3.740 \(\int \frac{\sqrt{x}}{a+c x^4} \, dx\)

Optimal. Leaf size=287 \[ -\frac{\log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} (-a)^{5/8} c^{3/8}}+\frac{\log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} (-a)^{5/8} c^{3/8}}+\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt{2} (-a)^{5/8} c^{3/8}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{2 \sqrt{2} (-a)^{5/8} c^{3/8}}+\frac{\tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{5/8} c^{3/8}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{5/8} c^{3/8}} \]

[Out]

ArcTan[1 - (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)]/(2*Sqrt[2]*(-a)^(5/8)*c^(3/8))
- ArcTan[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)]/(2*Sqrt[2]*(-a)^(5/8)*c^(3/8)
) + ArcTan[(c^(1/8)*Sqrt[x])/(-a)^(1/8)]/(2*(-a)^(5/8)*c^(3/8)) - ArcTanh[(c^(1/
8)*Sqrt[x])/(-a)^(1/8)]/(2*(-a)^(5/8)*c^(3/8)) - Log[(-a)^(1/4) - Sqrt[2]*(-a)^(
1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x]/(4*Sqrt[2]*(-a)^(5/8)*c^(3/8)) + Log[(-a)^(1/4
) + Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x]/(4*Sqrt[2]*(-a)^(5/8)*c^(3/8
))

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Rubi [A]  time = 0.493543, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.733 \[ -\frac{\log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} (-a)^{5/8} c^{3/8}}+\frac{\log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} (-a)^{5/8} c^{3/8}}+\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt{2} (-a)^{5/8} c^{3/8}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{2 \sqrt{2} (-a)^{5/8} c^{3/8}}+\frac{\tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{5/8} c^{3/8}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{5/8} c^{3/8}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]/(a + c*x^4),x]

[Out]

ArcTan[1 - (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)]/(2*Sqrt[2]*(-a)^(5/8)*c^(3/8))
- ArcTan[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)]/(2*Sqrt[2]*(-a)^(5/8)*c^(3/8)
) + ArcTan[(c^(1/8)*Sqrt[x])/(-a)^(1/8)]/(2*(-a)^(5/8)*c^(3/8)) - ArcTanh[(c^(1/
8)*Sqrt[x])/(-a)^(1/8)]/(2*(-a)^(5/8)*c^(3/8)) - Log[(-a)^(1/4) - Sqrt[2]*(-a)^(
1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x]/(4*Sqrt[2]*(-a)^(5/8)*c^(3/8)) + Log[(-a)^(1/4
) + Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x]/(4*Sqrt[2]*(-a)^(5/8)*c^(3/8
))

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Rubi in Sympy [A]  time = 108.745, size = 264, normalized size = 0.92 \[ - \frac{\sqrt{2} \log{\left (- \sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{8 c^{\frac{3}{8}} \left (- a\right )^{\frac{5}{8}}} + \frac{\sqrt{2} \log{\left (\sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{8 c^{\frac{3}{8}} \left (- a\right )^{\frac{5}{8}}} + \frac{\operatorname{atan}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{2 c^{\frac{3}{8}} \left (- a\right )^{\frac{5}{8}}} - \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} - 1 \right )}}{4 c^{\frac{3}{8}} \left (- a\right )^{\frac{5}{8}}} - \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} + 1 \right )}}{4 c^{\frac{3}{8}} \left (- a\right )^{\frac{5}{8}}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{2 c^{\frac{3}{8}} \left (- a\right )^{\frac{5}{8}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(2)*log(-sqrt(2)*c**(1/8)*sqrt(x)*(-a)**(1/8) + c**(1/4)*x + (-a)**(1/4))/(
8*c**(3/8)*(-a)**(5/8)) + sqrt(2)*log(sqrt(2)*c**(1/8)*sqrt(x)*(-a)**(1/8) + c**
(1/4)*x + (-a)**(1/4))/(8*c**(3/8)*(-a)**(5/8)) + atan(c**(1/8)*sqrt(x)/(-a)**(1
/8))/(2*c**(3/8)*(-a)**(5/8)) - sqrt(2)*atan(sqrt(2)*c**(1/8)*sqrt(x)/(-a)**(1/8
) - 1)/(4*c**(3/8)*(-a)**(5/8)) - sqrt(2)*atan(sqrt(2)*c**(1/8)*sqrt(x)/(-a)**(1
/8) + 1)/(4*c**(3/8)*(-a)**(5/8)) - atanh(c**(1/8)*sqrt(x)/(-a)**(1/8))/(2*c**(3
/8)*(-a)**(5/8))

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Mathematica [A]  time = 0.302655, size = 348, normalized size = 1.21 \[ -\frac{-\cos \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )+\cos \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )+\sin \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )-\sin \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )+2 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right )+2 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right )+2 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )-2 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )}{4 a^{5/8} c^{3/8}} \]

Antiderivative was successfully verified.

