3.1464 \(\int \frac{1}{x^2 \left (a+b x^8\right )} \, dx\)

Optimal. Leaf size=275 \[ \frac{\sqrt [8]{b} \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{9/8}}-\frac{\sqrt [8]{b} \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{9/8}}+\frac{\sqrt [8]{b} \tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{9/8}}-\frac{\sqrt [8]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt{2} (-a)^{9/8}}+\frac{\sqrt [8]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{4 \sqrt{2} (-a)^{9/8}}-\frac{\sqrt [8]{b} \tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{9/8}}-\frac{1}{a x} \]

[Out]

-(1/(a*x)) + (b^(1/8)*ArcTan[(b^(1/8)*x)/(-a)^(1/8)])/(4*(-a)^(9/8)) - (b^(1/8)*
ArcTan[1 - (Sqrt[2]*b^(1/8)*x)/(-a)^(1/8)])/(4*Sqrt[2]*(-a)^(9/8)) + (b^(1/8)*Ar
cTan[1 + (Sqrt[2]*b^(1/8)*x)/(-a)^(1/8)])/(4*Sqrt[2]*(-a)^(9/8)) - (b^(1/8)*ArcT
anh[(b^(1/8)*x)/(-a)^(1/8)])/(4*(-a)^(9/8)) + (b^(1/8)*Log[(-a)^(1/4) - Sqrt[2]*
(-a)^(1/8)*b^(1/8)*x + b^(1/4)*x^2])/(8*Sqrt[2]*(-a)^(9/8)) - (b^(1/8)*Log[(-a)^
(1/4) + Sqrt[2]*(-a)^(1/8)*b^(1/8)*x + b^(1/4)*x^2])/(8*Sqrt[2]*(-a)^(9/8))

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Rubi [A]  time = 0.503133, antiderivative size = 275, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.846 \[ \frac{\sqrt [8]{b} \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{9/8}}-\frac{\sqrt [8]{b} \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{9/8}}+\frac{\sqrt [8]{b} \tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{9/8}}-\frac{\sqrt [8]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt{2} (-a)^{9/8}}+\frac{\sqrt [8]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{4 \sqrt{2} (-a)^{9/8}}-\frac{\sqrt [8]{b} \tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{9/8}}-\frac{1}{a x} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(a + b*x^8)),x]

[Out]

-(1/(a*x)) + (b^(1/8)*ArcTan[(b^(1/8)*x)/(-a)^(1/8)])/(4*(-a)^(9/8)) - (b^(1/8)*
ArcTan[1 - (Sqrt[2]*b^(1/8)*x)/(-a)^(1/8)])/(4*Sqrt[2]*(-a)^(9/8)) + (b^(1/8)*Ar
cTan[1 + (Sqrt[2]*b^(1/8)*x)/(-a)^(1/8)])/(4*Sqrt[2]*(-a)^(9/8)) - (b^(1/8)*ArcT
anh[(b^(1/8)*x)/(-a)^(1/8)])/(4*(-a)^(9/8)) + (b^(1/8)*Log[(-a)^(1/4) - Sqrt[2]*
(-a)^(1/8)*b^(1/8)*x + b^(1/4)*x^2])/(8*Sqrt[2]*(-a)^(9/8)) - (b^(1/8)*Log[(-a)^
(1/4) + Sqrt[2]*(-a)^(1/8)*b^(1/8)*x + b^(1/4)*x^2])/(8*Sqrt[2]*(-a)^(9/8))

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Rubi in Sympy [A]  time = 108.675, size = 252, normalized size = 0.92 \[ \frac{\sqrt{2} \sqrt [8]{b} \log{\left (- \sqrt{2} \sqrt [8]{b} x \sqrt [8]{- a} + \sqrt [4]{b} x^{2} + \sqrt [4]{- a} \right )}}{16 \left (- a\right )^{\frac{9}{8}}} - \frac{\sqrt{2} \sqrt [8]{b} \log{\left (\sqrt{2} \sqrt [8]{b} x \sqrt [8]{- a} + \sqrt [4]{b} x^{2} + \sqrt [4]{- a} \right )}}{16 \left (- a\right )^{\frac{9}{8}}} + \frac{\sqrt [8]{b} \operatorname{atan}{\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{- a}} \right )}}{4 \left (- a\right )^{\frac{9}{8}}} + \frac{\sqrt{2} \sqrt [8]{b} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{- a}} - 1 \right )}}{8 \left (- a\right )^{\frac{9}{8}}} + \frac{\sqrt{2} \sqrt [8]{b} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{- a}} + 1 \right )}}{8 \left (- a\right )^{\frac{9}{8}}} - \frac{\sqrt [8]{b} \operatorname{atanh}{\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{- a}} \right )}}{4 \left (- a\right )^{\frac{9}{8}}} - \frac{1}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/(b*x**8+a),x)

