Optimal. Leaf size=319 \[ -\frac{7 i (1-i x)^{5/6} \sqrt [6]{1+i x}}{18 x^2}-\frac{(1-i x)^{5/6} \sqrt [6]{1+i x}}{3 x^3}+\frac{11 (1-i x)^{5/6} \sqrt [6]{1+i x}}{27 x}-\frac{19}{324} i \log \left (\frac{\sqrt [3]{1+i x}}{\sqrt [3]{1-i x}}-\frac{\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}+1\right )+\frac{19}{324} i \log \left (\frac{\sqrt [3]{1+i x}}{\sqrt [3]{1-i x}}+\frac{\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}+1\right )-\frac{19 i \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}}{\sqrt{3}}\right )}{54 \sqrt{3}}+\frac{19 i \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}}{\sqrt{3}}\right )}{54 \sqrt{3}}+\frac{19}{81} i \tanh ^{-1}\left (\frac{\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right ) \]
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Rubi [A] time = 0.200022, antiderivative size = 319, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 11, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.786, Rules used = {5062, 99, 151, 12, 93, 210, 634, 618, 204, 628, 206} \[ -\frac{7 i (1-i x)^{5/6} \sqrt [6]{1+i x}}{18 x^2}-\frac{(1-i x)^{5/6} \sqrt [6]{1+i x}}{3 x^3}+\frac{11 (1-i x)^{5/6} \sqrt [6]{1+i x}}{27 x}-\frac{19}{324} i \log \left (\frac{\sqrt [3]{1+i x}}{\sqrt [3]{1-i x}}-\frac{\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}+1\right )+\frac{19}{324} i \log \left (\frac{\sqrt [3]{1+i x}}{\sqrt [3]{1-i x}}+\frac{\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}+1\right )-\frac{19 i \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}}{\sqrt{3}}\right )}{54 \sqrt{3}}+\frac{19 i \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}}{\sqrt{3}}\right )}{54 \sqrt{3}}+\frac{19}{81} i \tanh ^{-1}\left (\frac{\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right ) \]
Antiderivative was successfully verified.
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Rule 5062
Rule 99
Rule 151
Rule 12
Rule 93
Rule 210
Rule 634
Rule 618
Rule 204
Rule 628
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{\frac{1}{3} i \tan ^{-1}(x)}}{x^4} \, dx &=\int \frac{\sqrt [6]{1+i x}}{\sqrt [6]{1-i x} x^4} \, dx\\ &=-\frac{(1-i x)^{5/6} \sqrt [6]{1+i x}}{3 x^3}+\frac{1}{3} \int \frac{\frac{7 i}{3}-2 x}{\sqrt [6]{1-i x} (1+i x)^{5/6} x^3} \, dx\\ &=-\frac{(1-i x)^{5/6} \sqrt [6]{1+i x}}{3 x^3}-\frac{7 i (1-i x)^{5/6} \sqrt [6]{1+i x}}{18 x^2}-\frac{1}{6} \int \frac{\frac{22}{9}+\frac{7 i x}{3}}{\sqrt [6]{1-i x} (1+i x)^{5/6} x^2} \, dx\\ &=-\frac{(1-i x)^{5/6} \sqrt [6]{1+i x}}{3 x^3}-\frac{7 i (1-i x)^{5/6} \sqrt [6]{1+i x}}{18 x^2}+\frac{11 (1-i x)^{5/6} \sqrt [6]{1+i x}}{27 x}+\frac{1}{6} \int -\frac{19 i}{27 \sqrt [6]{1-i x} (1+i x)^{5/6} x} \, dx\\ &=-\frac{(1-i x)^{5/6} \sqrt [6]{1+i x}}{3 x^3}-\frac{7 i (1-i x)^{5/6} \sqrt [6]{1+i x}}{18 x^2}+\frac{11 (1-i x)^{5/6} \sqrt [6]{1+i x}}{27 x}-\frac{19}{162} i \int \frac{1}{\sqrt [6]{1-i x} (1+i x)^{5/6} x} \, dx\\ &=-\frac{(1-i x)^{5/6} \sqrt [6]{1+i x}}{3 x^3}-\frac{7 i (1-i x)^{5/6} \sqrt [6]{1+i x}}{18 x^2}+\frac{11 (1-i x)^{5/6} \sqrt [6]{1+i x}}{27 x}-\frac{19}{27} i \operatorname{Subst}\left (\int \frac{1}{-1+x^6} \, dx,x,\frac{\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right )\\ &=-\frac{(1-i x)^{5/6} \sqrt [6]{1+i x}}{3 x^3}-\frac{7 i (1-i x)^{5/6} \sqrt [6]{1+i x}}{18 x^2}+\frac{11 (1-i x)^{5/6} \sqrt [6]{1+i x}}{27 x}+\frac{19}{81} i \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right )+\frac{19}{81} i \operatorname{Subst}\left (\int \frac{1-\frac{x}{2}}{1-x+x^2} \, dx,x,\frac{\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right )+\frac{19}{81} i \operatorname{Subst}\left (\int \frac{1+\frac{x}{2}}{1+x+x^2} \, dx,x,\frac{\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right )\\ &=-\frac{(1-i x)^{5/6} \sqrt [6]{1+i x}}{3 x^3}-\frac{7 i (1-i x)^{5/6} \sqrt [6]{1+i x}}{18 x^2}+\frac{11 (1-i x)^{5/6} \sqrt [6]{1+i x}}{27 x}+\frac{19}{81} i \tanh ^{-1}\left (\frac{\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right )-\frac{19}{324} i \operatorname{Subst}\left (\int \frac{-1+2 x}{1-x+x^2} \, dx,x,\frac{\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right )+\frac{19}{324} i \operatorname{Subst}\left (\int \frac{1+2 x}{1+x+x^2} \, dx,x,\frac{\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right )+\frac{19}{108} i \operatorname{Subst}\left (\int \frac{1}{1-x+x^2} \, dx,x,\frac{\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right )+\frac{19}{108} i \operatorname{Subst}\left (\int \frac{1}{1+x+x^2} \, dx,x,\frac{\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right )\\ &=-\frac{(1-i x)^{5/6} \sqrt [6]{1+i x}}{3 x^3}-\frac{7 i (1-i x)^{5/6} \sqrt [6]{1+i x}}{18 x^2}+\frac{11 (1-i x)^{5/6} \sqrt [6]{1+i x}}{27 x}+\frac{19}{81} i \tanh ^{-1}\left (\frac{\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right )-\frac{19}{324} i \log \left (1-\frac{\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}+\frac{\sqrt [3]{1+i x}}{\sqrt [3]{1-i x}}\right )+\frac{19}{324} i \log \left (1+\frac{\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}+\frac{\sqrt [3]{1+i x}}{\sqrt [3]{1-i x}}\right )-\frac{19}{54} i \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+\frac{2 \sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right )-\frac{19}{54} i \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right )\\ &=-\frac{(1-i x)^{5/6} \sqrt [6]{1+i x}}{3 x^3}-\frac{7 i (1-i x)^{5/6} \sqrt [6]{1+i x}}{18 x^2}+\frac{11 (1-i x)^{5/6} \sqrt [6]{1+i x}}{27 x}-\frac{19 i \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}}{\sqrt{3}}\right )}{54 \sqrt{3}}+\frac{19 i \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}}{\sqrt{3}}\right )}{54 \sqrt{3}}+\frac{19}{81} i \tanh ^{-1}\left (\frac{\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right )-\frac{19}{324} i \log \left (1-\frac{\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}+\frac{\sqrt [3]{1+i x}}{\sqrt [3]{1-i x}}\right )+\frac{19}{324} i \log \left (1+\frac{\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}+\frac{\sqrt [3]{1+i x}}{\sqrt [3]{1-i x}}\right )\\ \end{align*}
Mathematica [C] time = 0.022007, size = 81, normalized size = 0.25 \[ \frac{(1-i x)^{5/6} \left (38 i x^3 \text{Hypergeometric2F1}\left (\frac{5}{6},1,\frac{11}{6},\frac{x+i}{-x+i}\right )+5 \left (22 i x^3+43 x^2-39 i x-18\right )\right )}{270 (1+i x)^{5/6} x^3} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.056, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4}}\sqrt [3]{{(1+ix){\frac{1}{\sqrt{{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{\left (\frac{i \, x + 1}{\sqrt{x^{2} + 1}}\right )^{\frac{1}{3}}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73565, size = 733, normalized size = 2.3 \begin{align*} \frac{38 i \, x^{3} \log \left (\left (\frac{i \, \sqrt{x^{2} + 1}}{x + i}\right )^{\frac{1}{3}} + 1\right ) - 38 i \, x^{3} \log \left (\left (\frac{i \, \sqrt{x^{2} + 1}}{x + i}\right )^{\frac{1}{3}} - 1\right ) -{\left (19 \, \sqrt{3} x^{3} - 19 i \, x^{3}\right )} \log \left (\frac{1}{2} i \, \sqrt{3} + \left (\frac{i \, \sqrt{x^{2} + 1}}{x + i}\right )^{\frac{1}{3}} + \frac{1}{2}\right ) -{\left (19 \, \sqrt{3} x^{3} + 19 i \, x^{3}\right )} \log \left (\frac{1}{2} i \, \sqrt{3} + \left (\frac{i \, \sqrt{x^{2} + 1}}{x + i}\right )^{\frac{1}{3}} - \frac{1}{2}\right ) +{\left (19 \, \sqrt{3} x^{3} + 19 i \, x^{3}\right )} \log \left (-\frac{1}{2} i \, \sqrt{3} + \left (\frac{i \, \sqrt{x^{2} + 1}}{x + i}\right )^{\frac{1}{3}} + \frac{1}{2}\right ) +{\left (19 \, \sqrt{3} x^{3} - 19 i \, x^{3}\right )} \log \left (-\frac{1}{2} i \, \sqrt{3} + \left (\frac{i \, \sqrt{x^{2} + 1}}{x + i}\right )^{\frac{1}{3}} - \frac{1}{2}\right ) +{\left (-132 i \, x^{3} + 6 \, x^{2} - 18 i \, x - 108\right )} \left (\frac{i \, \sqrt{x^{2} + 1}}{x + i}\right )^{\frac{1}{3}}}{324 \, x^{3}} \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{\left (\frac{i \, x + 1}{\sqrt{x^{2} + 1}}\right )^{\frac{1}{3}}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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