Optimal. Leaf size=99 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt {\frac {1-2 x^2}{2 x^2+1}}}{x-1}\right )}{2 \sqrt [4]{2} 3^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt {\frac {1-2 x^2}{2 x^2+1}}}{x-1}\right )}{2 \sqrt [4]{2} 3^{3/4}} \]
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Rubi [F] time = 8.16, 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+2 x^2\right ) \left (-1+4 x-4 x^2+4 x^4\right )}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (-1-8 x+32 x^2-40 x^3+46 x^4-64 x^5+56 x^6-32 x^7+8 x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (-1+2 x^2\right ) \left (-1+4 x-4 x^2+4 x^4\right )}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (-1-8 x+32 x^2-40 x^3+46 x^4-64 x^5+56 x^6-32 x^7+8 x^8\right )} \, dx &=\frac {\sqrt {1-2 x^2} \int \frac {\left (-1+2 x^2\right ) \left (-1+4 x-4 x^2+4 x^4\right )}{\sqrt {1-2 x^2} \sqrt {1+2 x^2} \left (-1-8 x+32 x^2-40 x^3+46 x^4-64 x^5+56 x^6-32 x^7+8 x^8\right )} \, dx}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}}\\ &=-\frac {\sqrt {1-2 x^2} \int \frac {\sqrt {1-2 x^2} \left (-1+4 x-4 x^2+4 x^4\right )}{\sqrt {1+2 x^2} \left (-1-8 x+32 x^2-40 x^3+46 x^4-64 x^5+56 x^6-32 x^7+8 x^8\right )} \, dx}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}}\\ &=-\frac {\sqrt {1-2 x^2} \int \left (\frac {\sqrt {1-2 x^2}}{\sqrt {1+2 x^2} \left (1+8 x-32 x^2+40 x^3-46 x^4+64 x^5-56 x^6+32 x^7-8 x^8\right )}+\frac {4 x \sqrt {1-2 x^2}}{\sqrt {1+2 x^2} \left (-1-8 x+32 x^2-40 x^3+46 x^4-64 x^5+56 x^6-32 x^7+8 x^8\right )}-\frac {4 x^2 \sqrt {1-2 x^2}}{\sqrt {1+2 x^2} \left (-1-8 x+32 x^2-40 x^3+46 x^4-64 x^5+56 x^6-32 x^7+8 x^8\right )}+\frac {4 x^4 \sqrt {1-2 x^2}}{\sqrt {1+2 x^2} \left (-1-8 x+32 x^2-40 x^3+46 x^4-64 x^5+56 x^6-32 x^7+8 x^8\right )}\right ) \, dx}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}}\\ &=-\frac {\sqrt {1-2 x^2} \int \frac {\sqrt {1-2 x^2}}{\sqrt {1+2 x^2} \left (1+8 x-32 x^2+40 x^3-46 x^4+64 x^5-56 x^6+32 x^7-8 x^8\right )} \, dx}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}}-\frac {\left (4 \sqrt {1-2 x^2}\right ) \int \frac {x \sqrt {1-2 x^2}}{\sqrt {1+2 x^2} \left (-1-8 x+32 x^2-40 x^3+46 x^4-64 x^5+56 x^6-32 x^7+8 x^8\right )} \, dx}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}}+\frac {\left (4 \sqrt {1-2 x^2}\right ) \int \frac {x^2 \sqrt {1-2 x^2}}{\sqrt {1+2 x^2} \left (-1-8 x+32 x^2-40 x^3+46 x^4-64 x^5+56 x^6-32 x^7+8 x^8\right )} \, dx}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}}-\frac {\left (4 \sqrt {1-2 x^2}\right ) \int \frac {x^4 \sqrt {1-2 x^2}}{\sqrt {1+2 x^2} \left (-1-8 x+32 x^2-40 x^3+46 x^4-64 x^5+56 x^6-32 x^7+8 x^8\right )} \, dx}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}}\\ \end {align*}
