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]

-1/4*arctan(1-b^(1/4)*ln(c*x^n)*2^(1/2)/a^(1/4))/a^(3/4)/b^(1/4)/n*2^(1/2)+1/4*arctan(1+b^(1/4)*ln(c*x^n)*2^(1
/2)/a^(1/4))/a^(3/4)/b^(1/4)/n*2^(1/2)-1/8*ln(-a^(1/4)*b^(1/4)*ln(c*x^n)*2^(1/2)+a^(1/2)+ln(c*x^n)^2*b^(1/2))/
a^(3/4)/b^(1/4)/n*2^(1/2)+1/8*ln(a^(1/4)*b^(1/4)*ln(c*x^n)*2^(1/2)+a^(1/2)+ln(c*x^n)^2*b^(1/2))/a^(3/4)/b^(1/4
)/n*2^(1/2)

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Rubi [A]  time = 0.16, antiderivative size = 227, normalized size of antiderivative = 1.00, 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 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 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 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 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 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]

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*}

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Mathematica [A]  time = 0.07, 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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fricas [A]  time = 0.46, size = 195, normalized size = 0.86 \[ \left (-\frac {1}{a^{3} b n^{4}}\right )^{\frac {1}{4}} \arctan \left (\sqrt {a^{2} n^{2} \sqrt {-\frac {1}{a^{3} b n^{4}}} + n^{2} \log \relax (x)^{2} + 2 \, n \log \relax (c) \log \relax (x) + \log \relax (c)^{2}} a^{2} b n^{3} \left (-\frac {1}{a^{3} b n^{4}}\right )^{\frac {3}{4}} - {\left (a^{2} b n^{4} \log \relax (x) + a^{2} b n^{3} \log \relax (c)\right )} \left (-\frac {1}{a^{3} b n^{4}}\right )^{\frac {3}{4}}\right ) + \frac {1}{4} \, \left (-\frac {1}{a^{3} b n^{4}}\right )^{\frac {1}{4}} \log \left (a n \left (-\frac {1}{a^{3} b n^{4}}\right )^{\frac {1}{4}} + n \log \relax (x) + \log \relax (c)\right ) - \frac {1}{4} \, \left (-\frac {1}{a^{3} b n^{4}}\right )^{\frac {1}{4}} \log \left (-a n \left (-\frac {1}{a^{3} b n^{4}}\right )^{\frac {1}{4}} + n \log \relax (x) + \log \relax (c)\right ) \]

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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giac [A]  time = 0.22, size = 170, normalized size = 0.75 \[ -\frac {1}{2} \, \left (-\frac {1}{a^{3} b n^{4}}\right )^{\frac {1}{4}} \arctan \left (\frac {\pi b {\left (\mathrm {sgn}\relax (c) - 1\right )} + 2 \, \left (-a b^{3}\right )^{\frac {1}{4}}}{2 \, {\left (b n \log \relax (x) + b \log \left ({\left | c \right |}\right )\right )}}\right ) + \frac {1}{8} \, \left (-\frac {1}{a^{3} b n^{4}}\right )^{\frac {1}{4}} \log \left (\frac {1}{4} \, {\left (\pi b n {\left (\mathrm {sgn}\relax (x) - 1\right )} + \pi b {\left (\mathrm {sgn}\relax (c) - 1\right )}\right )}^{2} + {\left (b n \log \left ({\left | x \right |}\right ) + b \log \left ({\left | c \right |}\right ) + \left (-a b^{3}\right )^{\frac {1}{4}}\right )}^{2}\right ) - \frac {1}{8} \, \left (-\frac {1}{a^{3} b n^{4}}\right )^{\frac {1}{4}} \log \left (\frac {1}{4} \, {\left (\pi b n {\left (\mathrm {sgn}\relax (x) - 1\right )} + \pi b {\left (\mathrm {sgn}\relax (c) - 1\right )}\right )}^{2} + {\left (b n \log \left ({\left | x \right |}\right ) + b \log \left ({\left | c \right |}\right ) - \left (-a b^{3}\right )^{\frac {1}{4}}\right )}^{2}\right ) \]

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]

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

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maple [A]  time = 0.07, size = 168, normalized size = 0.74 \[ -\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \ln \left (c \,x^{n}\right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{4 a n}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \ln \left (c \,x^{n}\right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{4 a n}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {\ln \left (c \,x^{n}\right )^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (c \,x^{n}\right )+\sqrt {\frac {a}{b}}}{\ln \left (c \,x^{n}\right )^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (c \,x^{n}\right )+\sqrt {\frac {a}{b}}}\right )}{8 a n} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{b x \log \left (c x^{n}\right )^{4} + a x}\,{d x} \]

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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mupad [B]  time = 2.24, size = 95, normalized size = 0.42 \[ -\frac {\ln \left ({\left (-a\right )}^{1/4}+b^{1/4}\,\ln \left (c\,x^n\right )\right )-\ln \left ({\left (-a\right )}^{1/4}-b^{1/4}\,\ln \left (c\,x^n\right )\right )+\ln \left ({\left (-a\right )}^{1/4}-b^{1/4}\,\ln \left (c\,x^n\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}-\ln \left ({\left (-a\right )}^{1/4}+b^{1/4}\,\ln \left (c\,x^n\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{4\,{\left (-a\right )}^{3/4}\,b^{1/4}\,n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x + b*x*log(c*x^n)^4),x)

[Out]

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

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sympy [A]  time = 36.01, size = 246, normalized size = 1.08 \[ \begin {cases} \frac {\tilde {\infty } \log {\relax (x )}}{\log {\relax (c )}^{4}} & \text {for}\: a = 0 \wedge b = 0 \wedge n = 0 \\\frac {\log {\relax (x )}}{a + b \log {\relax (c )}^{4}} & \text {for}\: n = 0 \\\frac {\log {\relax (x )}}{a} & \text {for}\: b = 0 \\- \frac {1}{b \left (3 n^{4} \log {\relax (x )}^{3} + 9 n^{3} \log {\relax (c )} \log {\relax (x )}^{2} + 9 n^{2} \log {\relax (c )}^{2} \log {\relax (x )} + 3 n \log {\relax (c )}^{3}\right )} & \text {for}\: a = 0 \\- \frac {\sqrt [4]{-1} \sqrt [4]{\frac {1}{b}} \log {\left (- \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{b}} + n \log {\relax (x )} + \log {\relax (c )} \right )}}{4 a^{\frac {3}{4}} n} + \frac {\sqrt [4]{-1} \sqrt [4]{\frac {1}{b}} \log {\left (\sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{b}} + n \log {\relax (x )} + \log {\relax (c )} \right )}}{4 a^{\frac {3}{4}} n} - \frac {\sqrt [4]{-1} \sqrt [4]{\frac {1}{b}} \operatorname {atan}{\left (\frac {\left (-1\right )^{\frac {3}{4}} n \log {\relax (x )}}{\sqrt [4]{a} \sqrt [4]{\frac {1}{b}}} + \frac {\left (-1\right )^{\frac {3}{4}} \log {\relax (c )}}{\sqrt [4]{a} \sqrt [4]{\frac {1}{b}}} \right )}}{2 a^{\frac {3}{4}} n} & \text {otherwise} \end {cases} \]

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)), (log(x)/(a + b*log(c)**4), Eq(n, 0)), (log(x
)/a, Eq(b, 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)), (-(-1)**(1/4)*(1/b)**(1/4)*log(-(-1)**(1/4)*a**(1/4)*(1/b)**(1/4) + n*log(x) + log(c))/(4*a**(3/4
)*n) + (-1)**(1/4)*(1/b)**(1/4)*log((-1)**(1/4)*a**(1/4)*(1/b)**(1/4) + n*log(x) + log(c))/(4*a**(3/4)*n) - (-
1)**(1/4)*(1/b)**(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)*n), True))

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