Optimal. Leaf size=399 \[ \frac {\sqrt {2} \sqrt {2 x^4+2 x^2+1} x}{3 \left (\sqrt {2} x^2+1\right )}-\frac {\sqrt {2 x^4+2 x^2+1}}{3 x}-\frac {\tan ^{-1}\left (\frac {\sqrt {\frac {5}{3}} x}{\sqrt {2 x^4+2 x^2+1}}\right )}{3 \sqrt {15}}+\frac {\left (5-3 \sqrt {2}\right ) \left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{21\ 2^{3/4} \sqrt {2 x^4+2 x^2+1}}-\frac {\sqrt [4]{2} \left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{3 \sqrt {2 x^4+2 x^2+1}}+\frac {\left (3+\sqrt {2}\right )^2 \left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} \Pi \left (\frac {1}{24} \left (12-11 \sqrt {2}\right );2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{126 \sqrt [4]{2} \sqrt {2 x^4+2 x^2+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.35, antiderivative size = 399, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {1329, 1714, 1195, 1708, 1103, 1706} \[ \frac {\sqrt {2} \sqrt {2 x^4+2 x^2+1} x}{3 \left (\sqrt {2} x^2+1\right )}-\frac {\sqrt {2 x^4+2 x^2+1}}{3 x}-\frac {\tan ^{-1}\left (\frac {\sqrt {\frac {5}{3}} x}{\sqrt {2 x^4+2 x^2+1}}\right )}{3 \sqrt {15}}+\frac {\left (5-3 \sqrt {2}\right ) \left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{21\ 2^{3/4} \sqrt {2 x^4+2 x^2+1}}-\frac {\sqrt [4]{2} \left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{3 \sqrt {2 x^4+2 x^2+1}}+\frac {\left (3+\sqrt {2}\right )^2 \left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} \Pi \left (\frac {1}{24} \left (12-11 \sqrt {2}\right );2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{126 \sqrt [4]{2} \sqrt {2 x^4+2 x^2+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1103
Rule 1195
Rule 1329
Rule 1706
Rule 1708
Rule 1714
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (3+2 x^2\right ) \sqrt {1+2 x^2+2 x^4}} \, dx &=-\frac {\sqrt {1+2 x^2+2 x^4}}{3 x}+\frac {1}{3} \int \frac {-2+6 x^2+4 x^4}{\left (3+2 x^2\right ) \sqrt {1+2 x^2+2 x^4}} \, dx\\ &=-\frac {\sqrt {1+2 x^2+2 x^4}}{3 x}+\frac {1}{12} \int \frac {-8+12 \sqrt {2}+\left (24-4 \left (6-2 \sqrt {2}\right )\right ) x^2}{\left (3+2 x^2\right ) \sqrt {1+2 x^2+2 x^4}} \, dx-\frac {1}{3} \sqrt {2} \int \frac {1-\sqrt {2} x^2}{\sqrt {1+2 x^2+2 x^4}} \, dx\\ &=-\frac {\sqrt {1+2 x^2+2 x^4}}{3 x}+\frac {\sqrt {2} x \sqrt {1+2 x^2+2 x^4}}{3 \left (1+\sqrt {2} x^2\right )}-\frac {\sqrt [4]{2} \left (1+\sqrt {2} x^2\right ) \sqrt {\frac {1+2 x^2+2 x^4}{\left (1+\sqrt {2} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{3 \sqrt {1+2 x^2+2 x^4}}+\frac {1}{21} \left (2 \left (2+3 \sqrt {2}\right )\right ) \int \frac {1+\sqrt {2} x^2}{\left (3+2 x^2\right ) \sqrt {1+2 x^2+2 x^4}} \, dx+\frac {1}{21} \left (-6+5 \sqrt {2}\right ) \int \frac {1}{\sqrt {1+2 x^2+2 x^4}} \, dx\\ &=-\frac {\sqrt {1+2 x^2+2 x^4}}{3 x}+\frac {\sqrt {2} x \sqrt {1+2 x^2+2 x^4}}{3 \left (1+\sqrt {2} x^2\right )}-\frac {\tan ^{-1}\left (\frac {\sqrt {\frac {5}{3}} x}{\sqrt {1+2 x^2+2 x^4}}\right )}{3 \sqrt {15}}-\frac {\sqrt [4]{2} \left (1+\sqrt {2} x^2\right ) \sqrt {\frac {1+2 x^2+2 x^4}{\left (1+\sqrt {2} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{3 \sqrt {1+2 x^2+2 x^4}}+\frac {\left (5-3 \sqrt {2}\right ) \left (1+\sqrt {2} x^2\right ) \sqrt {\frac {1+2 x^2+2 x^4}{\left (1+\sqrt {2} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{21\ 2^{3/4} \sqrt {1+2 x^2+2 x^4}}+\frac {\left (3+\sqrt {2}\right )^2 \left (1+\sqrt {2} x^2\right ) \sqrt {\frac {1+2 x^2+2 x^4}{\left (1+\sqrt {2} x^2\right )^2}} \Pi \left (\frac {1}{24} \left (12-11 \sqrt {2}\right );2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{126 \sqrt [4]{2} \sqrt {1+2 x^2+2 x^4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.22, size = 147, normalized size = 0.37 \[ -\frac {i \left (\sqrt {1-i} x \sqrt {1+(1-i) x^2} \sqrt {1+(1+i) x^2} \left (-3 F\left (\left .i \sinh ^{-1}\left (\sqrt {1-i} x\right )\right |i\right )+3 E\left (\left .i \sinh ^{-1}\left (\sqrt {1-i} x\right )\right |i\right )-(1+i) \Pi \left (\frac {1}{3}+\frac {i}{3};\left .i \sinh ^{-1}\left (\sqrt {1-i} x\right )\right |i\right )\right )-3 i \left (2 x^4+2 x^2+1\right )\right )}{9 x \sqrt {2 x^4+2 x^2+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.31, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {2 \, x^{4} + 2 \, x^{2} + 1}}{4 \, x^{8} + 10 \, x^{6} + 8 \, x^{4} + 3 \, x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {2 \, x^{4} + 2 \, x^{2} + 1} {\left (2 \, x^{2} + 3\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.02, size = 178, normalized size = 0.45 \[ -\frac {2 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \EllipticPi \left (\sqrt {-1+i}\, x , \frac {1}{3}+\frac {i}{3}, \frac {\sqrt {-1-i}}{\sqrt {-1+i}}\right )}{9 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}-\frac {\sqrt {2 x^{4}+2 x^{2}+1}}{3 x}+\frac {\left (-\frac {1}{3}+\frac {i}{3}\right ) \sqrt {\left (1-i\right ) x^{2}+1}\, \sqrt {\left (1+i\right ) x^{2}+1}\, \left (-\EllipticE \left (\sqrt {-1+i}\, x , \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )+\EllipticF \left (\sqrt {-1+i}\, x , \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )\right )}{\sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {2 \, x^{4} + 2 \, x^{2} + 1} {\left (2 \, x^{2} + 3\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{x^2\,\left (2\,x^2+3\right )\,\sqrt {2\,x^4+2\,x^2+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \left (2 x^{2} + 3\right ) \sqrt {2 x^{4} + 2 x^{2} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________