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

Optimal. Leaf size=219 \[ -\frac{21 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{11/4} \sqrt [4]{c}}+\frac{21 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{11/4} \sqrt [4]{c}}-\frac{21 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{11/4} \sqrt [4]{c}}+\frac{21 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} a^{11/4} \sqrt [4]{c}}+\frac{7 x}{32 a^2 \left (a+c x^4\right )}+\frac{x}{8 a \left (a+c x^4\right )^2} \]

[Out]

x/(8*a*(a + c*x^4)^2) + (7*x)/(32*a^2*(a + c*x^4)) - (21*ArcTan[1 - (Sqrt[2]*c^(
1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(11/4)*c^(1/4)) + (21*ArcTan[1 + (Sqrt[2]*c^(1/4
)*x)/a^(1/4)])/(64*Sqrt[2]*a^(11/4)*c^(1/4)) - (21*Log[Sqrt[a] - Sqrt[2]*a^(1/4)
*c^(1/4)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^(11/4)*c^(1/4)) + (21*Log[Sqrt[a] + Sq
rt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^(11/4)*c^(1/4))

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Rubi [A]  time = 0.314379, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.778 \[ -\frac{21 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{11/4} \sqrt [4]{c}}+\frac{21 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{11/4} \sqrt [4]{c}}-\frac{21 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{11/4} \sqrt [4]{c}}+\frac{21 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} a^{11/4} \sqrt [4]{c}}+\frac{7 x}{32 a^2 \left (a+c x^4\right )}+\frac{x}{8 a \left (a+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^4)^(-3),x]

[Out]

x/(8*a*(a + c*x^4)^2) + (7*x)/(32*a^2*(a + c*x^4)) - (21*ArcTan[1 - (Sqrt[2]*c^(
1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(11/4)*c^(1/4)) + (21*ArcTan[1 + (Sqrt[2]*c^(1/4
)*x)/a^(1/4)])/(64*Sqrt[2]*a^(11/4)*c^(1/4)) - (21*Log[Sqrt[a] - Sqrt[2]*a^(1/4)
*c^(1/4)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^(11/4)*c^(1/4)) + (21*Log[Sqrt[a] + Sq
rt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^(11/4)*c^(1/4))

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Rubi in Sympy [A]  time = 59.9886, size = 207, normalized size = 0.95 \[ \frac{x}{8 a \left (a + c x^{4}\right )^{2}} + \frac{7 x}{32 a^{2} \left (a + c x^{4}\right )} - \frac{21 \sqrt{2} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x + \sqrt{a} + \sqrt{c} x^{2} \right )}}{256 a^{\frac{11}{4}} \sqrt [4]{c}} + \frac{21 \sqrt{2} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x + \sqrt{a} + \sqrt{c} x^{2} \right )}}{256 a^{\frac{11}{4}} \sqrt [4]{c}} - \frac{21 \sqrt{2} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{128 a^{\frac{11}{4}} \sqrt [4]{c}} + \frac{21 \sqrt{2} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{128 a^{\frac{11}{4}} \sqrt [4]{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/(8*a*(a + c*x**4)**2) + 7*x/(32*a**2*(a + c*x**4)) - 21*sqrt(2)*log(-sqrt(2)*a
**(1/4)*c**(1/4)*x + sqrt(a) + sqrt(c)*x**2)/(256*a**(11/4)*c**(1/4)) + 21*sqrt(
2)*log(sqrt(2)*a**(1/4)*c**(1/4)*x + sqrt(a) + sqrt(c)*x**2)/(256*a**(11/4)*c**(
1/4)) - 21*sqrt(2)*atan(1 - sqrt(2)*c**(1/4)*x/a**(1/4))/(128*a**(11/4)*c**(1/4)
) + 21*sqrt(2)*atan(1 + sqrt(2)*c**(1/4)*x/a**(1/4))/(128*a**(11/4)*c**(1/4))

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Mathematica [A]  time = 0.181054, size = 200, normalized size = 0.91 \[ \frac{\frac{32 a^{7/4} x}{\left (a+c x^4\right )^2}+\frac{56 a^{3/4} x}{a+c x^4}-\frac{21 \sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{\sqrt [4]{c}}+\frac{21 \sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{\sqrt [4]{c}}-\frac{42 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt [4]{c}}+\frac{42 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{c}}}{256 a^{11/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x^4)^(-3),x]

[Out]

