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

Optimal. Leaf size=297 \[ \frac{\sqrt [8]{c} \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} (-a)^{9/8}}-\frac{\sqrt [8]{c} \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} (-a)^{9/8}}-\frac{\sqrt [8]{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt{2} (-a)^{9/8}}+\frac{\sqrt [8]{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{2 \sqrt{2} (-a)^{9/8}}+\frac{\sqrt [8]{c} \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{9/8}}-\frac{\sqrt [8]{c} \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{9/8}}-\frac{2}{a \sqrt{x}} \]

[Out]

-2/(a*Sqrt[x]) - (c^(1/8)*ArcTan[1 - (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(2*S
qrt[2]*(-a)^(9/8)) + (c^(1/8)*ArcTan[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/
(2*Sqrt[2]*(-a)^(9/8)) + (c^(1/8)*ArcTan[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(2*(-a)^
(9/8)) - (c^(1/8)*ArcTanh[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(2*(-a)^(9/8)) + (c^(1/
8)*Log[(-a)^(1/4) - Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(4*Sqrt[2]*
(-a)^(9/8)) - (c^(1/8)*Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(
1/4)*x])/(4*Sqrt[2]*(-a)^(9/8))

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Rubi [A]  time = 0.619237, antiderivative size = 297, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 12, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.8 \[ \frac{\sqrt [8]{c} \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} (-a)^{9/8}}-\frac{\sqrt [8]{c} \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} (-a)^{9/8}}-\frac{\sqrt [8]{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt{2} (-a)^{9/8}}+\frac{\sqrt [8]{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{2 \sqrt{2} (-a)^{9/8}}+\frac{\sqrt [8]{c} \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{9/8}}-\frac{\sqrt [8]{c} \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{9/8}}-\frac{2}{a \sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(3/2)*(a + c*x^4)),x]

[Out]

-2/(a*Sqrt[x]) - (c^(1/8)*ArcTan[1 - (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(2*S
qrt[2]*(-a)^(9/8)) + (c^(1/8)*ArcTan[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/
(2*Sqrt[2]*(-a)^(9/8)) + (c^(1/8)*ArcTan[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(2*(-a)^
(9/8)) - (c^(1/8)*ArcTanh[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(2*(-a)^(9/8)) + (c^(1/
8)*Log[(-a)^(1/4) - Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(4*Sqrt[2]*
(-a)^(9/8)) - (c^(1/8)*Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(
1/4)*x])/(4*Sqrt[2]*(-a)^(9/8))

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Rubi in Sympy [A]  time = 113.016, size = 272, normalized size = 0.92 \[ \frac{\sqrt{2} \sqrt [8]{c} \log{\left (- \sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{8 \left (- a\right )^{\frac{9}{8}}} - \frac{\sqrt{2} \sqrt [8]{c} \log{\left (\sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{8 \left (- a\right )^{\frac{9}{8}}} + \frac{\sqrt [8]{c} \operatorname{atan}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{2 \left (- a\right )^{\frac{9}{8}}} + \frac{\sqrt{2} \sqrt [8]{c} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} - 1 \right )}}{4 \left (- a\right )^{\frac{9}{8}}} + \frac{\sqrt{2} \sqrt [8]{c} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} + 1 \right )}}{4 \left (- a\right )^{\frac{9}{8}}} - \frac{\sqrt [8]{c} \operatorname{atanh}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{2 \left (- a\right )^{\frac{9}{8}}} - \frac{2}{a \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2)*c**(1/8)*log(-sqrt(2)*c**(1/8)*sqrt(x)*(-a)**(1/8) + c**(1/4)*x + (-a)**
(1/4))/(8*(-a)**(9/8)) - sqrt(2)*c**(1/8)*log(sqrt(2)*c**(1/8)*sqrt(x)*(-a)**(1/
8) + c**(1/4)*x + (-a)**(1/4))/(8*(-a)**(9/8)) + c**(1/8)*atan(c**(1/8)*sqrt(x)/
(-a)**(1/8))/(2*(-a)**(9/8)) + sqrt(2)*c**(1/8)*atan(sqrt(2)*c**(1/8)*sqrt(x)/(-
a)**(1/8) - 1)/(4*(-a)**(9/8)) + sqrt(2)*c**(1/8)*atan(sqrt(2)*c**(1/8)*sqrt(x)/
(-a)**(1/8) + 1)/(4*(-a)**(9/8)) - c**(1/8)*atanh(c**(1/8)*sqrt(x)/(-a)**(1/8))/
(2*(-a)**(9/8)) - 2/(a*sqrt(x))

