3.1337 \(\int \frac{1}{\left (a+b x^6\right )^2} \, dx\)

Optimal. Leaf size=232 \[ -\frac{5 \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{11/6} \sqrt [6]{b}}+\frac{5 \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{11/6} \sqrt [6]{b}}+\frac{5 \tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{11/6} \sqrt [6]{b}}-\frac{5 \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{11/6} \sqrt [6]{b}}+\frac{5 \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{11/6} \sqrt [6]{b}}+\frac{x}{6 a \left (a+b x^6\right )} \]

[Out]

x/(6*a*(a + b*x^6)) + (5*ArcTan[(b^(1/6)*x)/a^(1/6)])/(18*a^(11/6)*b^(1/6)) - (5
*ArcTan[(Sqrt[3]*a^(1/6) - 2*b^(1/6)*x)/a^(1/6)])/(36*a^(11/6)*b^(1/6)) + (5*Arc
Tan[(Sqrt[3]*a^(1/6) + 2*b^(1/6)*x)/a^(1/6)])/(36*a^(11/6)*b^(1/6)) - (5*Log[a^(
1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2])/(24*Sqrt[3]*a^(11/6)*b^(1/6)) +
(5*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2])/(24*Sqrt[3]*a^(11/6)
*b^(1/6))

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Rubi [A]  time = 0.76468, antiderivative size = 232, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.778 \[ -\frac{5 \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{11/6} \sqrt [6]{b}}+\frac{5 \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{11/6} \sqrt [6]{b}}+\frac{5 \tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{11/6} \sqrt [6]{b}}-\frac{5 \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{11/6} \sqrt [6]{b}}+\frac{5 \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{11/6} \sqrt [6]{b}}+\frac{x}{6 a \left (a+b x^6\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^6)^(-2),x]

[Out]

x/(6*a*(a + b*x^6)) + (5*ArcTan[(b^(1/6)*x)/a^(1/6)])/(18*a^(11/6)*b^(1/6)) - (5
*ArcTan[(Sqrt[3]*a^(1/6) - 2*b^(1/6)*x)/a^(1/6)])/(36*a^(11/6)*b^(1/6)) + (5*Arc
Tan[(Sqrt[3]*a^(1/6) + 2*b^(1/6)*x)/a^(1/6)])/(36*a^(11/6)*b^(1/6)) - (5*Log[a^(
1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2])/(24*Sqrt[3]*a^(11/6)*b^(1/6)) +
(5*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2])/(24*Sqrt[3]*a^(11/6)
*b^(1/6))

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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/(b*x**6+a)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 0.253866, size = 192, normalized size = 0.83 \[ \frac{\frac{12 a^{5/6} x}{a+b x^6}-\frac{5 \sqrt{3} \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\sqrt [6]{b}}+\frac{5 \sqrt{3} \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\sqrt [6]{b}}+\frac{20 \tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{\sqrt [6]{b}}-\frac{10 \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{\sqrt [6]{b}}+\frac{10 \tan ^{-1}\left (\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}+\sqrt{3}\right )}{\sqrt [6]{b}}}{72 a^{11/6}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^6)^(-2),x]

[Out]

((12*a^(5/6)*x)/(a + b*x^6) + (20*ArcTan[(b^(1/6)*x)/a^(1/6)])/b^(1/6) - (10*Arc
Tan[Sqrt[3] - (2*b^(1/6)*x)/a^(1/6)])/b^(1/6) + (10*ArcTan[Sqrt[3] + (2*b^(1/6)*
x)/a^(1/6)])/b^(1/6) - (5*Sqrt[3]*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1
/3)*x^2])/b^(1/6) + (5*Sqrt[3]*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)
*x^2])/b^(1/6))/(72*a^(11/6))

