Optimal. Leaf size=131 \[ -\frac {9}{10 x^{5/2}}+\frac {9}{2 \sqrt {x}}+\frac {1}{2 x^{5/2} \left (1+x^2\right )}-\frac {9 \tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )}{4 \sqrt {2}}+\frac {9 \tan ^{-1}\left (1+\sqrt {2} \sqrt {x}\right )}{4 \sqrt {2}}+\frac {9 \log \left (1-\sqrt {2} \sqrt {x}+x\right )}{8 \sqrt {2}}-\frac {9 \log \left (1+\sqrt {2} \sqrt {x}+x\right )}{8 \sqrt {2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 9, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {296, 331, 335,
303, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {9 \text {ArcTan}\left (1-\sqrt {2} \sqrt {x}\right )}{4 \sqrt {2}}+\frac {9 \text {ArcTan}\left (\sqrt {2} \sqrt {x}+1\right )}{4 \sqrt {2}}-\frac {9}{10 x^{5/2}}+\frac {1}{2 x^{5/2} \left (x^2+1\right )}+\frac {9}{2 \sqrt {x}}+\frac {9 \log \left (x-\sqrt {2} \sqrt {x}+1\right )}{8 \sqrt {2}}-\frac {9 \log \left (x+\sqrt {2} \sqrt {x}+1\right )}{8 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 296
Rule 303
Rule 331
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {1}{x^{7/2} \left (1+x^2\right )^2} \, dx &=\frac {1}{2 x^{5/2} \left (1+x^2\right )}+\frac {9}{4} \int \frac {1}{x^{7/2} \left (1+x^2\right )} \, dx\\ &=-\frac {9}{10 x^{5/2}}+\frac {1}{2 x^{5/2} \left (1+x^2\right )}-\frac {9}{4} \int \frac {1}{x^{3/2} \left (1+x^2\right )} \, dx\\ &=-\frac {9}{10 x^{5/2}}+\frac {9}{2 \sqrt {x}}+\frac {1}{2 x^{5/2} \left (1+x^2\right )}+\frac {9}{4} \int \frac {\sqrt {x}}{1+x^2} \, dx\\ &=-\frac {9}{10 x^{5/2}}+\frac {9}{2 \sqrt {x}}+\frac {1}{2 x^{5/2} \left (1+x^2\right )}+\frac {9}{2} \text {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\sqrt {x}\right )\\ &=-\frac {9}{10 x^{5/2}}+\frac {9}{2 \sqrt {x}}+\frac {1}{2 x^{5/2} \left (1+x^2\right )}-\frac {9}{4} \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {x}\right )+\frac {9}{4} \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {x}\right )\\ &=-\frac {9}{10 x^{5/2}}+\frac {9}{2 \sqrt {x}}+\frac {1}{2 x^{5/2} \left (1+x^2\right )}+\frac {9}{8} \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {x}\right )+\frac {9}{8} \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {x}\right )+\frac {9 \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2}}+\frac {9 \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2}}\\ &=-\frac {9}{10 x^{5/2}}+\frac {9}{2 \sqrt {x}}+\frac {1}{2 x^{5/2} \left (1+x^2\right )}+\frac {9 \log \left (1-\sqrt {2} \sqrt {x}+x\right )}{8 \sqrt {2}}-\frac {9 \log \left (1+\sqrt {2} \sqrt {x}+x\right )}{8 \sqrt {2}}+\frac {9 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {x}\right )}{4 \sqrt {2}}-\frac {9 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {x}\right )}{4 \sqrt {2}}\\ &=-\frac {9}{10 x^{5/2}}+\frac {9}{2 \sqrt {x}}+\frac {1}{2 x^{5/2} \left (1+x^2\right )}-\frac {9 \tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )}{4 \sqrt {2}}+\frac {9 \tan ^{-1}\left (1+\sqrt {2} \sqrt {x}\right )}{4 \sqrt {2}}+\frac {9 \log \left (1-\sqrt {2} \sqrt {x}+x\right )}{8 \sqrt {2}}-\frac {9 \log \left (1+\sqrt {2} \sqrt {x}+x\right )}{8 \sqrt {2}}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 77, normalized size = 0.59 \begin {gather*} \frac {1}{40} \left (\frac {4 \left (-4+36 x^2+45 x^4\right )}{x^{5/2} \left (1+x^2\right )}+45 \sqrt {2} \tan ^{-1}\left (\frac {-1+x}{\sqrt {2} \sqrt {x}}\right )-45 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{1+x}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.46, size = 79, normalized size = 0.60
