\(\int \frac {1}{a x+b x \log ^3(c x^n)} \, dx\) [252]

Optimal result

Integrand size = 17, antiderivative size = 144 \[ \int \frac {1}{a x+b x \log ^3\left (c x^n\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \log \left (c x^n\right )}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} \sqrt [3]{b} n}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \log \left (c x^n\right )\right )}{3 a^{2/3} \sqrt [3]{b} n}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \log \left (c x^n\right )+b^{2/3} \log ^2\left (c x^n\right )\right )}{6 a^{2/3} \sqrt [3]{b} n} \]

[Out]

1/3*ln(a^(1/3)+b^(1/3)*ln(c*x^n))/a^(2/3)/b^(1/3)/n-1/6*ln(a^(2/3)-a^(1/3)*b^(1/3)*ln(c*x^n)+b^(2/3)*ln(c*x^n)
^2)/a^(2/3)/b^(1/3)/n-1/3*arctan(1/3*(a^(1/3)-2*b^(1/3)*ln(c*x^n))/a^(1/3)*3^(1/2))/a^(2/3)/b^(1/3)/n*3^(1/2)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {206, 31, 648, 631, 210, 642} \[ \int \frac {1}{a x+b x \log ^3\left (c x^n\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \log \left (c x^n\right )}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} \sqrt [3]{b} n}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \log \left (c x^n\right )+b^{2/3} \log ^2\left (c x^n\right )\right )}{6 a^{2/3} \sqrt [3]{b} n}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \log \left (c x^n\right )\right )}{3 a^{2/3} \sqrt [3]{b} n} \]

[In]

Int[(a*x + b*x*Log[c*x^n]^3)^(-1),x]

[Out]

-(ArcTan[(a^(1/3) - 2*b^(1/3)*Log[c*x^n])/(Sqrt[3]*a^(1/3))]/(Sqrt[3]*a^(2/3)*b^(1/3)*n)) + Log[a^(1/3) + b^(1
/3)*Log[c*x^n]]/(3*a^(2/3)*b^(1/3)*n) - Log[a^(2/3) - a^(1/3)*b^(1/3)*Log[c*x^n] + b^(2/3)*Log[c*x^n]^2]/(6*a^
(2/3)*b^(1/3)*n)

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 206

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

Rule 210

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

Rule 631

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 642

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 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{a+b x^3} \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = \frac {\text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,\log \left (c x^n\right )\right )}{3 a^{2/3} n}+\frac {\text {Subst}\left (\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\log \left (c x^n\right )\right )}{3 a^{2/3} n} \\ & = \frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \log \left (c x^n\right )\right )}{3 a^{2/3} \sqrt [3]{b} n}+\frac {\text {Subst}\left (\int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\log \left (c x^n\right )\right )}{2 \sqrt [3]{a} n}-\frac {\text {Subst}\left (\int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\log \left (c x^n\right )\right )}{6 a^{2/3} \sqrt [3]{b} n} \\ & = \frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \log \left (c x^n\right )\right )}{3 a^{2/3} \sqrt [3]{b} n}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \log \left (c x^n\right )+b^{2/3} \log ^2\left (c x^n\right )\right )}{6 a^{2/3} \sqrt [3]{b} n}+\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} \log \left (c x^n\right )}{\sqrt [3]{a}}\right )}{a^{2/3} \sqrt [3]{b} n} \\ & = -\frac {\tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \log \left (c x^n\right )}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} a^{2/3} \sqrt [3]{b} n}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \log \left (c x^n\right )\right )}{3 a^{2/3} \sqrt [3]{b} n}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \log \left (c x^n\right )+b^{2/3} \log ^2\left (c x^n\right )\right )}{6 a^{2/3} \sqrt [3]{b} n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.78 \[ \int \frac {1}{a x+b x \log ^3\left (c x^n\right )} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \log \left (c x^n\right )}{\sqrt [3]{a}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \log \left (c x^n\right )\right )+\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \log \left (c x^n\right )+b^{2/3} \log ^2\left (c x^n\right )\right )}{6 a^{2/3} \sqrt [3]{b} n} \]

[In]

Integrate[(a*x + b*x*Log[c*x^n]^3)^(-1),x]

[Out]

-1/6*(2*Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*Log[c*x^n])/a^(1/3))/Sqrt[3]] - 2*Log[a^(1/3) + b^(1/3)*Log[c*x^n]] + L
og[a^(2/3) - a^(1/3)*b^(1/3)*Log[c*x^n] + b^(2/3)*Log[c*x^n]^2])/(a^(2/3)*b^(1/3)*n)

