Optimal. Leaf size=45 \[ 2 \tan ^{-1}\left (\frac {x-1}{\sqrt [4]{\frac {x+1}{x^2-2}}}\right )-2 \tanh ^{-1}\left (\frac {x-1}{\sqrt [4]{\frac {x+1}{x^2-2}}}\right ) \]
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Rubi [F] time = 14.35, 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 {(-1+x)^2 \left (-10-8 x+5 x^2+5 x^3\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right ) \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {(-1+x)^2 \left (-10-8 x+5 x^2+5 x^3\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right ) \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )} \, dx &=\frac {(1+x)^{3/4} \int \frac {(-1+x)^2 \left (-10-8 x+5 x^2+5 x^3\right )}{(1+x)^{3/4} \sqrt [4]{-2+x^2} \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )} \, dx}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}\\ &=\frac {(1+x)^{3/4} \int \left (-\frac {10}{(1+x)^{3/4} \sqrt [4]{-2+x^2} \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )}+\frac {12 x}{(1+x)^{3/4} \sqrt [4]{-2+x^2} \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )}+\frac {11 x^2}{(1+x)^{3/4} \sqrt [4]{-2+x^2} \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )}-\frac {13 x^3}{(1+x)^{3/4} \sqrt [4]{-2+x^2} \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )}-\frac {5 x^4}{(1+x)^{3/4} \sqrt [4]{-2+x^2} \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )}+\frac {5 x^5}{(1+x)^{3/4} \sqrt [4]{-2+x^2} \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )}\right ) \, dx}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}\\ &=-\frac {\left (5 (1+x)^{3/4}\right ) \int \frac {x^4}{(1+x)^{3/4} \sqrt [4]{-2+x^2} \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )} \, dx}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}+\frac {\left (5 (1+x)^{3/4}\right ) \int \frac {x^5}{(1+x)^{3/4} \sqrt [4]{-2+x^2} \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )} \, dx}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}-\frac {\left (10 (1+x)^{3/4}\right ) \int \frac {1}{(1+x)^{3/4} \sqrt [4]{-2+x^2} \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )} \, dx}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}+\frac {\left (11 (1+x)^{3/4}\right ) \int \frac {x^2}{(1+x)^{3/4} \sqrt [4]{-2+x^2} \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )} \, dx}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}+\frac {\left (12 (1+x)^{3/4}\right ) \int \frac {x}{(1+x)^{3/4} \sqrt [4]{-2+x^2} \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )} \, dx}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}-\frac {\left (13 (1+x)^{3/4}\right ) \int \frac {x^3}{(1+x)^{3/4} \sqrt [4]{-2+x^2} \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )} \, dx}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}\\ &=-\frac {\left (20 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\left (-1+x^4\right )^4}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}+\frac {\left (20 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\left (-1+x^4\right )^5}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}-\frac {\left (40 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}+\frac {\left (44 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\left (-1+x^4\right )^2}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}+\frac {\left (48 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {-1+x^4}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}-\frac {\left (52 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\left (-1+x^4\right )^3}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}\\ &=-\frac {\left (20 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )}-\frac {4 x^4}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )}+\frac {6 x^8}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )}-\frac {4 x^{12}}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )}+\frac {x^{16}}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )}\right ) \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}+\frac {\left (20 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt [4]{-1-2 x^4+x^8} \left (16+x^4-56 x^8+72 x^{12}-39 x^{16}+10 x^{20}-x^{24}\right )}+\frac {5 x^4}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )}-\frac {10 x^8}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )}+\frac {10 x^{12}}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )}-\frac {5 x^{16}}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )}+\frac {x^{20}}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )}\right ) \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}-\frac {\left (40 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}+\frac {\left (44 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )}-\frac {2 x^4}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )}+\frac {x^8}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )}\right ) \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}+\frac {\left (48 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt [4]{-1-2 x^4+x^8} \left (16+x^4-56 x^8+72 x^{12}-39 x^{16}+10 x^{20}-x^{24}\right )}+\frac {x^4}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )}\right ) \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}-\frac {\left (52 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt [4]{-1-2 x^4+x^8} \left (16+x^4-56 x^8+72 x^{12}-39 x^{16}+10 x^{20}-x^{24}\right )}+\frac {3 x^4}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )}-\frac {3 x^8}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )}+\frac {x^{12}}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )}\right ) \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}\\ &=\frac {\left (20 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1-2 x^4+x^8} \left (16+x^4-56 x^8+72 x^{12}-39 x^{16}+10 x^{20}-x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}-\frac {\left (20 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}-\frac {\left (20 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^{16}}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}+\frac {\left (20 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^{20}}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}-\frac {\left (40 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}+\frac {\left (44 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}+\frac {\left (44 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^8}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}+\frac {\left (48 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1-2 x^4+x^8} \left (16+x^4-56 