Optimal. Leaf size=202 \[ \frac{29 a^2 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{96 x^2}-\frac{83 i a^3 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{192 x}+\frac{11}{64} a^4 \tan ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac{11}{64} a^4 \tanh ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac{7 i a \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{24 x^3}-\frac{\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{4 x^4} \]
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Rubi [A] time = 0.0807455, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5062, 99, 151, 12, 93, 298, 203, 206} \[ \frac{29 a^2 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{96 x^2}-\frac{83 i a^3 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{192 x}+\frac{11}{64} a^4 \tan ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac{11}{64} a^4 \tanh ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac{7 i a \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{24 x^3}-\frac{\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{4 x^4} \]
Antiderivative was successfully verified.
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Rule 5062
Rule 99
Rule 151
Rule 12
Rule 93
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{-\frac{1}{2} i \tan ^{-1}(a x)}}{x^5} \, dx &=\int \frac{\sqrt [4]{1-i a x}}{x^5 \sqrt [4]{1+i a x}} \, dx\\ &=-\frac{\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{4 x^4}+\frac{1}{4} \int \frac{-\frac{7 i a}{2}-3 a^2 x}{x^4 (1-i a x)^{3/4} \sqrt [4]{1+i a x}} \, dx\\ &=-\frac{\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{4 x^4}+\frac{7 i a \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{24 x^3}-\frac{1}{12} \int \frac{\frac{29 a^2}{4}-7 i a^3 x}{x^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}} \, dx\\ &=-\frac{\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{4 x^4}+\frac{7 i a \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{24 x^3}+\frac{29 a^2 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{96 x^2}+\frac{1}{24} \int \frac{\frac{83 i a^3}{8}+\frac{29 a^4 x}{4}}{x^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}} \, dx\\ &=-\frac{\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{4 x^4}+\frac{7 i a \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{24 x^3}+\frac{29 a^2 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{96 x^2}-\frac{83 i a^3 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{192 x}-\frac{1}{24} \int -\frac{33 a^4}{16 x (1-i a x)^{3/4} \sqrt [4]{1+i a x}} \, dx\\ &=-\frac{\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{4 x^4}+\frac{7 i a \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{24 x^3}+\frac{29 a^2 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{96 x^2}-\frac{83 i a^3 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{192 x}+\frac{1}{128} \left (11 a^4\right ) \int \frac{1}{x (1-i a x)^{3/4} \sqrt [4]{1+i a x}} \, dx\\ &=-\frac{\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{4 x^4}+\frac{7 i a \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{24 x^3}+\frac{29 a^2 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{96 x^2}-\frac{83 i a^3 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{192 x}+\frac{1}{32} \left (11 a^4\right ) \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{\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{4 x^4}+\frac{7 i a \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{24 x^3}+\frac{29 a^2 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{96 x^2}-\frac{83 i a^3 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{192 x}-\frac{1}{64} \left (11 a^4\right ) \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}{64} \left (11 a^4\right ) \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{\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{4 x^4}+\frac{7 i a \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{24 x^3}+\frac{29 a^2 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{96 x^2}-\frac{83 i a^3 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{192 x}+\frac{11}{64} a^4 \tan ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac{11}{64} a^4 \tanh ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )\\ \end{align*}
Mathematica [C] time = 0.0312894, size = 99, normalized size = 0.49 \[ \frac{\sqrt [4]{1-i a x} \left (-66 a^4 x^4 \text{Hypergeometric2F1}\left (\frac{1}{4},1,\frac{5}{4},\frac{a x+i}{-a x+i}\right )+83 a^4 x^4-25 i a^3 x^3+2 a^2 x^2+8 i a x-48\right )}{192 x^4 \sqrt [4]{1+i a x}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.146, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{5}}{\frac{1}{\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{1}{x^{5} \sqrt{\frac{i \, a x + 1}{\sqrt{a^{2} x^{2} + 1}}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.09505, size = 475, normalized size = 2.35 \begin{align*} -\frac{33 \, a^{4} x^{4} \log \left (\sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} + 1\right ) - 33 i \, a^{4} x^{4} \log \left (\sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} + i\right ) + 33 i \, a^{4} x^{4} \log \left (\sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} - i\right ) - 33 \, a^{4} x^{4} \log \left (\sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} - 1\right ) -{\left (-166 i \, a^{3} x^{3} + 116 \, a^{2} x^{2} + 112 i \, a x - 96\right )} \sqrt{a^{2} x^{2} + 1} \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}}{384 \, x^{4}} \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^{5} \sqrt{\frac{i \, a x + 1}{\sqrt{a^{2} x^{2} + 1}}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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