Optimal. Leaf size=39 \[ -\frac {\tan ^{-1}\left (\frac {1-2 x^3}{\sqrt {3}}\right )}{\sqrt {3}}+\log (x)-\frac {1}{6} \log \left (1-x^3+x^6\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {1608, 1488,
814, 648, 632, 210, 642} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {1-2 x^3}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{6} \log \left (x^6-x^3+1\right )+\log (x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 210
Rule 632
Rule 642
Rule 648
Rule 814
Rule 1488
Rule 1608
Rubi steps
\begin {align*} \int \frac {1+x^3}{x-x^4+x^7} \, dx &=\int \frac {1+x^3}{x \left (1-x^3+x^6\right )} \, dx\\ &=\frac {1}{3} \text {Subst}\left (\int \frac {1+x}{x \left (1-x+x^2\right )} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (\frac {1}{x}+\frac {2-x}{1-x+x^2}\right ) \, dx,x,x^3\right )\\ &=\log (x)+\frac {1}{3} \text {Subst}\left (\int \frac {2-x}{1-x+x^2} \, dx,x,x^3\right )\\ &=\log (x)-\frac {1}{6} \text {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,x^3\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,x^3\right )\\ &=\log (x)-\frac {1}{6} \log \left (1-x^3+x^6\right )-\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x^3\right )\\ &=\frac {\tan ^{-1}\left (\frac {-1+2 x^3}{\sqrt {3}}\right )}{\sqrt {3}}+\log (x)-\frac {1}{6} \log \left (1-x^3+x^6\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.01, size = 55, normalized size = 1.41 \begin {gather*} \log (x)-\frac {1}{3} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-2 \log (x-\text {$\#$1})+\log (x-\text {$\#$1}) \text {$\#$1}^3}{-1+2 \text {$\#$1}^3}\&\right ] \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.04, size = 35, normalized size = 0.90
method | result | size |
risch | \(\ln \left (x \right )-\frac {\ln \left (x^{6}-x^{3}+1\right )}{6}+\frac {\sqrt {3}\, \arctan \left (\frac {2 \left (x^{3}-\frac {1}{2}\right ) \sqrt {3}}{3}\right )}{3}\) | \(33\) |
default | \(\ln \left (x \right )-\frac {\ln \left (x^{6}-x^{3}+1\right )}{6}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{3}-1\right ) \sqrt {3}}{3}\right )}{3}\) | \(35\) |
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 = 0.33, size = 34, normalized size = 0.87 \begin {gather*} \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{3} - 1\right )}\right ) - \frac {1}{6} \, \log \left (x^{6} - x^{3} + 1\right ) + \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.05, size = 41, normalized size = 1.05 \begin {gather*} \log {\left (x \right )} - \frac {\log {\left (x^{6} - x^{3} + 1 \right )}}{6} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x^{3}}{3} - \frac {\sqrt {3}}{3} \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 4.83, size = 35, normalized size = 0.90 \begin {gather*} \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{3} - 1\right )}\right ) - \frac {1}{6} \, \log \left (x^{6} - x^{3} + 1\right ) + \log \left ({\left | x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.04, size = 36, normalized size = 0.92 \begin {gather*} \ln \left (x\right )-\frac {\ln \left (x^6-x^3+1\right )}{6}-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}}{3}-\frac {2\,\sqrt {3}\,x^3}{3}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________