Optimal. Leaf size=203 \[ \frac{287 i a^3 \sqrt [4]{1-i a x}}{24 \sqrt [4]{1+i a x}}+\frac{61 a^2 \sqrt [4]{1-i a x}}{24 x \sqrt [4]{1+i a x}}+\frac{55}{8} i a^3 \tan ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac{55}{8} i a^3 \tanh ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac{13 i a \sqrt [4]{1-i a x}}{12 x^2 \sqrt [4]{1+i a x}}-\frac{\sqrt [4]{1-i a x}}{3 x^3 \sqrt [4]{1+i a x}} \]
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Rubi [A] time = 0.0789914, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 10, 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} \[ \frac{287 i a^3 \sqrt [4]{1-i a x}}{24 \sqrt [4]{1+i a x}}+\frac{61 a^2 \sqrt [4]{1-i a x}}{24 x \sqrt [4]{1+i a x}}+\frac{55}{8} i a^3 \tan ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac{55}{8} i a^3 \tanh ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac{13 i a \sqrt [4]{1-i a x}}{12 x^2 \sqrt [4]{1+i a x}}-\frac{\sqrt [4]{1-i a x}}{3 x^3 \sqrt [4]{1+i a x}} \]
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^4} \, dx &=\int \frac{(1-i a x)^{5/4}}{x^4 (1+i a x)^{5/4}} \, dx\\ &=-\frac{\sqrt [4]{1-i a x}}{3 x^3 \sqrt [4]{1+i a x}}-\frac{1}{3} \int \frac{\frac{13 i a}{2}+6 a^2 x}{x^3 (1-i a x)^{3/4} (1+i a x)^{5/4}} \, dx\\ &=-\frac{\sqrt [4]{1-i a x}}{3 x^3 \sqrt [4]{1+i a x}}+\frac{13 i a \sqrt [4]{1-i a x}}{12 x^2 \sqrt [4]{1+i a x}}+\frac{1}{6} \int \frac{-\frac{61 a^2}{4}+13 i a^3 x}{x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}} \, dx\\ &=-\frac{\sqrt [4]{1-i a x}}{3 x^3 \sqrt [4]{1+i a x}}+\frac{13 i a \sqrt [4]{1-i a x}}{12 x^2 \sqrt [4]{1+i a x}}+\frac{61 a^2 \sqrt [4]{1-i a x}}{24 x \sqrt [4]{1+i a x}}-\frac{1}{6} \int \frac{-\frac{165 i a^3}{8}-\frac{61 a^4 x}{4}}{x (1-i a x)^{3/4} (1+i a x)^{5/4}} \, dx\\ &=\frac{287 i a^3 \sqrt [4]{1-i a x}}{24 \sqrt [4]{1+i a x}}-\frac{\sqrt [4]{1-i a x}}{3 x^3 \sqrt [4]{1+i a x}}+\frac{13 i a \sqrt [4]{1-i a x}}{12 x^2 \sqrt [4]{1+i a x}}+\frac{61 a^2 \sqrt [4]{1-i a x}}{24 x \sqrt [4]{1+i a x}}+\frac{i \int \frac{165 a^4}{16 x (1-i a x)^{3/4} \sqrt [4]{1+i a x}} \, dx}{3 a}\\ &=\frac{287 i a^3 \sqrt [4]{1-i a x}}{24 \sqrt [4]{1+i a x}}-\frac{\sqrt [4]{1-i a x}}{3 x^3 \sqrt [4]{1+i a x}}+\frac{13 i a \sqrt [4]{1-i a x}}{12 x^2 \sqrt [4]{1+i a x}}+\frac{61 a^2 \sqrt [4]{1-i a x}}{24 x \sqrt [4]{1+i a x}}+\frac{1}{16} \left (55 i a^3\right ) \int \frac{1}{x (1-i a x)^{3/4} \sqrt [4]{1+i a x}} \, dx\\ &=\frac{287 i a^3 \sqrt [4]{1-i a x}}{24 \sqrt [4]{1+i a x}}-\frac{\sqrt [4]{1-i a x}}{3 x^3 \sqrt [4]{1+i a x}}+\frac{13 i a \sqrt [4]{1-i a x}}{12 x^2 \sqrt [4]{1+i a x}}+\frac{61 a^2 \sqrt [4]{1-i a x}}{24 x \sqrt [4]{1+i a x}}+\frac{1}{4} \left (55 i a^3\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{287 i a^3 \sqrt [4]{1-i a x}}{24 \sqrt [4]{1+i a x}}-\frac{\sqrt [4]{1-i a x}}{3 x^3 \sqrt [4]{1+i a x}}+\frac{13 i a \sqrt [4]{1-i a x}}{12 x^2 \sqrt [4]{1+i a x}}+\frac{61 a^2 \sqrt [4]{1-i a x}}{24 x \sqrt [4]{1+i a x}}-\frac{1}{8} \left (55 i a^3\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}{8} \left (55 i a^3\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{287 i a^3 \sqrt [4]{1-i a x}}{24 \sqrt [4]{1+i a x}}-\frac{\sqrt [4]{1-i a x}}{3 x^3 \sqrt [4]{1+i a x}}+\frac{13 i a \sqrt [4]{1-i a x}}{12 x^2 \sqrt [4]{1+i a x}}+\frac{61 a^2 \sqrt [4]{1-i a x}}{24 x \sqrt [4]{1+i a x}}+\frac{55}{8} i a^3 \tan ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac{55}{8} i a^3 \tanh ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )\\ \end{align*}
Mathematica [C] time = 0.0249633, size = 93, normalized size = 0.46 \[ \frac{\sqrt [4]{1-i a x} \left (-330 i a^3 x^3 \text{Hypergeometric2F1}\left (\frac{1}{4},1,\frac{5}{4},\frac{a x+i}{-a x+i}\right )+287 i a^3 x^3+61 a^2 x^2+26 i a x-8\right )}{24 x^3 \sqrt [4]{1+i a x}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.174, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4}} \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^{4} \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 = 1.79354, size = 574, normalized size = 2.83 \begin{align*} \frac{{\left (574 \, a^{3} x^{3} - 122 i \, a^{2} x^{2} + 52 \, a x + 16 i\right )} \sqrt{a^{2} x^{2} + 1} \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} - 165 \,{\left (i \, a^{4} x^{4} + a^{3} x^{3}\right )} \log \left (\sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} + 1\right ) -{\left (165 \, a^{4} x^{4} - 165 i \, a^{3} x^{3}\right )} \log \left (\sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} + i\right ) +{\left (165 \, a^{4} x^{4} - 165 i \, a^{3} x^{3}\right )} \log \left (\sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} - i\right ) - 165 \,{\left (-i \, a^{4} x^{4} - a^{3} x^{3}\right )} \log \left (\sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} - 1\right )}{48 \,{\left (a x^{4} - i \, x^{3}\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^{4} \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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