Integrand size = 17, antiderivative size = 144 \[ \int \frac {1}{a x+b x \log ^3\left (c x^n\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \log \left (c x^n\right )}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} \sqrt [3]{b} n}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \log \left (c x^n\right )\right )}{3 a^{2/3} \sqrt [3]{b} n}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \log \left (c x^n\right )+b^{2/3} \log ^2\left (c x^n\right )\right )}{6 a^{2/3} \sqrt [3]{b} n} \] Output:
-1/3*arctan(1/3*(a^(1/3)-2*b^(1/3)*ln(c*x^n))*3^(1/2)/a^(1/3))*3^(1/2)/a^( 2/3)/b^(1/3)/n+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
Time = 0.12 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.78 \[ \int \frac {1}{a x+b x \log ^3\left (c x^n\right )} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \log \left (c x^n\right )}{\sqrt [3]{a}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \log \left (c x^n\right )\right )+\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \log \left (c x^n\right )+b^{2/3} \log ^2\left (c x^n\right )\right )}{6 a^{2/3} \sqrt [3]{b} n} \] Input:
Integrate[(a*x + b*x*Log[c*x^n]^3)^(-1),x]
Output:
-1/6*(2*Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*Log[c*x^n])/a^(1/3))/Sqrt[3]] - 2*L og[a^(1/3) + b^(1/3)*Log[c*x^n]] + Log[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)
Time = 0.34 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.94, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {3039, 750, 16, 1142, 25, 27, 1082, 217, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{a x+b x \log ^3\left (c x^n\right )} \, dx\) |
| \(\Big \downarrow \) 3039 |
| \(\displaystyle \frac {\int \frac {1}{b \log ^3\left (c x^n\right )+a}d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} \log \left (c x^n\right )}{b^{2/3} \log ^2\left (c x^n\right )-\sqrt [3]{a} \sqrt [3]{b} \log \left (c x^n\right )+a^{2/3}}d\log \left (c x^n\right )}{3 a^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{b} \log \left (c x^n\right )+\sqrt [3]{a}}d\log \left (c x^n\right )}{3 a^{2/3}}}{n}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} \log \left (c x^n\right )}{b^{2/3} \log ^2\left (c x^n\right )-\sqrt [3]{a} \sqrt [3]{b} \log \left (c x^n\right )+a^{2/3}}d\log \left (c x^n\right )}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \log \left (c x^n\right )\right )}{3 a^{2/3} \sqrt [3]{b}}}{n}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} \log ^2\left (c x^n\right )-\sqrt [3]{a} \sqrt [3]{b} \log \left (c x^n\right )+a^{2/3}}d\log \left (c x^n\right )-\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} \log \left (c x^n\right )\right )}{b^{2/3} \log ^2\left (c x^n\right )-\sqrt [3]{a} \sqrt [3]{b} \log \left (c x^n\right )+a^{2/3}}d\log \left (c x^n\right )}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \log \left (c x^n\right )\right )}{3 a^{2/3} \sqrt [3]{b}}}{n}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} \log ^2\left (c x^n\right )-\sqrt [3]{a} \sqrt [3]{b} \log \left (c x^n\right )+a^{2/3}}d\log \left (c x^n\right )+\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} \log \left (c x^n\right )\right )}{b^{2/3} \log ^2\left (c x^n\right )-\sqrt [3]{a} \sqrt [3]{b} \log \left (c x^n\right )+a^{2/3}}d\log \left (c x^n\right )}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \log \left (c x^n\right )\right )}{3 a^{2/3} \sqrt [3]{b}}}{n}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} \log ^2\left (c x^n\right )-\sqrt [3]{a} \sqrt [3]{b} \log \left (c x^n\right )+a^{2/3}}d\log \left (c x^n\right )+\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} \log \left (c x^n\right )}{b^{2/3} \log ^2\left (c x^n\right )-\sqrt [3]{a} \sqrt [3]{b} \log \left (c x^n\right )+a^{2/3}}d\log \left (c x^n\right )}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \log \left (c x^n\right )\right )}{3 a^{2/3} \sqrt [3]{b}}}{n}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} \log \left (c x^n\right )}{b^{2/3} \log ^2\left (c x^n\right )-\sqrt [3]{a} \sqrt [3]{b} \log \left (c x^n\right )+a^{2/3}}d\log \left (c x^n\right )+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} \log \left (c x^n\right )}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} \log \left (c x^n\right )}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \log \left (c x^n\right )\right )}{3 a^{2/3} \sqrt [3]{b}}}{n}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} \log \left (c x^n\right )}{b^{2/3} \log ^2\left (c x^n\right )-\sqrt [3]{a} \sqrt [3]{b} \log \left (c x^n\right )+a^{2/3}}d\log \left (c x^n\right )-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \log \left (c x^n\right )}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \log \left (c x^n\right )\right )}{3 a^{2/3} \sqrt [3]{b}}}{n}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \log \left (c x^n\right )+b^{2/3} \log ^2\left (c x^n\right )\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \log \left (c x^n\right )}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \log \left (c x^n\right )\right )}{3 a^{2/3} \sqrt [3]{b}}}{n}\) |
