3.865 \(\int \frac{1}{x \left (a+b x^2+c x^4\right )^2} \, dx\)

Optimal. Leaf size=122 \[ \frac{b \left (b^2-6 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right )^{3/2}}-\frac{\log \left (a+b x^2+c x^4\right )}{4 a^2}+\frac{\log (x)}{a^2}+\frac{-2 a c+b^2+b c x^2}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]

[Out]

(b^2 - 2*a*c + b*c*x^2)/(2*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (b*(b^2 - 6*a*
c)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*a^2*(b^2 - 4*a*c)^(3/2)) + Log[x
]/a^2 - Log[a + b*x^2 + c*x^4]/(4*a^2)

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Rubi [A]  time = 0.428213, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389 \[ \frac{b \left (b^2-6 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right )^{3/2}}-\frac{\log \left (a+b x^2+c x^4\right )}{4 a^2}+\frac{\log (x)}{a^2}+\frac{-2 a c+b^2+b c x^2}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a + b*x^2 + c*x^4)^2),x]

[Out]

(b^2 - 2*a*c + b*c*x^2)/(2*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (b*(b^2 - 6*a*
c)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*a^2*(b^2 - 4*a*c)^(3/2)) + Log[x
]/a^2 - Log[a + b*x^2 + c*x^4]/(4*a^2)

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Rubi in Sympy [A]  time = 44.701, size = 116, normalized size = 0.95 \[ \frac{- 2 a c + b^{2} + b c x^{2}}{2 a \left (- 4 a c + b^{2}\right ) \left (a + b x^{2} + c x^{4}\right )} + \frac{b \left (- 6 a c + b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{2 a^{2} \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} + \frac{\log{\left (x^{2} \right )}}{2 a^{2}} - \frac{\log{\left (a + b x^{2} + c x^{4} \right )}}{4 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(c*x**4+b*x**2+a)**2,x)

[Out]

(-2*a*c + b**2 + b*c*x**2)/(2*a*(-4*a*c + b**2)*(a + b*x**2 + c*x**4)) + b*(-6*a
*c + b**2)*atanh((b + 2*c*x**2)/sqrt(-4*a*c + b**2))/(2*a**2*(-4*a*c + b**2)**(3
/2)) + log(x**2)/(2*a**2) - log(a + b*x**2 + c*x**4)/(4*a**2)

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Mathematica [A]  time = 0.727698, size = 207, normalized size = 1.7 \[ \frac{\frac{2 a \left (-2 a c+b^2+b c x^2\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\left (b^2 \sqrt{b^2-4 a c}-4 a c \sqrt{b^2-4 a c}-6 a b c+b^3\right ) \log \left (-\sqrt{b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac{\left (-b^2 \sqrt{b^2-4 a c}+4 a c \sqrt{b^2-4 a c}-6 a b c+b^3\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}+4 \log (x)}{4 a^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(a + b*x^2 + c*x^4)^2),x]

[Out]

((2*a*(b^2 - 2*a*c + b*c*x^2))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + 4*Log[x] -
((b^3 - 6*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - 4*a*c*Sqrt[b^2 - 4*a*c])*Log[b - Sqrt[
b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2) + ((b^3 - 6*a*b*c - b^2*Sqrt[b^2 -
4*a*c] + 4*a*c*Sqrt[b^2 - 4*a*c])*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4
*a*c)^(3/2))/(4*a^2)