[Out]

sqrt(2)*b**(1/8)*log(-sqrt(2)*b**(1/8)*x*(-a)**(1/8) + b**(1/4)*x**2 + (-a)**(1/
4))/(16*(-a)**(9/8)) - sqrt(2)*b**(1/8)*log(sqrt(2)*b**(1/8)*x*(-a)**(1/8) + b**
(1/4)*x**2 + (-a)**(1/4))/(16*(-a)**(9/8)) + b**(1/8)*atan(b**(1/8)*x/(-a)**(1/8
))/(4*(-a)**(9/8)) + sqrt(2)*b**(1/8)*atan(sqrt(2)*b**(1/8)*x/(-a)**(1/8) - 1)/(
8*(-a)**(9/8)) + sqrt(2)*b**(1/8)*atan(sqrt(2)*b**(1/8)*x/(-a)**(1/8) + 1)/(8*(-
a)**(9/8)) - b**(1/8)*atanh(b**(1/8)*x/(-a)**(1/8))/(4*(-a)**(9/8)) - 1/(a*x)

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Mathematica [A]  time = 0.452668, size = 377, normalized size = 1.37 \[ -\frac{\sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )-\sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )+\sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )-\sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )+2 \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{b} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right )+2 \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{b} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right )-2 \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{b} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )+2 \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{b} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )+8 \sqrt [8]{a}}{8 a^{9/8} x} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^2*(a + b*x^8)),x]

[Out]

-(8*a^(1/8) + 2*b^(1/8)*x*ArcTan[(b^(1/8)*x*Sec[Pi/8])/a^(1/8) - Tan[Pi/8]]*Cos[
Pi/8] + 2*b^(1/8)*x*ArcTan[(b^(1/8)*x*Sec[Pi/8])/a^(1/8) + Tan[Pi/8]]*Cos[Pi/8]
+ b^(1/8)*x*Cos[Pi/8]*Log[a^(1/4) + b^(1/4)*x^2 - 2*a^(1/8)*b^(1/8)*x*Cos[Pi/8]]
- b^(1/8)*x*Cos[Pi/8]*Log[a^(1/4) + b^(1/4)*x^2 + 2*a^(1/8)*b^(1/8)*x*Cos[Pi/8]
] - 2*b^(1/8)*x*ArcTan[Cot[Pi/8] - (b^(1/8)*x*Csc[Pi/8])/a^(1/8)]*Sin[Pi/8] + 2*
b^(1/8)*x*ArcTan[Cot[Pi/8] + (b^(1/8)*x*Csc[Pi/8])/a^(1/8)]*Sin[Pi/8] + b^(1/8)*
x*Log[a^(1/4) + b^(1/4)*x^2 - 2*a^(1/8)*b^(1/8)*x*Sin[Pi/8]]*Sin[Pi/8] - b^(1/8)
*x*Log[a^(1/4) + b^(1/4)*x^2 + 2*a^(1/8)*b^(1/8)*x*Sin[Pi/8]]*Sin[Pi/8])/(8*a^(9
/8)*x)

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Maple [C]  time = 0.008, size = 36, normalized size = 0.1 \[ -{\frac{1}{8\,a}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}b+a \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{\it \_R}}}}-{\frac{1}{ax}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(b*x^8+a),x)

[Out]

-1/8/a*sum(1/_R*ln(x-_R),_R=RootOf(_Z^8*b+a))-1/a/x

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{b \int \frac{x^{6}}{b x^{8} + a}\,{d x}}{a} - \frac{1}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^8 + a)*x^2),x, algorithm="maxima")

