Optimal. Leaf size=152 \[ -\frac {2}{3 a^2 x^2}+\frac {1}{6 a x^2 \left (a+b x^6\right )}+\frac {2 \sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{7/3}}+\frac {2 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{9 a^{7/3}}-\frac {\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{9 a^{7/3}} \]
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Rubi [A]
time = 0.09, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 9, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {281, 296, 331,
298, 31, 648, 631, 210, 642} \begin {gather*} \frac {2 \sqrt [3]{b} \text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{7/3}}-\frac {\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{9 a^{7/3}}+\frac {2 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{9 a^{7/3}}-\frac {2}{3 a^2 x^2}+\frac {1}{6 a x^2 \left (a+b x^6\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 210
Rule 281
Rule 296
Rule 298
Rule 331
Rule 631
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (a+b x^6\right )^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 \left (a+b x^3\right )^2} \, dx,x,x^2\right )\\ &=\frac {1}{6 a x^2 \left (a+b x^6\right )}+\frac {2 \text {Subst}\left (\int \frac {1}{x^2 \left (a+b x^3\right )} \, dx,x,x^2\right )}{3 a}\\ &=-\frac {2}{3 a^2 x^2}+\frac {1}{6 a x^2 \left (a+b x^6\right )}-\frac {(2 b) \text {Subst}\left (\int \frac {x}{a+b x^3} \, dx,x,x^2\right )}{3 a^2}\\ &=-\frac {2}{3 a^2 x^2}+\frac {1}{6 a x^2 \left (a+b x^6\right )}+\frac {\left (2 b^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,x^2\right )}{9 a^{7/3}}-\frac {\left (2 b^{2/3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{a}+\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,x^2\right )}{9 a^{7/3}}\\ &=-\frac {2}{3 a^2 x^2}+\frac {1}{6 a x^2 \left (a+b x^6\right )}+\frac {2 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{9 a^{7/3}}-\frac {\sqrt [3]{b} \text {Subst}\left (\int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,x^2\right )}{9 a^{7/3}}-\frac {b^{2/3} \text {Subst}\left (\int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,x^2\right )}{3 a^2}\\ &=-\frac {2}{3 a^2 x^2}+\frac {1}{6 a x^2 \left (a+b x^6\right )}+\frac {2 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{9 a^{7/3}}-\frac {\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{9 a^{7/3}}-\frac {\left (2 \sqrt [3]{b}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x^2}{\sqrt [3]{a}}\right )}{3 a^{7/3}}\\ &=-\frac {2}{3 a^2 x^2}+\frac {1}{6 a x^2 \left (a+b x^6\right )}+\frac {2 \sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{7/3}}+\frac {2 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{9 a^{7/3}}-\frac {\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{9 a^{7/3}}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 208, normalized size = 1.37 \begin {gather*} \frac {-\frac {9 \sqrt [3]{a}}{x^2}-\frac {3 \sqrt [3]{a} b x^4}{a+b x^6}+4 \sqrt {3} \sqrt [3]{b} \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )+4 \sqrt {3} \sqrt [3]{b} \tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )+4 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )-2 \sqrt [3]{b} \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )-2 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{18 a^{7/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 126, normalized size = 0.83
method | result | size |
risch | \(\frac {-\frac {2 b \,x^{6}}{3 a^{2}}-\frac {1}{2 a}}{x^{2} \left (b \,x^{6}+a \right )}+\frac {2 \left (\munderset {\textit {\_R} =\RootOf \left (a^{7} \textit {\_Z}^{3}-b \right )}{\sum }\textit {\_R} \ln \left (\left (-7 a^{7} \textit {\_R}^{3}+6 b \right ) x^{2}-a^{5} \textit {\_R}^{2}\right )\right )}{9}\) | \(75\) |
default | \(-\frac {1}{2 a^{2} x^{2}}-\frac {b \left (\frac {x^{4}}{3 b \,x^{6}+3 a}-\frac {4 \ln \left (x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {2 \ln \left (x^{4}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{2}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{2}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{2 a^{2}}\) | \(126\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 134, normalized size = 0.88 \begin {gather*} -\frac {4 \, b x^{6} + 3 \, a}{6 \, {\left (a^{2} b x^{8} + a^{3} x^{2}\right )}} - \frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{2} - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {\log \left (x^{4} - x^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \, a^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {2 \, \log \left (x^{2} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 160, normalized size = 1.05 \begin {gather*} -\frac {12 \, b x^{6} + 4 \, \sqrt {3} {\left (b x^{8} + a x^{2}\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} x^{2} \left (\frac {b}{a}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + 2 \, {\left (b x^{8} + a x^{2}\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x^{4} - a x^{2} \left (\frac {b}{a}\right )^{\frac {2}{3}} + a \left (\frac {b}{a}\right )^{\frac {1}{3}}\right ) - 4 \, {\left (b x^{8} + a x^{2}\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x^{2} + a \left (\frac {b}{a}\right )^{\frac {2}{3}}\right ) + 9 \, a}{18 \, {\left (a^{2} b x^{8} + a^{3} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.23, size = 60, normalized size = 0.39 \begin {gather*} \frac {- 3 a - 4 b x^{6}}{6 a^{3} x^{2} + 6 a^{2} b x^{8}} + \operatorname {RootSum} {\left (729 t^{3} a^{7} - 8 b, \left ( t \mapsto t \log {\left (\frac {81 t^{2} a^{5}}{4 b} + x^{2} \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.72, size = 147, normalized size = 0.97 \begin {gather*} \frac {2 \, b \left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x^{2} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{3}} + \frac {2 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{2} + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{3} b} - \frac {4 \, b x^{6} + 3 \, a}{6 \, {\left (b x^{8} + a x^{2}\right )} a^{2}} - \frac {\left (-a b^{2}\right )^{\frac {2}{3}} \log \left (x^{4} + x^{2} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \, a^{3} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.21, size = 140, normalized size = 0.92 \begin {gather*} \frac {2\,b^{1/3}\,\ln \left (a^{1/3}+b^{1/3}\,x^2\right )}{9\,a^{7/3}}-\frac {\frac {1}{2\,a}+\frac {2\,b\,x^6}{3\,a^2}}{b\,x^8+a\,x^2}-\frac {2\,b^{1/3}\,\ln \left (6912\,a^7\,b^6-6912\,a^{20/3}\,b^{19/3}\,x^2\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{9\,a^{7/3}}+\frac {b^{1/3}\,\ln \left (6912\,a^7\,b^6+31104\,a^{20/3}\,b^{19/3}\,x^2\,\left (-\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )\right )\,\left (-\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )}{a^{7/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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