Optimal. Leaf size=98 \[ \frac {\log \left (\sqrt {2} x^2+\sqrt {2 x^4-7 x^2+8}-2 \sqrt {2}\right )}{2 \sqrt {2}}+\frac {\sqrt {2 x^4-7 x^2+8} \left (4 x^4+3 x^3-14 x^2-6 x+16\right )}{6 x^3}-\frac {\log (x)}{2 \sqrt {2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.14, antiderivative size = 93, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1662, 1590, 1251, 812, 838, 206} \begin {gather*} -\frac {\left (2-x^2\right ) \sqrt {2 x^4-7 x^2+8}}{2 x^2}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \left (2-x^2\right )}{\sqrt {2 x^4-7 x^2+8}}\right )}{2 \sqrt {2}}+\frac {\left (2 x^4-7 x^2+8\right )^{3/2}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 812
Rule 838
Rule 1251
Rule 1590
Rule 1662
Rubi steps
\begin {align*} \int \frac {\left (2+x^2\right ) \left (-4+x+2 x^2\right ) \sqrt {8-7 x^2+2 x^4}}{x^4} \, dx &=\int \frac {\left (2+x^2\right ) \sqrt {8-7 x^2+2 x^4}}{x^3} \, dx+\int \frac {\left (-8+2 x^4\right ) \sqrt {8-7 x^2+2 x^4}}{x^4} \, dx\\ &=\frac {\left (8-7 x^2+2 x^4\right )^{3/2}}{3 x^3}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {(2+x) \sqrt {8-7 x+2 x^2}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {\left (2-x^2\right ) \sqrt {8-7 x^2+2 x^4}}{2 x^2}+\frac {\left (8-7 x^2+2 x^4\right )^{3/2}}{3 x^3}-\frac {1}{4} \operatorname {Subst}\left (\int \frac {-2-x}{x \sqrt {8-7 x+2 x^2}} \, dx,x,x^2\right )\\ &=-\frac {\left (2-x^2\right ) \sqrt {8-7 x^2+2 x^4}}{2 x^2}+\frac {\left (8-7 x^2+2 x^4\right )^{3/2}}{3 x^3}-2 \operatorname {Subst}\left (\int \frac {1}{32-x^2} \, dx,x,\frac {8 \left (2-x^2\right )}{\sqrt {8-7 x^2+2 x^4}}\right )\\ &=-\frac {\left (2-x^2\right ) \sqrt {8-7 x^2+2 x^4}}{2 x^2}+\frac {\left (8-7 x^2+2 x^4\right )^{3/2}}{3 x^3}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \left (2-x^2\right )}{\sqrt {8-7 x^2+2 x^4}}\right )}{2 \sqrt {2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.25, size = 105, normalized size = 1.07 \begin {gather*} \frac {\sinh ^{-1}\left (\frac {4 x^2-7}{\sqrt {15}}\right )-\tanh ^{-1}\left (\frac {16-7 x^2}{4 \sqrt {2} \sqrt {2 x^4-7 x^2+8}}\right )}{4 \sqrt {2}}+\sqrt {2 x^4-7 x^2+8} \left (\frac {8}{3 x^3}-\frac {1}{x^2}+\frac {2 x}{3}-\frac {7}{3 x}+\frac {1}{2}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.70, size = 98, normalized size = 1.00 \begin {gather*} \frac {\sqrt {8-7 x^2+2 x^4} \left (16-6 x-14 x^2+3 x^3+4 x^4\right )}{6 x^3}-\frac {\log (x)}{2 \sqrt {2}}+\frac {\log \left (-2 \sqrt {2}+\sqrt {2} x^2+\sqrt {8-7 x^2+2 x^4}\right )}{2 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.69, size = 91, normalized size = 0.93 \begin {gather*} \frac {3 \, \sqrt {2} x^{3} \log \left (\frac {4 \, x^{4} + 2 \, \sqrt {2} \sqrt {2 \, x^{4} - 7 \, x^{2} + 8} {\left (x^{2} - 2\right )} - 15 \, x^{2} + 16}{x^{2}}\right ) + 4 \, {\left (4 \, x^{4} + 3 \, x^{3} - 14 \, x^{2} - 6 \, x + 16\right )} \sqrt {2 \, x^{4} - 7 \, x^{2} + 8}}{24 \, x^{3}} \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*} \int \frac {\sqrt {2 \, x^{4} - 7 \, x^{2} + 8} {\left (2 \, x^{2} + x - 4\right )} {\left (x^{2} + 2\right )}}{x^{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.64, size = 88, normalized size = 0.90
method | result | size |
trager | \(\frac {\sqrt {2 x^{4}-7 x^{2}+8}\, \left (4 x^{4}+3 x^{3}-14 x^{2}-6 x +16\right )}{6 x^{3}}-\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}-2\right ) x^{2}+\sqrt {2 x^{4}-7 x^{2}+8}+2 \RootOf \left (\textit {\_Z}^{2}-2\right )}{x}\right )}{4}\) | \(88\) |
elliptic | \(\frac {\sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {15}\, \left (x^{2}-\frac {7}{4}\right )}{15}\right )}{8}+\frac {\sqrt {2 x^{4}-7 x^{2}+8}}{2}-\frac {\sqrt {2 x^{4}-7 x^{2}+8}}{x^{2}}-\frac {\sqrt {2}\, \arctanh \left (\frac {\left (-7 x^{2}+16\right ) \sqrt {2}}{8 \sqrt {2 x^{4}-7 x^{2}+8}}\right )}{8}+\frac {\left (2 x^{4}-7 x^{2}+8\right )^{\frac {3}{2}}}{3 x^{3}}\) | \(104\) |
risch | \(-\frac {14 x^{6}+6 x^{5}-65 x^{4}-21 x^{3}+112 x^{2}+24 x -64}{3 x^{3} \sqrt {2 x^{4}-7 x^{2}+8}}+\frac {2 x \sqrt {2 x^{4}-7 x^{2}+8}}{3}+\frac {\sqrt {2 x^{4}-7 x^{2}+8}}{2}+\frac {\sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {15}\, \left (x^{2}-\frac {7}{4}\right )}{15}\right )}{8}-\frac {\sqrt {2}\, \arctanh \left (\frac {\left (-7 x^{2}+16\right ) \sqrt {2}}{8 \sqrt {2 x^{4}-7 x^{2}+8}}\right )}{8}\) | \(132\) |
default | \(\frac {2 x \sqrt {2 x^{4}-7 x^{2}+8}}{3}+\frac {8 \sqrt {2 x^{4}-7 x^{2}+8}}{3 x^{3}}-\frac {7 \sqrt {2 x^{4}-7 x^{2}+8}}{3 x}+\frac {\sqrt {2 x^{4}-7 x^{2}+8}}{16}+\frac {\sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {15}\, \left (x^{2}-\frac {7}{4}\right )}{15}\right )}{8}-\frac {\sqrt {2}\, \arctanh \left (\frac {\left (-7 x^{2}+16\right ) \sqrt {2}}{8 \sqrt {2 x^{4}-7 x^{2}+8}}\right )}{8}-\frac {\left (2 x^{4}-7 x^{2}+8\right )^{\frac {3}{2}}}{8 x^{2}}+\frac {\left (4 x^{2}-7\right ) \sqrt {2 x^{4}-7 x^{2}+8}}{16}\) | \(163\) |
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*} \int \frac {\sqrt {2 \, x^{4} - 7 \, x^{2} + 8} {\left (2 \, x^{2} + x - 4\right )} {\left (x^{2} + 2\right )}}{x^{4}}\,{d x} \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^2+2\right )\,\left (2\,x^2+x-4\right )\,\sqrt {2\,x^4-7\,x^2+8}}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{2} + 2\right ) \left (2 x^{2} + x - 4\right ) \sqrt {2 x^{4} - 7 x^{2} + 8}}{x^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________