3.253 \(\int \frac{1}{a x+b x \log ^4(c x^n)} \, dx\)

Optimal. Leaf size=227 \[ -\frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt{a}+\sqrt{b} \log ^2\left (c x^n\right )\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} n}+\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt{a}+\sqrt{b} \log ^2\left (c x^n\right )\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} n}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{b} n}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{b} n} \]

[Out]

-ArcTan[1 - (Sqrt[2]*b^(1/4)*Log[c*x^n])/a^(1/4)]/(2*Sqrt[2]*a^(3/4)*b^(1/4)*n) + ArcTan[1 + (Sqrt[2]*b^(1/4)*
Log[c*x^n])/a^(1/4)]/(2*Sqrt[2]*a^(3/4)*b^(1/4)*n) - Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Log[c*x^n] + Sqrt[b
]*Log[c*x^n]^2]/(4*Sqrt[2]*a^(3/4)*b^(1/4)*n) + Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Log[c*x^n] + Sqrt[b]*Log
[c*x^n]^2]/(4*Sqrt[2]*a^(3/4)*b^(1/4)*n)

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Rubi [A]  time = 0.162432, antiderivative size = 227, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353, Rules used = {211, 1165, 628, 1162, 617, 204} \[ -\frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt{a}+\sqrt{b} \log ^2\left (c x^n\right )\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} n}+\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt{a}+\sqrt{b} \log ^2\left (c x^n\right )\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} n}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{b} n}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{b} n} \]

Antiderivative was successfully verified.

[In]

Int[(a*x + b*x*Log[c*x^n]^4)^(-1),x]

[Out]

-ArcTan[1 - (Sqrt[2]*b^(1/4)*Log[c*x^n])/a^(1/4)]/(2*Sqrt[2]*a^(3/4)*b^(1/4)*n) + ArcTan[1 + (Sqrt[2]*b^(1/4)*
Log[c*x^n])/a^(1/4)]/(2*Sqrt[2]*a^(3/4)*b^(1/4)*n) - Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Log[c*x^n] + Sqrt[b
]*Log[c*x^n]^2]/(4*Sqrt[2]*a^(3/4)*b^(1/4)*n) + Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Log[c*x^n] + Sqrt[b]*Log
[c*x^n]^2]/(4*Sqrt[2]*a^(3/4)*b^(1/4)*n)

Rule 211

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

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]

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])

Rubi steps

\begin{align*} \int \frac{1}{a x+b x \log ^4\left (c x^n\right )} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{a+b x^4} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\sqrt{a}-\sqrt{b} x^2}{a+b x^4} \, dx,x,\log \left (c x^n\right )\right )}{2 \sqrt{a} n}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{a}+\sqrt{b} x^2}{a+b x^4} \, dx,x,\log \left (c x^n\right )\right )}{2 \sqrt{a} n}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\log \left (c x^n\right )\right )}{4 \sqrt{a} \sqrt{b} n}+\frac{\operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\log \left (c x^n\right )\right )}{4 \sqrt{a} \sqrt{b} n}-\frac{\operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\log \left (c x^n\right )\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} n}-\frac{\operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\log \left (c x^n\right )\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} n}\\ &=-\frac{\log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt{b} \log ^2\left (c x^n\right )\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} n}+\frac{\log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt{b} \log ^2\left (c x^n\right )\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} n}+\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{b} n}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{b} n}\\ &=-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{b} n}+\frac{\tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{b} n}-\frac{\log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt{b} \log ^2\left (c x^n\right )\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} n}+\frac{\log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt{b} \log ^2\left (c x^n\right )\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} n}\\ \end{align*}

Mathematica [A]  time = 0.06405, size = 167, normalized size = 0.74 \[ \frac{-\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt{a}+\sqrt{b} \log ^2\left (c x^n\right )\right )+\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt{a}+\sqrt{b} \log ^2\left (c x^n\right )\right )-2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{a}}\right )+2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} n} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*x + b*x*Log[c*x^n]^4)^(-1),x]

[Out]

(-2*ArcTan[1 - (Sqrt[2]*b^(1/4)*Log[c*x^n])/a^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*b^(1/4)*Log[c*x^n])/a^(1/4)] - Lo
g[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Log[c*x^n] + Sqrt[b]*Log[c*x^n]^2] + Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)
*Log[c*x^n] + Sqrt[b]*Log[c*x^n]^2])/(4*Sqrt[2]*a^(3/4)*b^(1/4)*n)