((32*a^(7/4)*x)/(a + c*x^4)^2 + (56*a^(3/4)*x)/(a + c*x^4) - (42*Sqrt[2]*ArcTan[
1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/c^(1/4) + (42*Sqrt[2]*ArcTan[1 + (Sqrt[2]*c^(1
/4)*x)/a^(1/4)])/c^(1/4) - (21*Sqrt[2]*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x +
Sqrt[c]*x^2])/c^(1/4) + (21*Sqrt[2]*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + S
qrt[c]*x^2])/c^(1/4))/(256*a^(11/4))

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Maple [A]  time = 0.006, size = 158, normalized size = 0.7 \[{\frac{x}{8\,a \left ( c{x}^{4}+a \right ) ^{2}}}+{\frac{7\,x}{32\,{a}^{2} \left ( c{x}^{4}+a \right ) }}+{\frac{21\,\sqrt{2}}{256\,{a}^{3}}\sqrt [4]{{\frac{a}{c}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }+{\frac{21\,\sqrt{2}}{128\,{a}^{3}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }+{\frac{21\,\sqrt{2}}{128\,{a}^{3}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x^4+a)^3,x)

[Out]

1/8*x/a/(c*x^4+a)^2+7/32*x/a^2/(c*x^4+a)+21/256/a^3*(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)))+21/128/a^3*(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x+1)+21/128/
a^3*(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(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((c*x^4 + a)^(-3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.300502, size = 293, normalized size = 1.34 \[ \frac{28 \, c x^{5} - 84 \,{\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )} \left (-\frac{1}{a^{11} c}\right )^{\frac{1}{4}} \arctan \left (\frac{a^{3} \left (-\frac{1}{a^{11} c}\right )^{\frac{1}{4}}}{x + \sqrt{a^{6} \sqrt{-\frac{1}{a^{11} c}} + x^{2}}}\right ) + 21 \,{\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )} \left (-\frac{1}{a^{11} c}\right )^{\frac{1}{4}} \log \left (a^{3} \left (-\frac{1}{a^{11} c}\right )^{\frac{1}{4}} + x\right ) - 21 \,{\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )} \left (-\frac{1}{a^{11} c}\right )^{\frac{1}{4}} \log \left (-a^{3} \left (-\frac{1}{a^{11} c}\right )^{\frac{1}{4}} + x\right ) + 44 \, a x}{128 \,{\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/128*(28*c*x^5 - 84*(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)*(-1/(a^11*c))^(1/4)*arcta
n(a^3*(-1/(a^11*c))^(1/4)/(x + sqrt(a^6*sqrt(-1/(a^11*c)) + x^2))) + 21*(a^2*c^2
*x^8 + 2*a^3*c*x^4 + a^4)*(-1/(a^11*c))^(1/4)*log(a^3*(-1/(a^11*c))^(1/4) + x) -
21*(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)*(-1/(a^11*c))^(1/4)*log(-a^3*(-1/(a^11*c))
^(1/4) + x) + 44*a*x)/(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)

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Sympy [A]  time = 5.04828, size = 63, normalized size = 0.29 \[ \frac{11 a x + 7 c x^{5}}{32 a^{4} + 64 a^{3} c x^{4} + 32 a^{2} c^{2} x^{8}} + \operatorname{RootSum}{\left (268435456 t^{4} a^{11} c + 194481, \left ( t \mapsto t \log{\left (\frac{128 t a^{3}}{21} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c*x**4+a)**3,x)

[Out]

(11*a*x + 7*c*x**5)/(32*a**4 + 64*a**3*c*x**4 + 32*a**2*c**2*x**8) + RootSum(268
435456*_t**4*a**11*c + 194481, Lambda(_t, _t*log(128*_t*a**3/21 + x)))

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GIAC/XCAS [A]  time = 0.266752, size = 275, normalized size = 1.26 \[ \frac{21 \, \sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{128 \, a^{3} c} + \frac{21 \, \sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{128 \, a^{3} c} + \frac{21 \, \sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{256 \, a^{3} c} - \frac{21 \, \sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{256 \, a^{3} c} + \frac{7 \, c x^{5} + 11 \, a x}{32 \,{\left (c x^{4} + a\right )}^{2} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

21/128*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*c) + 21/128*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*c) + 21/256*sqrt(2)*(a*c^3)^(1/4)*ln(x^2 + sqrt
(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a^3*c) - 21/256*sqrt(2)*(a*c^3)^(1/4)*ln(x^2 - s
qrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a^3*c) + 1/32*(7*c*x^5 + 11*a*x)/((c*x^4 + a)
^2*a^2)