3.1339 \(\int \frac{1}{x^2 \left (a+b x^6\right )^2} \, dx\)

Optimal. Leaf size=244 \[ -\frac{7 \sqrt [6]{b} \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{13/6}}+\frac{7 \sqrt [6]{b} \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{13/6}}-\frac{7 \sqrt [6]{b} \tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{13/6}}+\frac{7 \sqrt [6]{b} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{13/6}}-\frac{7 \sqrt [6]{b} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{13/6}}-\frac{7}{6 a^2 x}+\frac{1}{6 a x \left (a+b x^6\right )} \]

[Out]

-7/(6*a^2*x) + 1/(6*a*x*(a + b*x^6)) - (7*b^(1/6)*ArcTan[(b^(1/6)*x)/a^(1/6)])/(
18*a^(13/6)) + (7*b^(1/6)*ArcTan[(Sqrt[3]*a^(1/6) - 2*b^(1/6)*x)/a^(1/6)])/(36*a
^(13/6)) - (7*b^(1/6)*ArcTan[(Sqrt[3]*a^(1/6) + 2*b^(1/6)*x)/a^(1/6)])/(36*a^(13
/6)) - (7*b^(1/6)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2])/(24*Sq
rt[3]*a^(13/6)) + (7*b^(1/6)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x
^2])/(24*Sqrt[3]*a^(13/6))

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Rubi [A]  time = 1.1646, antiderivative size = 244, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615 \[ -\frac{7 \sqrt [6]{b} \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{13/6}}+\frac{7 \sqrt [6]{b} \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{13/6}}-\frac{7 \sqrt [6]{b} \tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{13/6}}+\frac{7 \sqrt [6]{b} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{13/6}}-\frac{7 \sqrt [6]{b} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{13/6}}-\frac{7}{6 a^2 x}+\frac{1}{6 a x \left (a+b x^6\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(a + b*x^6)^2),x]

[Out]

-7/(6*a^2*x) + 1/(6*a*x*(a + b*x^6)) - (7*b^(1/6)*ArcTan[(b^(1/6)*x)/a^(1/6)])/(
18*a^(13/6)) + (7*b^(1/6)*ArcTan[(Sqrt[3]*a^(1/6) - 2*b^(1/6)*x)/a^(1/6)])/(36*a
^(13/6)) - (7*b^(1/6)*ArcTan[(Sqrt[3]*a^(1/6) + 2*b^(1/6)*x)/a^(1/6)])/(36*a^(13
/6)) - (7*b^(1/6)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2])/(24*Sq
rt[3]*a^(13/6)) + (7*b^(1/6)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x
^2])/(24*Sqrt[3]*a^(13/6))

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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)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 0.361149, size = 205, normalized size = 0.84 \[ \frac{-7 \sqrt{3} \sqrt [6]{b} \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )+7 \sqrt{3} \sqrt [6]{b} \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )-\frac{12 \sqrt [6]{a} b x^5}{a+b x^6}-28 \sqrt [6]{b} \tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )+14 \sqrt [6]{b} \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )-14 \sqrt [6]{b} \tan ^{-1}\left (\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}+\sqrt{3}\right )-\frac{72 \sqrt [6]{a}}{x}}{72 a^{13/6}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^2*(a + b*x^6)^2),x]

[Out]

((-72*a^(1/6))/x - (12*a^(1/6)*b*x^5)/(a + b*x^6) - 28*b^(1/6)*ArcTan[(b^(1/6)*x
)/a^(1/6)] + 14*b^(1/6)*ArcTan[Sqrt[3] - (2*b^(1/6)*x)/a^(1/6)] - 14*b^(1/6)*Arc
Tan[Sqrt[3] + (2*b^(1/6)*x)/a^(1/6)] - 7*Sqrt[3]*b^(1/6)*Log[a^(1/3) - Sqrt[3]*a
^(1/6)*b^(1/6)*x + b^(1/3)*x^2] + 7*Sqrt[3]*b^(1/6)*Log[a^(1/3) + Sqrt[3]*a^(1/6
)*b^(1/6)*x + b^(1/3)*x^2])/(72*a^(13/6))

