Optimal. Leaf size=140 \[ \frac {3 \sqrt {x^4+1}}{5 x}-\frac {\sqrt {x^4+1}}{5 x^5}-\frac {3 \sqrt {x^4+1} x}{5 \left (x^2+1\right )}-\frac {3 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{10 \sqrt {x^4+1}}+\frac {3 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{5 \sqrt {x^4+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {325, 305, 220, 1196} \[ -\frac {3 \sqrt {x^4+1} x}{5 \left (x^2+1\right )}+\frac {3 \sqrt {x^4+1}}{5 x}-\frac {\sqrt {x^4+1}}{5 x^5}-\frac {3 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{10 \sqrt {x^4+1}}+\frac {3 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{5 \sqrt {x^4+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 220
Rule 305
Rule 325
Rule 1196
Rubi steps
\begin {align*} \int \frac {1}{x^6 \sqrt {1+x^4}} \, dx &=-\frac {\sqrt {1+x^4}}{5 x^5}-\frac {3}{5} \int \frac {1}{x^2 \sqrt {1+x^4}} \, dx\\ &=-\frac {\sqrt {1+x^4}}{5 x^5}+\frac {3 \sqrt {1+x^4}}{5 x}-\frac {3}{5} \int \frac {x^2}{\sqrt {1+x^4}} \, dx\\ &=-\frac {\sqrt {1+x^4}}{5 x^5}+\frac {3 \sqrt {1+x^4}}{5 x}-\frac {3}{5} \int \frac {1}{\sqrt {1+x^4}} \, dx+\frac {3}{5} \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx\\ &=-\frac {\sqrt {1+x^4}}{5 x^5}+\frac {3 \sqrt {1+x^4}}{5 x}-\frac {3 x \sqrt {1+x^4}}{5 \left (1+x^2\right )}+\frac {3 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{5 \sqrt {1+x^4}}-\frac {3 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{10 \sqrt {1+x^4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.00, size = 22, normalized size = 0.16 \[ -\frac {\, _2F_1\left (-\frac {5}{4},\frac {1}{2};-\frac {1}{4};-x^4\right )}{5 x^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.90, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x^{4} + 1}}{x^{10} + x^{6}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x^{4} + 1} x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.01, size = 107, normalized size = 0.76 \[ \frac {3 \sqrt {x^{4}+1}}{5 x}-\frac {\sqrt {x^{4}+1}}{5 x^{5}}-\frac {3 i \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \left (-\EllipticE \left (\left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) x , i\right )+\EllipticF \left (\left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) x , i\right )\right )}{5 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x^{4} + 1} x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^6\,\sqrt {x^4+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 1.12, size = 36, normalized size = 0.26 \[ \frac {\Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, \frac {1}{2} \\ - \frac {1}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 x^{5} \Gamma \left (- \frac {1}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________