\(\int \frac {1}{a x+\frac {b x}{\log ^4(c x^n)}} \, dx\) [257]

Optimal result

Integrand size = 17, antiderivative size = 233 \[ \int \frac {1}{a x+\frac {b x}{\log ^4\left (c x^n\right )}} \, dx=\frac {\sqrt [4]{b} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a} \log \left (c x^n\right )}{\sqrt [4]{b}}\right )}{2 \sqrt {2} a^{5/4} n}-\frac {\sqrt [4]{b} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{a} \log \left (c x^n\right )}{\sqrt [4]{b}}\right )}{2 \sqrt {2} a^{5/4} n}+\frac {\log (x)}{a}+\frac {\sqrt [4]{b} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt {a} \log ^2\left (c x^n\right )\right )}{4 \sqrt {2} a^{5/4} n}-\frac {\sqrt [4]{b} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt {a} \log ^2\left (c x^n\right )\right )}{4 \sqrt {2} a^{5/4} n} \]

[Out]

ln(x)/a-1/4*b^(1/4)*arctan(-1+a^(1/4)*ln(c*x^n)*2^(1/2)/b^(1/4))/a^(5/4)/n*2^(1/2)-1/4*b^(1/4)*arctan(1+a^(1/4
)*ln(c*x^n)*2^(1/2)/b^(1/4))/a^(5/4)/n*2^(1/2)+1/8*b^(1/4)*ln(-a^(1/4)*b^(1/4)*ln(c*x^n)*2^(1/2)+ln(c*x^n)^2*a
^(1/2)+b^(1/2))/a^(5/4)/n*2^(1/2)-1/8*b^(1/4)*ln(a^(1/4)*b^(1/4)*ln(c*x^n)*2^(1/2)+ln(c*x^n)^2*a^(1/2)+b^(1/2)
)/a^(5/4)/n*2^(1/2)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {327, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {1}{a x+\frac {b x}{\log ^4\left (c x^n\right )}} \, dx=\frac {\sqrt [4]{b} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a} \log \left (c x^n\right )}{\sqrt [4]{b}}\right )}{2 \sqrt {2} a^{5/4} n}-\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \log \left (c x^n\right )}{\sqrt [4]{b}}+1\right )}{2 \sqrt {2} a^{5/4} n}+\frac {\sqrt [4]{b} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt {a} \log ^2\left (c x^n\right )+\sqrt {b}\right )}{4 \sqrt {2} a^{5/4} n}-\frac {\sqrt [4]{b} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt {a} \log ^2\left (c x^n\right )+\sqrt {b}\right )}{4 \sqrt {2} a^{5/4} n}+\frac {\log (x)}{a} \]

[In]

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

[Out]

(b^(1/4)*ArcTan[1 - (Sqrt[2]*a^(1/4)*Log[c*x^n])/b^(1/4)])/(2*Sqrt[2]*a^(5/4)*n) - (b^(1/4)*ArcTan[1 + (Sqrt[2
]*a^(1/4)*Log[c*x^n])/b^(1/4)])/(2*Sqrt[2]*a^(5/4)*n) + Log[x]/a + (b^(1/4)*Log[Sqrt[b] - Sqrt[2]*a^(1/4)*b^(1
/4)*Log[c*x^n] + Sqrt[a]*Log[c*x^n]^2])/(4*Sqrt[2]*a^(5/4)*n) - (b^(1/4)*Log[Sqrt[b] + Sqrt[2]*a^(1/4)*b^(1/4)
*Log[c*x^n] + Sqrt[a]*Log[c*x^n]^2])/(4*Sqrt[2]*a^(5/4)*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 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

