Optimal. Leaf size=130 \[ -\frac {\sqrt {x^8+1}}{2 x^2}+\frac {\sqrt {x^8+1} x^2}{2 \left (x^4+1\right )}+\frac {\left (x^4+1\right ) \sqrt {\frac {x^8+1}{\left (x^4+1\right )^2}} F\left (2 \tan ^{-1}\left (x^2\right )|\frac {1}{2}\right )}{4 \sqrt {x^8+1}}-\frac {\left (x^4+1\right ) \sqrt {\frac {x^8+1}{\left (x^4+1\right )^2}} E\left (2 \tan ^{-1}\left (x^2\right )|\frac {1}{2}\right )}{2 \sqrt {x^8+1}} \]
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Rubi [A] time = 0.05, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {275, 325, 305, 220, 1196} \[ \frac {\sqrt {x^8+1} x^2}{2 \left (x^4+1\right )}-\frac {\sqrt {x^8+1}}{2 x^2}+\frac {\left (x^4+1\right ) \sqrt {\frac {x^8+1}{\left (x^4+1\right )^2}} F\left (2 \tan ^{-1}\left (x^2\right )|\frac {1}{2}\right )}{4 \sqrt {x^8+1}}-\frac {\left (x^4+1\right ) \sqrt {\frac {x^8+1}{\left (x^4+1\right )^2}} E\left (2 \tan ^{-1}\left (x^2\right )|\frac {1}{2}\right )}{2 \sqrt {x^8+1}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 275
Rule 305
Rule 325
Rule 1196
Rubi steps
\begin {align*} \int \frac {1}{x^3 \sqrt {1+x^8}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1+x^4}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {1+x^8}}{2 x^2}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+x^4}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {1+x^8}}{2 x^2}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,x^2\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1-x^2}{\sqrt {1+x^4}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {1+x^8}}{2 x^2}+\frac {x^2 \sqrt {1+x^8}}{2 \left (1+x^4\right )}-\frac {\left (1+x^4\right ) \sqrt {\frac {1+x^8}{\left (1+x^4\right )^2}} E\left (2 \tan ^{-1}\left (x^2\right )|\frac {1}{2}\right )}{2 \sqrt {1+x^8}}+\frac {\left (1+x^4\right ) \sqrt {\frac {1+x^8}{\left (1+x^4\right )^2}} F\left (2 \tan ^{-1}\left (x^2\right )|\frac {1}{2}\right )}{4 \sqrt {1+x^8}}\\ \end {align*}
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Mathematica [C] time = 0.00, size = 22, normalized size = 0.17 \[ -\frac {\, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-x^8\right )}{2 x^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x^{8} + 1}}{x^{11} + x^{3}}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {x^{8} + 1} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 30, normalized size = 0.23 \[ \frac {x^{6} \hypergeom \left (\left [\frac {1}{2}, \frac {3}{4}\right ], \left [\frac {7}{4}\right ], -x^{8}\right )}{6}-\frac {\sqrt {x^{8}+1}}{2 x^{2}} \]
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 \[ \int \frac {1}{\sqrt {x^{8} + 1} x^{3}}\,{d x} \]
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 \[ \int \frac {1}{x^3\,\sqrt {x^8+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.80, size = 32, normalized size = 0.25 \[ \frac {\Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{2} \\ \frac {3}{4} \end {matrix}\middle | {x^{8} e^{i \pi }} \right )}}{8 x^{2} \Gamma \left (\frac {3}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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