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

Optimal. Leaf size=214 \[ \frac{7 c^{3/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{11/4}}-\frac{7 c^{3/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{11/4}}+\frac{7 c^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{11/4}}-\frac{7 c^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{11/4}}-\frac{7}{12 a^2 x^3}+\frac{1}{4 a x^3 \left (a+c x^4\right )} \]

[Out]

-7/(12*a^2*x^3) + 1/(4*a*x^3*(a + c*x^4)) + (7*c^(3/4)*ArcTan[1 - (Sqrt[2]*c^(1/
4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(11/4)) - (7*c^(3/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)
/a^(1/4)])/(8*Sqrt[2]*a^(11/4)) + (7*c^(3/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/
4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(11/4)) - (7*c^(3/4)*Log[Sqrt[a] + Sqrt[2]*a^
(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(11/4))

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Rubi [A]  time = 0.280431, 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 \[ \frac{7 c^{3/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{11/4}}-\frac{7 c^{3/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{11/4}}+\frac{7 c^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{11/4}}-\frac{7 c^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{11/4}}-\frac{7}{12 a^2 x^3}+\frac{1}{4 a x^3 \left (a+c x^4\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*(a + c*x^4)^2),x]

[Out]

-7/(12*a^2*x^3) + 1/(4*a*x^3*(a + c*x^4)) + (7*c^(3/4)*ArcTan[1 - (Sqrt[2]*c^(1/
4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(11/4)) - (7*c^(3/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)
/a^(1/4)])/(8*Sqrt[2]*a^(11/4)) + (7*c^(3/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/
4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(11/4)) - (7*c^(3/4)*Log[Sqrt[a] + Sqrt[2]*a^
(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(11/4))

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Rubi in Sympy [A]  time = 58.7664, size = 204, normalized size = 0.95 \[ \frac{1}{4 a x^{3} \left (a + c x^{4}\right )} - \frac{7}{12 a^{2} x^{3}} + \frac{7 \sqrt{2} c^{\frac{3}{4}} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x + \sqrt{a} + \sqrt{c} x^{2} \right )}}{32 a^{\frac{11}{4}}} - \frac{7 \sqrt{2} c^{\frac{3}{4}} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x + \sqrt{a} + \sqrt{c} x^{2} \right )}}{32 a^{\frac{11}{4}}} + \frac{7 \sqrt{2} c^{\frac{3}{4}} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{16 a^{\frac{11}{4}}} - \frac{7 \sqrt{2} c^{\frac{3}{4}} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{16 a^{\frac{11}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(4*a*x**3*(a + c*x**4)) - 7/(12*a**2*x**3) + 7*sqrt(2)*c**(3/4)*log(-sqrt(2)*a
**(1/4)*c**(1/4)*x + sqrt(a) + sqrt(c)*x**2)/(32*a**(11/4)) - 7*sqrt(2)*c**(3/4)
*log(sqrt(2)*a**(1/4)*c**(1/4)*x + sqrt(a) + sqrt(c)*x**2)/(32*a**(11/4)) + 7*sq
rt(2)*c**(3/4)*atan(1 - sqrt(2)*c**(1/4)*x/a**(1/4))/(16*a**(11/4)) - 7*sqrt(2)*
c**(3/4)*atan(1 + sqrt(2)*c**(1/4)*x/a**(1/4))/(16*a**(11/4))

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Mathematica [A]  time = 0.32208, size = 194, normalized size = 0.91 \[ \frac{-\frac{24 a^{3/4} c x}{a+c x^4}-\frac{32 a^{3/4}}{x^3}+21 \sqrt{2} c^{3/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )-21 \sqrt{2} c^{3/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )+42 \sqrt{2} c^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )-42 \sqrt{2} c^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{96 a^{11/4}} \]

Antiderivative was successfully verified.

