\(\int \frac {1}{a x+b x \log ^4(c x^n)} \, dx\) [253]

Optimal result

Integrand size = 17, antiderivative size = 227 \[ \int \frac {1}{a x+b x \log ^4\left (c x^n\right )} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b} n}+\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b} n}-\frac {\log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt {b} \log ^2\left (c x^n\right )\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b} n}+\frac {\log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt {b} \log ^2\left (c x^n\right )\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b} n} \]

[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)

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {217, 1179, 642, 1176, 631, 210} \[ \int \frac {1}{a x+b x \log ^4\left (c x^n\right )} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b} n}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{a}}+1\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b} n}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt {a}+\sqrt {b} \log ^2\left (c x^n\right )\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b} n}+\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt {a}+\sqrt {b} \log ^2\left (c x^n\right )\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b} n} \]

[In]

Int[(a*x + b*x*Log[c*x^n]^4)^(-1),x]

[Out]

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

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 217

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 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 1176

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 1179

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*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = \frac {\text {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 {\text {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 {\text {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 {\text {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 {\text {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 {\text {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 {\text {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 {\text {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] (verified)

Time = 0.08 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.74 \[ \int \frac {1}{a x+b x \log ^4\left (c x^n\right )} \, dx=\frac {-2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{a}}\right )+2 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{a}}\right )-\log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt {b} \log ^2\left (c x^n\right )\right )+\log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt {b} \log ^2\left (c x^n\right )\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b} n} \]

[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)

Maple [C] (warning: unable to verify)

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

Time = 1.36 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.49

method	result	size
risch	\(\munderset {\textit {\_R} =\operatorname {RootOf}\left (256 a^{3} b \,n^{4} \textit {\_Z}^{4}+1\right )}{\sum }\textit {\_R} \ln \left (\ln \left (x^{n}\right )+4 a n \textit {\_R} -\frac {i \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+\ln \left (c \right )\right )\)	\(112\)
default	\(\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {\ln \left (c \,x^{n}\right )^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} \ln \left (c \,x^{n}\right ) \sqrt {2}+\sqrt {\frac {a}{b}}}{\ln \left (c \,x^{n}\right )^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} \ln \left (c \,x^{n}\right ) \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \ln \left (c \,x^{n}\right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \ln \left (c \,x^{n}\right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )\right )}{8 n a}\)	\(136\)

[In]

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

[Out]

sum(_R*ln(ln(x^n)+4*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(256*_Z^4*a^3*b*n^4+1))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.29 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.70 \[ \int \frac {1}{a x+b x \log ^4\left (c x^n\right )} \, dx=\frac {1}{4} \, \left (-\frac {1}{a^{3} b n^{4}}\right )^{\frac {1}{4}} \log \left (a n \left (-\frac {1}{a^{3} b n^{4}}\right )^{\frac {1}{4}} + n \log \left (x\right ) + \log \left (c\right )\right ) + \frac {1}{4} i \, \left (-\frac {1}{a^{3} b n^{4}}\right )^{\frac {1}{4}} \log \left (i \, a n \left (-\frac {1}{a^{3} b n^{4}}\right )^{\frac {1}{4}} + n \log \left (x\right ) + \log \left (c\right )\right ) - \frac {1}{4} i \, \left (-\frac {1}{a^{3} b n^{4}}\right )^{\frac {1}{4}} \log \left (-i \, a n \left (-\frac {1}{a^{3} b n^{4}}\right )^{\frac {1}{4}} + n \log \left (x\right ) + \log \left (c\right )\right ) - \frac {1}{4} \, \left (-\frac {1}{a^{3} b n^{4}}\right )^{\frac {1}{4}} \log \left (-a n \left (-\frac {1}{a^{3} b n^{4}}\right )^{\frac {1}{4}} + n \log \left (x\right ) + \log \left (c\right )\right ) \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 13.47 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.59 \[ \int \frac {1}{a x+b x \log ^4\left (c x^n\right )} \, dx=\begin {cases} \frac {\tilde {\infty } \log {\left (x \right )}}{\log {\left (c \right )}^{4}} & \text {for}\: a = 0 \wedge b = 0 \wedge n = 0 \\- \frac {1}{3 b n \log {\left (c x^{n} \right )}^{3}} & \text {for}\: a = 0 \\\frac {\log {\left (x \right )}}{a} & \text {for}\: b = 0 \\\frac {\log {\left (x \right )}}{a + b \log {\left (c \right )}^{4}} & \text {for}\: n = 0 \\- \frac {\sqrt [4]{- \frac {a}{b}} \log {\left (- \sqrt [4]{- \frac {a}{b}} + \log {\left (c x^{n} \right )} \right )}}{4 a n} + \frac {\sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt [4]{- \frac {a}{b}} + \log {\left (c x^{n} \right )} \right )}}{4 a n} + \frac {\sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\log {\left (c x^{n} \right )}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{2 a n} & \text {otherwise} \end {cases} \]