Input:
Int[(a*x + b*x*Log[c*x^n]^3)^(-1),x]
Output:
(Log[a^(1/3) + b^(1/3)*Log[c*x^n]]/(3*a^(2/3)*b^(1/3)) + (-((Sqrt[3]*ArcTa n[(1 - (2*b^(1/3)*Log[c*x^n])/a^(1/3))/Sqrt[3]])/b^(1/3)) - Log[a^(2/3) - a^(1/3)*b^(1/3)*Log[c*x^n] + b^(2/3)*Log[c*x^n]^2]/(2*b^(1/3)))/(3*a^(2/3) ))/n
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2) Int[1/ (Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[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]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[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])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[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]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[u_, x_Symbol] :> With[{lst = FunctionOfLog[Cancel[x*u], x]}, Simp[1/lst [[3]] Subst[Int[lst[[1]], x], x, Log[lst[[2]]]], x] /; !FalseQ[lst]] /; NonsumQ[u]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.47 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.78
| method | result | size |
| risch | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (27 a^{2} b \,n^{3} \textit {\_Z}^{3}-1\right )}{\sum }\textit {\_R} \ln \left (\ln \left (x^{n}\right )+3 a n \textit {\_R} +\frac {i \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right )}{2}-\frac {i \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+\frac {i \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right )\) | \(112\) |
| default | \(\frac {\frac {\ln \left (\ln \left (c \,x^{n}\right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\ln \left (c \,x^{n}\right )^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} \ln \left (c \,x^{n}\right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \ln \left (c \,x^{n}\right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}}{n}\) | \(115\) |
Input:
int(1/(a*x+b*x*ln(c*x^n)^3),x,method=_RETURNVERBOSE)
Output:
sum(_R*ln(ln(x^n)+3*a*n*_R+1/2*I*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-1/2*I*Pi*c sgn(I*c*x^n)*csgn(I*c)*csgn(I*x^n)-1/2*I*Pi*csgn(I*c*x^n)^3+1/2*I*Pi*csgn( I*c*x^n)^2*csgn(I*c)+ln(c)),_R=RootOf(27*_Z^3*a^2*b*n^3-1))
Time = 0.10 (sec) , antiderivative size = 480, normalized size of antiderivative = 3.33 \[ \int \frac {1}{a x+b x \log ^3\left (c x^n\right )} \, dx =\text {Too large to display} \] Input:
integrate(1/(a*x+b*x*log(c*x^n)^3),x, algorithm="fricas")
Output:
[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*lo g(c)*log(x)^2 + 3*b*n*log(c)^2*log(x) + b*log(c)^3 + a)) - (a^2*b)^(2/3)*l og(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)]
Time = 27.91 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.22 \[ \int \frac {1}{a x+b x \log ^3\left (c x^n\right )} \, dx=\begin {cases} \frac {\tilde {\infty } \log {\left (x \right )}}{\log {\left (c \right )}^{3}} & \text {for}\: a = 0 \wedge b = 0 \wedge n = 0 \\- \frac {1}{2 b n \log {\left (c x^{n} \right )}^{2}} & \text {for}\: a = 0 \\\frac {\log {\left (x \right )}}{a} & \text {for}\: b = 0 \\\frac {\log {\left (x \right )}}{a + b \log {\left (c \right )}^{3}} & \text {for}\: n = 0 \\- \frac {\sqrt [3]{- \frac {a}{b}} \log {\left (- \sqrt [3]{- \frac {a}{b}} + \log {\left (c x^{n} \right )} \right )}}{3 a n} + \frac {\sqrt [3]{- \frac {a}{b}} \log {\left (4 \left (- \frac {a}{b}\right )^{\frac {2}{3}} + 4 \sqrt [3]{- \frac {a}{b}} \log {\left (c x^{n} \right )} + 4 \log {\left (c x^{n} \right )}^{2} \right )}}{6 a n} + \frac {\sqrt {3} \sqrt [3]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {3}}{3} + \frac {2 \sqrt {3} \log {\left (c x^{n} \right )}}{3 \sqrt [3]{- \frac {a}{b}}} \right )}}{3 a n} & \text {otherwise} \end {cases} \] Input:
integrate(1/(a*x+b*x*ln(c*x**n)**3),x)
Output:
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)*lo g(c*x**n)/(3*(-a/b)**(1/3)))/(3*a*n), True))
\[ \int \frac {1}{a x+b x \log ^3\left (c x^n\right )} \, dx=\int { \frac {1}{b x \log \left (c x^{n}\right )^{3} + a x} \,d x } \] Input:
integrate(1/(a*x+b*x*log(c*x^n)^3),x, algorithm="maxima")
Output:
integrate(1/(b*x*log(c*x^n)^3 + a*x), x)
Leaf count of result is larger than twice the leaf count of optimal. 239 vs. \(2 (109) = 218\).