[Out]

-b*integrate(x^6/(b*x^8 + a), x)/a - 1/(a*x)

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Fricas [A]  time = 0.239827, size = 576, normalized size = 2.09 \[ -\frac{\sqrt{2}{\left (4 \, \sqrt{2} a x \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} \arctan \left (\frac{a^{8} \left (-\frac{b}{a^{9}}\right )^{\frac{7}{8}}}{b x + b \sqrt{-\frac{a^{7} \left (-\frac{b}{a^{9}}\right )^{\frac{3}{4}} - b x^{2}}{b}}}\right ) + \sqrt{2} a x \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} \log \left (a^{8} \left (-\frac{b}{a^{9}}\right )^{\frac{7}{8}} + b x\right ) - \sqrt{2} a x \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} \log \left (-a^{8} \left (-\frac{b}{a^{9}}\right )^{\frac{7}{8}} + b x\right ) + 4 \, a x \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} \arctan \left (\frac{a^{8} \left (-\frac{b}{a^{9}}\right )^{\frac{7}{8}}}{a^{8} \left (-\frac{b}{a^{9}}\right )^{\frac{7}{8}} + \sqrt{2} b x + \sqrt{2} b \sqrt{\frac{\sqrt{2} a^{8} x \left (-\frac{b}{a^{9}}\right )^{\frac{7}{8}} - a^{7} \left (-\frac{b}{a^{9}}\right )^{\frac{3}{4}} + b x^{2}}{b}}}\right ) + 4 \, a x \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} \arctan \left (-\frac{a^{8} \left (-\frac{b}{a^{9}}\right )^{\frac{7}{8}}}{a^{8} \left (-\frac{b}{a^{9}}\right )^{\frac{7}{8}} - \sqrt{2} b x - \sqrt{2} b \sqrt{-\frac{\sqrt{2} a^{8} x \left (-\frac{b}{a^{9}}\right )^{\frac{7}{8}} + a^{7} \left (-\frac{b}{a^{9}}\right )^{\frac{3}{4}} - b x^{2}}{b}}}\right ) + a x \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} \log \left (\sqrt{2} a^{8} x \left (-\frac{b}{a^{9}}\right )^{\frac{7}{8}} - a^{7} \left (-\frac{b}{a^{9}}\right )^{\frac{3}{4}} + b x^{2}\right ) - a x \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} \log \left (-\sqrt{2} a^{8} x \left (-\frac{b}{a^{9}}\right )^{\frac{7}{8}} - a^{7} \left (-\frac{b}{a^{9}}\right )^{\frac{3}{4}} + b x^{2}\right ) + 8 \, \sqrt{2}\right )}}{16 \, a x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^8 + a)*x^2),x, algorithm="fricas")

[Out]

-1/16*sqrt(2)*(4*sqrt(2)*a*x*(-b/a^9)^(1/8)*arctan(a^8*(-b/a^9)^(7/8)/(b*x + b*s
qrt(-(a^7*(-b/a^9)^(3/4) - b*x^2)/b))) + sqrt(2)*a*x*(-b/a^9)^(1/8)*log(a^8*(-b/
a^9)^(7/8) + b*x) - sqrt(2)*a*x*(-b/a^9)^(1/8)*log(-a^8*(-b/a^9)^(7/8) + b*x) +
4*a*x*(-b/a^9)^(1/8)*arctan(a^8*(-b/a^9)^(7/8)/(a^8*(-b/a^9)^(7/8) + sqrt(2)*b*x
+ sqrt(2)*b*sqrt((sqrt(2)*a^8*x*(-b/a^9)^(7/8) - a^7*(-b/a^9)^(3/4) + b*x^2)/b)
)) + 4*a*x*(-b/a^9)^(1/8)*arctan(-a^8*(-b/a^9)^(7/8)/(a^8*(-b/a^9)^(7/8) - sqrt(
2)*b*x - sqrt(2)*b*sqrt(-(sqrt(2)*a^8*x*(-b/a^9)^(7/8) + a^7*(-b/a^9)^(3/4) - b*
x^2)/b))) + a*x*(-b/a^9)^(1/8)*log(sqrt(2)*a^8*x*(-b/a^9)^(7/8) - a^7*(-b/a^9)^(
3/4) + b*x^2) - a*x*(-b/a^9)^(1/8)*log(-sqrt(2)*a^8*x*(-b/a^9)^(7/8) - a^7*(-b/a
^9)^(3/4) + b*x^2) + 8*sqrt(2))/(a*x)

