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

Optimal. Leaf size=202 \[ -\frac{3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{7/4} \sqrt [4]{c}}+\frac{3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{7/4} \sqrt [4]{c}}-\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} \sqrt [4]{c}}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} \sqrt [4]{c}}+\frac{x}{4 a \left (a+c x^4\right )} \]

[Out]

x/(4*a*(a + c*x^4)) - (3*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(
7/4)*c^(1/4)) + (3*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*c
^(1/4)) - (3*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]
*a^(7/4)*c^(1/4)) + (3*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(
16*Sqrt[2]*a^(7/4)*c^(1/4))

_______________________________________________________________________________________

Rubi [A]  time = 0.255305, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.778 \[ -\frac{3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{7/4} \sqrt [4]{c}}+\frac{3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{7/4} \sqrt [4]{c}}-\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} \sqrt [4]{c}}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} \sqrt [4]{c}}+\frac{x}{4 a \left (a+c x^4\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^4)^(-2),x]

[Out]

x/(4*a*(a + c*x^4)) - (3*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(
7/4)*c^(1/4)) + (3*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*c
^(1/4)) - (3*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]
*a^(7/4)*c^(1/4)) + (3*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(
16*Sqrt[2]*a^(7/4)*c^(1/4))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 55.0951, size = 190, normalized size = 0.94 \[ \frac{x}{4 a \left (a + c x^{4}\right )} - \frac{3 \sqrt{2} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x + \sqrt{a} + \sqrt{c} x^{2} \right )}}{32 a^{\frac{7}{4}} \sqrt [4]{c}} + \frac{3 \sqrt{2} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x + \sqrt{a} + \sqrt{c} x^{2} \right )}}{32 a^{\frac{7}{4}} \sqrt [4]{c}} - \frac{3 \sqrt{2} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{16 a^{\frac{7}{4}} \sqrt [4]{c}} + \frac{3 \sqrt{2} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{16 a^{\frac{7}{4}} \sqrt [4]{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/(4*a*(a + c*x**4)) - 3*sqrt(2)*log(-sqrt(2)*a**(1/4)*c**(1/4)*x + sqrt(a) + sq
rt(c)*x**2)/(32*a**(7/4)*c**(1/4)) + 3*sqrt(2)*log(sqrt(2)*a**(1/4)*c**(1/4)*x +
sqrt(a) + sqrt(c)*x**2)/(32*a**(7/4)*c**(1/4)) - 3*sqrt(2)*atan(1 - sqrt(2)*c**
(1/4)*x/a**(1/4))/(16*a**(7/4)*c**(1/4)) + 3*sqrt(2)*atan(1 + sqrt(2)*c**(1/4)*x
/a**(1/4))/(16*a**(7/4)*c**(1/4))

_______________________________________________________________________________________

Mathematica [A]  time = 0.242037, size = 183, normalized size = 0.91 \[ \frac{\frac{8 a^{3/4} x}{a+c x^4}-\frac{3 \sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{\sqrt [4]{c}}+\frac{3 \sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{\sqrt [4]{c}}-\frac{6 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt [4]{c}}+\frac{6 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{c}}}{32 a^{7/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x^4)^(-2),x]

[Out]

((8*a^(3/4)*x)/(a + c*x^4) - (6*Sqrt[2]*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])
/c^(1/4) + (6*Sqrt[2]*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/c^(1/4) - (3*Sqrt
[2]*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/c^(1/4) + (3*Sqrt[2]
*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/c^(1/4))/(32*a^(7/4))

_______________________________________________________________________________________

Maple [A]  time = 0.002, size = 143, normalized size = 0.7 \[{\frac{x}{4\,a \left ( c{x}^{4}+a \right ) }}+{\frac{3\,\sqrt{2}}{32\,{a}^{2}}\sqrt [4]{{\frac{a}{c}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }+{\frac{3\,\sqrt{2}}{16\,{a}^{2}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }+{\frac{3\,\sqrt{2}}{16\,{a}^{2}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x^4+a)^2,x)

