Optimal. Leaf size=61 \[ 2 \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^5+x^4-2 x}}\right )-2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {x^5+x^4-2 x}}{x^4+x^3-2}\right ) \]
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Rubi [F] time = 7.95, 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 (6+x^4\right ) \sqrt {-2 x+x^4+x^5}}{\left (-2+x^4\right ) \left (-2-x^3+x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (6+x^4\right ) \sqrt {-2 x+x^4+x^5}}{\left (-2+x^4\right ) \left (-2-x^3+x^4\right )} \, dx &=\frac {\sqrt {-2 x+x^4+x^5} \int \frac {\sqrt {x} \left (6+x^4\right ) \sqrt {-2+x^3+x^4}}{\left (-2+x^4\right ) \left (-2-x^3+x^4\right )} \, dx}{\sqrt {x} \sqrt {-2+x^3+x^4}}\\ &=\frac {\sqrt {-2 x+x^4+x^5} \int \left (\frac {\sqrt {x} \sqrt {-2+x^3+x^4}}{1+x}+\frac {\sqrt {x} \left (-1+3 x-x^2\right ) \sqrt {-2+x^3+x^4}}{-2+2 x-2 x^2+x^3}-\frac {4 x^{3/2} \sqrt {-2+x^3+x^4}}{-2+x^4}\right ) \, dx}{\sqrt {x} \sqrt {-2+x^3+x^4}}\\ &=\frac {\sqrt {-2 x+x^4+x^5} \int \frac {\sqrt {x} \sqrt {-2+x^3+x^4}}{1+x} \, dx}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\sqrt {-2 x+x^4+x^5} \int \frac {\sqrt {x} \left (-1+3 x-x^2\right ) \sqrt {-2+x^3+x^4}}{-2+2 x-2 x^2+x^3} \, dx}{\sqrt {x} \sqrt {-2+x^3+x^4}}-\frac {\left (4 \sqrt {-2 x+x^4+x^5}\right ) \int \frac {x^{3/2} \sqrt {-2+x^3+x^4}}{-2+x^4} \, dx}{\sqrt {x} \sqrt {-2+x^3+x^4}}\\ &=\frac {\left (2 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {-2+x^6+x^8}}{1+x^2} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (2 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (-1+3 x^2-x^4\right ) \sqrt {-2+x^6+x^8}}{-2+2 x^2-2 x^4+x^6} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}-\frac {\left (4 \sqrt {-2 x+x^4+x^5}\right ) \int \left (-\frac {x^{3/2} \sqrt {-2+x^3+x^4}}{2 \sqrt {2} \left (\sqrt {2}-x^2\right )}-\frac {x^{3/2} \sqrt {-2+x^3+x^4}}{2 \sqrt {2} \left (\sqrt {2}+x^2\right )}\right ) \, dx}{\sqrt {x} \sqrt {-2+x^3+x^4}}\\ &=\frac {\left (2 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \left (\sqrt {-2+x^6+x^8}-\frac {\sqrt {-2+x^6+x^8}}{1+x^2}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (2 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \left (-\sqrt {-2+x^6+x^8}-\frac {\left (2-x^2-x^4\right ) \sqrt {-2+x^6+x^8}}{-2+2 x^2-2 x^4+x^6}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (\sqrt {2} \sqrt {-2 x+x^4+x^5}\right ) \int \frac {x^{3/2} \sqrt {-2+x^3+x^4}}{\sqrt {2}-x^2} \, dx}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (\sqrt {2} \sqrt {-2 x+x^4+x^5}\right ) \int \frac {x^{3/2} \sqrt {-2+x^3+x^4}}{\sqrt {2}+x^2} \, dx}{\sqrt {x} \sqrt {-2+x^3+x^4}}\\ &=-\frac {\left (2 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{1+x^2} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}-\frac {\left (2 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\left (2-x^2-x^4\right ) \sqrt {-2+x^6+x^8}}{-2+2 x^2-2 x^4+x^6} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (\sqrt {2} \sqrt {-2 x+x^4+x^5}\right ) \int \left (\frac {i x^{3/2} \sqrt {-2+x^3+x^4}}{2 \sqrt [4]{2} \left (i \sqrt [4]{2}-x\right )}+\frac {i x^{3/2} \sqrt {-2+x^3+x^4}}{2 \sqrt [4]{2} \left (i \sqrt [4]{2}+x\right )}\right ) \, dx}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (\sqrt {2} \sqrt {-2 x+x^4+x^5}\right ) \int \left (\frac {x^{3/2} \sqrt {-2+x^3+x^4}}{2 \sqrt [4]{2} \left (\sqrt [4]{2}-x\right )}+\frac {x^{3/2} \sqrt {-2+x^3+x^4}}{2 \sqrt [4]{2} \left (\sqrt [4]{2}+x\right )}\right ) \, dx}{\sqrt {x} \sqrt {-2+x^3+x^4}}\\ &=-\frac {\left (2 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \left (\frac {i \sqrt {-2+x^6+x^8}}{2 (i-x)}+\frac {i \sqrt {-2+x^6+x^8}}{2 (i+x)}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}-\frac {\left (2 