Optimal. Leaf size=74 \[ \frac {\sqrt {-1+x^3} \left (12+10 x^2-24 x^3+15 x^4-10 x^5+12 x^6\right )}{60 x^5}-\frac {1}{4} \sqrt {\frac {3}{2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt {-1+x^3}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [F]
time = 0.87, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {\sqrt {-1+x^3} \left (2+x^3\right ) \left (-1-x^2+x^3\right )^2}{x^6 \left (-2-3 x^2+2 x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {\sqrt {-1+x^3} \left (2+x^3\right ) \left (-1-x^2+x^3\right )^2}{x^6 \left (-2-3 x^2+2 x^3\right )} \, dx &=\int \left (\frac {1}{2} \sqrt {-1+x^3}-\frac {\sqrt {-1+x^3}}{x^6}-\frac {\sqrt {-1+x^3}}{2 x^4}+\frac {\sqrt {-1+x^3}}{2 x^3}-\frac {\sqrt {-1+x^3}}{4 x^2}-\frac {\sqrt {-1+x^3}}{4 x}+\frac {3 (-1+x) \sqrt {-1+x^3}}{4 \left (-2-3 x^2+2 x^3\right )}\right ) \, dx\\ &=-\left (\frac {1}{4} \int \frac {\sqrt {-1+x^3}}{x^2} \, dx\right )-\frac {1}{4} \int \frac {\sqrt {-1+x^3}}{x} \, dx+\frac {1}{2} \int \sqrt {-1+x^3} \, dx-\frac {1}{2} \int \frac {\sqrt {-1+x^3}}{x^4} \, dx+\frac {1}{2} \int \frac {\sqrt {-1+x^3}}{x^3} \, dx+\frac {3}{4} \int \frac {(-1+x) \sqrt {-1+x^3}}{-2-3 x^2+2 x^3} \, dx-\int \frac {\sqrt {-1+x^3}}{x^6} \, dx\\ &=\frac {\sqrt {-1+x^3}}{5 x^5}-\frac {\sqrt {-1+x^3}}{4 x^2}+\frac {\sqrt {-1+x^3}}{4 x}+\frac {1}{5} x \sqrt {-1+x^3}-\frac {1}{12} \text {Subst}\left (\int \frac {\sqrt {-1+x}}{x} \, dx,x,x^3\right )-\frac {1}{6} \text {Subst}\left (\int \frac {\sqrt {-1+x}}{x^2} \, dx,x,x^3\right )-\frac {3}{10} \int \frac {1}{\sqrt {-1+x^3}} \, dx-\frac {3}{10} \int \frac {1}{x^3 \sqrt {-1+x^3}} \, dx+\frac {3}{8} \int \frac {1}{\sqrt {-1+x^3}} \, dx-\frac {3}{8} \int \frac {x}{\sqrt {-1+x^3}} \, dx+\frac {3}{4} \int \left (\frac {\sqrt {-1+x^3}}{2+3 x^2-2 x^3}+\frac {x \sqrt {-1+x^3}}{-2-3 x^2+2 x^3}\right ) \, dx\\ &=-\frac {1}{6} \sqrt {-1+x^3}+\frac {\sqrt {-1+x^3}}{5 x^5}+\frac {\sqrt {-1+x^3}}{6 x^3}-\frac {2 \sqrt {-1+x^3}}{5 x^2}+\frac {\sqrt {-1+x^3}}{4 x}+\frac {1}{5} x \sqrt {-1+x^3}-\frac {3^{3/4} \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right )|-7+4 \sqrt {3}\right )}{20 \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}-\frac {3}{40} \int \frac {1}{\sqrt {-1+x^3}} \, dx+\frac {3}{8} \int \frac {1+\sqrt {3}-x}{\sqrt {-1+x^3}} \, dx+\frac {3}{4} \int \frac {\sqrt {-1+x^3}}{2+3 x^2-2 x^3} \, dx+\frac {3}{4} \int \frac {x \sqrt {-1+x^3}}{-2-3 x^2+2 x^3} \, dx-\frac {1}{4} \left (3 \sqrt {\frac {1}{2} \left (2+\sqrt {3}\right )}\right ) \int \frac {1}{\sqrt {-1+x^3}} \, dx\\ &=-\frac {1}{6} \sqrt {-1+x^3}+\frac {3 \sqrt {-1+x^3}}{4 \left (1-\sqrt {3}-x\right )}+\frac {\sqrt {-1+x^3}}{5 x^5}+\frac {\sqrt {-1+x^3}}{6 x^3}-\frac {2 \sqrt {-1+x^3}}{5 x^2}+\frac {\sqrt {-1+x^3}}{4 x}+\frac {1}{5} x \sqrt {-1+x^3}-\frac {3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} E\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right )|-7+4 \sqrt {3}\right )}{8 \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {3^{3/4} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right )|-7+4 \sqrt {3}\right )}{2 \sqrt {2} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {3}{4} \int \frac {\sqrt {-1+x^3}}{2+3 x^2-2 x^3} \, dx+\frac {3}{4} \int \frac {x \sqrt {-1+x^3}}{-2-3 x^2+2 x^3} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.90, size = 74, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-1+x^3} \left (12+10 x^2-24 x^3+15 x^4-10 x^5+12 x^6\right )}{60 x^5}-\frac {1}{4} \sqrt {\frac {3}{2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt {-1+x^3}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 2.67, size = 364, normalized size = 4.92
method | result | size |
trager | \(\frac {\sqrt {x^{3}-1}\, \left (12 x^{6}-10 x^{5}+15 x^{4}-24 x^{3}+10 x^{2}+12\right )}{60 x^{5}}+\frac {\RootOf \left (\textit {\_Z}^{2}-6\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}-6\right ) x^{3}+3 \RootOf \left (\textit {\_Z}^{2}-6\right ) x^{2}-12 x \sqrt {x^{3}-1}-2 \RootOf \left (\textit {\_Z}^{2}-6\right )}{2 x^{3}-3 x^{2}-2}\right )}{16}\) | \(106\) |