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 {x^4}{b+a x^4} \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = \frac {\log (x)}{a}-\frac {b \text {Subst}\left (\int \frac {1}{b+a x^4} \, dx,x,\log \left (c x^n\right )\right )}{a n} \\ & = \frac {\log (x)}{a}-\frac {\sqrt {b} \text {Subst}\left (\int \frac {\sqrt {b}-\sqrt {a} x^2}{b+a x^4} \, dx,x,\log \left (c x^n\right )\right )}{2 a n}-\frac {\sqrt {b} \text {Subst}\left (\int \frac {\sqrt {b}+\sqrt {a} x^2}{b+a x^4} \, dx,x,\log \left (c x^n\right )\right )}{2 a n} \\ & = \frac {\log (x)}{a}+\frac {\sqrt [4]{b} \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{b}}{\sqrt [4]{a}}+2 x}{-\frac {\sqrt {b}}{\sqrt {a}}-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}-x^2} \, dx,x,\log \left (c x^n\right )\right )}{4 \sqrt {2} a^{5/4} n}+\frac {\sqrt [4]{b} \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{b}}{\sqrt [4]{a}}-2 x}{-\frac {\sqrt {b}}{\sqrt {a}}+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}-x^2} \, dx,x,\log \left (c x^n\right )\right )}{4 \sqrt {2} a^{5/4} n}-\frac {\sqrt {b} \text {Subst}\left (\int \frac {1}{\frac {\sqrt {b}}{\sqrt {a}}-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+x^2} \, dx,x,\log \left (c x^n\right )\right )}{4 a^{3/2} n}-\frac {\sqrt {b} \text {Subst}\left (\int \frac {1}{\frac {\sqrt {b}}{\sqrt {a}}+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+x^2} \, dx,x,\log \left (c x^n\right )\right )}{4 a^{3/2} n} \\ & = \frac {\log (x)}{a}+\frac {\sqrt [4]{b} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt {a} \log ^2\left (c x^n\right )\right )}{4 \sqrt {2} a^{5/4} n}-\frac {\sqrt [4]{b} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt {a} \log ^2\left (c x^n\right )\right )}{4 \sqrt {2} a^{5/4} n}-\frac {\sqrt [4]{b} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{a} \log \left (c x^n\right )}{\sqrt [4]{b}}\right )}{2 \sqrt {2} a^{5/4} n}+\frac {\sqrt [4]{b} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{a} \log \left (c x^n\right )}{\sqrt [4]{b}}\right )}{2 \sqrt {2} a^{5/4} n} \\ & = \frac {\sqrt [4]{b} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a} \log \left (c x^n\right )}{\sqrt [4]{b}}\right )}{2 \sqrt {2} a^{5/4} n}-\frac {\sqrt [4]{b} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{a} \log \left (c x^n\right )}{\sqrt [4]{b}}\right )}{2 \sqrt {2} a^{5/4} n}+\frac {\log (x)}{a}+\frac {\sqrt [4]{b} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt {a} \log ^2\left (c x^n\right )\right )}{4 \sqrt {2} a^{5/4} n}-\frac {\sqrt [4]{b} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt {a} \log ^2\left (c x^n\right )\right )}{4 \sqrt {2} a^{5/4} n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.89 \[ \int \frac {1}{a x+\frac {b x}{\log ^4\left (c x^n\right )}} \, dx=\frac {2 \sqrt {2} \sqrt [4]{b} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a} \log \left (c x^n\right )}{\sqrt [4]{b}}\right )-2 \sqrt {2} \sqrt [4]{b} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{a} \log \left (c x^n\right )}{\sqrt [4]{b}}\right )+8 \sqrt [4]{a} n \log (x)+\sqrt {2} \sqrt [4]{b} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt {a} \log ^2\left (c x^n\right )\right )-\sqrt {2} \sqrt [4]{b} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt {a} \log ^2\left (c x^n\right )\right )}{8 a^{5/4} n} \]

[In]

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

[Out]

(2*Sqrt[2]*b^(1/4)*ArcTan[1 - (Sqrt[2]*a^(1/4)*Log[c*x^n])/b^(1/4)] - 2*Sqrt[2]*b^(1/4)*ArcTan[1 + (Sqrt[2]*a^
(1/4)*Log[c*x^n])/b^(1/4)] + 8*a^(1/4)*n*Log[x] + Sqrt[2]*b^(1/4)*Log[Sqrt[b] - Sqrt[2]*a^(1/4)*b^(1/4)*Log[c*
x^n] + Sqrt[a]*Log[c*x^n]^2] - Sqrt[2]*b^(1/4)*Log[Sqrt[b] + Sqrt[2]*a^(1/4)*b^(1/4)*Log[c*x^n] + Sqrt[a]*Log[
c*x^n]^2])/(8*a^(5/4)*n)

Maple [C] (warning: unable to verify)

