Optimal. Leaf size=63 \[ -\frac {9 \sqrt {1-x^4}}{14 x^7}+\frac {1}{2 x^7 \sqrt {1-x^4}}-\frac {15 \sqrt {1-x^4}}{14 x^3}+\frac {15}{14} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {290, 325, 221} \[ -\frac {15 \sqrt {1-x^4}}{14 x^3}-\frac {9 \sqrt {1-x^4}}{14 x^7}+\frac {1}{2 x^7 \sqrt {1-x^4}}+\frac {15}{14} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 221
Rule 290
Rule 325
Rubi steps
\begin {align*} \int \frac {1}{x^8 \left (1-x^4\right )^{3/2}} \, dx &=\frac {1}{2 x^7 \sqrt {1-x^4}}+\frac {9}{2} \int \frac {1}{x^8 \sqrt {1-x^4}} \, dx\\ &=\frac {1}{2 x^7 \sqrt {1-x^4}}-\frac {9 \sqrt {1-x^4}}{14 x^7}+\frac {45}{14} \int \frac {1}{x^4 \sqrt {1-x^4}} \, dx\\ &=\frac {1}{2 x^7 \sqrt {1-x^4}}-\frac {9 \sqrt {1-x^4}}{14 x^7}-\frac {15 \sqrt {1-x^4}}{14 x^3}+\frac {15}{14} \int \frac {1}{\sqrt {1-x^4}} \, dx\\ &=\frac {1}{2 x^7 \sqrt {1-x^4}}-\frac {9 \sqrt {1-x^4}}{14 x^7}-\frac {15 \sqrt {1-x^4}}{14 x^3}+\frac {15}{14} F\left (\left .\sin ^{-1}(x)\right |-1\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.00, size = 20, normalized size = 0.32 \[ -\frac {\, _2F_1\left (-\frac {7}{4},\frac {3}{2};-\frac {3}{4};x^4\right )}{7 x^7} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.91, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-x^{4} + 1}}{x^{16} - 2 \, x^{12} + x^{8}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (-x^{4} + 1\right )}^{\frac {3}{2}} x^{8}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 73, normalized size = 1.16 \[ \frac {x}{2 \sqrt {-x^{4}+1}}+\frac {15 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \EllipticF \left (x , i\right )}{14 \sqrt {-x^{4}+1}}-\frac {4 \sqrt {-x^{4}+1}}{7 x^{3}}-\frac {\sqrt {-x^{4}+1}}{7 x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (-x^{4} + 1\right )}^{\frac {3}{2}} x^{8}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{x^8\,{\left (1-x^4\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.50, size = 37, normalized size = 0.59 \[ \frac {\Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, \frac {3}{2} \\ - \frac {3}{4} \end {matrix}\middle | {x^{4} e^{2 i \pi }} \right )}}{4 x^{7} \Gamma \left (- \frac {3}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________