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

[Out]

((-32*a^(3/4))/x^3 - (24*a^(3/4)*c*x)/(a + c*x^4) + 42*Sqrt[2]*c^(3/4)*ArcTan[1
- (Sqrt[2]*c^(1/4)*x)/a^(1/4)] - 42*Sqrt[2]*c^(3/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*
x)/a^(1/4)] + 21*Sqrt[2]*c^(3/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[
c]*x^2] - 21*Sqrt[2]*c^(3/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x
^2])/(96*a^(11/4))

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Maple [A]  time = 0.015, size = 155, normalized size = 0.7 \[ -{\frac{1}{3\,{x}^{3}{a}^{2}}}-{\frac{cx}{4\,{a}^{2} \left ( c{x}^{4}+a \right ) }}-{\frac{7\,c\sqrt{2}}{32\,{a}^{3}}\sqrt [4]{{\frac{a}{c}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }-{\frac{7\,c\sqrt{2}}{16\,{a}^{3}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }-{\frac{7\,c\sqrt{2}}{16\,{a}^{3}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(c*x^4+a)^2,x)

[Out]

-1/3/x^3/a^2-1/4/a^2*c*x/(c*x^4+a)-7/32/a^3*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)))-7/16/a^3*c
*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x+1)-7/16/a^3*c*(a/c)^(1/4)*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(1/((c*x^4 + a)^2*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.251383, size = 279, normalized size = 1.3 \[ -\frac{28 \, c x^{4} - 84 \,{\left (a^{2} c x^{7} + a^{3} x^{3}\right )} \left (-\frac{c^{3}}{a^{11}}\right )^{\frac{1}{4}} \arctan \left (\frac{a^{3} \left (-\frac{c^{3}}{a^{11}}\right )^{\frac{1}{4}}}{c x + c \sqrt{\frac{a^{6} \sqrt{-\frac{c^{3}}{a^{11}}} + c^{2} x^{2}}{c^{2}}}}\right ) + 21 \,{\left (a^{2} c x^{7} + a^{3} x^{3}\right )} \left (-\frac{c^{3}}{a^{11}}\right )^{\frac{1}{4}} \log \left (7 \, a^{3} \left (-\frac{c^{3}}{a^{11}}\right )^{\frac{1}{4}} + 7 \, c x\right ) - 21 \,{\left (a^{2} c x^{7} + a^{3} x^{3}\right )} \left (-\frac{c^{3}}{a^{11}}\right )^{\frac{1}{4}} \log \left (-7 \, a^{3} \left (-\frac{c^{3}}{a^{11}}\right )^{\frac{1}{4}} + 7 \, c x\right ) + 16 \, a}{48 \,{\left (a^{2} c x^{7} + a^{3} x^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/48*(28*c*x^4 - 84*(a^2*c*x^7 + a^3*x^3)*(-c^3/a^11)^(1/4)*arctan(a^3*(-c^3/a^
11)^(1/4)/(c*x + c*sqrt((a^6*sqrt(-c^3/a^11) + c^2*x^2)/c^2))) + 21*(a^2*c*x^7 +
a^3*x^3)*(-c^3/a^11)^(1/4)*log(7*a^3*(-c^3/a^11)^(1/4) + 7*c*x) - 21*(a^2*c*x^7
+ a^3*x^3)*(-c^3/a^11)^(1/4)*log(-7*a^3*(-c^3/a^11)^(1/4) + 7*c*x) + 16*a)/(a^2
*c*x^7 + a^3*x^3)

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Sympy [A]  time = 5.22274, size = 56, normalized size = 0.26 \[ - \frac{4 a + 7 c x^{4}}{12 a^{3} x^{3} + 12 a^{2} c x^{7}} + \operatorname{RootSum}{\left (65536 t^{4} a^{11} + 2401 c^{3}, \left ( t \mapsto t \log{\left (- \frac{16 t a^{3}}{7 c} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(4*a + 7*c*x**4)/(12*a**3*x**3 + 12*a**2*c*x**7) + RootSum(65536*_t**4*a**11 +
2401*c**3, Lambda(_t, _t*log(-16*_t*a**3/(7*c) + x)))

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GIAC/XCAS [A]  time = 0.234782, size = 258, normalized size = 1.21 \[ -\frac{c x}{4 \,{\left (c x^{4} + a\right )} a^{2}} - \frac{7 \, \sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{16 \, a^{3}} - \frac{7 \, \sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{16 \, a^{3}} - \frac{7 \, \sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{32 \, a^{3}} + \frac{7 \, \sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{32 \, a^{3}} - \frac{1}{3 \, a^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*c*x/((c*x^4 + a)*a^2) - 7/16*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^3 - 7/16*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^3 - 7/32*sqrt(2)*(a*c^3)^(1/4
)*ln(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/a^3 + 7/32*sqrt(2)*(a*c^3)^(1/4)*l
n(x^2 - sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/a^3 - 1/3/(a^2*x^3)