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

Time = 1.42 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.51

method	result	size
risch	\(\frac {\ln \left (x \right )}{a}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (256 n^{4} a^{5} \textit {\_Z}^{4}+b \right )}{\sum }\textit {\_R} \ln \left (\ln \left (x^{n}\right )-4 a n \textit {\_R} -\frac {i \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+\ln \left (c \right )\right )\right )\)	\(118\)
default	\(\frac {\frac {\ln \left (c \,x^{n}\right )}{a}-\frac {\left (\frac {b}{a}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {\ln \left (c \,x^{n}\right )^{2}+\left (\frac {b}{a}\right )^{\frac {1}{4}} \ln \left (c \,x^{n}\right ) \sqrt {2}+\sqrt {\frac {b}{a}}}{\ln \left (c \,x^{n}\right )^{2}-\left (\frac {b}{a}\right )^{\frac {1}{4}} \ln \left (c \,x^{n}\right ) \sqrt {2}+\sqrt {\frac {b}{a}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \ln \left (c \,x^{n}\right )}{\left (\frac {b}{a}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \ln \left (c \,x^{n}\right )}{\left (\frac {b}{a}\right )^{\frac {1}{4}}}+1\right )\right )}{8 a}}{n}\)	\(148\)

[In]

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

[Out]

1/a*ln(x)+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^5*n^4+b))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.31 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.67 \[ \int \frac {1}{a x+\frac {b x}{\log ^4\left (c x^n\right )}} \, dx=-\frac {a \left (-\frac {b}{a^{5} n^{4}}\right )^{\frac {1}{4}} \log \left (a n \left (-\frac {b}{a^{5} n^{4}}\right )^{\frac {1}{4}} + n \log \left (x\right ) + \log \left (c\right )\right ) + i \, a \left (-\frac {b}{a^{5} n^{4}}\right )^{\frac {1}{4}} \log \left (i \, a n \left (-\frac {b}{a^{5} n^{4}}\right )^{\frac {1}{4}} + n \log \left (x\right ) + \log \left (c\right )\right ) - i \, a \left (-\frac {b}{a^{5} n^{4}}\right )^{\frac {1}{4}} \log \left (-i \, a n \left (-\frac {b}{a^{5} n^{4}}\right )^{\frac {1}{4}} + n \log \left (x\right ) + \log \left (c\right )\right ) - a \left (-\frac {b}{a^{5} n^{4}}\right )^{\frac {1}{4}} \log \left (-a n \left (-\frac {b}{a^{5} n^{4}}\right )^{\frac {1}{4}} + n \log \left (x\right ) + \log \left (c\right )\right ) - 4 \, \log \left (x\right )}{4 \, a} \]

[In]

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

[Out]

-1/4*(a*(-b/(a^5*n^4))^(1/4)*log(a*n*(-b/(a^5*n^4))^(1/4) + n*log(x) + log(c)) + I*a*(-b/(a^5*n^4))^(1/4)*log(
I*a*n*(-b/(a^5*n^4))^(1/4) + n*log(x) + log(c)) - I*a*(-b/(a^5*n^4))^(1/4)*log(-I*a*n*(-b/(a^5*n^4))^(1/4) + n
*log(x) + log(c)) - a*(-b/(a^5*n^4))^(1/4)*log(-a*n*(-b/(a^5*n^4))^(1/4) + n*log(x) + log(c)) - 4*log(x))/a

Sympy [A] (verification not implemented)