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.97 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.78

method	result	size
risch	\(\munderset {\textit {\_R} =\operatorname {RootOf}\left (27 a^{2} b \,n^{3} \textit {\_Z}^{3}-1\right )}{\sum }\textit {\_R} \ln \left (\ln \left (x^{n}\right )+3 a n \textit {\_R} -\frac {i \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+\ln \left (c \right )\right )\)	\(112\)
default	\(\frac {\frac {\ln \left (\ln \left (c \,x^{n}\right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\ln \left (c \,x^{n}\right )^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} \ln \left (c \,x^{n}\right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \ln \left (c \,x^{n}\right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}}{n}\)	\(115\)

[In]

int(1/(a*x+b*x*ln(c*x^n)^3),x,method=_RETURNVERBOSE)

[Out]

sum(_R*ln(ln(x^n)+3*a*n*_R-1/2*I*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/2*I*Pi*csgn(I*c)*csgn(I*c*x^n)^2+1/2
*I*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-1/2*I*Pi*csgn(I*c*x^n)^3+ln(c)),_R=RootOf(27*_Z^3*a^2*b*n^3-1))

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 480, normalized size of antiderivative = 3.33 \[ \int \frac {1}{a x+b x \log ^3\left (c x^n\right )} \, dx=\left [\frac {3 \, \sqrt {\frac {1}{3}} a b \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, a b n^{3} \log \left (x\right )^{3} + 6 \, a b n^{2} \log \left (c\right ) \log \left (x\right )^{2} + 6 \, a b n \log \left (c\right )^{2} \log \left (x\right ) + 2 \, a b \log \left (c\right )^{3} - a^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b n^{2} \log \left (x\right )^{2} + 4 \, a b n \log \left (c\right ) \log \left (x\right ) + 2 \, a b \log \left (c\right )^{2} + \left (a^{2} b\right )^{\frac {2}{3}} {\left (n \log \left (x\right ) + \log \left (c\right )\right )} - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} - 3 \, \left (a^{2} b\right )^{\frac {1}{3}} {\left (a n \log \left (x\right ) + a \log \left (c\right )\right )}}{b n^{3} \log \left (x\right )^{3} + 3 \, b n^{2} \log \left (c\right ) \log \left (x\right )^{2} + 3 \, b n \log \left (c\right )^{2} \log \left (x\right ) + b \log \left (c\right )^{3} + a}\right ) - \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b n^{2} \log \left (x\right )^{2} + 2 \, a b n \log \left (c\right ) \log \left (x\right ) + a b \log \left (c\right )^{2} - \left (a^{2} b\right )^{\frac {2}{3}} {\left (n \log \left (x\right ) + \log \left (c\right )\right )} + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) + 2 \, \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b n \log \left (x\right ) + a b \log \left (c\right ) + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{6 \, a^{2} b n}, \frac {6 \, \sqrt {\frac {1}{3}} a b \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (a^{2} b\right )^{\frac {2}{3}} {\left (n \log \left (x\right ) + \log \left (c\right )\right )} - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{a^{2}}\right ) - \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b n^{2} \log \left (x\right )^{2} + 2 \, a b n \log \left (c\right ) \log \left (x\right ) + a b \log \left (c\right )^{2} - \left (a^{2} b\right )^{\frac {2}{3}} {\left (n \log \left (x\right ) + \log \left (c\right )\right )} + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) + 2 \, \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b n \log \left (x\right ) + a b \log \left (c\right ) + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{6 \, a^{2} b n}\right ] \]

[In]

integrate(1/(a*x+b*x*log(c*x^n)^3),x, algorithm="fricas")

[Out]

