Optimal. Leaf size=240 \[ \frac{269 i a^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{960 x^2}+\frac{11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{48 x^3}-\frac{611 a^4 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{1920 x}-\frac{31}{128} i a^5 \tan ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac{31}{128} i a^5 \tanh ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac{9 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{40 x^4}-\frac{(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{5 x^5} \]
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Rubi [A] time = 0.101977, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 11, 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, 212, 206, 203} \[ \frac{269 i a^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{960 x^2}+\frac{11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{48 x^3}-\frac{611 a^4 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{1920 x}-\frac{31}{128} i a^5 \tan ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac{31}{128} i a^5 \tanh ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac{9 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{40 x^4}-\frac{(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{5 x^5} \]
Antiderivative was successfully verified.
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Rule 5062
Rule 99
Rule 151
Rule 12
Rule 93
Rule 212
Rule 206
Rule 203
Rubi steps
\begin{align*} \int \frac{e^{\frac{1}{2} i \tan ^{-1}(a x)}}{x^6} \, dx &=\int \frac{\sqrt [4]{1+i a x}}{x^6 \sqrt [4]{1-i a x}} \, dx\\ &=-\frac{(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{5 x^5}+\frac{1}{5} \int \frac{\frac{9 i a}{2}-4 a^2 x}{x^5 \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx\\ &=-\frac{(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{5 x^5}-\frac{9 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{40 x^4}-\frac{1}{20} \int \frac{\frac{55 a^2}{4}+\frac{27}{2} i a^3 x}{x^4 \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx\\ &=-\frac{(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{5 x^5}-\frac{9 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{40 x^4}+\frac{11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{48 x^3}+\frac{1}{60} \int \frac{-\frac{269 i a^3}{8}+\frac{55 a^4 x}{2}}{x^3 \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx\\ &=-\frac{(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{5 x^5}-\frac{9 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{40 x^4}+\frac{11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{48 x^3}+\frac{269 i a^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{960 x^2}-\frac{1}{120} \int \frac{-\frac{611 a^4}{16}-\frac{269}{8} i a^5 x}{x^2 \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx\\ &=-\frac{(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{5 x^5}-\frac{9 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{40 x^4}+\frac{11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{48 x^3}+\frac{269 i a^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{960 x^2}-\frac{611 a^4 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{1920 x}+\frac{1}{120} \int \frac{465 i a^5}{32 x \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx\\ &=-\frac{(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{5 x^5}-\frac{9 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{40 x^4}+\frac{11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{48 x^3}+\frac{269 i a^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{960 x^2}-\frac{611 a^4 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{1920 x}+\frac{1}{256} \left (31 i a^5\right ) \int \frac{1}{x \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx\\ &=-\frac{(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{5 x^5}-\frac{9 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{40 x^4}+\frac{11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{48 x^3}+\frac{269 i a^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{960 x^2}-\frac{611 a^4 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{1920 x}+\frac{1}{64} \left (31 i a^5\right ) \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{(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{5 x^5}-\frac{9 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{40 x^4}+\frac{11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{48 x^3}+\frac{269 i a^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{960 x^2}-\frac{611 a^4 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{1920 x}-\frac{1}{128} \left (31 i a^5\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}{128} \left (31 i a^5\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-i a x)^{3/4} \sqrt [4]{1+i a x}}{5 x^5}-\frac{9 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{40 x^4}+\frac{11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{48 x^3}+\frac{269 i a^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{960 x^2}-\frac{611 a^4 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{1920 x}-\frac{31}{128} i a^5 \tan ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac{31}{128} i a^5 \tanh ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )\\ \end{align*}
Mathematica [C] time = 0.0318243, size = 111, normalized size = 0.46 \[ \frac{(1-i a x)^{3/4} \left (-310 i a^5 x^5 \text{Hypergeometric2F1}\left (\frac{3}{4},1,\frac{7}{4},\frac{a x+i}{-a x+i}\right )-611 i a^5 x^5-1149 a^4 x^4+978 i a^3 x^3+872 a^2 x^2-816 i a x-384\right )}{1920 x^5 (1+i a x)^{3/4}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.144, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{6}}\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^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74454, size = 497, normalized size = 2.07 \begin{align*} \frac{-465 i \, a^{5} x^{5} \log \left (\sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} + 1\right ) + 465 \, a^{5} x^{5} \log \left (\sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} + i\right ) - 465 \, a^{5} x^{5} \log \left (\sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} - i\right ) + 465 i \, a^{5} x^{5} \log \left (\sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} - 1\right ) +{\left (1222 i \, a^{5} x^{5} - 146 \, a^{4} x^{4} + 196 i \, a^{3} x^{3} + 16 \, a^{2} x^{2} - 96 i \, a x - 768\right )} \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}}{3840 \, x^{5}} \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^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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