Optimal. Leaf size=223 \[ -\frac{\sqrt [6]{b} \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt{3} a^{7/6}}+\frac{\sqrt [6]{b} \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt{3} a^{7/6}}-\frac{\sqrt [6]{b} \tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 a^{7/6}}+\frac{\sqrt [6]{b} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 a^{7/6}}-\frac{\sqrt [6]{b} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 a^{7/6}}-\frac{1}{a x} \]
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Rubi [A] time = 1.07774, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538 \[ -\frac{\sqrt [6]{b} \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt{3} a^{7/6}}+\frac{\sqrt [6]{b} \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt{3} a^{7/6}}-\frac{\sqrt [6]{b} \tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 a^{7/6}}+\frac{\sqrt [6]{b} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 a^{7/6}}-\frac{\sqrt [6]{b} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 a^{7/6}}-\frac{1}{a x} \]
Antiderivative was successfully verified.
[In] Int[1/(x^2*(a + b*x^6)),x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**2/(b*x**6+a),x)
[Out]
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Mathematica [A] time = 0.0892535, size = 189, normalized size = 0.85 \[ -\frac{\sqrt{3} \sqrt [6]{b} x \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )-\sqrt{3} \sqrt [6]{b} x \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )+4 \sqrt [6]{b} x \tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )-2 \sqrt [6]{b} x \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )+2 \sqrt [6]{b} x \tan ^{-1}\left (\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}+\sqrt{3}\right )+12 \sqrt [6]{a}}{12 a^{7/6} x} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^2*(a + b*x^6)),x]
[Out]
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Maple [A] time = 0.042, size = 172, normalized size = 0.8 \[ -{\frac{1}{3\,a}\arctan \left ({x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}-{\frac{b\sqrt{3}}{12\,{a}^{2}} \left ({\frac{a}{b}} \right ) ^{{\frac{5}{6}}}\ln \left ( \sqrt{3}\sqrt [6]{{\frac{a}{b}}}x-{x}^{2}-\sqrt [3]{{\frac{a}{b}}} \right ) }-{\frac{1}{6\,a}\arctan \left ( -\sqrt{3}+2\,{x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+{\frac{b\sqrt{3}}{12\,{a}^{2}} \left ({\frac{a}{b}} \right ) ^{{\frac{5}{6}}}\ln \left ({x}^{2}+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) }-{\frac{1}{6\,a}\arctan \left ( 2\,{x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+\sqrt{3} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}-{\frac{1}{ax}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^2/(b*x^6+a),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^6 + a)*x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.238735, size = 463, normalized size = 2.08 \[ -\frac{4 \, \sqrt{3} a x \left (-\frac{b}{a^{7}}\right )^{\frac{1}{6}} \arctan \left (\frac{\sqrt{3} a^{6} \left (-\frac{b}{a^{7}}\right )^{\frac{5}{6}}}{a^{6} \left (-\frac{b}{a^{7}}\right )^{\frac{5}{6}} + 2 \, b x + 2 \, b \sqrt{\frac{a^{6} x \left (-\frac{b}{a^{7}}\right )^{\frac{5}{6}} - a^{5} \left (-\frac{b}{a^{7}}\right )^{\frac{2}{3}} + b x^{2}}{b}}}\right ) + 4 \, \sqrt{3} a x \left (-\frac{b}{a^{7}}\right )^{\frac{1}{6}} \arctan \left (-\frac{\sqrt{3} a^{6} \left (-\frac{b}{a^{7}}\right )^{\frac{5}{6}}}{a^{6} \left (-\frac{b}{a^{7}}\right )^{\frac{5}{6}} - 2 \, b x - 2 \, b \sqrt{-\frac{a^{6} x \left (-\frac{b}{a^{7}}\right )^{\frac{5}{6}} + a^{5} \left (-\frac{b}{a^{7}}\right )^{\frac{2}{3}} - b x^{2}}{b}}}\right ) + a x \left (-\frac{b}{a^{7}}\right )^{\frac{1}{6}} \log \left (a^{6} x \left (-\frac{b}{a^{7}}\right )^{\frac{5}{6}} - a^{5} \left (-\frac{b}{a^{7}}\right )^{\frac{2}{3}} + b x^{2}\right ) - a x \left (-\frac{b}{a^{7}}\right )^{\frac{1}{6}} \log \left (-a^{6} x \left (-\frac{b}{a^{7}}\right )^{\frac{5}{6}} - a^{5} \left (-\frac{b}{a^{7}}\right )^{\frac{2}{3}} + b x^{2}\right ) + 2 \, a x \left (-\frac{b}{a^{7}}\right )^{\frac{1}{6}} \log \left (a^{6} \left (-\frac{b}{a^{7}}\right )^{\frac{5}{6}} + b x\right ) - 2 \, a x \left (-\frac{b}{a^{7}}\right )^{\frac{1}{6}} \log \left (-a^{6} \left (-\frac{b}{a^{7}}\right )^{\frac{5}{6}} + b x\right ) + 12}{12 \, a x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^6 + a)*x^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.52605, size = 29, normalized size = 0.13 \[ \operatorname{RootSum}{\left (46656 t^{6} a^{7} + b, \left ( t \mapsto t \log{\left (- \frac{7776 t^{5} a^{6}}{b} + x \right )} \right )\right )} - \frac{1}{a x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**2/(b*x**6+a),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^6 + a)*x^2),x, algorithm="giac")
[Out]