Optimal. Leaf size=106 \[ \frac {\sqrt [4]{-1+x^4+2 x^6} \left (5-19 x^4-20 x^6+104 x^8+38 x^{10}+20 x^{12}\right )}{45 x^9}+\sqrt [4]{2} \text {ArcTan}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4+2 x^6}}\right )-\sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4+2 x^6}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [F]
time = 1.88, 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 {\left (1+x^6\right ) \left (-1+2 x^6\right ) \left (-1+x^4+2 x^6\right )^{5/4}}{x^{10} \left (-1-x^4+2 x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {\left (1+x^6\right ) \left (-1+2 x^6\right ) \left (-1+x^4+2 x^6\right )^{5/4}}{x^{10} \left (-1-x^4+2 x^6\right )} \, dx &=\int \left (\frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^{10}}-\frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^6}+\frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^4}+\frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^2}+\frac {\left (-1+x^4+2 x^6\right )^{5/4}}{2 \left (-1+x^2\right )}+\frac {\left (-5-6 x^2\right ) \left (-1+x^4+2 x^6\right )^{5/4}}{2 \left (1+x^2+2 x^4\right )}\right ) \, dx\\ &=\frac {1}{2} \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{-1+x^2} \, dx+\frac {1}{2} \int \frac {\left (-5-6 x^2\right ) \left (-1+x^4+2 x^6\right )^{5/4}}{1+x^2+2 x^4} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^{10}} \, dx-\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^6} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^4} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^2} \, dx\\ &=\frac {1}{2} \int \left (\frac {\left (-1+x^4+2 x^6\right )^{5/4}}{2 (-1+x)}-\frac {\left (-1+x^4+2 x^6\right )^{5/4}}{2 (1+x)}\right ) \, dx+\frac {1}{2} \int \left (\frac {\left (-6+2 i \sqrt {7}\right ) \left (-1+x^4+2 x^6\right )^{5/4}}{1-i \sqrt {7}+4 x^2}+\frac {\left (-6-2 i \sqrt {7}\right ) \left (-1+x^4+2 x^6\right )^{5/4}}{1+i \sqrt {7}+4 x^2}\right ) \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^{10}} \, dx-\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^6} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^4} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^2} \, dx\\ &=\frac {1}{4} \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{-1+x} \, dx-\frac {1}{4} \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{1+x} \, dx+\left (-3-i \sqrt {7}\right ) \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{1+i \sqrt {7}+4 x^2} \, dx+\left (-3+i \sqrt {7}\right ) \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{1-i \sqrt {7}+4 x^2} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^{10}} \, dx-\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^6} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^4} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^2} \, dx\\ &=\frac {1}{4} \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{-1+x} \, dx-\frac {1}{4} \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{1+x} \, dx+\left (-3-i \sqrt {7}\right ) \int \left (\frac {\sqrt {-1-i \sqrt {7}} \left (-1+x^4+2 x^6\right )^{5/4}}{2 \left (1+i \sqrt {7}\right ) \left (\sqrt {-1-i \sqrt {7}}-2 x\right )}+\frac {\sqrt {-1-i \sqrt {7}} \left (-1+x^4+2 x^6\right )^{5/4}}{2 \left (1+i \sqrt {7}\right ) \left (\sqrt {-1-i \sqrt {7}}+2 x\right )}\right ) \, dx+\left (-3+i \sqrt {7}\right ) \int \left (\frac {\sqrt {-1+i \sqrt {7}} \left (-1+x^4+2 x^6\right )^{5/4}}{2 \left (1-i \sqrt {7}\right ) \left (\sqrt {-1+i \sqrt {7}}-2 x\right )}+\frac {\sqrt {-1+i \sqrt {7}} \left (-1+x^4+2 x^6\right )^{5/4}}{2 \left (1-i \sqrt {7}\right ) \left (\sqrt {-1+i \sqrt {7}}+2 x\right )}\right ) \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^{10}} \, dx-\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^6} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^4} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^2} \, dx\\ &=\frac {1}{4} \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{-1+x} \, dx-\frac {1}{4} \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{1+x} \, dx+\frac {\left (3-i \sqrt {7}\right ) \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{\sqrt {-1+i \sqrt {7}}-2 x} \, dx}{2 \sqrt {-1+i \sqrt {7}}}+\frac {\left (3-i \sqrt {7}\right ) \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{\sqrt {-1+i \sqrt {7}}+2 x} \, dx}{2 \sqrt {-1+i \sqrt {7}}}+\frac {\left (3+i \sqrt {7}\right ) \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{\sqrt {-1-i \sqrt {7}}-2 x} \, dx}{2 \sqrt {-1-i \sqrt {7}}}+\frac {\left (3+i \sqrt {7}\right ) \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{\sqrt {-1-i \sqrt {7}}+2 x} \, dx}{2 \sqrt {-1-i \sqrt {7}}}+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^{10}} \, dx-\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^6} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^4} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^2} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 1.45, size = 106, normalized size = 1.00 \begin {gather*} \frac {\sqrt [4]{-1+x^4+2 x^6} \left (5-19 x^4-20 x^6+104 x^8+38 x^{10}+20 x^{12}\right )}{45 x^9}+\sqrt [4]{2} \text {ArcTan}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4+2 x^6}}\right )-\sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4+2 x^6}}\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 = 118.29, size = 363, normalized size = 3.42
method | result | size |
trager | \(\frac {\left (2 x^{6}+x^{4}-1\right )^{\frac {1}{4}} \left (20 x^{12}+38 x^{10}+104 x^{8}-20 x^{6}-19 x^{4}+5\right )}{45 x^{9}}-\frac {\RootOf \left (\textit {\_Z}^{4}-2\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{6}+3 \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{4}+4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \left (2 x^{6}+x^{4}-1\right )^{\frac {1}{4}} x^{3}+4 \RootOf \left (\textit {\_Z}^{4}-2\right ) \sqrt {2 x^{6}+x^{4}-1}\, x^{2}+4 \left (2 x^{6}+x^{4}-1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}-2\right )^{3}}{\left (-1+x \right ) \left (1+x \right ) \left (2 x^{4}+x^{2}+1\right )}\right )}{2}+\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{6}+3 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{4}-4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \left (2 x^{6}+x^{4}-1\right )^{\frac {1}{4}} x^{3}-4 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \sqrt {2 x^{6}+x^{4}-1}\, x^{2}+4 \left (2 x^{6}+x^{4}-1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right )}{\left (-1+x \right ) \left (1+x \right ) \left (2 x^{4}+x^{2}+1\right )}\right )}{2}\) | \(363\) |
risch | \(\text {Expression too large to display}\) | \(1580\) |
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. 345 vs.
\(2 (90) = 180\).
time = 70.28, size = 345, normalized size = 3.25 \begin {gather*} -\frac {180 \cdot 8^{\frac {3}{4}} x^{9} \arctan \left (-\frac {16 \cdot 8^{\frac {1}{4}} {\left (2 \, x^{6} + x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 4 \cdot 8^{\frac {3}{4}} {\left (2 \, x^{6} + x^{4} - 1\right )}^{\frac {3}{4}} x - 2^{\frac {3}{4}} {\left (8 \cdot 8^{\frac {1}{4}} \sqrt {2 \, x^{6} + x^{4} - 1} x^{2} + 8^{\frac {3}{4}} {\left (2 \, x^{6} + 3 \, x^{4} - 1\right )}\right )}}{8 \, {\left (2 \, x^{6} - x^{4} - 1\right )}}\right ) + 45 \cdot 8^{\frac {3}{4}} x^{9} \log \left (\frac {4 \, \sqrt {2} {\left (2 \, x^{6} + x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 8^{\frac {3}{4}} \sqrt {2 \, x^{6} + x^{4} - 1} x^{2} + 4 \, {\left (2 \, x^{6} + x^{4} - 1\right )}^{\frac {3}{4}} x + 8^{\frac {1}{4}} {\left (2 \, x^{6} + 3 \, x^{4} - 1\right )}}{2 \, x^{6} - x^{4} - 1}\right ) - 45 \cdot 8^{\frac {3}{4}} x^{9} \log \left (\frac {4 \, \sqrt {2} {\left (2 \, x^{6} + x^{4} - 1\right )}^{\frac {1}{4}} x^{3} - 8^{\frac {3}{4}} \sqrt {2 \, x^{6} + x^{4} - 1} x^{2} + 4 \, {\left (2 \, x^{6} + x^{4} - 1\right )}^{\frac {3}{4}} x - 8^{\frac {1}{4}} {\left (2 \, x^{6} + 3 \, x^{4} - 1\right )}}{2 \, x^{6} - x^{4} - 1}\right ) - 16 \, {\left (20 \, x^{12} + 38 \, x^{10} + 104 \, x^{8} - 20 \, x^{6} - 19 \, x^{4} + 5\right )} {\left (2 \, x^{6} + x^{4} - 1\right )}^{\frac {1}{4}}}{720 \, x^{9}} \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 (x^6+1\right )\,\left (2\,x^6-1\right )\,{\left (2\,x^6+x^4-1\right )}^{5/4}}{x^{10}\,\left (-2\,x^6+x^4+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________