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Maple [B]  time = 0.322, size = 346, normalized size = 1.5 \[{\frac{x}{18\,{a}^{2}}\sqrt [3]{{\frac{a}{b}}} \left ({x}^{2}+\sqrt [3]{{\frac{a}{b}}} \right ) ^{-1}}+{\frac{5}{18\,{a}^{2}}\sqrt [6]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ) }-{\frac{x}{36\,{a}^{2}}\sqrt [3]{{\frac{a}{b}}} \left ({x}^{2}-\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) ^{-1}}+{\frac{\sqrt{3}}{36\,{a}^{2}}\sqrt{{\frac{a}{b}}} \left ({x}^{2}-\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) ^{-1}}-{\frac{5\,\sqrt{3}}{72\,{a}^{2}}\sqrt [6]{{\frac{a}{b}}}\ln \left ( \sqrt{3}\sqrt [6]{{\frac{a}{b}}}x-{x}^{2}-\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{5}{36\,{a}^{2}}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( -\sqrt{3}+2\,{x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ) }-{\frac{x}{36\,{a}^{2}}\sqrt [3]{{\frac{a}{b}}} \left ({x}^{2}+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) ^{-1}}-{\frac{\sqrt{3}}{36\,{a}^{2}}\sqrt{{\frac{a}{b}}} \left ({x}^{2}+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) ^{-1}}+{\frac{5\,\sqrt{3}}{72\,{a}^{2}}\sqrt [6]{{\frac{a}{b}}}\ln \left ({x}^{2}+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{5}{36\,{a}^{2}}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( 2\,{x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+\sqrt{3} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^6+a)^2,x)

[Out]

1/18*(a/b)^(1/3)/a^2*x/(x^2+(a/b)^(1/3))+5/18*(a/b)^(1/6)/a^2*arctan(x/(a/b)^(1/
6))-1/36/a^2/(x^2-3^(1/2)*(a/b)^(1/6)*x+(a/b)^(1/3))*x*(a/b)^(1/3)+1/36/a^2/(x^2
-3^(1/2)*(a/b)^(1/6)*x+(a/b)^(1/3))*(a/b)^(1/2)*3^(1/2)-5/72/a^2*3^(1/2)*(a/b)^(
1/6)*ln(3^(1/2)*(a/b)^(1/6)*x-x^2-(a/b)^(1/3))+5/36/a^2*(a/b)^(1/6)*arctan(-3^(1
/2)+2*x/(a/b)^(1/6))-1/36/a^2/(x^2+3^(1/2)*(a/b)^(1/6)*x+(a/b)^(1/3))*x*(a/b)^(1
/3)-1/36/a^2/(x^2+3^(1/2)*(a/b)^(1/6)*x+(a/b)^(1/3))*(a/b)^(1/2)*3^(1/2)+5/72/a^
2*3^(1/2)*(a/b)^(1/6)*ln(x^2+3^(1/2)*(a/b)^(1/6)*x+(a/b)^(1/3))+5/36/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((b*x^6 + a)^(-2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.242164, size = 556, normalized size = 2.4 \[ -\frac{20 \, \sqrt{3}{\left (a b x^{6} + a^{2}\right )} \left (-\frac{1}{a^{11} b}\right )^{\frac{1}{6}} \arctan \left (\frac{\sqrt{3} a^{2} \left (-\frac{1}{a^{11} b}\right )^{\frac{1}{6}}}{a^{2} \left (-\frac{1}{a^{11} b}\right )^{\frac{1}{6}} + 2 \, x + 2 \, \sqrt{a^{4} \left (-\frac{1}{a^{11} b}\right )^{\frac{1}{3}} + a^{2} x \left (-\frac{1}{a^{11} b}\right )^{\frac{1}{6}} + x^{2}}}\right ) + 20 \, \sqrt{3}{\left (a b x^{6} + a^{2}\right )} \left (-\frac{1}{a^{11} b}\right )^{\frac{1}{6}} \arctan \left (-\frac{\sqrt{3} a^{2} \left (-\frac{1}{a^{11} b}\right )^{\frac{1}{6}}}{a^{2} \left (-\frac{1}{a^{11} b}\right )^{\frac{1}{6}} - 2 \, x - 2 \, \sqrt{a^{4} \left (-\frac{1}{a^{11} b}\right )^{\frac{1}{3}} - a^{2} x \left (-\frac{1}{a^{11} b}\right )^{\frac{1}{6}} + x^{2}}}\right ) - 5 \,{\left (a b x^{6} + a^{2}\right )} \left (-\frac{1}{a^{11} b}\right )^{\frac{1}{6}} \log \left (a^{4} \left (-\frac{1}{a^{11} b}\right )^{\frac{1}{3}} + a^{2} x \left (-\frac{1}{a^{11} b}\right )^{\frac{1}{6}} + x^{2}\right ) + 5 \,{\left (a b x^{6} + a^{2}\right )} \left (-\frac{1}{a^{11} b}\right )^{\frac{1}{6}} \log \left (a^{4} \left (-\frac{1}{a^{11} b}\right )^{\frac{1}{3}} - a^{2} x \left (-\frac{1}{a^{11} b}\right )^{\frac{1}{6}} + x^{2}\right ) - 10 \,{\left (a b x^{6} + a^{2}\right )} \left (-\frac{1}{a^{11} b}\right )^{\frac{1}{6}} \log \left (a^{2} \left (-\frac{1}{a^{11} b}\right )^{\frac{1}{6}} + x\right ) + 10 \,{\left (a b x^{6} + a^{2}\right )} \left (-\frac{1}{a^{11} b}\right )^{\frac{1}{6}} \log \left (-a^{2} \left (-\frac{1}{a^{11} b}\right )^{\frac{1}{6}} + x\right ) - 12 \, x}{72 \,{\left (a b x^{6} + a^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/72*(20*sqrt(3)*(a*b*x^6 + a^2)*(-1/(a^11*b))^(1/6)*arctan(sqrt(3)*a^2*(-1/(a^
11*b))^(1/6)/(a^2*(-1/(a^11*b))^(1/6) + 2*x + 2*sqrt(a^4*(-1/(a^11*b))^(1/3) + a
^2*x*(-1/(a^11*b))^(1/6) + x^2))) + 20*sqrt(3)*(a*b*x^6 + a^2)*(-1/(a^11*b))^(1/
6)*arctan(-sqrt(3)*a^2*(-1/(a^11*b))^(1/6)/(a^2*(-1/(a^11*b))^(1/6) - 2*x - 2*sq
rt(a^4*(-1/(a^11*b))^(1/3) - a^2*x*(-1/(a^11*b))^(1/6) + x^2))) - 5*(a*b*x^6 + a
^2)*(-1/(a^11*b))^(1/6)*log(a^4*(-1/(a^11*b))^(1/3) + a^2*x*(-1/(a^11*b))^(1/6)
+ x^2) + 5*(a*b*x^6 + a^2)*(-1/(a^11*b))^(1/6)*log(a^4*(-1/(a^11*b))^(1/3) - a^2
*x*(-1/(a^11*b))^(1/6) + x^2) - 10*(a*b*x^6 + a^2)*(-1/(a^11*b))^(1/6)*log(a^2*(
-1/(a^11*b))^(1/6) + x) + 10*(a*b*x^6 + a^2)*(-1/(a^11*b))^(1/6)*log(-a^2*(-1/(a
^11*b))^(1/6) + x) - 12*x)/(a*b*x^6 + a^2)