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Maple [A]  time = 0.008, size = 168, normalized size = 0.7 \begin{align*}{\frac{\sqrt{2}}{8\,na}\sqrt [4]{{\frac{a}{b}}}\ln \left ({ \left ( \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{2}+\sqrt [4]{{\frac{a}{b}}}\ln \left ( c{x}^{n} \right ) \sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{2}-\sqrt [4]{{\frac{a}{b}}}\ln \left ( c{x}^{n} \right ) \sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{\sqrt{2}}{4\,na}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\ln \left ( c{x}^{n} \right ) \sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }-{\frac{\sqrt{2}}{4\,na}\sqrt [4]{{\frac{a}{b}}}\arctan \left ( -{\ln \left ( c{x}^{n} \right ) \sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x+b*x*ln(c*x^n)^4),x)

[Out]

1/8/n*(a/b)^(1/4)/a*2^(1/2)*ln((ln(c*x^n)^2+(a/b)^(1/4)*ln(c*x^n)*2^(1/2)+(a/b)^(1/2))/(ln(c*x^n)^2-(a/b)^(1/4
)*ln(c*x^n)*2^(1/2)+(a/b)^(1/2)))+1/4/n*(a/b)^(1/4)/a*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*ln(c*x^n)+1)-1/4/n*(a
/b)^(1/4)/a*2^(1/2)*arctan(-2^(1/2)/(a/b)^(1/4)*ln(c*x^n)+1)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{b x \log \left (c x^{n}\right )^{4} + a x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+b*x*log(c*x^n)^4),x, algorithm="maxima")

[Out]

integrate(1/(b*x*log(c*x^n)^4 + a*x), x)

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Fricas [A]  time = 2.01281, size = 502, normalized size = 2.21 \begin{align*} \left (-\frac{1}{a^{3} b n^{4}}\right )^{\frac{1}{4}} \arctan \left (\sqrt{a^{2} n^{2} \sqrt{-\frac{1}{a^{3} b n^{4}}} + n^{2} \log \left (x\right )^{2} + 2 \, n \log \left (c\right ) \log \left (x\right ) + \log \left (c\right )^{2}} a^{2} b n^{3} \left (-\frac{1}{a^{3} b n^{4}}\right )^{\frac{3}{4}} -{\left (a^{2} b n^{4} \log \left (x\right ) + a^{2} b n^{3} \log \left (c\right )\right )} \left (-\frac{1}{a^{3} b n^{4}}\right )^{\frac{3}{4}}\right ) + \frac{1}{4} \, \left (-\frac{1}{a^{3} b n^{4}}\right )^{\frac{1}{4}} \log \left (a n \left (-\frac{1}{a^{3} b n^{4}}\right )^{\frac{1}{4}} + n \log \left (x\right ) + \log \left (c\right )\right ) - \frac{1}{4} \, \left (-\frac{1}{a^{3} b n^{4}}\right )^{\frac{1}{4}} \log \left (-a n \left (-\frac{1}{a^{3} b n^{4}}\right )^{\frac{1}{4}} + n \log \left (x\right ) + \log \left (c\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+b*x*log(c*x^n)^4),x, algorithm="fricas")

[Out]

(-1/(a^3*b*n^4))^(1/4)*arctan(sqrt(a^2*n^2*sqrt(-1/(a^3*b*n^4)) + n^2*log(x)^2 + 2*n*log(c)*log(x) + log(c)^2)
*a^2*b*n^3*(-1/(a^3*b*n^4))^(3/4) - (a^2*b*n^4*log(x) + a^2*b*n^3*log(c))*(-1/(a^3*b*n^4))^(3/4)) + 1/4*(-1/(a
^3*b*n^4))^(1/4)*log(a*n*(-1/(a^3*b*n^4))^(1/4) + n*log(x) + log(c)) - 1/4*(-1/(a^3*b*n^4))^(1/4)*log(-a*n*(-1
/(a^3*b*n^4))^(1/4) + n*log(x) + log(c))

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Sympy [A]  time = 66.9285, size = 257, normalized size = 1.13 \begin{align*} \begin{cases} \frac{\tilde{\infty } \log{\left (x \right )}}{\log{\left (c \right )}^{4}} & \text{for}\: a = 0 \wedge b = 0 \wedge n = 0 \\- \frac{1}{b \left (3 n^{4} \log{\left (x \right )}^{3} + 9 n^{3} \log{\left (c \right )} \log{\left (x \right )}^{2} + 9 n^{2} \log{\left (c \right )}^{2} \log{\left (x \right )} + 3 n \log{\left (c \right )}^{3}\right )} & \text{for}\: a = 0 \\\frac{\log{\left (x \right )}}{a} & \text{for}\: b = 0 \\\frac{\log{\left (x \right )}}{a + b \log{\left (c \right )}^{4}} & \text{for}\: n = 0 \\- \frac{\sqrt [4]{-1} \log{\left (- \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac{1}{b}} + n \log{\left (x \right )} + \log{\left (c \right )} \right )}}{4 a^{\frac{3}{4}} b^{3} n \left (\frac{1}{b}\right )^{\frac{11}{4}}} + \frac{\sqrt [4]{-1} \log{\left (\sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac{1}{b}} + n \log{\left (x \right )} + \log{\left (c \right )} \right )}}{4 a^{\frac{3}{4}} b^{3} n \left (\frac{1}{b}\right )^{\frac{11}{4}}} - \frac{\sqrt [4]{-1} \operatorname{atan}{\left (\frac{\left (-1\right )^{\frac{3}{4}} n \log{\left (x \right )}}{\sqrt [4]{a} \sqrt [4]{\frac{1}{b}}} + \frac{\left (-1\right )^{\frac{3}{4}} \log{\left (c \right )}}{\sqrt [4]{a} \sqrt [4]{\frac{1}{b}}} \right )}}{2 a^{\frac{3}{4}} b^{3} n \left (\frac{1}{b}\right )^{\frac{11}{4}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+b*x*ln(c*x**n)**4),x)

