Optimal. Leaf size=319 \[ -\frac {(1-i x)^{5/6} \sqrt [6]{1+i x}}{3 x^3}-\frac {7 i (1-i x)^{5/6} \sqrt [6]{1+i x}}{18 x^2}+\frac {11 (1-i x)^{5/6} \sqrt [6]{1+i x}}{27 x}-\frac {19}{324} i \log \left (\frac {\sqrt [3]{1+i x}}{\sqrt [3]{1-i x}}-\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}+1\right )+\frac {19}{324} i \log \left (\frac {\sqrt [3]{1+i x}}{\sqrt [3]{1-i x}}+\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}+1\right )-\frac {19 i \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}}{\sqrt {3}}\right )}{54 \sqrt {3}}+\frac {19 i \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}}{\sqrt {3}}\right )}{54 \sqrt {3}}+\frac {19}{81} i \tanh ^{-1}\left (\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right ) \]
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Rubi [A] time = 0.20, antiderivative size = 319, normalized size of antiderivative = 1.00, 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 12
Rule 93
Rule 99
Rule 151
Rule 204
Rule 206
Rule 210
Rule 618
Rule 628
Rule 634
Rule 5062
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*}
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Mathematica [C] time = 0.02, size = 81, normalized size = 0.25 \[ \frac {(1-i x)^{5/6} \left (38 i x^3 \, _2F_1\left (\frac {5}{6},1;\frac {11}{6};\frac {x+i}{i-x}\right )+5 \left (22 i x^3+43 x^2-39 i x-18\right )\right )}{270 (1+i x)^{5/6} x^3} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 1.96, size = 244, normalized size = 0.76 \[ \frac {38 i \, x^{3} \log \left (\left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} + 1\right ) - 38 i \, x^{3} \log \left (\left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} - 1\right ) - {\left (19 \, \sqrt {3} x^{3} - 19 i \, x^{3}\right )} \log \left (\frac {1}{2} i \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} + \frac {1}{2}\right ) - {\left (19 \, \sqrt {3} x^{3} + 19 i \, x^{3}\right )} \log \left (\frac {1}{2} i \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} - \frac {1}{2}\right ) + {\left (19 \, \sqrt {3} x^{3} + 19 i \, x^{3}\right )} \log \left (-\frac {1}{2} i \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} + \frac {1}{2}\right ) + {\left (19 \, \sqrt {3} x^{3} - 19 i \, x^{3}\right )} \log \left (-\frac {1}{2} i \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} - \frac {1}{2}\right ) + {\left (-132 i \, x^{3} + 6 \, x^{2} - 18 i \, x - 108\right )} \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}}}{324 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {i \, x + 1}{\sqrt {x^{2} + 1}}\right )^{\frac {1}{3}}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {i x +1}{\sqrt {x^{2}+1}}\right )^{\frac {1}{3}}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {i \, x + 1}{\sqrt {x^{2} + 1}}\right )^{\frac {1}{3}}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (\frac {1+x\,1{}\mathrm {i}}{\sqrt {x^2+1}}\right )}^{1/3}}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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