Optimal. Leaf size=203 \[ -\frac {1}{2 a x^2}+\frac {\sqrt [4]{b} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4}}-\frac {\sqrt [4]{b} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4}}-\frac {\sqrt [4]{b} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt {b} x^4\right )}{8 \sqrt {2} a^{5/4}}+\frac {\sqrt [4]{b} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt {b} x^4\right )}{8 \sqrt {2} a^{5/4}} \]
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Rubi [A]
time = 0.12, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 8, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {281, 331, 303,
1176, 631, 210, 1179, 642} \begin {gather*} \frac {\sqrt [4]{b} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4}}-\frac {\sqrt [4]{b} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{5/4}}-\frac {\sqrt [4]{b} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt {a}+\sqrt {b} x^4\right )}{8 \sqrt {2} a^{5/4}}+\frac {\sqrt [4]{b} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt {a}+\sqrt {b} x^4\right )}{8 \sqrt {2} a^{5/4}}-\frac {1}{2 a x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 281
Rule 303
Rule 331
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (a+b x^8\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 \left (a+b x^4\right )} \, dx,x,x^2\right )\\ &=-\frac {1}{2 a x^2}-\frac {b \text {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,x^2\right )}{2 a}\\ &=-\frac {1}{2 a x^2}+\frac {\sqrt {b} \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,x^2\right )}{4 a}-\frac {\sqrt {b} \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,x^2\right )}{4 a}\\ &=-\frac {1}{2 a x^2}-\frac {\text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,x^2\right )}{8 a}-\frac {\text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,x^2\right )}{8 a}-\frac {\sqrt [4]{b} \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,x^2\right )}{8 \sqrt {2} a^{5/4}}-\frac {\sqrt [4]{b} \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,x^2\right )}{8 \sqrt {2} a^{5/4}}\\ &=-\frac {1}{2 a x^2}-\frac {\sqrt [4]{b} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt {b} x^4\right )}{8 \sqrt {2} a^{5/4}}+\frac {\sqrt [4]{b} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt {b} x^4\right )}{8 \sqrt {2} a^{5/4}}-\frac {\sqrt [4]{b} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4}}+\frac {\sqrt [4]{b} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4}}\\ &=-\frac {1}{2 a x^2}+\frac {\sqrt [4]{b} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4}}-\frac {\sqrt [4]{b} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4}}-\frac {\sqrt [4]{b} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt {b} x^4\right )}{8 \sqrt {2} a^{5/4}}+\frac {\sqrt [4]{b} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt {b} x^4\right )}{8 \sqrt {2} a^{5/4}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 385, normalized size = 1.90 \begin {gather*} \frac {-8 \sqrt [4]{a}+2 \sqrt {2} \sqrt [4]{b} x^2 \tan ^{-1}\left (\cot \left (\frac {\pi }{8}\right )-\frac {\sqrt [8]{b} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}\right )+2 \sqrt {2} \sqrt [4]{b} x^2 \tan ^{-1}\left (\cot \left (\frac {\pi }{8}\right )+\frac {\sqrt [8]{b} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}\right )-2 \sqrt {2} \sqrt [4]{b} x^2 \tan ^{-1}\left (\frac {\sqrt [8]{b} x \sec \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac {\pi }{8}\right )\right )+2 \sqrt {2} \sqrt [4]{b} x^2 \tan ^{-1}\left (\frac {\sqrt [8]{b} x \sec \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac {\pi }{8}\right )\right )-\sqrt {2} \sqrt [4]{b} x^2 \log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2-2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac {\pi }{8}\right )\right )-\sqrt {2} \sqrt [4]{b} x^2 \log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2+2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac {\pi }{8}\right )\right )+\sqrt {2} \sqrt [4]{b} x^2 \log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2-2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac {\pi }{8}\right )\right )+\sqrt {2} \sqrt [4]{b} x^2 \log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2+2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac {\pi }{8}\right )\right )}{16 a^{5/4} x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 119, normalized size = 0.59
method | result | size |
risch | \(-\frac {1}{2 a \,x^{2}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (a^{5} \textit {\_Z}^{4}+b \right )}{\sum }\textit {\_R} \ln \left (\left (-9 \textit {\_R}^{4} a^{5}-8 b \right ) x^{2}-a^{4} \textit {\_R}^{3}\right )\right )}{8}\) | \(53\) |
default | \(-\frac {1}{2 a \,x^{2}}-\frac {\sqrt {2}\, \left (\ln \left (\frac {x^{4}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x^{2} \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{4}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x^{2} \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x^{2}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x^{2}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{16 a \left (\frac {a}{b}\right )^{\frac {1}{4}}}\) | \(119\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 191, normalized size = 0.94 \begin {gather*} -\frac {b {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {b} x^{4} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x^{2} + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {b} x^{4} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x^{2} + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{16 \, a} - \frac {1}{2 \, a x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 146, normalized size = 0.72 \begin {gather*} \frac {4 \, a x^{2} \left (-\frac {b}{a^{5}}\right )^{\frac {1}{4}} \arctan \left (-\frac {a b x^{2} \left (-\frac {b}{a^{5}}\right )^{\frac {1}{4}} - \sqrt {b^{2} x^{4} - a^{3} b \sqrt {-\frac {b}{a^{5}}}} a \left (-\frac {b}{a^{5}}\right )^{\frac {1}{4}}}{b}\right ) - a x^{2} \left (-\frac {b}{a^{5}}\right )^{\frac {1}{4}} \log \left (a^{4} \left (-\frac {b}{a^{5}}\right )^{\frac {3}{4}} + b x^{2}\right ) + a x^{2} \left (-\frac {b}{a^{5}}\right )^{\frac {1}{4}} \log \left (-a^{4} \left (-\frac {b}{a^{5}}\right )^{\frac {3}{4}} + b x^{2}\right ) - 4}{8 \, a x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.12, size = 34, normalized size = 0.17 \begin {gather*} \operatorname {RootSum} {\left (4096 t^{4} a^{5} + b, \left ( t \mapsto t \log {\left (- \frac {512 t^{3} a^{4}}{b} + x^{2} \right )} \right )\right )} - \frac {1}{2 a x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.86, size = 195, normalized size = 0.96 \begin {gather*} -\frac {\sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} x^{4} \arctan \left (\frac {\sqrt {2} {\left (2 \, x^{2} + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{2}} - \frac {\sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} x^{4} \arctan \left (\frac {\sqrt {2} {\left (2 \, x^{2} - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{2}} - \frac {\sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} x^{4} \log \left (x^{4} + \sqrt {2} x^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{16 \, a^{2}} + \frac {\sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} x^{4} \log \left (x^{4} - \sqrt {2} x^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{16 \, a^{2}} - \frac {1}{2 \, a x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.11, size = 55, normalized size = 0.27 \begin {gather*} \frac {{\left (-b\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-b\right )}^{1/4}\,x^2}{a^{1/4}}\right )}{4\,a^{5/4}}-\frac {{\left (-b\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/4}\,x^2}{a^{1/4}}\right )}{4\,a^{5/4}}-\frac {1}{2\,a\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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