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

Optimal. Leaf size=214 \[ -\frac{5 \sqrt [4]{c} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{9/4}}+\frac{5 \sqrt [4]{c} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{9/4}}+\frac{5 \sqrt [4]{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{9/4}}-\frac{5 \sqrt [4]{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{9/4}}-\frac{5}{4 a^2 x}+\frac{1}{4 a x \left (a+c x^4\right )} \]

[Out]

-5/(4*a^2*x) + 1/(4*a*x*(a + c*x^4)) + (5*c^(1/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)
/a^(1/4)])/(8*Sqrt[2]*a^(9/4)) - (5*c^(1/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/
4)])/(8*Sqrt[2]*a^(9/4)) - (5*c^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x +
Sqrt[c]*x^2])/(16*Sqrt[2]*a^(9/4)) + (5*c^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^
(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(9/4))

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Rubi [A]  time = 0.283516, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615 \[ -\frac{5 \sqrt [4]{c} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{9/4}}+\frac{5 \sqrt [4]{c} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{9/4}}+\frac{5 \sqrt [4]{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{9/4}}-\frac{5 \sqrt [4]{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{9/4}}-\frac{5}{4 a^2 x}+\frac{1}{4 a x \left (a+c x^4\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(a + c*x^4)^2),x]

[Out]

-5/(4*a^2*x) + 1/(4*a*x*(a + c*x^4)) + (5*c^(1/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)
/a^(1/4)])/(8*Sqrt[2]*a^(9/4)) - (5*c^(1/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/
4)])/(8*Sqrt[2]*a^(9/4)) - (5*c^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x +
Sqrt[c]*x^2])/(16*Sqrt[2]*a^(9/4)) + (5*c^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^
(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(9/4))

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Rubi in Sympy [A]  time = 59.8355, size = 201, normalized size = 0.94 \[ \frac{1}{4 a x \left (a + c x^{4}\right )} - \frac{5}{4 a^{2} x} - \frac{5 \sqrt{2} \sqrt [4]{c} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x + \sqrt{a} + \sqrt{c} x^{2} \right )}}{32 a^{\frac{9}{4}}} + \frac{5 \sqrt{2} \sqrt [4]{c} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x + \sqrt{a} + \sqrt{c} x^{2} \right )}}{32 a^{\frac{9}{4}}} + \frac{5 \sqrt{2} \sqrt [4]{c} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{16 a^{\frac{9}{4}}} - \frac{5 \sqrt{2} \sqrt [4]{c} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{16 a^{\frac{9}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(4*a*x*(a + c*x**4)) - 5/(4*a**2*x) - 5*sqrt(2)*c**(1/4)*log(-sqrt(2)*a**(1/4)
*c**(1/4)*x + sqrt(a) + sqrt(c)*x**2)/(32*a**(9/4)) + 5*sqrt(2)*c**(1/4)*log(sqr
t(2)*a**(1/4)*c**(1/4)*x + sqrt(a) + sqrt(c)*x**2)/(32*a**(9/4)) + 5*sqrt(2)*c**
(1/4)*atan(1 - sqrt(2)*c**(1/4)*x/a**(1/4))/(16*a**(9/4)) - 5*sqrt(2)*c**(1/4)*a
tan(1 + sqrt(2)*c**(1/4)*x/a**(1/4))/(16*a**(9/4))

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Mathematica [A]  time = 0.321301, size = 196, normalized size = 0.92 \[ \frac{-5 \sqrt{2} \sqrt [4]{c} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )+5 \sqrt{2} \sqrt [4]{c} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )-\frac{8 \sqrt [4]{a} c x^3}{a+c x^4}+10 \sqrt{2} \sqrt [4]{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )-10 \sqrt{2} \sqrt [4]{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )-\frac{32 \sqrt [4]{a}}{x}}{32 a^{9/4}} \]

Antiderivative was successfully verified.

