Optimal. Leaf size=157 \[ -\frac {\sqrt {\sqrt [4]{2}-1} \tan ^{-1}\left (\frac {x}{\sqrt {\sqrt [4]{2}-1}}\right )}{4\ 2^{3/4}}-\frac {i \sqrt {1-i \sqrt [4]{2}} \tanh ^{-1}\left (\frac {x}{\sqrt {1-i \sqrt [4]{2}}}\right )}{4\ 2^{3/4}}+\frac {i \sqrt {1+i \sqrt [4]{2}} \tanh ^{-1}\left (\frac {x}{\sqrt {1+i \sqrt [4]{2}}}\right )}{4\ 2^{3/4}}+\frac {\sqrt {1+\sqrt [4]{2}} \tanh ^{-1}\left (\frac {x}{\sqrt {1+\sqrt [4]{2}}}\right )}{4\ 2^{3/4}} \]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6740, 206, 203, 1972, 208} \begin {gather*} -\frac {\sqrt {\sqrt [4]{2}-1} \tan ^{-1}\left (\frac {x}{\sqrt {\sqrt [4]{2}-1}}\right )}{4\ 2^{3/4}}-\frac {i \sqrt {1-i \sqrt [4]{2}} \tanh ^{-1}\left (\frac {x}{\sqrt {1-i \sqrt [4]{2}}}\right )}{4\ 2^{3/4}}+\frac {i \sqrt {1+i \sqrt [4]{2}} \tanh ^{-1}\left (\frac {x}{\sqrt {1+i \sqrt [4]{2}}}\right )}{4\ 2^{3/4}}+\frac {\sqrt {1+\sqrt [4]{2}} \tanh ^{-1}\left (\frac {x}{\sqrt {1+\sqrt [4]{2}}}\right )}{4\ 2^{3/4}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 206
Rule 208
Rule 1972
Rule 6740
Rubi steps
\begin {align*} \int \frac {x^2}{2-\left (1-x^2\right )^4} \, dx &=\int \left (\frac {\sqrt [4]{2}+\sqrt {2}}{8 \left (1+\sqrt [4]{2}-x^2\right )}+\frac {\sqrt [4]{2}-\sqrt {2}}{8 \left (-1+\sqrt [4]{2}+x^2\right )}+\frac {\sqrt [4]{2}+i \sqrt {2}}{8 \left (\sqrt [4]{2}-i \left (1-x^2\right )\right )}+\frac {\sqrt [4]{2}-i \sqrt {2}}{8 \left (\sqrt [4]{2}+i \left (1-x^2\right )\right )}\right ) \, dx\\ &=\frac {\left (1-\sqrt [4]{2}\right ) \int \frac {1}{-1+\sqrt [4]{2}+x^2} \, dx}{4\ 2^{3/4}}+\frac {\left (1-i \sqrt [4]{2}\right ) \int \frac {1}{\sqrt [4]{2}+i \left (1-x^2\right )} \, dx}{4\ 2^{3/4}}+\frac {\left (1+i \sqrt [4]{2}\right ) \int \frac {1}{\sqrt [4]{2}-i \left (1-x^2\right )} \, dx}{4\ 2^{3/4}}+\frac {\left (1+\sqrt [4]{2}\right ) \int \frac {1}{1+\sqrt [4]{2}-x^2} \, dx}{4\ 2^{3/4}}\\ &=-\frac {\sqrt {-1+\sqrt [4]{2}} \tan ^{-1}\left (\frac {x}{\sqrt {-1+\sqrt [4]{2}}}\right )}{4\ 2^{3/4}}+\frac {\sqrt {1+\sqrt [4]{2}} \tanh ^{-1}\left (\frac {x}{\sqrt {1+\sqrt [4]{2}}}\right )}{4\ 2^{3/4}}+\frac {\left (1-i \sqrt [4]{2}\right ) \int \frac {1}{i+\sqrt [4]{2}-i x^2} \, dx}{4\ 2^{3/4}}+\frac {\left (1+i \sqrt [4]{2}\right ) \int \frac {1}{-i+\sqrt [4]{2}+i x^2} \, dx}{4\ 2^{3/4}}\\ &=-\frac {\sqrt {-1+\sqrt [4]{2}} \tan ^{-1}\left (\frac {x}{\sqrt {-1+\sqrt [4]{2}}}\right )}{4\ 2^{3/4}}-\frac {i \sqrt {1-i \sqrt [4]{2}} \tanh ^{-1}\left (\frac {x}{\sqrt {1-i \sqrt [4]{2}}}\right )}{4\ 2^{3/4}}+\frac {i \sqrt {1+i \sqrt [4]{2}} \tanh ^{-1}\left (\frac {x}{\sqrt {1+i \sqrt [4]{2}}}\right )}{4\ 2^{3/4}}+\frac {\sqrt {1+\sqrt [4]{2}} \tanh ^{-1}\left (\frac {x}{\sqrt {1+\sqrt [4]{2}}}\right )}{4\ 2^{3/4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.02, size = 61, normalized size = 0.39 \begin {gather*} -\frac {1}{8} \text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6+6 \text {$\#$1}^4-4 \text {$\#$1}^2-1\&,\frac {\text {$\#$1} \log (x-\text {$\#$1})}{\text {$\#$1}^6-3 \text {$\#$1}^4+3 \text {$\#$1}^2-1}\&\right ] \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2}{2-\left (1-x^2\right )^4} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 4.05, size = 1546, normalized size = 9.85
result too large to display
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 {x^{2}}{{\left (x^{2} - 1\right )}^{4} - 2}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.02, size = 56, normalized size = 0.36 \begin {gather*} -\frac {\RootOf \left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+6 \textit {\_Z}^{4}-4 \textit {\_Z}^{2}-1\right )^{2} \ln \left (-\RootOf \left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+6 \textit {\_Z}^{4}-4 \textit {\_Z}^{2}-1\right )+x \right )}{8 \left (\RootOf \left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+6 \textit {\_Z}^{4}-4 \textit {\_Z}^{2}-1\right )^{7}-3 \RootOf \left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+6 \textit {\_Z}^{4}-4 \textit {\_Z}^{2}-1\right )^{5}+3 \RootOf \left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+6 \textit {\_Z}^{4}-4 \textit {\_Z}^{2}-1\right )^{3}-\RootOf \left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+6 \textit {\_Z}^{4}-4 \textit {\_Z}^{2}-1\right )\right )} \end {gather*}
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 {x^{2}}{{\left (x^{2} - 1\right )}^{4} - 2}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.80, size = 142, normalized size = 0.90 \begin {gather*} \sum _{k=1}^8\ln \left (-\mathrm {root}\left (z^8-\frac {z^4}{16384}-\frac {z^2}{1048576}-\frac {1}{1073741824},z,k\right )\,\left (56\,x+\mathrm {root}\left (z^8-\frac {z^4}{16384}-\frac {z^2}{1048576}-\frac {1}{1073741824},z,k\right )\,\left (\mathrm {root}\left (z^8-\frac {z^4}{16384}-\frac {z^2}{1048576}-\frac {1}{1073741824},z,k\right )\,\left (4096\,x+{\mathrm {root}\left (z^8-\frac {z^4}{16384}-\frac {z^2}{1048576}-\frac {1}{1073741824},z,k\right )}^2\,\left (262144\,x-{\mathrm {root}\left (z^8-\frac {z^4}{16384}-\frac {z^2}{1048576}-\frac {1}{1073741824},z,k\right )}^2\,x\,67108864\right )\right )+256\right )\right )-1\right )\,\mathrm {root}\left (z^8-\frac {z^4}{16384}-\frac {z^2}{1048576}-\frac {1}{1073741824},z,k\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.23, size = 41, normalized size = 0.26 \begin {gather*} - \operatorname {RootSum} {\left (1073741824 t^{8} - 65536 t^{4} - 1024 t^{2} - 1, \left (t \mapsto t \log {\left (- \frac {67108864 t^{7}}{3} + \frac {262144 t^{5}}{3} + \frac {40 t}{3} + x \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________