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Mathematica [A]  time = 0.444393, size = 435, normalized size = 1.46 \[ -\frac{\sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )-\sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )+\sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )-\sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )+2 \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right )+2 \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right )-2 \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )+2 \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )+8 \sqrt [8]{a}}{4 a^{9/8} \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

-(8*a^(1/8) + 2*c^(1/8)*Sqrt[x]*ArcTan[(c^(1/8)*Sqrt[x]*Sec[Pi/8])/a^(1/8) - Tan
[Pi/8]]*Cos[Pi/8] + 2*c^(1/8)*Sqrt[x]*ArcTan[(c^(1/8)*Sqrt[x]*Sec[Pi/8])/a^(1/8)
+ Tan[Pi/8]]*Cos[Pi/8] + c^(1/8)*Sqrt[x]*Cos[Pi/8]*Log[a^(1/4) + c^(1/4)*x - 2*
a^(1/8)*c^(1/8)*Sqrt[x]*Cos[Pi/8]] - c^(1/8)*Sqrt[x]*Cos[Pi/8]*Log[a^(1/4) + c^(
1/4)*x + 2*a^(1/8)*c^(1/8)*Sqrt[x]*Cos[Pi/8]] - 2*c^(1/8)*Sqrt[x]*ArcTan[Cot[Pi/
8] - (c^(1/8)*Sqrt[x]*Csc[Pi/8])/a^(1/8)]*Sin[Pi/8] + 2*c^(1/8)*Sqrt[x]*ArcTan[C
ot[Pi/8] + (c^(1/8)*Sqrt[x]*Csc[Pi/8])/a^(1/8)]*Sin[Pi/8] + c^(1/8)*Sqrt[x]*Log[
a^(1/4) + c^(1/4)*x - 2*a^(1/8)*c^(1/8)*Sqrt[x]*Sin[Pi/8]]*Sin[Pi/8] - c^(1/8)*S
qrt[x]*Log[a^(1/4) + c^(1/4)*x + 2*a^(1/8)*c^(1/8)*Sqrt[x]*Sin[Pi/8]]*Sin[Pi/8])
/(4*a^(9/8)*Sqrt[x])

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Maple [C]  time = 0.013, size = 38, normalized size = 0.1 \[ -{\frac{1}{4\,a}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}c+a \right ) }{\frac{1}{{\it \_R}}\ln \left ( \sqrt{x}-{\it \_R} \right ) }}-2\,{\frac{1}{a\sqrt{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(3/2)/(c*x^4+a),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -c \int \frac{x^{\frac{5}{2}}}{a c x^{4} + a^{2}}\,{d x} - \frac{2}{a \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-c*integrate(x^(5/2)/(a*c*x^4 + a^2), x) - 2/(a*sqrt(x))