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

[Out]

-(2*ArcTan[Cot[Pi/8] - (c^(1/8)*Sqrt[x]*Csc[Pi/8])/a^(1/8)]*Cos[Pi/8] - 2*ArcTan
[Cot[Pi/8] + (c^(1/8)*Sqrt[x]*Csc[Pi/8])/a^(1/8)]*Cos[Pi/8] - Cos[Pi/8]*Log[a^(1
/4) + c^(1/4)*x - 2*a^(1/8)*c^(1/8)*Sqrt[x]*Sin[Pi/8]] + Cos[Pi/8]*Log[a^(1/4) +
c^(1/4)*x + 2*a^(1/8)*c^(1/8)*Sqrt[x]*Sin[Pi/8]] + 2*ArcTan[(c^(1/8)*Sqrt[x]*Se
c[Pi/8])/a^(1/8) - Tan[Pi/8]]*Sin[Pi/8] + 2*ArcTan[(c^(1/8)*Sqrt[x]*Sec[Pi/8])/a
^(1/8) + Tan[Pi/8]]*Sin[Pi/8] + Log[a^(1/4) + c^(1/4)*x - 2*a^(1/8)*c^(1/8)*Sqrt
[x]*Cos[Pi/8]]*Sin[Pi/8] - Log[a^(1/4) + c^(1/4)*x + 2*a^(1/8)*c^(1/8)*Sqrt[x]*C
os[Pi/8]]*Sin[Pi/8])/(4*a^(5/8)*c^(3/8))

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Maple [C]  time = 0.009, size = 29, normalized size = 0.1 \[{\frac{1}{4\,c}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}c+a \right ) }{\frac{1}{{{\it \_R}}^{5}}\ln \left ( \sqrt{x}-{\it \_R} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)/(c*x^4+a),x)

[Out]

1/4/c*sum(1/_R^5*ln(x^(1/2)-_R),_R=RootOf(_Z^8*c+a))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x}}{c x^{4} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(x)/(c*x^4 + a), x)