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Maple [B]  time = 0.028, size = 405, normalized size = 3.3 \[ -{\frac{bc{x}^{2}}{2\,a \left ( c{x}^{4}+b{x}^{2}+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{c}{ \left ( 4\,ac-{b}^{2} \right ) \left ( c{x}^{4}+b{x}^{2}+a \right ) }}-{\frac{{b}^{2}}{2\,a \left ( c{x}^{4}+b{x}^{2}+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-{\frac{c\ln \left ( \left ( 4\,ac-{b}^{2} \right ) \left ( c{x}^{4}+b{x}^{2}+a \right ) \right ) }{ \left ( 4\,ac-{b}^{2} \right ) a}}+{\frac{\ln \left ( \left ( 4\,ac-{b}^{2} \right ) \left ( c{x}^{4}+b{x}^{2}+a \right ) \right ){b}^{2}}{4\,{a}^{2} \left ( 4\,ac-{b}^{2} \right ) }}-3\,{\frac{bc}{a\sqrt{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}}\arctan \left ({\frac{2\, \left ( 4\,ac-{b}^{2} \right ) c{x}^{2}+ \left ( 4\,ac-{b}^{2} \right ) b}{\sqrt{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}}} \right ) }+{\frac{{b}^{3}}{2\,{a}^{2}}\arctan \left ({(2\, \left ( 4\,ac-{b}^{2} \right ) c{x}^{2}+ \left ( 4\,ac-{b}^{2} \right ) b){\frac{1}{\sqrt{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}}}} \right ){\frac{1}{\sqrt{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}}}}+{\frac{\ln \left ( x \right ) }{{a}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(c*x^4+b*x^2+a)^2,x)

[Out]

-1/2/a/(c*x^4+b*x^2+a)*b*c/(4*a*c-b^2)*x^2+1/(c*x^4+b*x^2+a)/(4*a*c-b^2)*c-1/2/a
/(c*x^4+b*x^2+a)/(4*a*c-b^2)*b^2-1/a/(4*a*c-b^2)*c*ln((4*a*c-b^2)*(c*x^4+b*x^2+a
))+1/4/a^2/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^4+b*x^2+a))*b^2-3/a/(64*a^3*c^3-48*a^
2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*arctan((2*(4*a*c-b^2)*c*x^2+(4*a*c-b^2)*b)/(64*a
^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))*b*c+1/2/a^2/(64*a^3*c^3-48*a^2*b^2*
c^2+12*a*b^4*c-b^6)^(1/2)*arctan((2*(4*a*c-b^2)*c*x^2+(4*a*c-b^2)*b)/(64*a^3*c^3
-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))*b^3+ln(x)/a^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/((c*x^4 + b*x^2 + a)^2*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.346798, size = 1, normalized size = 0.01 \[ \left [\frac{{\left ({\left (b^{3} c - 6 \, a b c^{2}\right )} x^{4} + a b^{3} - 6 \, a^{2} b c +{\left (b^{4} - 6 \, a b^{2} c\right )} x^{2}\right )} \log \left (\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} +{\left (2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) +{\left (2 \, a b c x^{2} + 2 \, a b^{2} - 4 \, a^{2} c -{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c +{\left (b^{3} - 4 \, a b c\right )} x^{2}\right )} \log \left (c x^{4} + b x^{2} + a\right ) + 4 \,{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c +{\left (b^{3} - 4 \, a b c\right )} x^{2}\right )} \log \left (x\right )\right )} \sqrt{b^{2} - 4 \, a c}}{4 \,{\left (a^{3} b^{2} - 4 \, a^{4} c +{\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} x^{4} +{\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x^{2}\right )} \sqrt{b^{2} - 4 \, a c}}, -\frac{2 \,{\left ({\left (b^{3} c - 6 \, a b c^{2}\right )} x^{4} + a b^{3} - 6 \, a^{2} b c +{\left (b^{4} - 6 \, a b^{2} c\right )} x^{2}\right )} \arctan \left (-\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) -{\left (2 \, a b c x^{2} + 2 \, a b^{2} - 4 \, a^{2} c -{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c +{\left (b^{3} - 4 \, a b c\right )} x^{2}\right )} \log \left (c x^{4} + b x^{2} + a\right ) + 4 \,{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c +{\left (b^{3} - 4 \, a b c\right )} x^{2}\right )} \log \left (x\right )\right )} \sqrt{-b^{2} + 4 \, a c}}{4 \,{\left (a^{3} b^{2} - 4 \, a^{4} c +{\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} x^{4} +{\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x^{2}\right )} \sqrt{-b^{2} + 4 \, a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + b*x^2 + a)^2*x),x, algorithm="fricas")

[Out]

