Optimal. Leaf size=116 \[ -\frac {(1-x)^{2/3} \sqrt [3]{1+x}}{3 x}-\frac {(1-x)^{2/3} (1+x)^{4/3}}{2 x^2}+\frac {2 \text {ArcTan}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{1+x}}\right )}{3 \sqrt {3}}-\frac {\log (x)}{9}+\frac {1}{3} \log \left (\sqrt [3]{1-x}-\sqrt [3]{1+x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6261, 98, 96,
93} \begin {gather*} \frac {2 \text {ArcTan}\left (\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{x+1}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {(1-x)^{2/3} (x+1)^{4/3}}{2 x^2}-\frac {(1-x)^{2/3} \sqrt [3]{x+1}}{3 x}-\frac {\log (x)}{9}+\frac {1}{3} \log \left (\sqrt [3]{1-x}-\sqrt [3]{x+1}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 93
Rule 96
Rule 98
Rule 6261
Rubi steps
\begin {align*} \int \frac {e^{\frac {2}{3} \tanh ^{-1}(x)}}{x^3} \, dx &=\int \frac {\sqrt [3]{1+x}}{\sqrt [3]{1-x} x^3} \, dx\\ &=-\frac {(1-x)^{2/3} (1+x)^{4/3}}{2 x^2}+\frac {1}{3} \int \frac {\sqrt [3]{1+x}}{\sqrt [3]{1-x} x^2} \, dx\\ &=-\frac {(1-x)^{2/3} \sqrt [3]{1+x}}{3 x}-\frac {(1-x)^{2/3} (1+x)^{4/3}}{2 x^2}+\frac {2}{9} \int \frac {1}{\sqrt [3]{1-x} x (1+x)^{2/3}} \, dx\\ &=-\frac {(1-x)^{2/3} \sqrt [3]{1+x}}{3 x}-\frac {(1-x)^{2/3} (1+x)^{4/3}}{2 x^2}+\frac {2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{1+x}}\right )}{3 \sqrt {3}}-\frac {\log (x)}{9}+\frac {1}{3} \log \left (\sqrt [3]{1-x}-\sqrt [3]{1+x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.01, size = 57, normalized size = 0.49 \begin {gather*} -\frac {(1-x)^{2/3} \left (3+8 x+5 x^2+2 x^2 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {1-x}{1+x}\right )\right )}{6 x^2 (1+x)^{2/3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (\frac {1+x}{\sqrt {-x^{2}+1}}\right )^{\frac {2}{3}}}{x^{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.37, size = 166, normalized size = 1.43 \begin {gather*} -\frac {4 \, \sqrt {3} x^{2} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {2}{3}} + \frac {1}{3} \, \sqrt {3}\right ) - 4 \, x^{2} \log \left (\left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {2}{3}} - 1\right ) + 2 \, x^{2} \log \left (\frac {{\left (x - 1\right )} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {2}{3}} + x - \sqrt {-x^{2} + 1} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}} - 1}{x - 1}\right ) - 3 \, {\left (5 \, x^{2} - 2 \, x - 3\right )} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {2}{3}}}{18 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (\frac {x+1}{\sqrt {1-x^2}}\right )}^{2/3}}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________