3.1326 \(\int \frac{1}{x^2 \left (a+b x^6\right )} \, dx\)

Optimal. Leaf size=223 \[ -\frac{\sqrt [6]{b} \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt{3} a^{7/6}}+\frac{\sqrt [6]{b} \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt{3} a^{7/6}}-\frac{\sqrt [6]{b} \tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 a^{7/6}}+\frac{\sqrt [6]{b} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 a^{7/6}}-\frac{\sqrt [6]{b} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 a^{7/6}}-\frac{1}{a x} \]

[Out]

-(1/(a*x)) - (b^(1/6)*ArcTan[(b^(1/6)*x)/a^(1/6)])/(3*a^(7/6)) + (b^(1/6)*ArcTan
[(Sqrt[3]*a^(1/6) - 2*b^(1/6)*x)/a^(1/6)])/(6*a^(7/6)) - (b^(1/6)*ArcTan[(Sqrt[3
]*a^(1/6) + 2*b^(1/6)*x)/a^(1/6)])/(6*a^(7/6)) - (b^(1/6)*Log[a^(1/3) - Sqrt[3]*
a^(1/6)*b^(1/6)*x + b^(1/3)*x^2])/(4*Sqrt[3]*a^(7/6)) + (b^(1/6)*Log[a^(1/3) + S
qrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2])/(4*Sqrt[3]*a^(7/6))

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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]

-(1/(a*x)) - (b^(1/6)*ArcTan[(b^(1/6)*x)/a^(1/6)])/(3*a^(7/6)) + (b^(1/6)*ArcTan
[(Sqrt[3]*a^(1/6) - 2*b^(1/6)*x)/a^(1/6)])/(6*a^(7/6)) - (b^(1/6)*ArcTan[(Sqrt[3
]*a^(1/6) + 2*b^(1/6)*x)/a^(1/6)])/(6*a^(7/6)) - (b^(1/6)*Log[a^(1/3) - Sqrt[3]*
a^(1/6)*b^(1/6)*x + b^(1/3)*x^2])/(4*Sqrt[3]*a^(7/6)) + (b^(1/6)*Log[a^(1/3) + S
qrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2])/(4*Sqrt[3]*a^(7/6))

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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]

Timed 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]

-(12*a^(1/6) + 4*b^(1/6)*x*ArcTan[(b^(1/6)*x)/a^(1/6)] - 2*b^(1/6)*x*ArcTan[Sqrt
[3] - (2*b^(1/6)*x)/a^(1/6)] + 2*b^(1/6)*x*ArcTan[Sqrt[3] + (2*b^(1/6)*x)/a^(1/6
)] + Sqrt[3]*b^(1/6)*x*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2] -
Sqrt[3]*b^(1/6)*x*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2])/(12*a^
(7/6)*x)

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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]

-1/3/a/(a/b)^(1/6)*arctan(x/(a/b)^(1/6))-1/12*b/a^2*3^(1/2)*(a/b)^(5/6)*ln(3^(1/
2)*(a/b)^(1/6)*x-x^2-(a/b)^(1/3))-1/6/a/(a/b)^(1/6)*arctan(-3^(1/2)+2*x/(a/b)^(1
/6))+1/12*b/a^2*3^(1/2)*(a/b)^(5/6)*ln(x^2+3^(1/2)*(a/b)^(1/6)*x+(a/b)^(1/3))-1/
6/a/(a/b)^(1/6)*arctan(2*x/(a/b)^(1/6)+3^(1/2))-1/a/x

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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]

Exception raised: ValueError

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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]

-1/12*(4*sqrt(3)*a*x*(-b/a^7)^(1/6)*arctan(sqrt(3)*a^6*(-b/a^7)^(5/6)/(a^6*(-b/a
^7)^(5/6) + 2*b*x + 2*b*sqrt((a^6*x*(-b/a^7)^(5/6) - a^5*(-b/a^7)^(2/3) + b*x^2)
/b))) + 4*sqrt(3)*a*x*(-b/a^7)^(1/6)*arctan(-sqrt(3)*a^6*(-b/a^7)^(5/6)/(a^6*(-b
/a^7)^(5/6) - 2*b*x - 2*b*sqrt(-(a^6*x*(-b/a^7)^(5/6) + a^5*(-b/a^7)^(2/3) - b*x
^2)/b))) + a*x*(-b/a^7)^(1/6)*log(a^6*x*(-b/a^7)^(5/6) - a^5*(-b/a^7)^(2/3) + b*
x^2) - a*x*(-b/a^7)^(1/6)*log(-a^6*x*(-b/a^7)^(5/6) - a^5*(-b/a^7)^(2/3) + b*x^2
) + 2*a*x*(-b/a^7)^(1/6)*log(a^6*(-b/a^7)^(5/6) + b*x) - 2*a*x*(-b/a^7)^(1/6)*lo
g(-a^6*(-b/a^7)^(5/6) + b*x) + 12)/(a*x)

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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]

RootSum(46656*_t**6*a**7 + b, Lambda(_t, _t*log(-7776*_t**5*a**6/b + x))) - 1/(a
*x)

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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]

Exception raised: NotImplementedError