[1/6*(3*sqrt(1/3)*a*b*sqrt(-(a^2*b)^(1/3)/b)*log((2*a*b*n^3*log(x)^3 + 6*a*b*n^2*log(c)*log(x)^2 + 6*a*b*n*log
(c)^2*log(x) + 2*a*b*log(c)^3 - a^2 + 3*sqrt(1/3)*(2*a*b*n^2*log(x)^2 + 4*a*b*n*log(c)*log(x) + 2*a*b*log(c)^2
+ (a^2*b)^(2/3)*(n*log(x) + log(c)) - (a^2*b)^(1/3)*a)*sqrt(-(a^2*b)^(1/3)/b) - 3*(a^2*b)^(1/3)*(a*n*log(x) +
a*log(c)))/(b*n^3*log(x)^3 + 3*b*n^2*log(c)*log(x)^2 + 3*b*n*log(c)^2*log(x) + b*log(c)^3 + a)) - (a^2*b)^(2/
3)*log(a*b*n^2*log(x)^2 + 2*a*b*n*log(c)*log(x) + a*b*log(c)^2 - (a^2*b)^(2/3)*(n*log(x) + log(c)) + (a^2*b)^(
1/3)*a) + 2*(a^2*b)^(2/3)*log(a*b*n*log(x) + a*b*log(c) + (a^2*b)^(2/3)))/(a^2*b*n), 1/6*(6*sqrt(1/3)*a*b*sqrt
((a^2*b)^(1/3)/b)*arctan(sqrt(1/3)*(2*(a^2*b)^(2/3)*(n*log(x) + log(c)) - (a^2*b)^(1/3)*a)*sqrt((a^2*b)^(1/3)/
b)/a^2) - (a^2*b)^(2/3)*log(a*b*n^2*log(x)^2 + 2*a*b*n*log(c)*log(x) + a*b*log(c)^2 - (a^2*b)^(2/3)*(n*log(x)
+ log(c)) + (a^2*b)^(1/3)*a) + 2*(a^2*b)^(2/3)*log(a*b*n*log(x) + a*b*log(c) + (a^2*b)^(2/3)))/(a^2*b*n)]

Sympy [A] (verification not implemented)

Time = 25.99 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.22 \[ \int \frac {1}{a x+b x \log ^3\left (c x^n\right )} \, dx=\begin {cases} \frac {\tilde {\infty } \log {\left (x \right )}}{\log {\left (c \right )}^{3}} & \text {for}\: a = 0 \wedge b = 0 \wedge n = 0 \\- \frac {1}{2 b n \log {\left (c x^{n} \right )}^{2}} & \text {for}\: a = 0 \\\frac {\log {\left (x \right )}}{a} & \text {for}\: b = 0 \\\frac {\log {\left (x \right )}}{a + b \log {\left (c \right )}^{3}} & \text {for}\: n = 0 \\- \frac {\sqrt [3]{- \frac {a}{b}} \log {\left (- \sqrt [3]{- \frac {a}{b}} + \log {\left (c x^{n} \right )} \right )}}{3 a n} + \frac {\sqrt [3]{- \frac {a}{b}} \log {\left (4 \left (- \frac {a}{b}\right )^{\frac {2}{3}} + 4 \sqrt [3]{- \frac {a}{b}} \log {\left (c x^{n} \right )} + 4 \log {\left (c x^{n} \right )}^{2} \right )}}{6 a n} + \frac {\sqrt {3} \sqrt [3]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {3}}{3} + \frac {2 \sqrt {3} \log {\left (c x^{n} \right )}}{3 \sqrt [3]{- \frac {a}{b}}} \right )}}{3 a n} & \text {otherwise} \end {cases} \]

[In]

integrate(1/(a*x+b*x*ln(c*x**n)**3),x)

[Out]

Piecewise((zoo*log(x)/log(c)**3, Eq(a, 0) & Eq(b, 0) & Eq(n, 0)), (-1/(2*b*n*log(c*x**n)**2), Eq(a, 0)), (log(
x)/a, Eq(b, 0)), (log(x)/(a + b*log(c)**3), Eq(n, 0)), (-(-a/b)**(1/3)*log(-(-a/b)**(1/3) + log(c*x**n))/(3*a*
n) + (-a/b)**(1/3)*log(4*(-a/b)**(2/3) + 4*(-a/b)**(1/3)*log(c*x**n) + 4*log(c*x**n)**2)/(6*a*n) + sqrt(3)*(-a
/b)**(1/3)*atan(sqrt(3)/3 + 2*sqrt(3)*log(c*x**n)/(3*(-a/b)**(1/3)))/(3*a*n), True))

Maxima [F]

\[ \int \frac {1}{a x+b x \log ^3\left (c x^n\right )} \, dx=\int { \frac {1}{b x \log \left (c x^{n}\right )^{3} + a x} \,d x } \]

[In]

integrate(1/(a*x+b*x*log(c*x^n)^3),x, algorithm="maxima")

[Out]

integrate(1/(b*x*log(c*x^n)^3 + a*x), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 239 vs. \(2 (109) = 218\).