method | result | size |
derivativedivides | \(\frac {x^{\frac {3}{2}}}{2 x^{2}+2}+\frac {9 \sqrt {2}\, \left (\ln \left (\frac {1+x -\sqrt {2}\, \sqrt {x}}{1+x +\sqrt {2}\, \sqrt {x}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {x}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {x}\right )\right )}{16}-\frac {2}{5 x^{\frac {5}{2}}}+\frac {4}{\sqrt {x}}\) | \(79\) |
default | \(\frac {x^{\frac {3}{2}}}{2 x^{2}+2}+\frac {9 \sqrt {2}\, \left (\ln \left (\frac {1+x -\sqrt {2}\, \sqrt {x}}{1+x +\sqrt {2}\, \sqrt {x}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {x}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {x}\right )\right )}{16}-\frac {2}{5 x^{\frac {5}{2}}}+\frac {4}{\sqrt {x}}\) | \(79\) |
risch | \(\frac {45 x^{4}+36 x^{2}-4}{10 \left (x^{2}+1\right ) x^{\frac {5}{2}}}+\frac {9 \arctan \left (1+\sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{8}+\frac {9 \arctan \left (-1+\sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{8}+\frac {9 \sqrt {2}\, \ln \left (\frac {1+x -\sqrt {2}\, \sqrt {x}}{1+x +\sqrt {2}\, \sqrt {x}}\right )}{16}\) | \(86\) |
meijerg | \(-\frac {2 \left (-45 x^{4}-36 x^{2}+4\right )}{5 x^{\frac {5}{2}} \left (4 x^{2}+4\right )}+\frac {9 x^{\frac {3}{2}} \left (\frac {\sqrt {2}\, \ln \left (1-\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}+\sqrt {x^{2}}\right )}{2 \left (x^{2}\right )^{\frac {3}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}{2-\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}\right )}{\left (x^{2}\right )^{\frac {3}{4}}}-\frac {\sqrt {2}\, \ln \left (1+\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}+\sqrt {x^{2}}\right )}{2 \left (x^{2}\right )^{\frac {3}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}{2+\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}\right )}{\left (x^{2}\right )^{\frac {3}{4}}}\right )}{8}\) | \(157\) |
trager | \(\frac {45 x^{4}+36 x^{2}-4}{10 \left (x^{2}+1\right ) x^{\frac {5}{2}}}+\frac {9 \RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{5} x -\RootOf \left (\textit {\_Z}^{4}+1\right )^{5}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3}-\RootOf \left (\textit {\_Z}^{4}+1\right ) x +4 \sqrt {x}-\RootOf \left (\textit {\_Z}^{4}+1\right )}{\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x -\RootOf \left (\textit {\_Z}^{4}+1\right )^{2}-x -1}\right )}{8}-\frac {9 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{4}+1\right )^{5} x +\RootOf \left (\textit {\_Z}^{4}+1\right )^{5}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x -\RootOf \left (\textit {\_Z}^{4}+1\right ) x +4 \sqrt {x}-\RootOf \left (\textit {\_Z}^{4}+1\right )}{\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x -\RootOf \left (\textit {\_Z}^{4}+1\right )^{2}+x +1}\right )}{8}\) | \(208\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.56, size = 97, normalized size = 0.74 \begin {gather*} \frac {9}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {x}\right )}\right ) + \frac {9}{8} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {x}\right )}\right ) - \frac {9}{16} \, \sqrt {2} \log \left (\sqrt {2} \sqrt {x} + x + 1\right ) + \frac {9}{16} \, \sqrt {2} \log \left (-\sqrt {2} \sqrt {x} + x + 1\right ) + \frac {45 \, x^{4} + 36 \, x^{2} - 4}{10 \, {\left (x^{\frac {9}{2}} + x^{\frac {5}{2}}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.62, size = 163, normalized size = 1.24 \begin {gather*} -\frac {180 \, \sqrt {2} {\left (x^{5} + x^{3}\right )} \arctan \left (\sqrt {2} \sqrt {\sqrt {2} \sqrt {x} + x + 1} - \sqrt {2} \sqrt {x} - 1\right ) + 180 \, \sqrt {2} {\left (x^{5} + x^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {-4 \, \sqrt {2} \sqrt {x} + 4 \, x + 4} - \sqrt {2} \sqrt {x} + 1\right ) + 45 \, \sqrt {2} {\left (x^{5} + x^{3}\right )} \log \left (4 \, \sqrt {2} \sqrt {x} + 4 \, x + 4\right ) - 45 \, \sqrt {2} {\left (x^{5} + x^{3}\right )} \log \left (-4 \, \sqrt {2} \sqrt {x} + 4 \, x + 4\right ) - 8 \, {\left (45 \, x^{4} + 36 \, x^{2} - 4\right )} \sqrt {x}}{80 \, {\left (x^{5} + x^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 384 vs.
\(2 (121) = 242\).
time = 2.64, size = 384, normalized size = 2.93 \begin {gather*} \frac {45 \sqrt {2} x^{\frac {9}{2}} \log {\left (- 4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{80 x^{\frac {9}{2}} + 80 x^{\frac {5}{2}}} - \frac {45 \sqrt {2} x^{\frac {9}{2}} \log {\left (4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{80 x^{\frac {9}{2}} + 80 x^{\frac {5}{2}}} + \frac {90 \sqrt {2} x^{\frac {9}{2}} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} - 1 \right )}}{80 x^{\frac {9}{2}} + 80 x^{\frac {5}{2}}} + \frac {90 \sqrt {2} x^{\frac {9}{2}} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} + 1 \right )}}{80 x^{\frac {9}{2}} + 80 x^{\frac {5}{2}}} + \frac {45 \sqrt {2} x^{\frac {5}{2}} \log {\left (- 4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{80 x^{\frac {9}{2}} + 80 x^{\frac {5}{2}}} - \frac {45 \sqrt {2} x^{\frac {5}{2}} \log {\left (4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{80 x^{\frac {9}{2}} + 80 x^{\frac {5}{2}}} + \frac {90 \sqrt {2} x^{\frac {5}{2}} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} - 1 \right )}}{80 x^{\frac {9}{2}} + 80 x^{\frac {5}{2}}} + \frac {90 \sqrt {2} x^{\frac {5}{2}} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} + 1 \right )}}{80 x^{\frac {9}{2}} + 80 x^{\frac {5}{2}}} + \frac {360 x^{4}}{80 x^{\frac {9}{2}} + 80 x^{\frac {5}{2}}} + \frac {288 x^{2}}{80 x^{\frac {9}{2}} + 80 x^{\frac {5}{2}}} - \frac {32}{80 x^{\frac {9}{2}} + 80 x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.98, size = 98, normalized size = 0.75 \begin {gather*} \frac {9}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {x}\right )}\right ) + \frac {9}{8} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {x}\right )}\right ) - \frac {9}{16} \, \sqrt {2} \log \left (\sqrt {2} \sqrt {x} + x + 1\right ) + \frac {9}{16} \, \sqrt {2} \log \left (-\sqrt {2} \sqrt {x} + x + 1\right ) + \frac {x^{\frac {3}{2}}}{2 \, {\left (x^{2} + 1\right )}} + \frac {2 \, {\left (10 \, x^{2} - 1\right )}}{5 \, x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 59, normalized size = 0.45 \begin {gather*} \frac {\frac {9\,x^4}{2}+\frac {18\,x^2}{5}-\frac {2}{5}}{x^{5/2}+x^{9/2}}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {9}{8}-\frac {9}{8}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {9}{8}+\frac {9}{8}{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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