Optimal. Leaf size=30 \[ \frac {\sqrt {-1+x^6}}{x^2}-\tanh ^{-1}\left (\frac {x^2}{\sqrt {-1+x^6}}\right ) \]
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Rubi [F]
time = 0.68, 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^6} \left (2+x^6\right )}{x^3 \left (-1-x^4+x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {-1+x^6} \left (2+x^6\right )}{x^3 \left (-1-x^4+x^6\right )} \, dx &=\int \left (-\frac {2 \sqrt {-1+x^6}}{x^3}+\frac {x \left (2-3 x^2\right ) \sqrt {-1+x^6}}{1+x^4-x^6}\right ) \, dx\\ &=-\left (2 \int \frac {\sqrt {-1+x^6}}{x^3} \, dx\right )+\int \frac {x \left (2-3 x^2\right ) \sqrt {-1+x^6}}{1+x^4-x^6} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {(2-3 x) \sqrt {-1+x^3}}{1+x^2-x^3} \, dx,x,x^2\right )-\text {Subst}\left (\int \frac {\sqrt {-1+x^3}}{x^2} \, dx,x,x^2\right )\\ &=\frac {\sqrt {-1+x^6}}{x^2}+\frac {1}{2} \text {Subst}\left (\int \left (-\frac {2 \sqrt {-1+x^3}}{-1-x^2+x^3}+\frac {3 x \sqrt {-1+x^3}}{-1-x^2+x^3}\right ) \, dx,x,x^2\right )-\frac {3}{2} \text {Subst}\left (\int \frac {x}{\sqrt {-1+x^3}} \, dx,x,x^2\right )\\ &=\frac {\sqrt {-1+x^6}}{x^2}+\frac {3}{2} \text {Subst}\left (\int \frac {1+\sqrt {3}-x}{\sqrt {-1+x^3}} \, dx,x,x^2\right )+\frac {3}{2} \text {Subst}\left (\int \frac {x \sqrt {-1+x^3}}{-1-x^2+x^3} \, dx,x,x^2\right )-\left (3 \sqrt {\frac {1}{2} \left (2+\sqrt {3}\right )}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^3}} \, dx,x,x^2\right )-\text {Subst}\left (\int \frac {\sqrt {-1+x^3}}{-1-x^2+x^3} \, dx,x,x^2\right )\\ &=\frac {\sqrt {-1+x^6}}{x^2}+\frac {3 \sqrt {-1+x^6}}{1-\sqrt {3}-x^2}-\frac {3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1-\sqrt {3}-x^2\right )^2}} E\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-x^2}{1-\sqrt {3}-x^2}\right )|-7+4 \sqrt {3}\right )}{2 \sqrt {-\frac {1-x^2}{\left (1-\sqrt {3}-x^2\right )^2}} \sqrt {-1+x^6}}+\frac {\sqrt {2} 3^{3/4} \left (1-x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1-\sqrt {3}-x^2\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-x^2}{1-\sqrt {3}-x^2}\right )|-7+4 \sqrt {3}\right )}{\sqrt {-\frac {1-x^2}{\left (1-\sqrt {3}-x^2\right )^2}} \sqrt {-1+x^6}}+\frac {3}{2} \text {Subst}\left (\int \frac {x \sqrt {-1+x^3}}{-1-x^2+x^3} \, dx,x,x^2\right )-\text {Subst}\left (\int \frac {\sqrt {-1+x^3}}{-1-x^2+x^3} \, dx,x,x^2\right )\\ \end {align*}
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Mathematica [A]
time = 4.87, size = 30, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-1+x^6}}{x^2}-\tanh ^{-1}\left (\frac {x^2}{\sqrt {-1+x^6}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.70, size = 53, normalized size = 1.77
method | result | size |
trager | \(\frac {\sqrt {x^{6}-1}}{x^{2}}+\frac {\ln \left (\frac {-x^{6}-x^{4}+2 x^{2} \sqrt {x^{6}-1}+1}{x^{6}-x^{4}-1}\right )}{2}\) | \(53\) |
risch | \(\frac {\sqrt {x^{6}-1}}{x^{2}}+\frac {\ln \left (-\frac {-x^{6}-x^{4}+2 x^{2} \sqrt {x^{6}-1}+1}{x^{6}-x^{4}-1}\right )}{2}\) | \(54\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 53 vs.
\(2 (26) = 52\).
time = 0.38, size = 53, normalized size = 1.77 \begin {gather*} \frac {x^{2} \log \left (\frac {x^{6} + x^{4} - 2 \, \sqrt {x^{6} - 1} x^{2} - 1}{x^{6} - x^{4} - 1}\right ) + 2 \, \sqrt {x^{6} - 1}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {\sqrt {x^6-1}\,\left (x^6+2\right )}{x^3\,\left (-x^6+x^4+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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