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \left (\frac {2 \sqrt {-2+x^6+x^8}}{-2+2 x^2-2 x^4+x^6}-\frac {x^2 \sqrt {-2+x^6+x^8}}{-2+2 x^2-2 x^4+x^6}-\frac {x^4 \sqrt {-2+x^6+x^8}}{-2+2 x^2-2 x^4+x^6}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (i \sqrt {-2 x+x^4+x^5}\right ) \int \frac {x^{3/2} \sqrt {-2+x^3+x^4}}{i \sqrt [4]{2}-x} \, dx}{2^{3/4} \sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (i \sqrt {-2 x+x^4+x^5}\right ) \int \frac {x^{3/2} \sqrt {-2+x^3+x^4}}{i \sqrt [4]{2}+x} \, dx}{2^{3/4} \sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\sqrt {-2 x+x^4+x^5} \int \frac {x^{3/2} \sqrt {-2+x^3+x^4}}{\sqrt [4]{2}-x} \, dx}{2^{3/4} \sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\sqrt {-2 x+x^4+x^5} \int \frac {x^{3/2} \sqrt {-2+x^3+x^4}}{\sqrt [4]{2}+x} \, dx}{2^{3/4} \sqrt {x} \sqrt {-2+x^3+x^4}}\\ &=-\frac {\left (i \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{i-x} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}-\frac {\left (i \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{i+x} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (2 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {-2+x^6+x^8}}{-2+2 x^2-2 x^4+x^6} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (2 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt {-2+x^6+x^8}}{-2+2 x^2-2 x^4+x^6} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}-\frac {\left (4 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{-2+2 x^2-2 x^4+x^6} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (i \sqrt [4]{2} \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt {-2+x^6+x^8}}{i \sqrt [4]{2}-x^2} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (i \sqrt [4]{2} \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt {-2+x^6+x^8}}{i \sqrt [4]{2}+x^2} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (\sqrt [4]{2} \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt {-2+x^6+x^8}}{\sqrt [4]{2}-x^2} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (\sqrt [4]{2} \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt {-2+x^6+x^8}}{\sqrt [4]{2}+x^2} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}\\ &=-\frac {\left (i \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{i-x} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}-\frac {\left (i \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{i+x} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (2 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {-2+x^6+x^8}}{-2+2 x^2-2 x^4+x^6} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (2 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt {-2+x^6+x^8}}{-2+2 x^2-2 x^4+x^6} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}-\frac {\left (4 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{-2+2 x^2-2 x^4+x^6} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (i \sqrt [4]{2} \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \left (-i \sqrt [4]{2} \sqrt {-2+x^6+x^8}-x^2 \sqrt {-2+x^6+x^8}-\frac {\sqrt {2} \sqrt {-2+x^6+x^8}}{i \sqrt [4]{2}-x^2}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (i \sqrt [4]{2} \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \left (-i \sqrt [4]{2} \sqrt {-2+x^6+x^8}+x^2 \sqrt {-2+x^6+x^8}-\frac {\sqrt {2} \sqrt {-2+x^6+x^8}}{i \sqrt [4]{2}+x^2}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (\sqrt [4]{2} \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \left (-\sqrt [4]{2} \sqrt {-2+x^6+x^8}-x^2 \sqrt {-2+x^6+x^8}+\frac {\sqrt {2} \sqrt {-2+x^6+x^8}}{\sqrt [4]{2}-x^2}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (\sqrt [4]{2} \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \left (-\sqrt [4]{2} \sqrt {-2+x^6+x^8}+x^2 \sqrt {-2+x^6+x^8}+\frac {\sqrt {2} \sqrt {-2+x^6+x^8}}{\sqrt [4]{2}+x^2}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}\\ &=-\frac {\left (i \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{i-x} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}-\frac {\left (i \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{i+x} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (2 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {-2+x^6+x^8}}{-2+2 x^2-2 x^4+x^6} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (2 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt {-2+x^6+x^8}}{-2+2 x^2-2 x^4+x^6} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}-\frac {\left (4 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{-2+2 x^2-2 x^4+x^6} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}-\frac {\left (i 2^{3/4} \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{i \sqrt [4]{2}-x^2} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}-\frac {\left (i 2^{3/4} \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{i \sqrt [4]{2}+x^2} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (2^{3/4} \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{\sqrt [4]{2}-x^2} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (2^{3/4} \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{\sqrt [4]{2}+x^2} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}\\ &=-\frac {\left (i \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{i-x} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}-\frac {\left (i \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{i+x} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (2 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {-2+x^6+x^8}}{-2+2 x^2-2 x^4+x^6} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (2 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt {-2+x^6+x^8}}{-2+2 x^2-2 x^4+x^6} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}-\frac {\left (4 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{-2+2 x^2-2 x^4+x^6} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}-\frac {\left (i 2^{3/4} \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \left (-\frac {(-1)^{3/4} \sqrt {-2+x^6+x^8}}{2 \sqrt [8]{2} \left (\sqrt [4]{-1} \sqrt [8]{2}-x\right )}-\frac {(-1)^{3/4} \sqrt {-2+x^6+x^8}}{2 \sqrt [8]{2} \left (\sqrt [4]{-1} \sqrt [8]{2}+x\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}-\frac {\left (i 2^{3/4} \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \left (-\frac {\sqrt [4]{-1} \sqrt {-2+x^6+x^8}}{2 \sqrt [8]{2} \left (-(-1)^{3/4} \sqrt [8]{2}-x\right )}-\frac {\sqrt [4]{-1} \sqrt {-2+x^6+x^8}}{2 \sqrt [8]{2} \left (-(-1)^{3/4} \sqrt [8]{2}+x\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (2^{3/4} \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \left (\frac {i \sqrt {-2+x^6+x^8}}{2 \sqrt [8]{2} \left (i \sqrt [8]{2}-x\right )}+\frac {i \sqrt {-2+x^6+x^8}}{2 \sqrt [8]{2} \left (i \sqrt [8]{2}+x\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (2^{3/4} \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \left (\frac {\sqrt {-2+x^6+x^8}}{2 \sqrt [8]{2} \left (\sqrt [8]{2}-x\right )}+\frac {\sqrt {-2+x^6+x^8}}{2 \sqrt [8]{2} \left (\sqrt [8]{2}+x\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}\\ &=-\frac {\left (i \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{i-x} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}-\frac {\left (i \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{i+x} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (2 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {-2+x^6+x^8}}{-2+2 x^2-2 x^4+x^6} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (2 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt {-2+x^6+x^8}}{-2+2 x^2-2 