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

Optimal. Leaf size=195 \[ \frac{c^{3/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{7/4}}-\frac{c^{3/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{7/4}}+\frac{c^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{7/4}}-\frac{c^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{7/4}}-\frac{1}{3 a x^3} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.239959, antiderivative size = 195, 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 \[ \frac{c^{3/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{7/4}}-\frac{c^{3/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{7/4}}+\frac{c^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{7/4}}-\frac{c^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{7/4}}-\frac{1}{3 a x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*(a + c*x^4)),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 52.7651, size = 180, normalized size = 0.92 \[ - \frac{1}{3 a x^{3}} + \frac{\sqrt{2} c^{\frac{3}{4}} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x + \sqrt{a} + \sqrt{c} x^{2} \right )}}{8 a^{\frac{7}{4}}} - \frac{\sqrt{2} c^{\frac{3}{4}} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x + \sqrt{a} + \sqrt{c} x^{2} \right )}}{8 a^{\frac{7}{4}}} + \frac{\sqrt{2} c^{\frac{3}{4}} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{4 a^{\frac{7}{4}}} - \frac{\sqrt{2} c^{\frac{3}{4}} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{4 a^{\frac{7}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(3*a*x**3) + sqrt(2)*c**(3/4)*log(-sqrt(2)*a**(1/4)*c**(1/4)*x + sqrt(a) + sq
rt(c)*x**2)/(8*a**(7/4)) - sqrt(2)*c**(3/4)*log(sqrt(2)*a**(1/4)*c**(1/4)*x + sq
rt(a) + sqrt(c)*x**2)/(8*a**(7/4)) + sqrt(2)*c**(3/4)*atan(1 - sqrt(2)*c**(1/4)*
x/a**(1/4))/(4*a**(7/4)) - sqrt(2)*c**(3/4)*atan(1 + sqrt(2)*c**(1/4)*x/a**(1/4)
)/(4*a**(7/4))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0664477, size = 188, normalized size = 0.96 \[ \frac{-8 a^{3/4}+6 \sqrt{2} c^{3/4} x^3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )-6 \sqrt{2} c^{3/4} x^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )+3 \sqrt{2} c^{3/4} x^3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )-3 \sqrt{2} c^{3/4} x^3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{24 a^{7/4} x^3} \]

Antiderivative was successfully verified.

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

[Out]

(-8*a^(3/4) + 6*Sqrt[2]*c^(3/4)*x^3*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)] - 6*
Sqrt[2]*c^(3/4)*x^3*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + 3*Sqrt[2]*c^(3/4)*
x^3*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2] - 3*Sqrt[2]*c^(3/4)*x
^3*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(24*a^(7/4)*x^3)

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 139, normalized size = 0.7 \[ -{\frac{c\sqrt{2}}{8\,{a}^{2}}\sqrt [4]{{\frac{a}{c}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }-{\frac{c\sqrt{2}}{4\,{a}^{2}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }-{\frac{c\sqrt{2}}{4\,{a}^{2}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }-{\frac{1}{3\,a{x}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(c*x^4+a),x)

[Out]

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

_______________________________________________________________________________________

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^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.249701, size = 200, normalized size = 1.03 \[ \frac{12 \, a x^{3} \left (-\frac{c^{3}}{a^{7}}\right )^{\frac{1}{4}} \arctan \left (\frac{a^{2} \left (-\frac{c^{3}}{a^{7}}\right )^{\frac{1}{4}}}{c x + c \sqrt{\frac{a^{4} \sqrt{-\frac{c^{3}}{a^{7}}} + c^{2} x^{2}}{c^{2}}}}\right ) - 3 \, a x^{3} \left (-\frac{c^{3}}{a^{7}}\right )^{\frac{1}{4}} \log \left (a^{2} \left (-\frac{c^{3}}{a^{7}}\right )^{\frac{1}{4}} + c x\right ) + 3 \, a x^{3} \left (-\frac{c^{3}}{a^{7}}\right )^{\frac{1}{4}} \log \left (-a^{2} \left (-\frac{c^{3}}{a^{7}}\right )^{\frac{1}{4}} + c x\right ) - 4}{12 \, a x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 1.58711, size = 32, normalized size = 0.16 \[ \operatorname{RootSum}{\left (256 t^{4} a^{7} + c^{3}, \left ( t \mapsto t \log{\left (- \frac{4 t a^{2}}{c} + x \right )} \right )\right )} - \frac{1}{3 a x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**4/(c*x**4+a),x)

[Out]

RootSum(256*_t**4*a**7 + c**3, Lambda(_t, _t*log(-4*_t*a**2/c + x))) - 1/(3*a*x*
*3)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.22824, size = 236, normalized size = 1.21 \[ -\frac{\sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{4 \, a^{2}} - \frac{\sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{4 \, a^{2}} - \frac{\sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{8 \, a^{2}} + \frac{\sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{8 \, a^{2}} - \frac{1}{3 \, a x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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