Time = 0.12 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.66 \[ \int \frac {1}{a x+b x \log ^3\left (c x^n\right )} \, dx=\frac {1}{3} \, \sqrt {3} \left (\frac {1}{a^{2} b n^{3}}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} \pi b {\left (\mathrm {sgn}\left (c\right ) - 1\right )} - 2 \, b n \log \left (x\right ) - 2 \, b \log \left ({\left | c \right |}\right ) - 2 \, \left (a b^{2}\right )^{\frac {1}{3}}}{2 \, \sqrt {3} b n \log \left (x\right ) + \pi b {\left (\mathrm {sgn}\left (c\right ) - 1\right )} + 2 \, \sqrt {3} b \log \left ({\left | c \right |}\right ) - 2 \, \sqrt {3} \left (a b^{2}\right )^{\frac {1}{3}}}\right ) + \frac {1}{6} \, \left (\frac {1}{a^{2} b n^{3}}\right )^{\frac {1}{3}} \log \left (\frac {1}{4} \, {\left (\pi b n {\left (\mathrm {sgn}\left (x\right ) - 1\right )} + \pi b {\left (\mathrm {sgn}\left (c\right ) - 1\right )}\right )}^{2} + {\left (b n \log \left ({\left | x \right |}\right ) + b \log \left ({\left | c \right |}\right ) + \left (a b^{2}\right )^{\frac {1}{3}}\right )}^{2}\right ) - \frac {1}{6} \, \left (\frac {1}{a^{2} b n^{3}}\right )^{\frac {1}{3}} \log \left ({\left (\sqrt {3} \pi b {\left (\mathrm {sgn}\left (c\right ) - 1\right )} - 2 \, b n \log \left (x\right ) - 2 \, b \log \left ({\left | c \right |}\right ) - 2 \, \left (a b^{2}\right )^{\frac {1}{3}}\right )}^{2} + {\left (2 \, \sqrt {3} b n \log \left (x\right ) + \pi b {\left (\mathrm {sgn}\left (c\right ) - 1\right )} + 2 \, \sqrt {3} b \log \left ({\left | c \right |}\right ) - 2 \, \sqrt {3} \left (a b^{2}\right )^{\frac {1}{3}}\right )}^{2}\right ) \] Input:
integrate(1/(a*x+b*x*log(c*x^n)^3),x, algorithm="giac")
Output:
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)
Time = 28.24 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.06 \[ \int \frac {1}{a x+b x \log ^3\left (c x^n\right )} \, dx=\frac {\ln \left (\frac {3\,a^{1/3}\,n}{b^{4/3}\,x^2}+\frac {3\,n\,\ln \left (c\,x^n\right )}{b\,x^2}\right )}{3\,a^{2/3}\,b^{1/3}\,n}+\frac {\ln \left (\frac {3\,n\,\ln \left (c\,x^n\right )}{b\,x^2}+\frac {3\,a^{1/3}\,n\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,b^{4/3}\,x^2}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^{2/3}\,b^{1/3}\,n}-\frac {\ln \left (\frac {3\,n\,\ln \left (c\,x^n\right )}{b\,x^2}-\frac {3\,a^{1/3}\,n\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,b^{4/3}\,x^2}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^{2/3}\,b^{1/3}\,n} \] Input:
int(1/(a*x + b*x*log(c*x^n)^3),x)
Output:
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)*1 i + 1))/(6*a^(2/3)*b^(1/3)*n)
Time = 0.15 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.62 \[ \int \frac {1}{a x+b x \log ^3\left (c x^n\right )} \, dx=\frac {-2 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} \mathrm {log}\left (x^{n} c \right )}{a^{\frac {1}{3}} \sqrt {3}}\right )-\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} \mathrm {log}\left (x^{n} c \right )+b^{\frac {2}{3}} \mathrm {log}\left (x^{n} c \right )^{2}\right )+2 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} \mathrm {log}\left (x^{n} c \right )\right )}{6 a^{\frac {2}{3}} b^{\frac {1}{3}} n} \] Input:
int(1/(a*x+b*x*log(c*x^n)^3),x)
Output:
(a**(1/3)*( - 2*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*log(x**n*c))/(a**(1/3) *sqrt(3))) - log(a**(2/3) - b**(1/3)*a**(1/3)*log(x**n*c) + b**(2/3)*log(x **n*c)**2) + 2*log(a**(1/3) + b**(1/3)*log(x**n*c))))/(6*b**(1/3)*a*n)