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Maple [A]  time = 0.017, size = 190, normalized size = 0.8 \[ -{\frac{1}{x{a}^{2}}}-{\frac{b{x}^{5}}{6\,{a}^{2} \left ( b{x}^{6}+a \right ) }}-{\frac{7}{18\,{a}^{2}}\arctan \left ({x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}-{\frac{7\,b\sqrt{3}}{72\,{a}^{3}} \left ({\frac{a}{b}} \right ) ^{{\frac{5}{6}}}\ln \left ( \sqrt{3}\sqrt [6]{{\frac{a}{b}}}x-{x}^{2}-\sqrt [3]{{\frac{a}{b}}} \right ) }-{\frac{7}{36\,{a}^{2}}\arctan \left ( -\sqrt{3}+2\,{x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+{\frac{7\,b\sqrt{3}}{72\,{a}^{3}} \left ({\frac{a}{b}} \right ) ^{{\frac{5}{6}}}\ln \left ({x}^{2}+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) }-{\frac{7}{36\,{a}^{2}}\arctan \left ( 2\,{x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+\sqrt{3} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(b*x^6+a)^2,x)

[Out]

-1/a^2/x-1/6*b/a^2*x^5/(b*x^6+a)-7/18/a^2/(a/b)^(1/6)*arctan(x/(a/b)^(1/6))-7/72
*b/a^3*3^(1/2)*(a/b)^(5/6)*ln(3^(1/2)*(a/b)^(1/6)*x-x^2-(a/b)^(1/3))-7/36/a^2/(a
/b)^(1/6)*arctan(-3^(1/2)+2*x/(a/b)^(1/6))+7/72*b/a^3*3^(1/2)*(a/b)^(5/6)*ln(x^2
+3^(1/2)*(a/b)^(1/6)*x+(a/b)^(1/3))-7/36/a^2/(a/b)^(1/6)*arctan(2*x/(a/b)^(1/6)+
3^(1/2))

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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)^2*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.249689, size = 594, normalized size = 2.43 \[ -\frac{84 \, b x^{6} + 28 \, \sqrt{3}{\left (a^{2} b x^{7} + a^{3} x\right )} \left (-\frac{b}{a^{13}}\right )^{\frac{1}{6}} \arctan \left (\frac{\sqrt{3} a^{11} \left (-\frac{b}{a^{13}}\right )^{\frac{5}{6}}}{a^{11} \left (-\frac{b}{a^{13}}\right )^{\frac{5}{6}} + 2 \, b x + 2 \, b \sqrt{\frac{a^{11} x \left (-\frac{b}{a^{13}}\right )^{\frac{5}{6}} - a^{9} \left (-\frac{b}{a^{13}}\right )^{\frac{2}{3}} + b x^{2}}{b}}}\right ) + 28 \, \sqrt{3}{\left (a^{2} b x^{7} + a^{3} x\right )} \left (-\frac{b}{a^{13}}\right )^{\frac{1}{6}} \arctan \left (-\frac{\sqrt{3} a^{11} \left (-\frac{b}{a^{13}}\right )^{\frac{5}{6}}}{a^{11} \left (-\frac{b}{a^{13}}\right )^{\frac{5}{6}} - 2 \, b x - 2 \, b \sqrt{-\frac{a^{11} x \left (-\frac{b}{a^{13}}\right )^{\frac{5}{6}} + a^{9} \left (-\frac{b}{a^{13}}\right )^{\frac{2}{3}} - b x^{2}}{b}}}\right ) + 7 \,{\left (a^{2} b x^{7} + a^{3} x\right )} \left (-\frac{b}{a^{13}}\right )^{\frac{1}{6}} \log \left (16807 \, a^{11} x \left (-\frac{b}{a^{13}}\right )^{\frac{5}{6}} - 16807 \, a^{9} \left (-\frac{b}{a^{13}}\right )^{\frac{2}{3}} + 16807 \, b x^{2}\right ) - 7 \,{\left (a^{2} b x^{7} + a^{3} x\right )} \left (-\frac{b}{a^{13}}\right )^{\frac{1}{6}} \log \left (-16807 \, a^{11} x \left (-\frac{b}{a^{13}}\right )^{\frac{5}{6}} - 16807 \, a^{9} \left (-\frac{b}{a^{13}}\right )^{\frac{2}{3}} + 16807 \, b x^{2}\right ) + 14 \,{\left (a^{2} b x^{7} + a^{3} x\right )} \left (-\frac{b}{a^{13}}\right )^{\frac{1}{6}} \log \left (16807 \, a^{11} \left (-\frac{b}{a^{13}}\right )^{\frac{5}{6}} + 16807 \, b x\right ) - 14 \,{\left (a^{2} b x^{7} + a^{3} x\right )} \left (-\frac{b}{a^{13}}\right )^{\frac{1}{6}} \log \left (-16807 \, a^{11} \left (-\frac{b}{a^{13}}\right )^{\frac{5}{6}} + 16807 \, b x\right ) + 72 \, a}{72 \,{\left (a^{2} b x^{7} + a^{3} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^6 + a)^2*x^2),x, algorithm="fricas")