[1/4*(((b^3*c - 6*a*b*c^2)*x^4 + a*b^3 - 6*a^2*b*c + (b^4 - 6*a*b^2*c)*x^2)*log(
(b^3 - 4*a*b*c + 2*(b^2*c - 4*a*c^2)*x^2 + (2*c^2*x^4 + 2*b*c*x^2 + b^2 - 2*a*c)
*sqrt(b^2 - 4*a*c))/(c*x^4 + b*x^2 + a)) + (2*a*b*c*x^2 + 2*a*b^2 - 4*a^2*c - ((
b^2*c - 4*a*c^2)*x^4 + a*b^2 - 4*a^2*c + (b^3 - 4*a*b*c)*x^2)*log(c*x^4 + b*x^2
+ a) + 4*((b^2*c - 4*a*c^2)*x^4 + a*b^2 - 4*a^2*c + (b^3 - 4*a*b*c)*x^2)*log(x))
*sqrt(b^2 - 4*a*c))/((a^3*b^2 - 4*a^4*c + (a^2*b^2*c - 4*a^3*c^2)*x^4 + (a^2*b^3
- 4*a^3*b*c)*x^2)*sqrt(b^2 - 4*a*c)), -1/4*(2*((b^3*c - 6*a*b*c^2)*x^4 + a*b^3
- 6*a^2*b*c + (b^4 - 6*a*b^2*c)*x^2)*arctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b
^2 - 4*a*c)) - (2*a*b*c*x^2 + 2*a*b^2 - 4*a^2*c - ((b^2*c - 4*a*c^2)*x^4 + a*b^2
- 4*a^2*c + (b^3 - 4*a*b*c)*x^2)*log(c*x^4 + b*x^2 + a) + 4*((b^2*c - 4*a*c^2)*
x^4 + a*b^2 - 4*a^2*c + (b^3 - 4*a*b*c)*x^2)*log(x))*sqrt(-b^2 + 4*a*c))/((a^3*b
^2 - 4*a^4*c + (a^2*b^2*c - 4*a^3*c^2)*x^4 + (a^2*b^3 - 4*a^3*b*c)*x^2)*sqrt(-b^
2 + 4*a*c))]

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Sympy [A]  time = 104.571, size = 772, normalized size = 6.33 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(c*x**4+b*x**2+a)**2,x)

[Out]

(-b*sqrt(-(4*a*c - b**2)**3)*(6*a*c - b**2)/(4*a**2*(64*a**3*c**3 - 48*a**2*b**2
*c**2 + 12*a*b**4*c - b**6)) - 1/(4*a**2))*log(x**2 + (-32*a**4*c**2*(-b*sqrt(-(
4*a*c - b**2)**3)*(6*a*c - b**2)/(4*a**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*
a*b**4*c - b**6)) - 1/(4*a**2)) + 16*a**3*b**2*c*(-b*sqrt(-(4*a*c - b**2)**3)*(6
*a*c - b**2)/(4*a**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) -
1/(4*a**2)) - 2*a**2*b**4*(-b*sqrt(-(4*a*c - b**2)**3)*(6*a*c - b**2)/(4*a**2*(6
4*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - 1/(4*a**2)) - 8*a**2*c*
*2 + 7*a*b**2*c - b**4)/(6*a*b*c**2 - b**3*c)) + (b*sqrt(-(4*a*c - b**2)**3)*(6*
a*c - b**2)/(4*a**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - 1
/(4*a**2))*log(x**2 + (-32*a**4*c**2*(b*sqrt(-(4*a*c - b**2)**3)*(6*a*c - b**2)/
(4*a**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - 1/(4*a**2)) +
16*a**3*b**2*c*(b*sqrt(-(4*a*c - b**2)**3)*(6*a*c - b**2)/(4*a**2*(64*a**3*c**3
- 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - 1/(4*a**2)) - 2*a**2*b**4*(b*sqrt(
-(4*a*c - b**2)**3)*(6*a*c - b**2)/(4*a**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 1
2*a*b**4*c - b**6)) - 1/(4*a**2)) - 8*a**2*c**2 + 7*a*b**2*c - b**4)/(6*a*b*c**2
- b**3*c)) - (-2*a*c + b**2 + b*c*x**2)/(8*a**3*c - 2*a**2*b**2 + x**4*(8*a**2*
c**2 - 2*a*b**2*c) + x**2*(8*a**2*b*c - 2*a*b**3)) + log(x)/a**2

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + b*x^2 + a)^2*x),x, algorithm="giac")

[Out]

Exception raised: TypeError