x^4+x^6} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}-\frac {\left (4 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{-2+2 x^2-2 x^4+x^6} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}-\frac {\left ((1-i) \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{-(-1)^{3/4} \sqrt [8]{2}-x} \, dx,x,\sqrt {x}\right )}{2^{7/8} \sqrt {x} \sqrt {-2+x^3+x^4}}-\frac {\left ((1-i) \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{-(-1)^{3/4} \sqrt [8]{2}+x} \, dx,x,\sqrt {x}\right )}{2^{7/8} \sqrt {x} \sqrt {-2+x^3+x^4}}-\frac {\left ((1+i) \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{\sqrt [4]{-1} \sqrt [8]{2}-x} \, dx,x,\sqrt {x}\right )}{2^{7/8} \sqrt {x} \sqrt {-2+x^3+x^4}}-\frac {\left ((1+i) \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{\sqrt [4]{-1} \sqrt [8]{2}+x} \, dx,x,\sqrt {x}\right )}{2^{7/8} \sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (i \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{i \sqrt [8]{2}-x} \, dx,x,\sqrt {x}\right )}{2^{3/8} \sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (i \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{i \sqrt [8]{2}+x} \, dx,x,\sqrt {x}\right )}{2^{3/8} \sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\sqrt {-2 x+x^4+x^5} \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{\sqrt [8]{2}-x} \, dx,x,\sqrt {x}\right )}{2^{3/8} \sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\sqrt {-2 x+x^4+x^5} \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{\sqrt [8]{2}+x} \, dx,x,\sqrt {x}\right )}{2^{3/8} \sqrt {x} \sqrt {-2+x^3+x^4}}\\ \end {align*}
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Mathematica [F] time = 1.94, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (6+x^4\right ) \sqrt {-2 x+x^4+x^5}}{\left (-2+x^4\right ) \left (-2-x^3+x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.17, size = 61, normalized size = 1.00 \begin {gather*} 2 \tanh ^{-1}\left (\frac {x^2}{\sqrt {-2 x+x^4+x^5}}\right )-2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {-2 x+x^4+x^5}}{-2+x^3+x^4}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.50, size = 121, normalized size = 1.98 \begin {gather*} \frac {1}{2} \, \sqrt {2} \log \left (\frac {x^{8} + 14 \, x^{7} + 17 \, x^{6} - 4 \, x^{4} - 28 \, x^{3} - 4 \, \sqrt {2} {\left (x^{5} + 3 \, x^{4} - 2 \, x\right )} \sqrt {x^{5} + x^{4} - 2 \, x} + 4}{x^{8} - 2 \, x^{7} + x^{6} - 4 \, x^{4} + 4 \, x^{3} + 4}\right ) + \log \left (\frac {x^{4} + 2 \, x^{3} + 2 \, \sqrt {x^{5} + x^{4} - 2 \, x} x - 2}{x^{4} - 2}\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 {\sqrt {x^{5} + x^{4} - 2 \, x} {\left (x^{4} + 6\right )}}{{\left (x^{4} - x^{3} - 2\right )} {\left (x^{4} - 2\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.64, size = 111, normalized size = 1.82
method | result | size |
trager | \(\ln \left (\frac {x^{4}+2 x^{3}+2 x \sqrt {x^{5}+x^{4}-2 x}-2}{x^{4}-2}\right )+\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}-2\right ) x^{4}-3 \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{3}+4 x \sqrt {x^{5}+x^{4}-2 x}+2 \RootOf \left (\textit {\_Z}^{2}-2\right )}{\left (1+x \right ) \left (x^{3}-2 x^{2}+2 x -2\right )}\right )\) | \(111\) |
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 {\sqrt {x^{5} + x^{4} - 2 \, x} {\left (x^{4} + 6\right )}}{{\left (x^{4} - x^{3} - 2\right )} {\left (x^{4} - 2\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.44, size = 81, normalized size = 1.33 \begin {gather*} \ln \left (\frac {2\,x\,\sqrt {x\,\left (x^4+x^3-2\right )}+2\,x^3+x^4-2}{x^4-2}\right )+\sqrt {2}\,\ln \left (\frac {3\,x^3+x^4-2\,\sqrt {2}\,x\,\sqrt {x\,\left (x^4+x^3-2\right )}-2}{-x^4+x^3+2}\right ) \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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