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Fricas [A]  time = 0.25896, size = 606, normalized size = 2.11 \[ -\frac{1}{8} \, \sqrt{2}{\left (4 \, \sqrt{2} \left (-\frac{1}{a^{5} c^{3}}\right )^{\frac{1}{8}} \arctan \left (\frac{a^{2} c \left (-\frac{1}{a^{5} c^{3}}\right )^{\frac{3}{8}}}{\sqrt{a^{4} c^{2} \left (-\frac{1}{a^{5} c^{3}}\right )^{\frac{3}{4}} + x} + \sqrt{x}}\right ) + \sqrt{2} \left (-\frac{1}{a^{5} c^{3}}\right )^{\frac{1}{8}} \log \left (a^{2} c \left (-\frac{1}{a^{5} c^{3}}\right )^{\frac{3}{8}} + \sqrt{x}\right ) - \sqrt{2} \left (-\frac{1}{a^{5} c^{3}}\right )^{\frac{1}{8}} \log \left (-a^{2} c \left (-\frac{1}{a^{5} c^{3}}\right )^{\frac{3}{8}} + \sqrt{x}\right ) - 4 \, \left (-\frac{1}{a^{5} c^{3}}\right )^{\frac{1}{8}} \arctan \left (\frac{a^{2} c \left (-\frac{1}{a^{5} c^{3}}\right )^{\frac{3}{8}}}{a^{2} c \left (-\frac{1}{a^{5} c^{3}}\right )^{\frac{3}{8}} + \sqrt{2} \sqrt{x} + \sqrt{2 \, a^{4} c^{2} \left (-\frac{1}{a^{5} c^{3}}\right )^{\frac{3}{4}} + 2 \, \sqrt{2} a^{2} c \sqrt{x} \left (-\frac{1}{a^{5} c^{3}}\right )^{\frac{3}{8}} + 2 \, x}}\right ) - 4 \, \left (-\frac{1}{a^{5} c^{3}}\right )^{\frac{1}{8}} \arctan \left (-\frac{a^{2} c \left (-\frac{1}{a^{5} c^{3}}\right )^{\frac{3}{8}}}{a^{2} c \left (-\frac{1}{a^{5} c^{3}}\right )^{\frac{3}{8}} - \sqrt{2} \sqrt{x} - \sqrt{2 \, a^{4} c^{2} \left (-\frac{1}{a^{5} c^{3}}\right )^{\frac{3}{4}} - 2 \, \sqrt{2} a^{2} c \sqrt{x} \left (-\frac{1}{a^{5} c^{3}}\right )^{\frac{3}{8}} + 2 \, x}}\right ) - \left (-\frac{1}{a^{5} c^{3}}\right )^{\frac{1}{8}} \log \left (2 \, a^{4} c^{2} \left (-\frac{1}{a^{5} c^{3}}\right )^{\frac{3}{4}} + 2 \, \sqrt{2} a^{2} c \sqrt{x} \left (-\frac{1}{a^{5} c^{3}}\right )^{\frac{3}{8}} + 2 \, x\right ) + \left (-\frac{1}{a^{5} c^{3}}\right )^{\frac{1}{8}} \log \left (2 \, a^{4} c^{2} \left (-\frac{1}{a^{5} c^{3}}\right )^{\frac{3}{4}} - 2 \, \sqrt{2} a^{2} c \sqrt{x} \left (-\frac{1}{a^{5} c^{3}}\right )^{\frac{3}{8}} + 2 \, x\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*sqrt(2)*(4*sqrt(2)*(-1/(a^5*c^3))^(1/8)*arctan(a^2*c*(-1/(a^5*c^3))^(3/8)/(
sqrt(a^4*c^2*(-1/(a^5*c^3))^(3/4) + x) + sqrt(x))) + sqrt(2)*(-1/(a^5*c^3))^(1/8
)*log(a^2*c*(-1/(a^5*c^3))^(3/8) + sqrt(x)) - sqrt(2)*(-1/(a^5*c^3))^(1/8)*log(-
a^2*c*(-1/(a^5*c^3))^(3/8) + sqrt(x)) - 4*(-1/(a^5*c^3))^(1/8)*arctan(a^2*c*(-1/
(a^5*c^3))^(3/8)/(a^2*c*(-1/(a^5*c^3))^(3/8) + sqrt(2)*sqrt(x) + sqrt(2*a^4*c^2*
(-1/(a^5*c^3))^(3/4) + 2*sqrt(2)*a^2*c*sqrt(x)*(-1/(a^5*c^3))^(3/8) + 2*x))) - 4
*(-1/(a^5*c^3))^(1/8)*arctan(-a^2*c*(-1/(a^5*c^3))^(3/8)/(a^2*c*(-1/(a^5*c^3))^(
3/8) - sqrt(2)*sqrt(x) - sqrt(2*a^4*c^2*(-1/(a^5*c^3))^(3/4) - 2*sqrt(2)*a^2*c*s
qrt(x)*(-1/(a^5*c^3))^(3/8) + 2*x))) - (-1/(a^5*c^3))^(1/8)*log(2*a^4*c^2*(-1/(a
^5*c^3))^(3/4) + 2*sqrt(2)*a^2*c*sqrt(x)*(-1/(a^5*c^3))^(3/8) + 2*x) + (-1/(a^5*
c^3))^(1/8)*log(2*a^4*c^2*(-1/(a^5*c^3))^(3/4) - 2*sqrt(2)*a^2*c*sqrt(x)*(-1/(a^
5*c^3))^(3/8) + 2*x))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.300316, size = 590, normalized size = 2.06 \[ -\frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{3}{8}} \arctan \left (\frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + 2 \, \sqrt{x}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{4 \, a} - \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{3}{8}} \arctan \left (-\frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} - 2 \, \sqrt{x}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{4 \, a} + \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{3}{8}} \arctan \left (\frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + 2 \, \sqrt{x}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{4 \, a} + \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{3}{8}} \arctan \left (-\frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} - 2 \, \sqrt{x}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{4 \, a} + \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{3}{8}}{\rm ln}\left (\sqrt{x} \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{8 \, a} - \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{3}{8}}{\rm ln}\left (-\sqrt{x} \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{8 \, a} - \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{3}{8}}{\rm ln}\left (\sqrt{x} \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{8 \, a} + \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{3}{8}}{\rm ln}\left (-\sqrt{x} \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{8 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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