[Out]

1/4*x/a/(c*x^4+a)+3/32/a^2*(1/c*a)^(1/4)*2^(1/2)*ln((x^2+(1/c*a)^(1/4)*x*2^(1/2)
+(1/c*a)^(1/2))/(x^2-(1/c*a)^(1/4)*x*2^(1/2)+(1/c*a)^(1/2)))+3/16/a^2*(1/c*a)^(1
/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x+1)+3/16/a^2*(1/c*a)^(1/4)*2^(1/2)*arc
tan(2^(1/2)/(1/c*a)^(1/4)*x-1)

_______________________________________________________________________________________

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((c*x^4 + a)^(-2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.281534, size = 213, normalized size = 1.05 \[ -\frac{12 \,{\left (a c x^{4} + a^{2}\right )} \left (-\frac{1}{a^{7} c}\right )^{\frac{1}{4}} \arctan \left (\frac{a^{2} \left (-\frac{1}{a^{7} c}\right )^{\frac{1}{4}}}{x + \sqrt{a^{4} \sqrt{-\frac{1}{a^{7} c}} + x^{2}}}\right ) - 3 \,{\left (a c x^{4} + a^{2}\right )} \left (-\frac{1}{a^{7} c}\right )^{\frac{1}{4}} \log \left (a^{2} \left (-\frac{1}{a^{7} c}\right )^{\frac{1}{4}} + x\right ) + 3 \,{\left (a c x^{4} + a^{2}\right )} \left (-\frac{1}{a^{7} c}\right )^{\frac{1}{4}} \log \left (-a^{2} \left (-\frac{1}{a^{7} c}\right )^{\frac{1}{4}} + x\right ) - 4 \, x}{16 \,{\left (a c x^{4} + a^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/16*(12*(a*c*x^4 + a^2)*(-1/(a^7*c))^(1/4)*arctan(a^2*(-1/(a^7*c))^(1/4)/(x +
sqrt(a^4*sqrt(-1/(a^7*c)) + x^2))) - 3*(a*c*x^4 + a^2)*(-1/(a^7*c))^(1/4)*log(a^
2*(-1/(a^7*c))^(1/4) + x) + 3*(a*c*x^4 + a^2)*(-1/(a^7*c))^(1/4)*log(-a^2*(-1/(a
^7*c))^(1/4) + x) - 4*x)/(a*c*x^4 + a^2)

_______________________________________________________________________________________

Sympy [A]  time = 2.01766, size = 39, normalized size = 0.19 \[ \frac{x}{4 a^{2} + 4 a c x^{4}} + \operatorname{RootSum}{\left (65536 t^{4} a^{7} c + 81, \left ( t \mapsto t \log{\left (\frac{16 t a^{2}}{3} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c*x**4+a)**2,x)

[Out]

x/(4*a**2 + 4*a*c*x**4) + RootSum(65536*_t**4*a**7*c + 81, Lambda(_t, _t*log(16*
_t*a**2/3 + x)))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.261857, size = 262, normalized size = 1.3 \[ \frac{x}{4 \,{\left (c x^{4} + a\right )} a} + \frac{3 \, \sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{16 \, a^{2} c} + \frac{3 \, \sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{16 \, a^{2} c} + \frac{3 \, \sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{32 \, a^{2} c} - \frac{3 \, \sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{32 \, a^{2} c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*x/((c*x^4 + a)*a) + 3/16*sqrt(2)*(a*c^3)^(1/4)*arctan(1/2*sqrt(2)*(2*x + sqr
t(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a^2*c) + 3/16*sqrt(2)*(a*c^3)^(1/4)*arctan(1/2*s
qrt(2)*(2*x - sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a^2*c) + 3/32*sqrt(2)*(a*c^3)^(
1/4)*ln(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a^2*c) - 3/32*sqrt(2)*(a*c^3)^
(1/4)*ln(x^2 - sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a^2*c)