3.684 \(\int \frac{1}{x^2 \left (a+c x^4\right )^3} \, dx\)

Optimal. Leaf size=233 \[ -\frac{45 \sqrt [4]{c} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{13/4}}+\frac{45 \sqrt [4]{c} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{13/4}}+\frac{45 \sqrt [4]{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{13/4}}-\frac{45 \sqrt [4]{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} a^{13/4}}-\frac{45}{32 a^3 x}+\frac{9}{32 a^2 x \left (a+c x^4\right )}+\frac{1}{8 a x \left (a+c x^4\right )^2} \]

[Out]

-45/(32*a^3*x) + 1/(8*a*x*(a + c*x^4)^2) + 9/(32*a^2*x*(a + c*x^4)) + (45*c^(1/4
)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(13/4)) - (45*c^(1/4)*A
rcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(13/4)) - (45*c^(1/4)*Log[
Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^(13/4)) + (45
*c^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^
(13/4))

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Rubi [A]  time = 0.327615, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615 \[ -\frac{45 \sqrt [4]{c} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{13/4}}+\frac{45 \sqrt [4]{c} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{13/4}}+\frac{45 \sqrt [4]{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{13/4}}-\frac{45 \sqrt [4]{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} a^{13/4}}-\frac{45}{32 a^3 x}+\frac{9}{32 a^2 x \left (a+c x^4\right )}+\frac{1}{8 a x \left (a+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(a + c*x^4)^3),x]

[Out]

-45/(32*a^3*x) + 1/(8*a*x*(a + c*x^4)^2) + 9/(32*a^2*x*(a + c*x^4)) + (45*c^(1/4
)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(13/4)) - (45*c^(1/4)*A
rcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(13/4)) - (45*c^(1/4)*Log[
Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^(13/4)) + (45
*c^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^
(13/4))

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Rubi in Sympy [A]  time = 66.7529, size = 218, normalized size = 0.94 \[ \frac{1}{8 a x \left (a + c x^{4}\right )^{2}} + \frac{9}{32 a^{2} x \left (a + c x^{4}\right )} - \frac{45}{32 a^{3} x} - \frac{45 \sqrt{2} \sqrt [4]{c} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x + \sqrt{a} + \sqrt{c} x^{2} \right )}}{256 a^{\frac{13}{4}}} + \frac{45 \sqrt{2} \sqrt [4]{c} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x + \sqrt{a} + \sqrt{c} x^{2} \right )}}{256 a^{\frac{13}{4}}} + \frac{45 \sqrt{2} \sqrt [4]{c} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{128 a^{\frac{13}{4}}} - \frac{45 \sqrt{2} \sqrt [4]{c} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{128 a^{\frac{13}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/(c*x**4+a)**3,x)

[Out]

1/(8*a*x*(a + c*x**4)**2) + 9/(32*a**2*x*(a + c*x**4)) - 45/(32*a**3*x) - 45*sqr
t(2)*c**(1/4)*log(-sqrt(2)*a**(1/4)*c**(1/4)*x + sqrt(a) + sqrt(c)*x**2)/(256*a*
*(13/4)) + 45*sqrt(2)*c**(1/4)*log(sqrt(2)*a**(1/4)*c**(1/4)*x + sqrt(a) + sqrt(
c)*x**2)/(256*a**(13/4)) + 45*sqrt(2)*c**(1/4)*atan(1 - sqrt(2)*c**(1/4)*x/a**(1
/4))/(128*a**(13/4)) - 45*sqrt(2)*c**(1/4)*atan(1 + sqrt(2)*c**(1/4)*x/a**(1/4))
/(128*a**(13/4))

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Mathematica [A]  time = 0.194921, size = 216, normalized size = 0.93 \[ \frac{-\frac{32 a^{5/4} c x^3}{\left (a+c x^4\right )^2}-45 \sqrt{2} \sqrt [4]{c} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )+45 \sqrt{2} \sqrt [4]{c} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )-\frac{104 \sqrt [4]{a} c x^3}{a+c x^4}+90 \sqrt{2} \sqrt [4]{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )-90 \sqrt{2} \sqrt [4]{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )-\frac{256 \sqrt [4]{a}}{x}}{256 a^{13/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^2*(a + c*x^4)^3),x]

[Out]

((-256*a^(1/4))/x - (32*a^(5/4)*c*x^3)/(a + c*x^4)^2 - (104*a^(1/4)*c*x^3)/(a +
c*x^4) + 90*Sqrt[2]*c^(1/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)] - 90*Sqrt[2]
*c^(1/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)] - 45*Sqrt[2]*c^(1/4)*Log[Sqrt[a
] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2] + 45*Sqrt[2]*c^(1/4)*Log[Sqrt[a] +
Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(256*a^(13/4))

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Maple [A]  time = 0.02, size = 174, normalized size = 0.8 \[ -{\frac{1}{{a}^{3}x}}-{\frac{13\,{c}^{2}{x}^{7}}{32\,{a}^{3} \left ( c{x}^{4}+a \right ) ^{2}}}-{\frac{17\,c{x}^{3}}{32\,{a}^{2} \left ( c{x}^{4}+a \right ) ^{2}}}-{\frac{45\,\sqrt{2}}{256\,{a}^{3}}\ln \left ({1 \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-{\frac{45\,\sqrt{2}}{128\,{a}^{3}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-{\frac{45\,\sqrt{2}}{128\,{a}^{3}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(c*x^4+a)^3,x)

