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

Optimal. Leaf size=202 \[ -\frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{3/4} c^{5/4}}+\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{3/4} c^{5/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{3/4} c^{5/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{3/4} c^{5/4}}-\frac{x}{4 c \left (a+c x^4\right )} \]

[Out]

-x/(4*c*(a + c*x^4)) - ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)]/(8*Sqrt[2]*a^(3/4
)*c^(5/4)) + ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)]/(8*Sqrt[2]*a^(3/4)*c^(5/4))
- Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2]/(16*Sqrt[2]*a^(3/4)*c^
(5/4)) + Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2]/(16*Sqrt[2]*a^(3
/4)*c^(5/4))

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Rubi [A]  time = 0.241701, antiderivative size = 202, 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 \[ -\frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{3/4} c^{5/4}}+\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{3/4} c^{5/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{3/4} c^{5/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{3/4} c^{5/4}}-\frac{x}{4 c \left (a+c x^4\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a + c*x^4)^2,x]

[Out]

-x/(4*c*(a + c*x^4)) - ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)]/(8*Sqrt[2]*a^(3/4
)*c^(5/4)) + ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)]/(8*Sqrt[2]*a^(3/4)*c^(5/4))
- Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2]/(16*Sqrt[2]*a^(3/4)*c^
(5/4)) + Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2]/(16*Sqrt[2]*a^(3
/4)*c^(5/4))

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Rubi in Sympy [A]  time = 52.8674, size = 184, normalized size = 0.91 \[ - \frac{x}{4 c \left (a + c x^{4}\right )} - \frac{\sqrt{2} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x + \sqrt{a} + \sqrt{c} x^{2} \right )}}{32 a^{\frac{3}{4}} c^{\frac{5}{4}}} + \frac{\sqrt{2} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x + \sqrt{a} + \sqrt{c} x^{2} \right )}}{32 a^{\frac{3}{4}} c^{\frac{5}{4}}} - \frac{\sqrt{2} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{16 a^{\frac{3}{4}} c^{\frac{5}{4}}} + \frac{\sqrt{2} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{16 a^{\frac{3}{4}} c^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x/(4*c*(a + c*x**4)) - sqrt(2)*log(-sqrt(2)*a**(1/4)*c**(1/4)*x + sqrt(a) + sqr
t(c)*x**2)/(32*a**(3/4)*c**(5/4)) + sqrt(2)*log(sqrt(2)*a**(1/4)*c**(1/4)*x + sq
rt(a) + sqrt(c)*x**2)/(32*a**(3/4)*c**(5/4)) - sqrt(2)*atan(1 - sqrt(2)*c**(1/4)
*x/a**(1/4))/(16*a**(3/4)*c**(5/4)) + sqrt(2)*atan(1 + sqrt(2)*c**(1/4)*x/a**(1/
4))/(16*a**(3/4)*c**(5/4))

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Mathematica [A]  time = 0.311557, size = 182, normalized size = 0.9 \[ \frac{-\frac{\sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{a^{3/4}}+\frac{\sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{a^{3/4}}-\frac{2 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{3/4}}+\frac{2 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{a^{3/4}}-\frac{8 \sqrt [4]{c} x}{a+c x^4}}{32 c^{5/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(a + c*x^4)^2,x]

[Out]

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

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Maple [A]  time = 0.011, size = 152, normalized size = 0.8 \[ -{\frac{x}{4\,c \left ( c{x}^{4}+a \right ) }}+{\frac{\sqrt{2}}{32\,ac}\sqrt [4]{{\frac{a}{c}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }+{\frac{\sqrt{2}}{16\,ac}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }+{\frac{\sqrt{2}}{16\,ac}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(c*x^4+a)^2,x)

[Out]

-1/4*x/c/(c*x^4+a)+1/32/c*(a/c)^(1/4)/a*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)))+1/16/c*(a/c)^(1/4)/a*2^(1/2)
*arctan(2^(1/2)/(a/c)^(1/4)*x+1)+1/16/c*(a/c)^(1/4)/a*2^(1/2)*arctan(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(x^4/(c*x^4 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.249044, size = 217, normalized size = 1.07 \[ -\frac{4 \,{\left (c^{2} x^{4} + a c\right )} \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{1}{4}} \arctan \left (\frac{a c \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{1}{4}}}{x + \sqrt{a^{2} c^{2} \sqrt{-\frac{1}{a^{3} c^{5}}} + x^{2}}}\right ) -{\left (c^{2} x^{4} + a c\right )} \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{1}{4}} \log \left (a c \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{1}{4}} + x\right ) +{\left (c^{2} x^{4} + a c\right )} \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{1}{4}} \log \left (-a c \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{1}{4}} + x\right ) + 4 \, x}{16 \,{\left (c^{2} x^{4} + a c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.79667, size = 39, normalized size = 0.19 \[ - \frac{x}{4 a c + 4 c^{2} x^{4}} + \operatorname{RootSum}{\left (65536 t^{4} a^{3} c^{5} + 1, \left ( t \mapsto t \log{\left (16 t a c + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(c*x**4+a)**2,x)

[Out]

-x/(4*a*c + 4*c**2*x**4) + RootSum(65536*_t**4*a**3*c**5 + 1, Lambda(_t, _t*log(
16*_t*a*c + x)))

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GIAC/XCAS [A]  time = 0.224315, size = 262, normalized size = 1.3 \[ -\frac{x}{4 \,{\left (c x^{4} + a\right )} c} + \frac{\sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{16 \, a c^{2}} + \frac{\sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{16 \, a c^{2}} + \frac{\sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{32 \, a c^{2}} - \frac{\sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{32 \, a c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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