x^8+72 x^{12}-39 x^{16}+10 x^{20}-x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}+\frac {\left (48 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}-\frac {\left (52 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1-2 x^4+x^8} \left (16+x^4-56 x^8+72 x^{12}-39 x^{16}+10 x^{20}-x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}-\frac {\left (52 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^{12}}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}+\frac {\left (80 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}+\frac {\left (80 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^{12}}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}-\frac {\left (88 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}+\frac {\left (100 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}-\frac {\left (100 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^{16}}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}-\frac {\left (120 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^8}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}-\frac {\left (156 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}+\frac {\left (156 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^8}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}-\frac {\left (200 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^8}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}+\frac {\left (200 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^{12}}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}\\ \end {align*}
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Mathematica [F] time = 2.34, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(-1+x)^2 \left (-10-8 x+5 x^2+5 x^3\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right ) \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 4.50, size = 45, normalized size = 1.00 \begin {gather*} 2 \tan ^{-1}\left (\frac {x-1}{\sqrt [4]{\frac {x+1}{x^2-2}}}\right )-2 \tanh ^{-1}\left (\frac {x-1}{\sqrt [4]{\frac {x+1}{x^2-2}}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 46.52, size = 256, normalized size = 5.69 \begin {gather*} -\arctan \left (\frac {2 \, {\left ({\left (x^{3} - x^{2} - 2 \, x + 2\right )} \left (\frac {x + 1}{x^{2} - 2}\right )^{\frac {3}{4}} + {\left (x^{5} - 3 \, x^{4} + x^{3} + 5 \, x^{2} - 6 \, x + 2\right )} \left (\frac {x + 1}{x^{2} - 2}\right )^{\frac {1}{4}}\right )}}{x^{6} - 4 \, x^{5} + 4 \, x^{4} + 4 \, x^{3} - 11 \, x^{2} + 7 \, x - 3}\right ) + \log \left (\frac {x^{6} - 4 \, x^{5} + 4 \, x^{4} + 4 \, x^{3} - 11 \, x^{2} - 2 \, {\left (x^{3} - x^{2} - 2 \, x + 2\right )} \left (\frac {x + 1}{x^{2} - 2}\right )^{\frac {3}{4}} + 2 \, {\left (x^{4} - 2 \, x^{3} - x^{2} + 4 \, x - 2\right )} \sqrt {\frac {x + 1}{x^{2} - 2}} - 2 \, {\left (x^{5} - 3 \, x^{4} + x^{3} + 5 \, x^{2} - 6 \, x + 2\right )} \left (\frac {x + 1}{x^{2} - 2}\right )^{\frac {1}{4}} + 9 \, x - 1}{x^{6} - 4 \, x^{5} + 4 \, x^{4} + 4 \, x^{3} - 11 \, x^{2} + 7 \, x - 3}\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 (5 \, x^{3} + 5 \, x^{2} - 8 \, x - 10\right )} {\left (x - 1\right )}^{2}}{{\left (x^{6} - 4 \, x^{5} + 4 \, x^{4} + 4 \, x^{3} - 11 \, x^{2} + 7 \, x - 3\right )} {\left (x^{2} - 2\right )} \left (\frac {x + 1}{x^{2} - 2}\right )^{\frac {3}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 5.13, size = 805, normalized size = 17.89 \begin {gather*} -\ln \left (-\frac {2 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {3}{4}} x^{3}+2 \sqrt {-\frac {-1-x}{x^{2}-2}}\, x^{4}+2 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}} x^{5}+x^{6}-2 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {3}{4}} x^{2}-4 \sqrt {-\frac {-1-x}{x^{2}-2}}\, x^{3}-6 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}} x^{4}-4 x^{5}-4 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {3}{4}} x -2 \sqrt {-\frac {-1-x}{x^{2}-2}}\, x^{2}+2 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}} x^{3}+4 x^{4}+4 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {3}{4}}+8 \sqrt {-\frac {-1-x}{x^{2}-2}}\, x +10 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}} x^{2}+4 x^{3}-4 \sqrt {-\frac {-1-x}{x^{2}-2}}-12 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}} x -11 x^{2}+4 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}}+9 x -1}{x^{6}-4 x^{5}+4 x^{4}+4 x^{3}-11 x^{2}+7 x -3}\right )+\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \sqrt {-\frac {-1-x}{x^{2}-2}}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{6}-4 \sqrt {-\frac {-1-x}{x^{2}-2}}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+4 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{5}-2 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {3}{4}} x^{3}+2 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}} x^{5}-2 \sqrt {-\frac {-1-x}{x^{2}-2}}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}-4 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {3}{4}} x^{2}-6 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}} x^{4}+8 \sqrt {-\frac {-1-x}{x^{2}-2}}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x -4 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+4 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {3}{4}} x +2 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}} x^{3}-4 \sqrt {-\frac {-1-x}{x^{2}-2}}\, \RootOf \left (\textit {\_Z}^{2}+1\right )+11 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}-4 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {3}{4}}+10 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}} x^{2}-9 \RootOf \left (\textit {\_Z}^{2}+1\right ) x -12 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}} x +\RootOf \left (\textit {\_Z}^{2}+1\right )+4 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}}}{x^{6}-4 x^{5}+4 x^{4}+4 x^{3}-11 x^{2}+7 x -3}\right ) \end {gather*}
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 (5 \, x^{3} + 5 \, x^{2} - 8 \, x - 10\right )} {\left (x - 1\right )}^{2}}{{\left (x^{6} - 4 \, x^{5} + 4 \, x^{4} + 4 \, x^{3} - 11 \, x^{2} + 7 \, x - 3\right )} {\left (x^{2} - 2\right )} \left (\frac {x + 1}{x^{2} - 2}\right )^{\frac {3}{4}}}\,{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 = -0.02 \begin {gather*} \int -\frac {{\left (x-1\right )}^2\,\left (-5\,x^3-5\,x^2+8\,x+10\right )}{\left (x^2-2\right )\,{\left (\frac {x+1}{x^2-2}\right )}^{3/4}\,\left (x^6-4\,x^5+4\,x^4+4\,x^3-11\,x^2+7\,x-3\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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