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Fricas [A]  time = 0.260167, size = 608, normalized size = 2.05 \[ -\frac{\sqrt{2}{\left (4 \, \sqrt{2} a \sqrt{x} \left (-\frac{c}{a^{9}}\right )^{\frac{1}{8}} \arctan \left (\frac{a^{8} \left (-\frac{c}{a^{9}}\right )^{\frac{7}{8}}}{c \sqrt{x} + \sqrt{-a^{7} c \left (-\frac{c}{a^{9}}\right )^{\frac{3}{4}} + c^{2} x}}\right ) + \sqrt{2} a \sqrt{x} \left (-\frac{c}{a^{9}}\right )^{\frac{1}{8}} \log \left (a^{8} \left (-\frac{c}{a^{9}}\right )^{\frac{7}{8}} + c \sqrt{x}\right ) - \sqrt{2} a \sqrt{x} \left (-\frac{c}{a^{9}}\right )^{\frac{1}{8}} \log \left (-a^{8} \left (-\frac{c}{a^{9}}\right )^{\frac{7}{8}} + c \sqrt{x}\right ) + 4 \, a \sqrt{x} \left (-\frac{c}{a^{9}}\right )^{\frac{1}{8}} \arctan \left (\frac{a^{8} \left (-\frac{c}{a^{9}}\right )^{\frac{7}{8}}}{a^{8} \left (-\frac{c}{a^{9}}\right )^{\frac{7}{8}} + \sqrt{2} c \sqrt{x} + \sqrt{2 \, \sqrt{2} a^{8} c \sqrt{x} \left (-\frac{c}{a^{9}}\right )^{\frac{7}{8}} - 2 \, a^{7} c \left (-\frac{c}{a^{9}}\right )^{\frac{3}{4}} + 2 \, c^{2} x}}\right ) + 4 \, a \sqrt{x} \left (-\frac{c}{a^{9}}\right )^{\frac{1}{8}} \arctan \left (-\frac{a^{8} \left (-\frac{c}{a^{9}}\right )^{\frac{7}{8}}}{a^{8} \left (-\frac{c}{a^{9}}\right )^{\frac{7}{8}} - \sqrt{2} c \sqrt{x} - \sqrt{-2 \, \sqrt{2} a^{8} c \sqrt{x} \left (-\frac{c}{a^{9}}\right )^{\frac{7}{8}} - 2 \, a^{7} c \left (-\frac{c}{a^{9}}\right )^{\frac{3}{4}} + 2 \, c^{2} x}}\right ) + a \sqrt{x} \left (-\frac{c}{a^{9}}\right )^{\frac{1}{8}} \log \left (2 \, \sqrt{2} a^{8} c \sqrt{x} \left (-\frac{c}{a^{9}}\right )^{\frac{7}{8}} - 2 \, a^{7} c \left (-\frac{c}{a^{9}}\right )^{\frac{3}{4}} + 2 \, c^{2} x\right ) - a \sqrt{x} \left (-\frac{c}{a^{9}}\right )^{\frac{1}{8}} \log \left (-2 \, \sqrt{2} a^{8} c \sqrt{x} \left (-\frac{c}{a^{9}}\right )^{\frac{7}{8}} - 2 \, a^{7} c \left (-\frac{c}{a^{9}}\right )^{\frac{3}{4}} + 2 \, c^{2} x\right ) + 8 \, \sqrt{2}\right )}}{8 \, a \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*sqrt(2)*(4*sqrt(2)*a*sqrt(x)*(-c/a^9)^(1/8)*arctan(a^8*(-c/a^9)^(7/8)/(c*sq
rt(x) + sqrt(-a^7*c*(-c/a^9)^(3/4) + c^2*x))) + sqrt(2)*a*sqrt(x)*(-c/a^9)^(1/8)
*log(a^8*(-c/a^9)^(7/8) + c*sqrt(x)) - sqrt(2)*a*sqrt(x)*(-c/a^9)^(1/8)*log(-a^8
*(-c/a^9)^(7/8) + c*sqrt(x)) + 4*a*sqrt(x)*(-c/a^9)^(1/8)*arctan(a^8*(-c/a^9)^(7
/8)/(a^8*(-c/a^9)^(7/8) + sqrt(2)*c*sqrt(x) + sqrt(2*sqrt(2)*a^8*c*sqrt(x)*(-c/a
^9)^(7/8) - 2*a^7*c*(-c/a^9)^(3/4) + 2*c^2*x))) + 4*a*sqrt(x)*(-c/a^9)^(1/8)*arc
tan(-a^8*(-c/a^9)^(7/8)/(a^8*(-c/a^9)^(7/8) - sqrt(2)*c*sqrt(x) - sqrt(-2*sqrt(2
)*a^8*c*sqrt(x)*(-c/a^9)^(7/8) - 2*a^7*c*(-c/a^9)^(3/4) + 2*c^2*x))) + a*sqrt(x)
*(-c/a^9)^(1/8)*log(2*sqrt(2)*a^8*c*sqrt(x)*(-c/a^9)^(7/8) - 2*a^7*c*(-c/a^9)^(3
/4) + 2*c^2*x) - a*sqrt(x)*(-c/a^9)^(1/8)*log(-2*sqrt(2)*a^8*c*sqrt(x)*(-c/a^9)^
