Optimal. Leaf size=233 \[ \frac{113 a^2 \sqrt [4]{1-i a x}}{96 x^2 \sqrt [4]{1+i a x}}+\frac{2467 a^4 \sqrt [4]{1-i a x}}{192 \sqrt [4]{1+i a x}}-\frac{521 i a^3 \sqrt [4]{1-i a x}}{192 x \sqrt [4]{1+i a x}}+\frac{475}{64} a^4 \tan ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac{475}{64} a^4 \tanh ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac{17 i a \sqrt [4]{1-i a x}}{24 x^3 \sqrt [4]{1+i a x}}-\frac{\sqrt [4]{1-i a x}}{4 x^4 \sqrt [4]{1+i a x}} \]
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Rubi [A] time = 0.0989286, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.562, Rules used = {5062, 98, 151, 155, 12, 93, 298, 203, 206} \[ \frac{113 a^2 \sqrt [4]{1-i a x}}{96 x^2 \sqrt [4]{1+i a x}}+\frac{2467 a^4 \sqrt [4]{1-i a x}}{192 \sqrt [4]{1+i a x}}-\frac{521 i a^3 \sqrt [4]{1-i a x}}{192 x \sqrt [4]{1+i a x}}+\frac{475}{64} a^4 \tan ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac{475}{64} a^4 \tanh ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac{17 i a \sqrt [4]{1-i a x}}{24 x^3 \sqrt [4]{1+i a x}}-\frac{\sqrt [4]{1-i a x}}{4 x^4 \sqrt [4]{1+i a x}} \]
Antiderivative was successfully verified.
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Rule 5062
Rule 98
Rule 151
Rule 155
Rule 12
Rule 93
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{-\frac{5}{2} i \tan ^{-1}(a x)}}{x^5} \, dx &=\int \frac{(1-i a x)^{5/4}}{x^5 (1+i a x)^{5/4}} \, dx\\ &=-\frac{\sqrt [4]{1-i a x}}{4 x^4 \sqrt [4]{1+i a x}}-\frac{1}{4} \int \frac{\frac{17 i a}{2}+8 a^2 x}{x^4 (1-i a x)^{3/4} (1+i a x)^{5/4}} \, dx\\ &=-\frac{\sqrt [4]{1-i a x}}{4 x^4 \sqrt [4]{1+i a x}}+\frac{17 i a \sqrt [4]{1-i a x}}{24 x^3 \sqrt [4]{1+i a x}}+\frac{1}{12} \int \frac{-\frac{113 a^2}{4}+\frac{51}{2} i a^3 x}{x^3 (1-i a x)^{3/4} (1+i a x)^{5/4}} \, dx\\ &=-\frac{\sqrt [4]{1-i a x}}{4 x^4 \sqrt [4]{1+i a x}}+\frac{17 i a \sqrt [4]{1-i a x}}{24 x^3 \sqrt [4]{1+i a x}}+\frac{113 a^2 \sqrt [4]{1-i a x}}{96 x^2 \sqrt [4]{1+i a x}}-\frac{1}{24} \int \frac{-\frac{521 i a^3}{8}-\frac{113 a^4 x}{2}}{x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}} \, dx\\ &=-\frac{\sqrt [4]{1-i a x}}{4 x^4 \sqrt [4]{1+i a x}}+\frac{17 i a \sqrt [4]{1-i a x}}{24 x^3 \sqrt [4]{1+i a x}}+\frac{113 a^2 \sqrt [4]{1-i a x}}{96 x^2 \sqrt [4]{1+i a x}}-\frac{521 i a^3 \sqrt [4]{1-i a x}}{192 x \sqrt [4]{1+i a x}}+\frac{1}{24} \int \frac{\frac{1425 a^4}{16}-\frac{521}{8} i a^5 x}{x (1-i a x)^{3/4} (1+i a x)^{5/4}} \, dx\\ &=\frac{2467 a^4 \sqrt [4]{1-i a x}}{192 \sqrt [4]{1+i a x}}-\frac{\sqrt [4]{1-i a x}}{4 x^4 \sqrt [4]{1+i a x}}+\frac{17 i a \sqrt [4]{1-i a x}}{24 x^3 \sqrt [4]{1+i a x}}+\frac{113 a^2 \sqrt [4]{1-i a x}}{96 x^2 \sqrt [4]{1+i a x}}-\frac{521 i a^3 \sqrt [4]{1-i a x}}{192 x \sqrt [4]{1+i a x}}-\frac{i \int \frac{1425 i a^5}{32 x (1-i a x)^{3/4} \sqrt [4]{1+i a x}} \, dx}{12 a}\\ &=\frac{2467 a^4 \sqrt [4]{1-i a x}}{192 \sqrt [4]{1+i a x}}-\frac{\sqrt [4]{1-i a x}}{4 x^4 \sqrt [4]{1+i a x}}+\frac{17 i a \sqrt [4]{1-i a x}}{24 x^3 \sqrt [4]{1+i a x}}+\frac{113 a^2 \sqrt [4]{1-i a x}}{96 x^2 \sqrt [4]{1+i a x}}-\frac{521 i a^3 \sqrt [4]{1-i a x}}{192 x \sqrt [4]{1+i a x}}+\frac{1}{128} \left (475 a^4\right ) \int \frac{1}{x (1-i a x)^{3/4} \sqrt [4]{1+i a x}} \, dx\\ &=\frac{2467 a^4 \sqrt [4]{1-i a x}}{192 \sqrt [4]{1+i a x}}-\frac{\sqrt [4]{1-i a x}}{4 x^4 \sqrt [4]{1+i a x}}+\frac{17 i a \sqrt [4]{1-i a x}}{24 x^3 \sqrt [4]{1+i a x}}+\frac{113 a^2 \sqrt [4]{1-i a x}}{96 x^2 \sqrt [4]{1+i a x}}-\frac{521 i a^3 \sqrt [4]{1-i a x}}{192 x \sqrt [4]{1+i a x}}+\frac{1}{32} \left (475 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{2467 a^4 \sqrt [4]{1-i a x}}{192 \sqrt [4]{1+i a x}}-\frac{\sqrt [4]{1-i a x}}{4 x^4 \sqrt [4]{1+i a x}}+\frac{17 i a \sqrt [4]{1-i a x}}{24 x^3 \sqrt [4]{1+i a x}}+\frac{113 a^2 \sqrt [4]{1-i a x}}{96 x^2 \sqrt [4]{1+i a x}}-\frac{521 i a^3 \sqrt [4]{1-i a x}}{192 x \sqrt [4]{1+i a x}}-\frac{1}{64} \left (475 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 (475 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{2467 a^4 \sqrt [4]{1-i a x}}{192 \sqrt [4]{1+i a x}}-\frac{\sqrt [4]{1-i a x}}{4 x^4 \sqrt [4]{1+i a x}}+\frac{17 i a \sqrt [4]{1-i a x}}{24 x^3 \sqrt [4]{1+i a x}}+\frac{113 a^2 \sqrt [4]{1-i a x}}{96 x^2 \sqrt [4]{1+i a x}}-\frac{521 i a^3 \sqrt [4]{1-i a x}}{192 x \sqrt [4]{1+i a x}}+\frac{475}{64} a^4 \tan ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac{475}{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 = 0.0325974, size = 99, normalized size = 0.42 \[ \frac{\sqrt [4]{1-i a x} \left (-2850 a^4 x^4 \text{Hypergeometric2F1}\left (\frac{1}{4},1,\frac{5}{4},\frac{a x+i}{-a x+i}\right )+2467 a^4 x^4-521 i a^3 x^3+226 a^2 x^2+136 i a x-48\right )}{192 x^4 \sqrt [4]{1+i a x}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.179, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{5}} \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^{5} \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 [A] time = 1.75454, size = 610, normalized size = 2.62 \begin{align*} \frac{{\left (-4934 i \, a^{4} x^{4} - 1042 \, a^{3} x^{3} - 452 i \, a^{2} x^{2} + 272 \, a x + 96 i\right )} \sqrt{a^{2} x^{2} + 1} \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} -{\left (1425 \, a^{5} x^{5} - 1425 i \, a^{4} x^{4}\right )} \log \left (\sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} + 1\right ) - 1425 \,{\left (-i \, a^{5} x^{5} - a^{4} x^{4}\right )} \log \left (\sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} + i\right ) - 1425 \,{\left (i \, a^{5} x^{5} + a^{4} x^{4}\right )} \log \left (\sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} - i\right ) +{\left (1425 \, a^{5} x^{5} - 1425 i \, a^{4} x^{4}\right )} \log \left (\sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} - 1\right )}{384 \,{\left (a x^{5} - i \, x^{4}\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^{5} \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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