[Out]

-1/a^3/x-13/32*c^2/a^3/(c*x^4+a)^2*x^7-17/32*c/a^2/(c*x^4+a)^2*x^3-45/256/a^3/(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)))-45/128/a^3/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*
x+1)-45/128/a^3/(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)^3*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.251099, size = 324, normalized size = 1.39 \[ -\frac{180 \, c^{2} x^{8} + 324 \, a c x^{4} + 180 \,{\left (a^{3} c^{2} x^{9} + 2 \, a^{4} c x^{5} + a^{5} x\right )} \left (-\frac{c}{a^{13}}\right )^{\frac{1}{4}} \arctan \left (\frac{a^{10} \left (-\frac{c}{a^{13}}\right )^{\frac{3}{4}}}{c x + c \sqrt{-\frac{a^{7} \sqrt{-\frac{c}{a^{13}}} - c x^{2}}{c}}}\right ) + 45 \,{\left (a^{3} c^{2} x^{9} + 2 \, a^{4} c x^{5} + a^{5} x\right )} \left (-\frac{c}{a^{13}}\right )^{\frac{1}{4}} \log \left (91125 \, a^{10} \left (-\frac{c}{a^{13}}\right )^{\frac{3}{4}} + 91125 \, c x\right ) - 45 \,{\left (a^{3} c^{2} x^{9} + 2 \, a^{4} c x^{5} + a^{5} x\right )} \left (-\frac{c}{a^{13}}\right )^{\frac{1}{4}} \log \left (-91125 \, a^{10} \left (-\frac{c}{a^{13}}\right )^{\frac{3}{4}} + 91125 \, c x\right ) + 128 \, a^{2}}{128 \,{\left (a^{3} c^{2} x^{9} + 2 \, a^{4} c x^{5} + a^{5} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + a)^3*x^2),x, algorithm="fricas")

[Out]

-1/128*(180*c^2*x^8 + 324*a*c*x^4 + 180*(a^3*c^2*x^9 + 2*a^4*c*x^5 + a^5*x)*(-c/
a^13)^(1/4)*arctan(a^10*(-c/a^13)^(3/4)/(c*x + c*sqrt(-(a^7*sqrt(-c/a^13) - c*x^
2)/c))) + 45*(a^3*c^2*x^9 + 2*a^4*c*x^5 + a^5*x)*(-c/a^13)^(1/4)*log(91125*a^10*
(-c/a^13)^(3/4) + 91125*c*x) - 45*(a^3*c^2*x^9 + 2*a^4*c*x^5 + a^5*x)*(-c/a^13)^
(1/4)*log(-91125*a^10*(-c/a^13)^(3/4) + 91125*c*x) + 128*a^2)/(a^3*c^2*x^9 + 2*a
^4*c*x^5 + a^5*x)

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Sympy [A]  time = 18.6368, size = 78, normalized size = 0.33 \[ - \frac{32 a^{2} + 81 a c x^{4} + 45 c^{2} x^{8}}{32 a^{5} x + 64 a^{4} c x^{5} + 32 a^{3} c^{2} x^{9}} + \operatorname{RootSum}{\left (268435456 t^{4} a^{13} + 4100625 c, \left ( t \mapsto t \log{\left (- \frac{2097152 t^{3} a^{10}}{91125 c} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**2/(c*x**4+a)**3,x)

[Out]

-(32*a**2 + 81*a*c*x**4 + 45*c**2*x**8)/(32*a**5*x + 64*a**4*c*x**5 + 32*a**3*c*
*2*x**9) + RootSum(268435456*_t**4*a**13 + 4100625*c, Lambda(_t, _t*log(-2097152
*_t**3*a**10/(91125*c) + x)))

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GIAC/XCAS [A]  time = 0.223603, size = 293, normalized size = 1.26 \[ -\frac{45 \, \sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{128 \, a^{4} c^{2}} - \frac{45 \, \sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{128 \, a^{4} c^{2}} + \frac{45 \, \sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{256 \, a^{4} c^{2}} - \frac{45 \, \sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{256 \, a^{4} c^{2}} - \frac{13 \, c^{2} x^{7} + 17 \, a c x^{3}}{32 \,{\left (c x^{4} + a\right )}^{2} a^{3}} - \frac{1}{a^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + a)^3*x^2),x, algorithm="giac")

[Out]

-45/128*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^4*c^2) - 45/128*sqrt(2)*(a*c^3)^(3/4)*arctan(1/2*sqrt(2)*(2*x - sqr
t(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a^4*c^2) + 45/256*sqrt(2)*(a*c^3)^(3/4)*ln(x^2 +
sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a^4*c^2) - 45/256*sqrt(2)*(a*c^3)^(3/4)*ln(
x^2 - sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a^4*c^2) - 1/32*(13*c^2*x^7 + 17*a*c*x
^3)/((c*x^4 + a)^2*a^3) - 1/(a^3*x)