Time = 15.02 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.98 \[ \int \frac {1}{a x+\frac {b x}{\log ^4\left (c x^n\right )}} \, dx=\begin {cases} \tilde {\infty } \log {\left (c \right )}^{4} \log {\left (x \right )} & \text {for}\: a = 0 \wedge b = 0 \wedge n = 0 \\\frac {\begin {cases} - \frac {\log {\left (\frac {x^{- n}}{c} \right )}^{5}}{5 n} + \frac {\log {\left (c x^{n} \right )}^{5}}{5 n} & \text {for}\: \frac {1}{\left |{c x^{n}}\right |} < 1 \wedge \left |{c x^{n}}\right | < 1 \\\frac {\log {\left (c x^{n} \right )}^{5}}{5 n} & \text {for}\: \left |{c x^{n}}\right | < 1 \\- \frac {\log {\left (\frac {x^{- n}}{c} \right )}^{5}}{5 n} & \text {for}\: \frac {1}{\left |{c x^{n}}\right |} < 1 \\- \frac {24 {G_{6, 6}^{6, 0}\left (\begin {matrix} & 1, 1, 1, 1, 1, 1 \\0, 0, 0, 0, 0, 0 & \end {matrix} \middle | {c x^{n}} \right )}}{n} + \frac {24 {G_{6, 6}^{0, 6}\left (\begin {matrix} 1, 1, 1, 1, 1, 1 & \\ & 0, 0, 0, 0, 0, 0 \end {matrix} \middle | {c x^{n}} \right )}}{n} & \text {otherwise} \end {cases}}{b} & \text {for}\: a = 0 \\\frac {\log {\left (c \right )}^{4} \log {\left (x \right )}}{a \log {\left (c \right )}^{4} + b} & \text {for}\: n = 0 \\\frac {\log {\left (x \right )}}{a} & \text {for}\: b = 0 \\\frac {\sqrt [4]{- \frac {b}{a}} \log {\left (- \sqrt [4]{- \frac {b}{a}} + \log {\left (c x^{n} \right )} \right )}}{4 a n} - \frac {\sqrt [4]{- \frac {b}{a}} \log {\left (\sqrt [4]{- \frac {b}{a}} + \log {\left (c x^{n} \right )} \right )}}{4 a n} - \frac {\sqrt [4]{- \frac {b}{a}} \operatorname {atan}{\left (\frac {\log {\left (c x^{n} \right )}}{\sqrt [4]{- \frac {b}{a}}} \right )}}{2 a n} + \frac {\log {\left (c x^{n} \right )}}{a n} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((zoo*log(c)**4*log(x), Eq(a, 0) & Eq(b, 0) & Eq(n, 0)), (Piecewise((-log(1/(c*x**n))**5/(5*n) + log(
c*x**n)**5/(5*n), (Abs(c*x**n) < 1) & (1/Abs(c*x**n) < 1)), (log(c*x**n)**5/(5*n), Abs(c*x**n) < 1), (-log(1/(
c*x**n))**5/(5*n), 1/Abs(c*x**n) < 1), (-24*meijerg(((), (1, 1, 1, 1, 1, 1)), ((0, 0, 0, 0, 0, 0), ()), c*x**n
)/n + 24*meijerg(((1, 1, 1, 1, 1, 1), ()), ((), (0, 0, 0, 0, 0, 0)), c*x**n)/n, True))/b, Eq(a, 0)), (log(c)**
4*log(x)/(a*log(c)**4 + b), Eq(n, 0)), (log(x)/a, Eq(b, 0)), ((-b/a)**(1/4)*log(-(-b/a)**(1/4) + log(c*x**n))/
(4*a*n) - (-b/a)**(1/4)*log((-b/a)**(1/4) + log(c*x**n))/(4*a*n) - (-b/a)**(1/4)*atan(log(c*x**n)/(-b/a)**(1/4
))/(2*a*n) + log(c*x**n)/(a*n), True))

Maxima [F]

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

[In]

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

[Out]

-b*integrate(1/(4*a^2*x*log(c)^3*log(x^n) + 6*a^2*x*log(c)^2*log(x^n)^2 + 4*a^2*x*log(c)*log(x^n)^3 + a^2*x*lo
g(x^n)^4 + (a^2*log(c)^4 + a*b)*x), x) + log(x)/a