(7/8) - 2*a^7*c*(-c/a^9)^(3/4) + 2*c^2*x) + 8*sqrt(2))/(a*sqrt(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(1/x**(3/2)/(c*x**4+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.29686, size = 612, normalized size = 2.06 \[ -\frac{c \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{7}{8}} \arctan \left (\frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + 2 \, \sqrt{x}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{4 \, a^{2}} - \frac{c \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{7}{8}} \arctan \left (-\frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} - 2 \, \sqrt{x}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{4 \, a^{2}} - \frac{c \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{7}{8}} \arctan \left (\frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + 2 \, \sqrt{x}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{4 \, a^{2}} - \frac{c \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{7}{8}} \arctan \left (-\frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} - 2 \, \sqrt{x}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{4 \, a^{2}} + \frac{c \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{7}{8}}{\rm ln}\left (\sqrt{x} \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{8 \, a^{2}} - \frac{c \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{7}{8}}{\rm ln}\left (-\sqrt{x} \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{8 \, a^{2}} + \frac{c \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{7}{8}}{\rm ln}\left (\sqrt{x} \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{8 \, a^{2}} - \frac{c \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{7}{8}}{\rm ln}\left (-\sqrt{x} \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{8 \, a^{2}} - \frac{2}{a \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*c*sqrt(sqrt(2) + 2)*(a/c)^(7/8)*arctan((sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + 2*
sqrt(x))/(sqrt(sqrt(2) + 2)*(a/c)^(1/8)))/a^2 - 1/4*c*sqrt(sqrt(2) + 2)*(a/c)^(7
/8)*arctan(-(sqrt(-sqrt(2) + 2)*(a/c)^(1/8) - 2*sqrt(x))/(sqrt(sqrt(2) + 2)*(a/c
)^(1/8)))/a^2 - 1/4*c*sqrt(-sqrt(2) + 2)*(a/c)^(7/8)*arctan((sqrt(sqrt(2) + 2)*(
a/c)^(1/8) + 2*sqrt(x))/(sqrt(-sqrt(2) + 2)*(a/c)^(1/8)))/a^2 - 1/4*c*sqrt(-sqrt
(2) + 2)*(a/c)^(7/8)*arctan(-(sqrt(sqrt(2) + 2)*(a/c)^(1/8) - 2*sqrt(x))/(sqrt(-
sqrt(2) + 2)*(a/c)^(1/8)))/a^2 + 1/8*c*sqrt(sqrt(2) + 2)*(a/c)^(7/8)*ln(sqrt(x)*
sqrt(sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/a^2 - 1/8*c*sqrt(sqrt(2) + 2)*(
a/c)^(7/8)*ln(-sqrt(x)*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/a^2 + 1/
8*c*sqrt(-sqrt(2) + 2)*(a/c)^(7/8)*ln(sqrt(x)*sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + x
+ (a/c)^(1/4))/a^2 - 1/8*c*sqrt(-sqrt(2) + 2)*(a/c)^(7/8)*ln(-sqrt(x)*sqrt(-sqr
t(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/a^2 - 2/(a*sqrt(x))