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

Optimal. Leaf size=193 \[ -\frac{\sqrt [4]{c} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{5/4}}+\frac{\sqrt [4]{c} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{5/4}}+\frac{\sqrt [4]{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{5/4}}-\frac{\sqrt [4]{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{5/4}}-\frac{1}{a x} \]

[Out]

-(1/(a*x)) + (c^(1/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(5/4
)) - (c^(1/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(5/4)) - (c^
(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*a^(5/4)
) + (c^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*
a^(5/4))

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Rubi [A]  time = 0.243608, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538 \[ -\frac{\sqrt [4]{c} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{5/4}}+\frac{\sqrt [4]{c} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{5/4}}+\frac{\sqrt [4]{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{5/4}}-\frac{\sqrt [4]{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{5/4}}-\frac{1}{a x} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(a + c*x^4)),x]

[Out]

-(1/(a*x)) + (c^(1/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(5/4
)) - (c^(1/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(5/4)) - (c^
(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*a^(5/4)
) + (c^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*
a^(5/4))

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Rubi in Sympy [A]  time = 54.7804, size = 177, normalized size = 0.92 \[ - \frac{1}{a x} - \frac{\sqrt{2} \sqrt [4]{c} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x + \sqrt{a} + \sqrt{c} x^{2} \right )}}{8 a^{\frac{5}{4}}} + \frac{\sqrt{2} \sqrt [4]{c} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x + \sqrt{a} + \sqrt{c} x^{2} \right )}}{8 a^{\frac{5}{4}}} + \frac{\sqrt{2} \sqrt [4]{c} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{4 a^{\frac{5}{4}}} - \frac{\sqrt{2} \sqrt [4]{c} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{4 a^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(a*x) - sqrt(2)*c**(1/4)*log(-sqrt(2)*a**(1/4)*c**(1/4)*x + sqrt(a) + sqrt(c)
*x**2)/(8*a**(5/4)) + sqrt(2)*c**(1/4)*log(sqrt(2)*a**(1/4)*c**(1/4)*x + sqrt(a)
+ sqrt(c)*x**2)/(8*a**(5/4)) + sqrt(2)*c**(1/4)*atan(1 - sqrt(2)*c**(1/4)*x/a**
(1/4))/(4*a**(5/4)) - sqrt(2)*c**(1/4)*atan(1 + sqrt(2)*c**(1/4)*x/a**(1/4))/(4*
a**(5/4))

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Mathematica [A]  time = 0.0583748, size = 179, normalized size = 0.93 \[ \frac{-\sqrt{2} \sqrt [4]{c} x \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )+\sqrt{2} \sqrt [4]{c} x \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )+2 \sqrt{2} \sqrt [4]{c} x \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )-2 \sqrt{2} \sqrt [4]{c} x \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )-8 \sqrt [4]{a}}{8 a^{5/4} x} \]

Antiderivative was successfully verified.

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

[Out]

(-8*a^(1/4) + 2*Sqrt[2]*c^(1/4)*x*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)] - 2*Sq
rt[2]*c^(1/4)*x*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)] - Sqrt[2]*c^(1/4)*x*Log[
Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2] + Sqrt[2]*c^(1/4)*x*Log[Sqrt[
a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(8*a^(5/4)*x)

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Maple [A]  time = 0.006, size = 136, normalized size = 0.7 \[ -{\frac{\sqrt{2}}{8\,a}\ln \left ({1 \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-{\frac{\sqrt{2}}{4\,a}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-{\frac{\sqrt{2}}{4\,a}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-{\frac{1}{ax}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(c*x^4+a),x)

[Out]

-1/8/a/(a/c)^(1/4)*2^(1/2)*ln((x^2-(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2))/(x^2+(a/c)
^(1/4)*x*2^(1/2)+(a/c)^(1/2)))-1/4/a/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1
/4)*x+1)-1/4/a/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x-1)-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/((c*x^4 + a)*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.256528, size = 171, normalized size = 0.89 \[ -\frac{4 \, a x \left (-\frac{c}{a^{5}}\right )^{\frac{1}{4}} \arctan \left (\frac{a^{4} \left (-\frac{c}{a^{5}}\right )^{\frac{3}{4}}}{c x + c \sqrt{-\frac{a^{3} \sqrt{-\frac{c}{a^{5}}} - c x^{2}}{c}}}\right ) + a x \left (-\frac{c}{a^{5}}\right )^{\frac{1}{4}} \log \left (a^{4} \left (-\frac{c}{a^{5}}\right )^{\frac{3}{4}} + c x\right ) - a x \left (-\frac{c}{a^{5}}\right )^{\frac{1}{4}} \log \left (-a^{4} \left (-\frac{c}{a^{5}}\right )^{\frac{3}{4}} + c x\right ) + 4}{4 \, a x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(4*a*x*(-c/a^5)^(1/4)*arctan(a^4*(-c/a^5)^(3/4)/(c*x + c*sqrt(-(a^3*sqrt(-c
/a^5) - c*x^2)/c))) + a*x*(-c/a^5)^(1/4)*log(a^4*(-c/a^5)^(3/4) + c*x) - a*x*(-c
/a^5)^(1/4)*log(-a^4*(-c/a^5)^(3/4) + c*x) + 4)/(a*x)

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Sympy [A]  time = 1.37081, size = 29, normalized size = 0.15 \[ \operatorname{RootSum}{\left (256 t^{4} a^{5} + c, \left ( t \mapsto t \log{\left (- \frac{64 t^{3} a^{4}}{c} + x \right )} \right )\right )} - \frac{1}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**2/(c*x**4+a),x)

[Out]

RootSum(256*_t**4*a**5 + c, Lambda(_t, _t*log(-64*_t**3*a**4/c + x))) - 1/(a*x)

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GIAC/XCAS [A]  time = 0.224642, size = 252, normalized size = 1.31 \[ -\frac{\sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{4 \, a^{2} c^{2}} - \frac{\sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{4 \, a^{2} c^{2}} + \frac{\sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{8 \, a^{2} c^{2}} - \frac{\sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{8 \, a^{2} c^{2}} - \frac{1}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*sqrt(2)*(a*c^3)^(3/4)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/c)^(1/4))/(a/c)^
(1/4))/(a^2*c^2) - 1/4*sqrt(2)*(a*c^3)^(3/4)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(
a/c)^(1/4))/(a/c)^(1/4))/(a^2*c^2) + 1/8*sqrt(2)*(a*c^3)^(3/4)*ln(x^2 + sqrt(2)*
x*(a/c)^(1/4) + sqrt(a/c))/(a^2*c^2) - 1/8*sqrt(2)*(a*c^3)^(3/4)*ln(x^2 - sqrt(2
)*x*(a/c)^(1/4) + sqrt(a/c))/(a^2*c^2) - 1/(a*x)