[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/(3*b*n*log(c*x**n)**3), Eq(a, 0)), (log(
x)/a, Eq(b, 0)), (log(x)/(a + b*log(c)**4), Eq(n, 0)), (-(-a/b)**(1/4)*log(-(-a/b)**(1/4) + log(c*x**n))/(4*a*
n) + (-a/b)**(1/4)*log((-a/b)**(1/4) + log(c*x**n))/(4*a*n) + (-a/b)**(1/4)*atan(log(c*x**n)/(-a/b)**(1/4))/(2
*a*n), True))

Maxima [F]

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

[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)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.75 \[ \int \frac {1}{a x+b x \log ^4\left (c x^n\right )} \, dx=-\frac {1}{2} \, \left (-\frac {1}{a^{3} b n^{4}}\right )^{\frac {1}{4}} \arctan \left (\frac {\pi b {\left (\mathrm {sgn}\left (c\right ) - 1\right )} + 2 \, \left (-a b^{3}\right )^{\frac {1}{4}}}{2 \, {\left (b n \log \left (x\right ) + b \log \left ({\left | c \right |}\right )\right )}}\right ) + \frac {1}{8} \, \left (-\frac {1}{a^{3} b n^{4}}\right )^{\frac {1}{4}} \log \left (\frac {1}{4} \, {\left (\pi b n {\left (\mathrm {sgn}\left (x\right ) - 1\right )} + \pi b {\left (\mathrm {sgn}\left (c\right ) - 1\right )}\right )}^{2} + {\left (b n \log \left ({\left | x \right |}\right ) + b \log \left ({\left | c \right |}\right ) + \left (-a b^{3}\right )^{\frac {1}{4}}\right )}^{2}\right ) - \frac {1}{8} \, \left (-\frac {1}{a^{3} b n^{4}}\right )^{\frac {1}{4}} \log \left (\frac {1}{4} \, {\left (\pi b n {\left (\mathrm {sgn}\left (x\right ) - 1\right )} + \pi b {\left (\mathrm {sgn}\left (c\right ) - 1\right )}\right )}^{2} + {\left (b n \log \left ({\left | x \right |}\right ) + b \log \left ({\left | c \right |}\right ) - \left (-a b^{3}\right )^{\frac {1}{4}}\right )}^{2}\right ) \]

[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)

Mupad [B] (verification not implemented)

Time = 3.41 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.42 \[ \int \frac {1}{a x+b x \log ^4\left (c x^n\right )} \, dx=-\frac {\ln \left ({\left (-a\right )}^{1/4}+b^{1/4}\,\ln \left (c\,x^n\right )\right )-\ln \left ({\left (-a\right )}^{1/4}-b^{1/4}\,\ln \left (c\,x^n\right )\right )+\ln \left ({\left (-a\right )}^{1/4}-b^{1/4}\,\ln \left (c\,x^n\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}-\ln \left ({\left (-a\right )}^{1/4}+b^{1/4}\,\ln \left (c\,x^n\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{4\,{\left (-a\right )}^{3/4}\,b^{1/4}\,n} \]

[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)