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.76 \[ \int \frac {1}{a x+\frac {b x}{\log ^4\left (c x^n\right )}} \, dx=\frac {\log \left (x\right )}{a} - \frac {4 \, \left (-\frac {b n^{12}}{a}\right )^{\frac {1}{4}} \arctan \left (\frac {\pi a {\left (\mathrm {sgn}\left (c\right ) - 1\right )} - 2 \, \left (-a^{3} b\right )^{\frac {1}{4}}}{2 \, {\left (a n \log \left (x\right ) + a \log \left ({\left | c \right |}\right )\right )}}\right ) + \left (-\frac {b n^{12}}{a}\right )^{\frac {1}{4}} \log \left (\frac {1}{4} \, {\left (\pi a n {\left (\mathrm {sgn}\left (x\right ) - 1\right )} + \pi a {\left (\mathrm {sgn}\left (c\right ) - 1\right )}\right )}^{2} + {\left (a n \log \left ({\left | x \right |}\right ) + a \log \left ({\left | c \right |}\right ) + \left (-a^{3} b\right )^{\frac {1}{4}}\right )}^{2}\right ) - \left (-\frac {b n^{12}}{a}\right )^{\frac {1}{4}} \log \left (\frac {1}{4} \, {\left (\pi a n {\left (\mathrm {sgn}\left (x\right ) - 1\right )} + \pi a {\left (\mathrm {sgn}\left (c\right ) - 1\right )}\right )}^{2} + {\left (a n \log \left ({\left | x \right |}\right ) + a \log \left ({\left | c \right |}\right ) - \left (-a^{3} b\right )^{\frac {1}{4}}\right )}^{2}\right )}{8 \, a n^{4}} \]

[In]

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

[Out]

log(x)/a - 1/8*(4*(-b*n^12/a)^(1/4)*arctan(1/2*(pi*a*(sgn(c) - 1) - 2*(-a^3*b)^(1/4))/(a*n*log(x) + a*log(abs(
c)))) + (-b*n^12/a)^(1/4)*log(1/4*(pi*a*n*(sgn(x) - 1) + pi*a*(sgn(c) - 1))^2 + (a*n*log(abs(x)) + a*log(abs(c
)) + (-a^3*b)^(1/4))^2) - (-b*n^12/a)^(1/4)*log(1/4*(pi*a*n*(sgn(x) - 1) + pi*a*(sgn(c) - 1))^2 + (a*n*log(abs
(x)) + a*log(abs(c)) - (-a^3*b)^(1/4))^2))/(a*n^4)

Mupad [B] (verification not implemented)

Time = 3.40 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.76 \[ \int \frac {1}{a x+\frac {b x}{\log ^4\left (c x^n\right )}} \, dx=\frac {\ln \left (x\right )}{a}+\frac {{\left (-b\right )}^{1/4}\,\left (\ln \left (-\frac {{\left (-b\right )}^{5/2}}{a^{11/2}\,x^3}-\frac {{\left (-b\right )}^{9/4}\,\ln \left (c\,x^n\right )\,1{}\mathrm {i}}{a^{21/4}\,x^3}\right )\,1{}\mathrm {i}-\ln \left (-\frac {{\left (-b\right )}^{5/2}}{a^{11/2}\,x^3}+\frac {{\left (-b\right )}^{9/4}\,\ln \left (c\,x^n\right )\,1{}\mathrm {i}}{a^{21/4}\,x^3}\right )\,1{}\mathrm {i}\right )}{4\,a^{5/4}\,n}-\frac {{\left (-b\right )}^{1/4}\,\ln \left (\frac {{\left (-b\right )}^{5/2}+a^{1/4}\,{\left (-b\right )}^{9/4}\,\ln \left (c\,x^n\right )}{x^3}\right )}{4\,a^{5/4}\,n}+\frac {{\left (-b\right )}^{1/4}\,\ln \left (\frac {{\left (-b\right )}^{5/2}-a^{1/4}\,{\left (-b\right )}^{9/4}\,\ln \left (c\,x^n\right )}{x^3}\right )}{4\,a^{5/4}\,n} \]

[In]

int(1/(a*x + (b*x)/log(c*x^n)^4),x)

[Out]

log(x)/a + ((-b)^(1/4)*(log(- (-b)^(5/2)/(a^(11/2)*x^3) - ((-b)^(9/4)*log(c*x^n)*1i)/(a^(21/4)*x^3))*1i - log(
((-b)^(9/4)*log(c*x^n)*1i)/(a^(21/4)*x^3) - (-b)^(5/2)/(a^(11/2)*x^3))*1i))/(4*a^(5/4)*n) - ((-b)^(1/4)*log(((
-b)^(5/2) + a^(1/4)*(-b)^(9/4)*log(c*x^n))/x^3))/(4*a^(5/4)*n) + ((-b)^(1/4)*log(((-b)^(5/2) - a^(1/4)*(-b)^(9
/4)*log(c*x^n))/x^3))/(4*a^(5/4)*n)