[Out]

Piecewise((zoo*log(x)/log(c)**4, Eq(a, 0) & Eq(b, 0) & Eq(n, 0)), (-1/(b*(3*n**4*log(x)**3 + 9*n**3*log(c)*log
(x)**2 + 9*n**2*log(c)**2*log(x) + 3*n*log(c)**3)), Eq(a, 0)), (log(x)/a, Eq(b, 0)), (log(x)/(a + b*log(c)**4)
, Eq(n, 0)), (-(-1)**(1/4)*log(-(-1)**(1/4)*a**(1/4)*(1/b)**(1/4) + n*log(x) + log(c))/(4*a**(3/4)*b**3*n*(1/b
)**(11/4)) + (-1)**(1/4)*log((-1)**(1/4)*a**(1/4)*(1/b)**(1/4) + n*log(x) + log(c))/(4*a**(3/4)*b**3*n*(1/b)**
(11/4)) - (-1)**(1/4)*atan((-1)**(3/4)*n*log(x)/(a**(1/4)*(1/b)**(1/4)) + (-1)**(3/4)*log(c)/(a**(1/4)*(1/b)**
(1/4)))/(2*a**(3/4)*b**3*n*(1/b)**(11/4)), True))

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Giac [A]  time = 1.43187, size = 271, normalized size = 1.19 \begin{align*} \frac{1}{4} \, i \left (-\frac{1}{a^{3} b n^{4}}\right )^{\frac{1}{4}} \log \left (b i n \log \left (x\right ) + b i \log \left (c\right ) - \left (-a b^{3}\right )^{\frac{1}{4}}\right ) - \frac{1}{4} \, i \left (-\frac{1}{a^{3} b n^{4}}\right )^{\frac{1}{4}} \log \left (-b i n \log \left (x\right ) - b i \log \left (c\right ) - \left (-a b^{3}\right )^{\frac{1}{4}}\right ) + \frac{1}{8} \, \left (-\frac{1}{a^{3} b n^{4}}\right )^{\frac{1}{4}} \log \left (\frac{1}{4} \,{\left (\pi b n{\left (\mathrm{sgn}\left (x\right ) - 1\right )} + \pi b{\left (\mathrm{sgn}\left (c\right ) - 1\right )}\right )}^{2} +{\left (b n \log \left ({\left | x \right |}\right ) + b \log \left ({\left | c \right |}\right ) + \left (-a b^{3}\right )^{\frac{1}{4}}\right )}^{2}\right ) - \frac{1}{8} \, \left (-\frac{1}{a^{3} b n^{4}}\right )^{\frac{1}{4}} \log \left (\frac{1}{4} \,{\left (\pi b n{\left (\mathrm{sgn}\left (x\right ) - 1\right )} + \pi b{\left (\mathrm{sgn}\left (c\right ) - 1\right )}\right )}^{2} +{\left (b n \log \left ({\left | x \right |}\right ) + b \log \left ({\left | c \right |}\right ) - \left (-a b^{3}\right )^{\frac{1}{4}}\right )}^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+b*x*log(c*x^n)^4),x, algorithm="giac")

[Out]

1/4*i*(-1/(a^3*b*n^4))^(1/4)*log(b*i*n*log(x) + b*i*log(c) - (-a*b^3)^(1/4)) - 1/4*i*(-1/(a^3*b*n^4))^(1/4)*lo
g(-b*i*n*log(x) - b*i*log(c) - (-a*b^3)^(1/4)) + 1/8*(-1/(a^3*b*n^4))^(1/4)*log(1/4*(pi*b*n*(sgn(x) - 1) + pi*
b*(sgn(c) - 1))^2 + (b*n*log(abs(x)) + b*log(abs(c)) + (-a*b^3)^(1/4))^2) - 1/8*(-1/(a^3*b*n^4))^(1/4)*log(1/4
*(pi*b*n*(sgn(x) - 1) + pi*b*(sgn(c) - 1))^2 + (b*n*log(abs(x)) + b*log(abs(c)) - (-a*b^3)^(1/4))^2)