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

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

[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)*ArcTan[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)*L
og[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 = 0.161181, 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, Rules used = {290, 325, 297, 1162, 617, 204, 1165, 628} \[ \frac{9}{32 a^2 x \left (a+c x^4\right )}-\frac{45 \sqrt [4]{c} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{13/4}}+\frac{45 \sqrt [4]{c} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{13/4}}+\frac{45 \sqrt [4]{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{13/4}}-\frac{45 \sqrt [4]{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} a^{13/4}}-\frac{45}{32 a^3 x}+\frac{1}{8 a x \left (a+c x^4\right )^2} \]

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)*ArcTan[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)*L
og[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^(13/4))

Rule 290

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(
a*c*n*(p + 1)), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[
{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 297

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{1}{x^2 \left (a+c x^4\right )^3} \, dx &=\frac{1}{8 a x \left (a+c x^4\right )^2}+\frac{9 \int \frac{1}{x^2 \left (a+c x^4\right )^2} \, dx}{8 a}\\ &=\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 \int \frac{1}{x^2 \left (a+c x^4\right )} \, dx}{32 a^2}\\ &=-\frac{45}{32 a^3 x}+\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 c) \int \frac{x^2}{a+c x^4} \, dx}{32 a^3}\\ &=-\frac{45}{32 a^3 x}+\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{\left (45 \sqrt{c}\right ) \int \frac{\sqrt{a}-\sqrt{c} x^2}{a+c x^4} \, dx}{64 a^3}-\frac{\left (45 \sqrt{c}\right ) \int \frac{\sqrt{a}+\sqrt{c} x^2}{a+c x^4} \, dx}{64 a^3}\\ &=-\frac{45}{32 a^3 x}+\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 \int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{128 a^3}-\frac{45 \int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{128 a^3}-\frac{\left (45 \sqrt [4]{c}\right ) \int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{128 \sqrt{2} a^{13/4}}-\frac{\left (45 \sqrt [4]{c}\right ) \int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{128 \sqrt{2} a^{13/4}}\\ &=-\frac{45}{32 a^3 x}+\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 \sqrt [4]{c} \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{13/4}}+\frac{45 \sqrt [4]{c} \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{13/4}}-\frac{\left (45 \sqrt [4]{c}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{13/4}}+\frac{\left (45 \sqrt [4]{c}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{13/4}}\\ &=-\frac{45}{32 a^3 x}+\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 \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 (1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{13/4}}-\frac{45 \sqrt [4]{c} \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{13/4}}+\frac{45 \sqrt [4]{c} \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{13/4}}\\ \end{align*}

Mathematica [A]  time = 0.0968345, size = 216, normalized size = 0.93 \[ \frac{-\frac{32 a^{5/4} c x^3}{\left (a+c x^4\right )^2}-\frac{104 \sqrt [4]{a} c x^3}{a+c x^4}-45 \sqrt{2} \sqrt [4]{c} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )+45 \sqrt{2} \sqrt [4]{c} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )+90 \sqrt{2} \sqrt [4]{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )-90 \sqrt{2} \sqrt [4]{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )-\frac{256 \sqrt [4]{a}}{x}}{256 a^{13/4}} \]

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)*Ar
cTan[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 = 0.015, size = 174, normalized size = 0.8 \begin{align*} -{\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 ({ \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}}}}}}-{\frac{1}{{a}^{3}x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(c*x^4+a)^3,x)

[Out]

-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)-1/a^3/x

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.81915, size = 585, normalized size = 2.51 \begin{align*} -\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 (-a^{3} x \left (-\frac{c}{a^{13}}\right )^{\frac{1}{4}} + a^{3} \sqrt{-\frac{a^{7} \sqrt{-\frac{c}{a^{13}}} - c x^{2}}{c}} \left (-\frac{c}{a^{13}}\right )^{\frac{1}{4}}\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(c*x^4+a)^3,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^3*x*(-c/
a^13)^(1/4) + a^3*sqrt(-(a^7*sqrt(-c/a^13) - c*x^2)/c)*(-c/a^13)^(1/4)) + 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 = 4.19152, size = 78, normalized size = 0.33 \begin{align*} - \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 )} \end{align*}

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 [A]  time = 1.14777, size = 293, normalized size = 1.26 \begin{align*} -\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}} \log \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}} \log \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(c*x^4+a)^3,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*s
qrt(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) + 45/256*sqrt(2)*(a
*c^3)^(3/4)*log(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)*log(x^2 - sq
rt(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)