4]{1+i a x}} \, dx\\ &=-\frac{10 i a \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-\frac{(1-i a x)^{5/4}}{x \sqrt [4]{1+i a x}}-(10 i a) \operatorname{Subst}\left (\int \frac{x^2}{-1+x^4} \, dx,x,\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )\\ &=-\frac{10 i a \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-\frac{(1-i a x)^{5/4}}{x \sqrt [4]{1+i a x}}+(5 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 )-(5 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{10 i a \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-\frac{(1-i a x)^{5/4}}{x \sqrt [4]{1+i a x}}-5 i a \tan ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+5 i a \tanh ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )\\ \end{align*}
Mathematica [C] time = 0.0165903, size = 69, normalized size = 0.57 \[ \frac{i \sqrt [4]{1-i a x} \left (10 a x \text{Hypergeometric2F1}\left (\frac{1}{4},1,\frac{5}{4},\frac{a x+i}{-a x+i}\right )-9 a x+i\right )}{x \sqrt [4]{1+i a x}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.175, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ({(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) ^{-{\frac{5}{2}}}}\, 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{1}{x^{2} \left (\frac{i \, a x + 1}{\sqrt{a^{2} x^{2} + 1}}\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.81714, size = 491, normalized size = 4.06 \begin{align*} -\frac{\sqrt{a^{2} x^{2} + 1}{\left (18 \, a x - 2 i\right )} \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} + 5 \,{\left (-i \, a^{2} x^{2} - a x\right )} \log \left (\sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} + 1\right ) -{\left (5 \, a^{2} x^{2} - 5 i \, a x\right )} \log \left (\sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} + i\right ) +{\left (5 \, a^{2} x^{2} - 5 i \, a x\right )} \log \left (\sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} - i\right ) + 5 \,{\left (i \, a^{2} x^{2} + a x\right )} \log \left (\sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} - 1\right )}{2 \,{\left (a x^{2} - i \, x\right )}} \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{1}{x^{2} \left (\frac{i \, a x + 1}{\sqrt{a^{2} x^{2} + 1}}\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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