Optimal. Leaf size=111 \[ -\frac {(1-i x)^{2/3} \sqrt [3]{1+i x}}{x}+i \log \left (\sqrt [3]{1-i x}-\sqrt [3]{1+i x}\right )-\frac {1}{3} i \log (x)+\frac {2 i \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1-i x}}{\sqrt {3} \sqrt [3]{1+i x}}\right )}{\sqrt {3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {5062, 94, 91} \[ -\frac {(1-i x)^{2/3} \sqrt [3]{1+i x}}{x}+i \log \left (\sqrt [3]{1-i x}-\sqrt [3]{1+i x}\right )-\frac {1}{3} i \log (x)+\frac {2 i \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1-i x}}{\sqrt {3} \sqrt [3]{1+i x}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 91
Rule 94
Rule 5062
Rubi steps
\begin {align*} \int \frac {e^{\frac {2}{3} i \tan ^{-1}(x)}}{x^2} \, dx &=\int \frac {\sqrt [3]{1+i x}}{\sqrt [3]{1-i x} x^2} \, dx\\ &=-\frac {(1-i x)^{2/3} \sqrt [3]{1+i x}}{x}+\frac {2}{3} i \int \frac {1}{\sqrt [3]{1-i x} (1+i x)^{2/3} x} \, dx\\ &=-\frac {(1-i x)^{2/3} \sqrt [3]{1+i x}}{x}+\frac {2 i \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1-i x}}{\sqrt {3} \sqrt [3]{1+i x}}\right )}{\sqrt {3}}+i \log \left (\sqrt [3]{1-i x}-\sqrt [3]{1+i x}\right )-\frac {1}{3} i \log (x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.01, size = 59, normalized size = 0.53 \[ -\frac {i (1-i x)^{2/3} \left (x \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {x+i}{i-x}\right )+x-i\right )}{(1+i x)^{2/3} x} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.63, size = 120, normalized size = 1.08 \[ \frac {{\left (\sqrt {3} x - i \, x\right )} \log \left (\left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {2}{3}} + \frac {1}{2} i \, \sqrt {3} + \frac {1}{2}\right ) - {\left (\sqrt {3} x + i \, x\right )} \log \left (\left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {2}{3}} - \frac {1}{2} i \, \sqrt {3} + \frac {1}{2}\right ) + 2 i \, x \log \left (\left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {2}{3}} - 1\right ) - 3 \, {\left (-i \, x + 1\right )} \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {2}{3}}}{3 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {i \, x + 1}{\sqrt {x^{2} + 1}}\right )^{\frac {2}{3}}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {i x +1}{\sqrt {x^{2}+1}}\right )^{\frac {2}{3}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {i \, x + 1}{\sqrt {x^{2} + 1}}\right )^{\frac {2}{3}}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (\frac {1+x\,1{}\mathrm {i}}{\sqrt {x^2+1}}\right )}^{2/3}}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________