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Sympy [A]  time = 1.65977, size = 29, normalized size = 0.11 \[ \operatorname{RootSum}{\left (16777216 t^{8} a^{9} + b, \left ( t \mapsto t \log{\left (- \frac{2097152 t^{7} a^{8}}{b} + x \right )} \right )\right )} - \frac{1}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**2/(b*x**8+a),x)

[Out]

RootSum(16777216*_t**8*a**9 + b, Lambda(_t, _t*log(-2097152*_t**7*a**8/b + x)))
- 1/(a*x)

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GIAC/XCAS [A]  time = 0.261067, size = 601, normalized size = 2.19 \[ -\frac{b \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{7}{8}} \arctan \left (\frac{2 \, x + \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{8 \, a^{2}} - \frac{b \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{7}{8}} \arctan \left (\frac{2 \, x - \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{8 \, a^{2}} - \frac{b \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{7}{8}} \arctan \left (\frac{2 \, x + \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{8 \, a^{2}} - \frac{b \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{7}{8}} \arctan \left (\frac{2 \, x - \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{8 \, a^{2}} + \frac{b \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{7}{8}}{\rm ln}\left (x^{2} + x \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} + \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}{16 \, a^{2}} - \frac{b \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{7}{8}}{\rm ln}\left (x^{2} - x \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} + \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}{16 \, a^{2}} + \frac{b \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{7}{8}}{\rm ln}\left (x^{2} + x \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} + \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}{16 \, a^{2}} - \frac{b \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{7}{8}}{\rm ln}\left (x^{2} - x \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} + \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}{16 \, a^{2}} - \frac{1}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^8 + a)*x^2),x, algorithm="giac")

[Out]

-1/8*b*sqrt(sqrt(2) + 2)*(a/b)^(7/8)*arctan((2*x + sqrt(-sqrt(2) + 2)*(a/b)^(1/8
))/(sqrt(sqrt(2) + 2)*(a/b)^(1/8)))/a^2 - 1/8*b*sqrt(sqrt(2) + 2)*(a/b)^(7/8)*ar
ctan((2*x - sqrt(-sqrt(2) + 2)*(a/b)^(1/8))/(sqrt(sqrt(2) + 2)*(a/b)^(1/8)))/a^2
- 1/8*b*sqrt(-sqrt(2) + 2)*(a/b)^(7/8)*arctan((2*x + sqrt(sqrt(2) + 2)*(a/b)^(1
/8))/(sqrt(-sqrt(2) + 2)*(a/b)^(1/8)))/a^2 - 1/8*b*sqrt(-sqrt(2) + 2)*(a/b)^(7/8
)*arctan((2*x - sqrt(sqrt(2) + 2)*(a/b)^(1/8))/(sqrt(-sqrt(2) + 2)*(a/b)^(1/8)))
/a^2 + 1/16*b*sqrt(sqrt(2) + 2)*(a/b)^(7/8)*ln(x^2 + x*sqrt(sqrt(2) + 2)*(a/b)^(
1/8) + (a/b)^(1/4))/a^2 - 1/16*b*sqrt(sqrt(2) + 2)*(a/b)^(7/8)*ln(x^2 - x*sqrt(s
qrt(2) + 2)*(a/b)^(1/8) + (a/b)^(1/4))/a^2 + 1/16*b*sqrt(-sqrt(2) + 2)*(a/b)^(7/
8)*ln(x^2 + x*sqrt(-sqrt(2) + 2)*(a/b)^(1/8) + (a/b)^(1/4))/a^2 - 1/16*b*sqrt(-s
qrt(2) + 2)*(a/b)^(7/8)*ln(x^2 - x*sqrt(-sqrt(2) + 2)*(a/b)^(1/8) + (a/b)^(1/4))
/a^2 - 1/(a*x)