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Sympy [A]  time = 4.28605, size = 39, normalized size = 0.17 \[ \frac{x}{6 a^{2} + 6 a b x^{6}} + \operatorname{RootSum}{\left (2176782336 t^{6} a^{11} b + 15625, \left ( t \mapsto t \log{\left (\frac{36 t a^{2}}{5} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x**6+a)**2,x)

[Out]

x/(6*a**2 + 6*a*b*x**6) + RootSum(2176782336*_t**6*a**11*b + 15625, Lambda(_t, _
t*log(36*_t*a**2/5 + x)))

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GIAC/XCAS [A]  time = 0.221906, size = 277, normalized size = 1.19 \[ \frac{x}{6 \,{\left (b x^{6} + a\right )} a} + \frac{5 \, \sqrt{3} \left (a b^{5}\right )^{\frac{1}{6}}{\rm ln}\left (x^{2} + \sqrt{3} x \left (\frac{a}{b}\right )^{\frac{1}{6}} + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right )}{72 \, a^{2} b} - \frac{5 \, \sqrt{3} \left (a b^{5}\right )^{\frac{1}{6}}{\rm ln}\left (x^{2} - \sqrt{3} x \left (\frac{a}{b}\right )^{\frac{1}{6}} + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right )}{72 \, a^{2} b} + \frac{5 \, \left (a b^{5}\right )^{\frac{1}{6}} \arctan \left (\frac{2 \, x + \sqrt{3} \left (\frac{a}{b}\right )^{\frac{1}{6}}}{\left (\frac{a}{b}\right )^{\frac{1}{6}}}\right )}{36 \, a^{2} b} + \frac{5 \, \left (a b^{5}\right )^{\frac{1}{6}} \arctan \left (\frac{2 \, x - \sqrt{3} \left (\frac{a}{b}\right )^{\frac{1}{6}}}{\left (\frac{a}{b}\right )^{\frac{1}{6}}}\right )}{36 \, a^{2} b} + \frac{5 \, \left (a b^{5}\right )^{\frac{1}{6}} \arctan \left (\frac{x}{\left (\frac{a}{b}\right )^{\frac{1}{6}}}\right )}{18 \, a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^6 + a)^(-2),x, algorithm="giac")

[Out]

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