Optimal. Leaf size=253 \[ -\frac {(1-i x)^{5/6} \sqrt [6]{1+i x}}{x}+\frac {1}{6} i \log \left (\frac {\sqrt [3]{1+i x}}{\sqrt [3]{1-i x}}-\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}+1\right )-\frac {1}{6} i \log \left (\frac {\sqrt [3]{1+i x}}{\sqrt [3]{1-i x}}+\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}+1\right )+\frac {i \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {i \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {2}{3} i \tanh ^{-1}\left (\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.16, antiderivative size = 253, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {5062, 94, 93, 210, 634, 618, 204, 628, 206} \[ -\frac {(1-i x)^{5/6} \sqrt [6]{1+i x}}{x}+\frac {1}{6} i \log \left (\frac {\sqrt [3]{1+i x}}{\sqrt [3]{1-i x}}-\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}+1\right )-\frac {1}{6} i \log \left (\frac {\sqrt [3]{1+i x}}{\sqrt [3]{1-i x}}+\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}+1\right )+\frac {i \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {i \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {2}{3} i \tanh ^{-1}\left (\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 93
Rule 94
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^2} \, dx &=\int \frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x} x^2} \, dx\\ &=-\frac {(1-i x)^{5/6} \sqrt [6]{1+i x}}{x}+\frac {1}{3} 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}}{x}+2 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}}{x}-\frac {2}{3} 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 {2}{3} 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 {2}{3} 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}}{x}-\frac {2}{3} i \tanh ^{-1}\left (\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right )+\frac {1}{6} 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 {1}{6} 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 {1}{2} 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}{2} 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}}{x}-\frac {2}{3} i \tanh ^{-1}\left (\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right )+\frac {1}{6} 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 {1}{6} 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 )+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 )+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}}{x}+\frac {i \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {i \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {2}{3} i \tanh ^{-1}\left (\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right )+\frac {1}{6} 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 {1}{6} 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 = 0.02, size = 64, normalized size = 0.25 \[ -\frac {i (1-i x)^{5/6} \left (2 x \, _2F_1\left (\frac {5}{6},1;\frac {11}{6};\frac {x+i}{i-x}\right )+5 x-5 i\right )}{5 (1+i x)^{5/6} x} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 3.17, size = 211, normalized size = 0.83 \[ \frac {{\left (\sqrt {3} x - i \, x\right )} \log \left (\frac {1}{2} i \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} + \frac {1}{2}\right ) + {\left (\sqrt {3} x + i \, x\right )} \log \left (\frac {1}{2} i \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} - \frac {1}{2}\right ) - {\left (\sqrt {3} x + i \, x\right )} \log \left (-\frac {1}{2} i \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} + \frac {1}{2}\right ) - {\left (\sqrt {3} x - i \, x\right )} \log \left (-\frac {1}{2} i \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} - \frac {1}{2}\right ) - 2 i \, x \log \left (\left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} + 1\right ) + 2 i \, x \log \left (\left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} - 1\right ) - 6 \, {\left (-i \, x + 1\right )} \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}}}{6 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {i \, x + 1}{\sqrt {x^{2} + 1}}\right )^{\frac {1}{3}}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {i x +1}{\sqrt {x^{2}+1}}\right )^{\frac {1}{3}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {i \, x + 1}{\sqrt {x^{2} + 1}}\right )^{\frac {1}{3}}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (\frac {1+x\,1{}\mathrm {i}}{\sqrt {x^2+1}}\right )}^{1/3}}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt [3]{\frac {i \left (x - i\right )}{\sqrt {x^{2} + 1}}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________