risch | \(\frac {12 x^{9}-10 x^{8}+15 x^{7}-36 x^{6}+20 x^{5}-15 x^{4}+36 x^{3}-10 x^{2}-12}{60 x^{5} \sqrt {x^{3}-1}}+\frac {3 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{8 \sqrt {x^{3}-1}}+\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (2 \textit {\_Z}^{3}-3 \textit {\_Z}^{2}-2\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha \left (-2 \underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha +1\right ) \left (-3-i \sqrt {3}\right ) \sqrt {\frac {-1+x}{-3-i \sqrt {3}}}\, \sqrt {\frac {2 x +1-i \sqrt {3}}{3-i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x +1}{3+i \sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\underline {\hspace {1.25 ex}}\alpha ^{2}+\frac {\underline {\hspace {1.25 ex}}\alpha }{2}+\frac {1}{2}-\frac {i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}}{3}+\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha }{6}+\frac {i \sqrt {3}}{6}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}\right )}{16}\) | \(351\) |
default | \(\frac {x \sqrt {x^{3}-1}}{5}+\frac {3 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{8 \sqrt {x^{3}-1}}+\frac {\sqrt {x^{3}-1}}{4 x}-\frac {2 \sqrt {x^{3}-1}}{5 x^{2}}-\frac {\sqrt {x^{3}-1}}{6}+\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (2 \textit {\_Z}^{3}-3 \textit {\_Z}^{2}-2\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha \left (-2 \underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha +1\right ) \left (-3-i \sqrt {3}\right ) \sqrt {\frac {-1+x}{-3-i \sqrt {3}}}\, \sqrt {\frac {2 x +1-i \sqrt {3}}{3-i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x +1}{3+i \sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\underline {\hspace {1.25 ex}}\alpha ^{2}+\frac {\underline {\hspace {1.25 ex}}\alpha }{2}+\frac {1}{2}-\frac {i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}}{3}+\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha }{6}+\frac {i \sqrt {3}}{6}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}\right )}{16}+\frac {\sqrt {x^{3}-1}}{6 x^{3}}+\frac {\sqrt {x^{3}-1}}{5 x^{5}}\) | \(364\) |
elliptic | \(\frac {x \sqrt {x^{3}-1}}{5}+\frac {3 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{8 \sqrt {x^{3}-1}}+\frac {\sqrt {x^{3}-1}}{4 x}-\frac {2 \sqrt {x^{3}-1}}{5 x^{2}}-\frac {\sqrt {x^{3}-1}}{6}+\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (2 \textit {\_Z}^{3}-3 \textit {\_Z}^{2}-2\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha \left (-2 \underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha +1\right ) \left (-3-i \sqrt {3}\right ) \sqrt {\frac {-1+x}{-3-i \sqrt {3}}}\, \sqrt {\frac {2 x +1-i \sqrt {3}}{3-i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x +1}{3+i \sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\underline {\hspace {1.25 ex}}\alpha ^{2}+\frac {\underline {\hspace {1.25 ex}}\alpha }{2}+\frac {1}{2}-\frac {i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}}{3}+\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha }{6}+\frac {i \sqrt {3}}{6}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}\right )}{16}+\frac {\sqrt {x^{3}-1}}{6 x^{3}}+\frac {\sqrt {x^{3}-1}}{5 x^{5}}\) | \(364\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 141 vs.
\(2 (59) = 118\).
time = 0.46, size = 141, normalized size = 1.91 \begin {gather*} \frac {15 \, \sqrt {3} \sqrt {2} x^{5} \log \left (-\frac {4 \, x^{6} + 36 \, x^{5} + 9 \, x^{4} - 8 \, x^{3} - 4 \, \sqrt {3} \sqrt {2} {\left (2 \, x^{4} + 3 \, x^{3} - 2 \, x\right )} \sqrt {x^{3} - 1} - 36 \, x^{2} + 4}{4 \, x^{6} - 12 \, x^{5} + 9 \, x^{4} - 8 \, x^{3} + 12 \, x^{2} + 4}\right ) + 8 \, {\left (12 \, x^{6} - 10 \, x^{5} + 15 \, x^{4} - 24 \, x^{3} + 10 \, x^{2} + 12\right )} \sqrt {x^{3} - 1}}{480 \, x^{5}} \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 [B]
time = 1.88, size = 117, normalized size = 1.58 \begin {gather*} \frac {x\,\sqrt {x^3-1}}{5}-\frac {\sqrt {x^3-1}}{6}+\frac {\sqrt {x^3-1}}{4\,x}-\frac {2\,\sqrt {x^3-1}}{5\,x^2}+\frac {\sqrt {x^3-1}}{6\,x^3}+\frac {\sqrt {x^3-1}}{5\,x^5}+\frac {\sqrt {2}\,\sqrt {3}\,\ln \left (\frac {3\,x^2+2\,x^3-2\,\sqrt {6}\,x\,\sqrt {x^3-1}-2}{-12\,x^3+18\,x^2+12}\right )}{16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________