Time = 0.33 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.66 \[ \int \frac {1}{a x+b x \log ^3\left (c x^n\right )} \, dx=\frac {1}{3} \, \sqrt {3} \left (\frac {1}{a^{2} b n^{3}}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} \pi b {\left (\mathrm {sgn}\left (c\right ) - 1\right )} - 2 \, b n \log \left (x\right ) - 2 \, b \log \left ({\left | c \right |}\right ) - 2 \, \left (a b^{2}\right )^{\frac {1}{3}}}{2 \, \sqrt {3} b n \log \left (x\right ) + \pi b {\left (\mathrm {sgn}\left (c\right ) - 1\right )} + 2 \, \sqrt {3} b \log \left ({\left | c \right |}\right ) - 2 \, \sqrt {3} \left (a b^{2}\right )^{\frac {1}{3}}}\right ) + \frac {1}{6} \, \left (\frac {1}{a^{2} b n^{3}}\right )^{\frac {1}{3}} \log \left (\frac {1}{4} \, {\left (\pi b n {\left (\mathrm {sgn}\left (x\right ) - 1\right )} + \pi b {\left (\mathrm {sgn}\left (c\right ) - 1\right )}\right )}^{2} + {\left (b n \log \left ({\left | x \right |}\right ) + b \log \left ({\left | c \right |}\right ) + \left (a b^{2}\right )^{\frac {1}{3}}\right )}^{2}\right ) - \frac {1}{6} \, \left (\frac {1}{a^{2} b n^{3}}\right )^{\frac {1}{3}} \log \left ({\left (\sqrt {3} \pi b {\left (\mathrm {sgn}\left (c\right ) - 1\right )} - 2 \, b n \log \left (x\right ) - 2 \, b \log \left ({\left | c \right |}\right ) - 2 \, \left (a b^{2}\right )^{\frac {1}{3}}\right )}^{2} + {\left (2 \, \sqrt {3} b n \log \left (x\right ) + \pi b {\left (\mathrm {sgn}\left (c\right ) - 1\right )} + 2 \, \sqrt {3} b \log \left ({\left | c \right |}\right ) - 2 \, \sqrt {3} \left (a b^{2}\right )^{\frac {1}{3}}\right )}^{2}\right ) \]

[In]

integrate(1/(a*x+b*x*log(c*x^n)^3),x, algorithm="giac")

[Out]

1/3*sqrt(3)*(1/(a^2*b*n^3))^(1/3)*arctan((sqrt(3)*pi*b*(sgn(c) - 1) - 2*b*n*log(x) - 2*b*log(abs(c)) - 2*(a*b^
2)^(1/3))/(2*sqrt(3)*b*n*log(x) + pi*b*(sgn(c) - 1) + 2*sqrt(3)*b*log(abs(c)) - 2*sqrt(3)*(a*b^2)^(1/3))) + 1/
6*(1/(a^2*b*n^3))^(1/3)*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^2)^(1/3))^2) - 1/6*(1/(a^2*b*n^3))^(1/3)*log((sqrt(3)*pi*b*(sgn(c) - 1) - 2*b*n*log(x) - 2*b*log(abs(c
)) - 2*(a*b^2)^(1/3))^2 + (2*sqrt(3)*b*n*log(x) + pi*b*(sgn(c) - 1) + 2*sqrt(3)*b*log(abs(c)) - 2*sqrt(3)*(a*b
^2)^(1/3))^2)

Mupad [B] (verification not implemented)

Time = 3.52 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.06 \[ \int \frac {1}{a x+b x \log ^3\left (c x^n\right )} \, dx=\frac {\ln \left (\frac {3\,a^{1/3}\,n}{b^{4/3}\,x^2}+\frac {3\,n\,\ln \left (c\,x^n\right )}{b\,x^2}\right )}{3\,a^{2/3}\,b^{1/3}\,n}+\frac {\ln \left (\frac {3\,n\,\ln \left (c\,x^n\right )}{b\,x^2}+\frac {3\,a^{1/3}\,n\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,b^{4/3}\,x^2}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^{2/3}\,b^{1/3}\,n}-\frac {\ln \left (\frac {3\,n\,\ln \left (c\,x^n\right )}{b\,x^2}-\frac {3\,a^{1/3}\,n\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,b^{4/3}\,x^2}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^{2/3}\,b^{1/3}\,n} \]

[In]

int(1/(a*x + b*x*log(c*x^n)^3),x)

[Out]

log((3*a^(1/3)*n)/(b^(4/3)*x^2) + (3*n*log(c*x^n))/(b*x^2))/(3*a^(2/3)*b^(1/3)*n) + (log((3*n*log(c*x^n))/(b*x
^2) + (3*a^(1/3)*n*(3^(1/2)*1i - 1))/(2*b^(4/3)*x^2))*(3^(1/2)*1i - 1))/(6*a^(2/3)*b^(1/3)*n) - (log((3*n*log(
c*x^n))/(b*x^2) - (3*a^(1/3)*n*(3^(1/2)*1i + 1))/(2*b^(4/3)*x^2))*(3^(1/2)*1i + 1))/(6*a^(2/3)*b^(1/3)*n)