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

[Out]

((-32*a^(1/4))/x - (8*a^(1/4)*c*x^3)/(a + c*x^4) + 10*Sqrt[2]*c^(1/4)*ArcTan[1 -
(Sqrt[2]*c^(1/4)*x)/a^(1/4)] - 10*Sqrt[2]*c^(1/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x
)/a^(1/4)] - 5*Sqrt[2]*c^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]
*x^2] + 5*Sqrt[2]*c^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2]
)/(32*a^(9/4))

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Maple [A]  time = 0.015, size = 154, normalized size = 0.7 \[ -{\frac{1}{x{a}^{2}}}-{\frac{c{x}^{3}}{4\,{a}^{2} \left ( c{x}^{4}+a \right ) }}-{\frac{5\,\sqrt{2}}{32\,{a}^{2}}\ln \left ({1 \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-{\frac{5\,\sqrt{2}}{16\,{a}^{2}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-{\frac{5\,\sqrt{2}}{16\,{a}^{2}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(c*x^4+a)^2,x)

[Out]

-1/a^2/x-1/4/a^2*c*x^3/(c*x^4+a)-5/32/a^2/(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)))-5/16/a^2/(a/c)
^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x+1)-5/16/a^2/(a/c)^(1/4)*2^(1/2)*arct
an(2^(1/2)/(a/c)^(1/4)*x-1)

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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)^2*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.261435, size = 250, normalized size = 1.17 \[ -\frac{20 \, c x^{4} + 20 \,{\left (a^{2} c x^{5} + a^{3} x\right )} \left (-\frac{c}{a^{9}}\right )^{\frac{1}{4}} \arctan \left (\frac{a^{7} \left (-\frac{c}{a^{9}}\right )^{\frac{3}{4}}}{c x + c \sqrt{-\frac{a^{5} \sqrt{-\frac{c}{a^{9}}} - c x^{2}}{c}}}\right ) + 5 \,{\left (a^{2} c x^{5} + a^{3} x\right )} \left (-\frac{c}{a^{9}}\right )^{\frac{1}{4}} \log \left (125 \, a^{7} \left (-\frac{c}{a^{9}}\right )^{\frac{3}{4}} + 125 \, c x\right ) - 5 \,{\left (a^{2} c x^{5} + a^{3} x\right )} \left (-\frac{c}{a^{9}}\right )^{\frac{1}{4}} \log \left (-125 \, a^{7} \left (-\frac{c}{a^{9}}\right )^{\frac{3}{4}} + 125 \, c x\right ) + 16 \, a}{16 \,{\left (a^{2} c x^{5} + a^{3} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/16*(20*c*x^4 + 20*(a^2*c*x^5 + a^3*x)*(-c/a^9)^(1/4)*arctan(a^7*(-c/a^9)^(3/4
)/(c*x + c*sqrt(-(a^5*sqrt(-c/a^9) - c*x^2)/c))) + 5*(a^2*c*x^5 + a^3*x)*(-c/a^9
)^(1/4)*log(125*a^7*(-c/a^9)^(3/4) + 125*c*x) - 5*(a^2*c*x^5 + a^3*x)*(-c/a^9)^(
1/4)*log(-125*a^7*(-c/a^9)^(3/4) + 125*c*x) + 16*a)/(a^2*c*x^5 + a^3*x)

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Sympy [A]  time = 2.73878, size = 54, normalized size = 0.25 \[ - \frac{4 a + 5 c x^{4}}{4 a^{3} x + 4 a^{2} c x^{5}} + \operatorname{RootSum}{\left (65536 t^{4} a^{9} + 625 c, \left ( t \mapsto t \log{\left (- \frac{4096 t^{3} a^{7}}{125 c} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(4*a + 5*c*x**4)/(4*a**3*x + 4*a**2*c*x**5) + RootSum(65536*_t**4*a**9 + 625*c,
Lambda(_t, _t*log(-4096*_t**3*a**7/(125*c) + x)))

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GIAC/XCAS [A]  time = 0.240104, size = 277, normalized size = 1.29 \[ -\frac{5 \, c x^{4} + 4 \, a}{4 \,{\left (c x^{5} + a x\right )} a^{2}} - \frac{5 \, \sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{16 \, a^{3} c^{2}} - \frac{5 \, \sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{16 \, a^{3} c^{2}} + \frac{5 \, \sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{32 \, a^{3} c^{2}} - \frac{5 \, \sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{32 \, a^{3} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(5*c*x^4 + 4*a)/((c*x^5 + a*x)*a^2) - 5/16*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^3*c^2) - 5/16*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^3*c^2) +
5/32*sqrt(2)*(a*c^3)^(3/4)*ln(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a^3*c^2
) - 5/32*sqrt(2)*(a*c^3)^(3/4)*ln(x^2 - sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a^3*
c^2)