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Mathematica [C] time = 60.49, size = 64755, normalized size = 654.09 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.89, size = 99, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt {\frac {1-2 x^2}{1+2 x^2}}}{-1+x}\right )}{2 \sqrt [4]{2} 3^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt {\frac {1-2 x^2}{1+2 x^2}}}{-1+x}\right )}{2 \sqrt [4]{2} 3^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.79, size = 708, normalized size = 7.15 \begin {gather*} \frac {1}{108} \cdot 54^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (54^{\frac {3}{4}} {\left (8 \, x^{8} - 32 \, x^{7} + 48 \, x^{6} - 48 \, x^{5} + 62 \, x^{4} - 40 \, x^{3} + 10 \, x^{2} - 12 \, x + 7\right )} + 3 \cdot 54^{\frac {1}{4}} {\left (8 \, x^{8} - 32 \, x^{7} + 8 \, x^{6} + 32 \, x^{5} + 22 \, x^{4} - 40 \, x^{3} + 20 \, x^{2} - 32 \, x + 17\right )}\right )} \sqrt {7 \, \sqrt {6} - 12} + 20 \, {\left (54^{\frac {3}{4}} {\left (4 \, x^{7} - 12 \, x^{6} + 16 \, x^{5} - 16 \, x^{4} + 13 \, x^{3} - 7 \, x^{2} + 3 \, x - 1\right )} - 9 \cdot 54^{\frac {1}{4}} {\left (4 \, x^{5} - 4 \, x^{4} - x + 1\right )}\right )} \sqrt {-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}}}{90 \, {\left (8 \, x^{8} - 32 \, x^{7} + 56 \, x^{6} - 64 \, x^{5} + 46 \, x^{4} - 40 \, x^{3} + 32 \, x^{2} - 8 \, x - 1\right )}}\right ) - \frac {1}{432} \cdot 54^{\frac {3}{4}} \log \left (-\frac {54^{\frac {3}{4}} {\left (8 \, x^{8} - 32 \, x^{7} + 8 \, x^{6} + 32 \, x^{5} + 22 \, x^{4} - 40 \, x^{3} + 20 \, x^{2} - 32 \, x + 17\right )} + 36 \, {\left (24 \, x^{7} - 72 \, x^{6} + 84 \, x^{5} - 84 \, x^{4} + 78 \, x^{3} - 42 \, x^{2} + \sqrt {6} {\left (4 \, x^{7} - 12 \, x^{6} + 4 \, x^{5} - 4 \, x^{4} + 13 \, x^{3} - 7 \, x^{2} + 6 \, x - 4\right )} + 21 \, x - 9\right )} \sqrt {-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}} + 18 \cdot 54^{\frac {1}{4}} {\left (8 \, x^{8} - 32 \, x^{7} + 48 \, x^{6} - 48 \, x^{5} + 62 \, x^{4} - 40 \, x^{3} + 10 \, x^{2} - 12 \, x + 7\right )}}{8 \, x^{8} - 32 \, x^{7} + 56 \, x^{6} - 64 \, x^{5} + 46 \, x^{4} - 40 \, x^{3} + 32 \, x^{2} - 8 \, x - 1}\right ) + \frac {1}{432} \cdot 54^{\frac {3}{4}} \log \left (\frac {54^{\frac {3}{4}} {\left (8 \, x^{8} - 32 \, x^{7} + 8 \, x^{6} + 32 \, x^{5} + 22 \, x^{4} - 40 \, x^{3} + 20 \, x^{2} - 32 \, x + 17\right )} - 36 \, {\left (24 \, x^{7} - 72 \, x^{6} + 84 \, x^{5} - 84 \, x^{4} + 78 \, x^{3} - 42 \, x^{2} + \sqrt {6} {\left (4 \, x^{7} - 12 \, x^{6} + 4 \, x^{5} - 4 \, x^{4} + 13 \, x^{3} - 7 \, x^{2} + 6 \, x - 4\right )} + 21 \, x - 9\right )} \sqrt {-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}} + 18 \cdot 54^{\frac {1}{4}} {\left (8 \, x^{8} - 32 \, x^{7} + 48 \, x^{6} - 48 \, x^{5} + 62 \, x^{4} - 40 \, x^{3} + 10 \, x^{2} - 12 \, x + 7\right )}}{8 \, x^{8} - 32 \, x^{7} + 56 \, x^{6} - 64 \, x^{5} + 46 \, x^{4} - 40 \, x^{3} + 32 \, x^{2} - 8 \, x - 1}\right ) \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*} \int \frac {{\left (4 \, x^{4} - 4 \, x^{2} + 4 \, x - 1\right )} {\left (2 \, x^{2} - 1\right )}}{{\left (8 \, x^{8} - 32 \, x^{7} + 56 \, x^{6} - 64 \, x^{5} + 46 \, x^{4} - 40 \, x^{3} + 32 \, x^{2} - 8 \, x - 1\right )} {\left (2 \, x^{2} + 1\right )} \sqrt {-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.08, size = 867, normalized size = 8.76 \[\frac {\left (2 x^{2}-1\right ) \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (8 \textit {\_Z}^{8}-32 \textit {\_Z}^{7}+56 \textit {\_Z}^{6}-64 \textit {\_Z}^{5}+46 \textit {\_Z}^{4}-40 \textit {\_Z}^{3}+32 \textit {\_Z}^{2}-8 \textit {\_Z} -1\right )}{\sum }\frac {\left (2 \underline {\hspace {1.25 ex}}\alpha ^{3}-2 \underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha -1\right ) \left (8 \sqrt {2}\, \sqrt {-4 \underline {\hspace {1.25 ex}}\alpha ^{4}+1}\, \sqrt {-2 x^{2}+1}\, \sqrt {2 x^{2}+1}\, \EllipticPi \left (\sqrt {2}\, x , -32 \underline {\hspace {1.25 ex}}\alpha ^{7}+132 \underline {\hspace {1.25 ex}}\alpha ^{6}-240 \underline {\hspace {1.25 ex}}\alpha ^{5}+284 \underline {\hspace {1.25 ex}}\alpha ^{4}-216 \underline {\hspace {1.25 ex}}\alpha ^{3}+183 \underline {\hspace {1.25 ex}}\alpha ^{2}-148 \underline {\hspace {1.25 ex}}\alpha +48, \frac {\sqrt {-2}\, \sqrt {2}}{2}\right ) \underline {\hspace {1.25 ex}}\alpha ^{7}-32 \sqrt {2}\, \sqrt {-4 \underline {\hspace {1.25 ex}}\alpha ^{4}+1}\, \sqrt {-2 x^{2}+1}\, \sqrt {2 x^{2}+1}\, \EllipticPi \left (\sqrt {2}\, x , -32 \underline {\hspace {1.25 ex}}\alpha ^{7}+132 \underline {\hspace {1.25 ex}}\alpha ^{6}-240 \underline {\hspace {1.25 ex}}\alpha ^{5}+284 \underline {\hspace {1.25 ex}}\alpha ^{4}-216 \underline {\hspace {1.25 ex}}\alpha ^{3}+183 \underline {\hspace {1.25 ex}}\alpha ^{2}-148 \underline {\hspace {1.25 ex}}\alpha +48, \frac {\sqrt {-2}\, \sqrt {2}}{2}\right ) \underline {\hspace {1.25 ex}}\alpha ^{6}+56 \sqrt {2}\, \sqrt {-4 \underline {\hspace {1.25 ex}}\alpha ^{4}+1}\, \sqrt {-2 x^{2}+1}\, \sqrt {2 x^{2}+1}\, \EllipticPi \left (\sqrt {2}\, x , -32 \underline {\hspace {1.25 ex}}\alpha ^{7}+132 \underline {\hspace {1.25 ex}}\alpha ^{6}-240 \underline {\hspace {1.25 ex}}\alpha ^{5}+284 \underline {\hspace {1.25 ex}}\alpha ^{4}-216 \underline {\hspace {1.25 ex}}\alpha ^{3}+183 \underline {\hspace {1.25 ex}}\alpha ^{2}-148 \underline {\hspace {1.25 ex}}\alpha +48, \frac {\sqrt {-2}\, \sqrt {2}}{2}\right ) \underline {\hspace {1.25 ex}}\alpha ^{5}-64 \sqrt {2}\, \sqrt {-4 \underline {\hspace {1.25 ex}}\alpha ^{4}+1}\, \sqrt {-2 x^{2}+1}\, \sqrt {2 x^{2}+1}\, \EllipticPi \left (\sqrt {2}\, x , -32 \underline {\hspace {1.25 ex}}\alpha ^{7}+132 \underline {\hspace {1.25 ex}}\alpha ^{6}-240 \underline {\hspace {1.25 ex}}\alpha ^{5}+284 \underline {\hspace {1.25 ex}}\alpha ^{4}-216 \underline {\hspace {1.25 ex}}\alpha ^{3}+183 \underline {\hspace {1.25 ex}}\alpha ^{2}-148 \underline {\hspace {1.25 ex}}\alpha +48, \frac {\sqrt {-2}\, \sqrt {2}}{2}\right ) \underline {\hspace {1.25 ex}}\alpha ^{4}+46 \sqrt {2}\, \sqrt {-4 \underline {\hspace {1.25 ex}}\alpha ^{4}+1}\, \sqrt {-2 x^{2}+1}\, \sqrt {2 x^{2}+1}\, \EllipticPi \left (\sqrt {2}\, x , -32 \underline {\hspace {1.25 ex}}\alpha ^{7}+132 \underline {\hspace {1.25 ex}}\alpha ^{6}-240 \underline {\hspace {1.25 ex}}\alpha ^{5}+284 \underline {\hspace {1.25 ex}}\alpha ^{4}-216 \underline {\hspace {1.25 ex}}\alpha ^{3}+183 \underline {\hspace {1.25 ex}}\alpha ^{2}-148 \underline {\hspace {1.25 ex}}\alpha +48, \frac {\sqrt {-2}\, \sqrt {2}}{2}\right ) \underline {\hspace {1.25 ex}}\alpha ^{3}-40 \sqrt {2}\, \sqrt {-4 \underline {\hspace {1.25 ex}}\alpha ^{4}+1}\, \sqrt {-2 x^{2}+1}\, \sqrt {2 x^{2}+1}\, \EllipticPi \left (\sqrt {2}\, x , -32 \underline {\hspace {1.25 ex}}\alpha ^{7}+132 \underline {\hspace {1.25 ex}}\alpha ^{6}-240 \underline {\hspace {1.25 ex}}\alpha ^{5}+284 \underline {\hspace {1.25 ex}}\alpha ^{4}-216 \underline {\hspace {1.25 ex}}\alpha ^{3}+183 \underline {\hspace {1.25 ex}}\alpha ^{2}-148 \underline {\hspace {1.25 ex}}\alpha +48, \frac {\sqrt {-2}\, \sqrt {2}}{2}\right ) \underline {\hspace {1.25 ex}}\alpha ^{2}+32 \sqrt {2}\, \sqrt {-4 \underline {\hspace {1.25 ex}}\alpha ^{4}+1}\, \sqrt {-2 x^{2}+1}\, \sqrt {2 x^{2}+1}\, \EllipticPi \left (\sqrt {2}\, x , -32 \underline {\hspace {1.25 ex}}\alpha ^{7}+132 \underline {\hspace {1.25 ex}}\alpha ^{6}-240 \underline {\hspace {1.25 ex}}\alpha ^{5}+284 \underline {\hspace {1.25 ex}}\alpha ^{4}-216 \underline {\hspace {1.25 ex}}\alpha ^{3}+183 \underline {\hspace {1.25 ex}}\alpha ^{2}-148 \underline {\hspace {1.25 ex}}\alpha +48, \frac {\sqrt {-2}\, \sqrt {2}}{2}\right ) \underline {\hspace {1.25 ex}}\alpha -8 \sqrt {2}\, \sqrt {-2 x^{2}+1}\, \sqrt {2 x^{2}+1}\, \EllipticPi \left (\sqrt {2}\, x , -32 \underline {\hspace {1.25 ex}}\alpha ^{7}+132 \underline {\hspace {1.25 ex}}\alpha ^{6}-240 \underline {\hspace {1.25 ex}}\alpha ^{5}+284 \underline {\hspace {1.25 ex}}\alpha ^{4}-216 \underline {\hspace {1.25 ex}}\alpha ^{3}+183 \underline {\hspace {1.25 ex}}\alpha ^{2}-148 \underline {\hspace {1.25 ex}}\alpha +48, \frac {\sqrt {-2}\, \sqrt {2}}{2}\right ) \sqrt {-4 \underline {\hspace {1.25 ex}}\alpha ^{4}+1}-\arctanh \left (\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{2} \left (32 \underline {\hspace {1.25 ex}}\alpha ^{7}-132 \underline {\hspace {1.25 ex}}\alpha ^{6}+240 \underline {\hspace {1.25 ex}}\alpha ^{5}-284 \underline {\hspace {1.25 ex}}\alpha ^{4}+216 \underline {\hspace {1.25 ex}}\alpha ^{3}-183 \underline {\hspace {1.25 ex}}\alpha ^{2}+2 x^{2}+148 \underline {\hspace {1.25 ex}}\alpha -48\right )}{\sqrt {-4 \underline {\hspace {1.25 ex}}\alpha ^{4}+1}\, \sqrt {-4 x^{4}+1}}\right ) \sqrt {-4 x^{4}+1}\right )}{\sqrt {-4 \underline {\hspace {1.25 ex}}\alpha ^{4}+1}\, \sqrt {-4 x^{4}+1}}\right )}{24 \sqrt {-\frac {2 x^{2}-1}{2 x^{2}+1}}\, \sqrt {-\left (2 x^{2}+1\right ) \left (2 x^{2}-1\right )}}\]
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*} \int \frac {{\left (4 \, x^{4} - 4 \, x^{2} + 4 \, x - 1\right )} {\left (2 \, x^{2} - 1\right )}}{{\left (8 \, x^{8} - 32 \, x^{7} + 56 \, x^{6} - 64 \, x^{5} + 46 \, x^{4} - 40 \, x^{3} + 32 \, x^{2} - 8 \, x - 1\right )} {\left (2 \, x^{2} + 1\right )} \sqrt {-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}}}\,{d x} \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 = -0.01 \begin {gather*} \int -\frac {\left (2\,x^2-1\right )\,\left (4\,x^4-4\,x^2+4\,x-1\right )}{\left (2\,x^2+1\right )\,\sqrt {-\frac {2\,x^2-1}{2\,x^2+1}}\,\left (-8\,x^8+32\,x^7-56\,x^6+64\,x^5-46\,x^4+40\,x^3-32\,x^2+8\,x+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] 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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