[Out]

-1/72*(84*b*x^6 + 28*sqrt(3)*(a^2*b*x^7 + a^3*x)*(-b/a^13)^(1/6)*arctan(sqrt(3)*
a^11*(-b/a^13)^(5/6)/(a^11*(-b/a^13)^(5/6) + 2*b*x + 2*b*sqrt((a^11*x*(-b/a^13)^
(5/6) - a^9*(-b/a^13)^(2/3) + b*x^2)/b))) + 28*sqrt(3)*(a^2*b*x^7 + a^3*x)*(-b/a
^13)^(1/6)*arctan(-sqrt(3)*a^11*(-b/a^13)^(5/6)/(a^11*(-b/a^13)^(5/6) - 2*b*x -
2*b*sqrt(-(a^11*x*(-b/a^13)^(5/6) + a^9*(-b/a^13)^(2/3) - b*x^2)/b))) + 7*(a^2*b
*x^7 + a^3*x)*(-b/a^13)^(1/6)*log(16807*a^11*x*(-b/a^13)^(5/6) - 16807*a^9*(-b/a
^13)^(2/3) + 16807*b*x^2) - 7*(a^2*b*x^7 + a^3*x)*(-b/a^13)^(1/6)*log(-16807*a^1
1*x*(-b/a^13)^(5/6) - 16807*a^9*(-b/a^13)^(2/3) + 16807*b*x^2) + 14*(a^2*b*x^7 +
a^3*x)*(-b/a^13)^(1/6)*log(16807*a^11*(-b/a^13)^(5/6) + 16807*b*x) - 14*(a^2*b*
x^7 + a^3*x)*(-b/a^13)^(1/6)*log(-16807*a^11*(-b/a^13)^(5/6) + 16807*b*x) + 72*a
)/(a^2*b*x^7 + a^3*x)

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Sympy [A]  time = 11.6451, size = 54, normalized size = 0.22 \[ - \frac{6 a + 7 b x^{6}}{6 a^{3} x + 6 a^{2} b x^{7}} + \operatorname{RootSum}{\left (2176782336 t^{6} a^{13} + 117649 b, \left ( t \mapsto t \log{\left (- \frac{60466176 t^{5} a^{11}}{16807 b} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**2/(b*x**6+a)**2,x)

[Out]

-(6*a + 7*b*x**6)/(6*a**3*x + 6*a**2*b*x**7) + RootSum(2176782336*_t**6*a**13 +
117649*b, Lambda(_t, _t*log(-60466176*_t**5*a**11/(16807*b) + 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)^2*x^2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError