Optimal. Leaf size=43 \[ -\frac {\tan ^{-1}\left (\frac {1+x}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {1}{12} \log (2-x)-\frac {1}{24} \log \left (4+2 x+x^2\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {206, 31, 648,
632, 210, 642} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {x+1}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {1}{24} \log \left (x^2+2 x+4\right )+\frac {1}{12} \log (2-x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 206
Rule 210
Rule 632
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {1}{-8+x^3} \, dx &=\frac {1}{12} \int \frac {1}{-2+x} \, dx+\frac {1}{12} \int \frac {-4-x}{4+2 x+x^2} \, dx\\ &=\frac {1}{12} \log (2-x)-\frac {1}{24} \int \frac {2+2 x}{4+2 x+x^2} \, dx-\frac {1}{4} \int \frac {1}{4+2 x+x^2} \, dx\\ &=\frac {1}{12} \log (2-x)-\frac {1}{24} \log \left (4+2 x+x^2\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{-12-x^2} \, dx,x,2+2 x\right )\\ &=-\frac {\tan ^{-1}\left (\frac {1+x}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {1}{12} \log (2-x)-\frac {1}{24} \log \left (4+2 x+x^2\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 43, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {1+x}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {1}{12} \log (2-x)-\frac {1}{24} \log \left (4+2 x+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.05, size = 35, normalized size = 0.81
method | result | size |
risch | \(-\frac {\ln \left (x^{2}+2 x +4\right )}{24}-\frac {\arctan \left (\frac {\left (1+x \right ) \sqrt {3}}{3}\right ) \sqrt {3}}{12}+\frac {\ln \left (-2+x \right )}{12}\) | \(33\) |
default | \(\frac {\ln \left (-2+x \right )}{12}-\frac {\ln \left (x^{2}+2 x +4\right )}{24}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2+2 x \right ) \sqrt {3}}{6}\right )}{12}\) | \(35\) |
meijerg | \(\frac {x \left (\ln \left (1-\frac {\left (x^{3}\right )^{\frac {1}{3}}}{2}\right )-\frac {\ln \left (1+\frac {\left (x^{3}\right )^{\frac {1}{3}}}{2}+\frac {\left (x^{3}\right )^{\frac {2}{3}}}{4}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{3}\right )^{\frac {1}{3}}}{4+\left (x^{3}\right )^{\frac {1}{3}}}\right )\right )}{12 \left (x^{3}\right )^{\frac {1}{3}}}\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 2.51, size = 32, normalized size = 0.74 \begin {gather*} -\frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (x + 1\right )}\right ) - \frac {1}{24} \, \log \left (x^{2} + 2 \, x + 4\right ) + \frac {1}{12} \, \log \left (x - 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.49, size = 32, normalized size = 0.74 \begin {gather*} -\frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (x + 1\right )}\right ) - \frac {1}{24} \, \log \left (x^{2} + 2 \, x + 4\right ) + \frac {1}{12} \, \log \left (x - 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.05, size = 41, normalized size = 0.95 \begin {gather*} \frac {\log {\left (x - 2 \right )}}{12} - \frac {\log {\left (x^{2} + 2 x + 4 \right )}}{24} - \frac {\sqrt {3} \operatorname {atan}{\left (\frac {\sqrt {3} x}{3} + \frac {\sqrt {3}}{3} \right )}}{12} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.19, size = 33, normalized size = 0.77 \begin {gather*} -\frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (x + 1\right )}\right ) - \frac {1}{24} \, \log \left (x^{2} + 2 \, x + 4\right ) + \frac {1}{12} \, \log \left ({\left | x - 2 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.09, size = 46, normalized size = 1.07 \begin {gather*} \frac {\ln \left (x-2\right )}{12}+\ln \left (x+1-\sqrt {3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{24}+\frac {\sqrt {3}\,1{}\mathrm {i}}{24}\right )-\ln \left (x+1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{24}+\frac {\sqrt {3}\,1{}\mathrm {i}}{24}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________