Optimal. Leaf size=71 \[ -\frac {21 \sqrt {1-x^4}}{10 x}-\frac {7 \sqrt {1-x^4}}{10 x^5}+\frac {1}{2 x^5 \sqrt {1-x^4}}+\frac {21}{10} F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac {21}{10} E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {290, 325, 307, 221, 1181, 424} \[ -\frac {21 \sqrt {1-x^4}}{10 x}-\frac {7 \sqrt {1-x^4}}{10 x^5}+\frac {1}{2 x^5 \sqrt {1-x^4}}+\frac {21}{10} F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac {21}{10} E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 221
Rule 290
Rule 307
Rule 325
Rule 424
Rule 1181
Rubi steps
\begin {align*} \int \frac {1}{x^6 \left (1-x^4\right )^{3/2}} \, dx &=\frac {1}{2 x^5 \sqrt {1-x^4}}+\frac {7}{2} \int \frac {1}{x^6 \sqrt {1-x^4}} \, dx\\ &=\frac {1}{2 x^5 \sqrt {1-x^4}}-\frac {7 \sqrt {1-x^4}}{10 x^5}+\frac {21}{10} \int \frac {1}{x^2 \sqrt {1-x^4}} \, dx\\ &=\frac {1}{2 x^5 \sqrt {1-x^4}}-\frac {7 \sqrt {1-x^4}}{10 x^5}-\frac {21 \sqrt {1-x^4}}{10 x}-\frac {21}{10} \int \frac {x^2}{\sqrt {1-x^4}} \, dx\\ &=\frac {1}{2 x^5 \sqrt {1-x^4}}-\frac {7 \sqrt {1-x^4}}{10 x^5}-\frac {21 \sqrt {1-x^4}}{10 x}+\frac {21}{10} \int \frac {1}{\sqrt {1-x^4}} \, dx-\frac {21}{10} \int \frac {1+x^2}{\sqrt {1-x^4}} \, dx\\ &=\frac {1}{2 x^5 \sqrt {1-x^4}}-\frac {7 \sqrt {1-x^4}}{10 x^5}-\frac {21 \sqrt {1-x^4}}{10 x}+\frac {21}{10} F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac {21}{10} \int \frac {\sqrt {1+x^2}}{\sqrt {1-x^2}} \, dx\\ &=\frac {1}{2 x^5 \sqrt {1-x^4}}-\frac {7 \sqrt {1-x^4}}{10 x^5}-\frac {21 \sqrt {1-x^4}}{10 x}-\frac {21}{10} E\left (\left .\sin ^{-1}(x)\right |-1\right )+\frac {21}{10} F\left (\left .\sin ^{-1}(x)\right |-1\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.01, size = 20, normalized size = 0.28 \[ -\frac {\, _2F_1\left (-\frac {5}{4},\frac {3}{2};-\frac {1}{4};x^4\right )}{5 x^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.21, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-x^{4} + 1}}{x^{14} - 2 \, x^{10} + x^{6}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (-x^{4} + 1\right )}^{\frac {3}{2}} x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 82, normalized size = 1.15 \[ \frac {x^{3}}{2 \sqrt {-x^{4}+1}}-\frac {8 \sqrt {-x^{4}+1}}{5 x}-\frac {\sqrt {-x^{4}+1}}{5 x^{5}}+\frac {21 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (-\EllipticE \left (x , i\right )+\EllipticF \left (x , i\right )\right )}{10 \sqrt {-x^{4}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (-x^{4} + 1\right )}^{\frac {3}{2}} x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^6\,{\left (1-x^4\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.48, size = 37, normalized size = 0.52 \[ \frac {\Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, \frac {3}{2} \\ - \frac {1}{4} \end {matrix}\middle | {x^{4} e^{2 i \pi }} \right )}}{4 x^{5} \Gamma \left (- \frac {1}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________