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

Optimal. Leaf size=233 \[ \frac{\sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt{a} \log ^2\left (c x^n\right )+\sqrt{b}\right )}{4 \sqrt{2} a^{5/4} n}-\frac{\sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt{a} \log ^2\left (c x^n\right )+\sqrt{b}\right )}{4 \sqrt{2} a^{5/4} n}+\frac{\sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a} \log \left (c x^n\right )}{\sqrt [4]{b}}\right )}{2 \sqrt{2} a^{5/4} n}-\frac{\sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{a} \log \left (c x^n\right )}{\sqrt [4]{b}}+1\right )}{2 \sqrt{2} a^{5/4} n}+\frac{\log (x)}{a} \]

[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)

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Rubi [A]  time = 0.182802, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412, Rules used = {321, 211, 1165, 628, 1162, 617, 204} \[ \frac{\sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt{a} \log ^2\left (c x^n\right )+\sqrt{b}\right )}{4 \sqrt{2} a^{5/4} n}-\frac{\sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt{a} \log ^2\left (c x^n\right )+\sqrt{b}\right )}{4 \sqrt{2} a^{5/4} n}+\frac{\sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a} \log \left (c x^n\right )}{\sqrt [4]{b}}\right )}{2 \sqrt{2} a^{5/4} n}-\frac{\sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{a} \log \left (c x^n\right )}{\sqrt [4]{b}}+1\right )}{2 \sqrt{2} a^{5/4} n}+\frac{\log (x)}{a} \]

Antiderivative was successfully verified.

[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 321

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 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 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]

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 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 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])

Rubi steps

\begin{align*} \int \frac{1}{a x+\frac{b x}{\log ^4\left (c x^n\right )}} \, dx &=\frac{\operatorname{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 \operatorname{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} \operatorname{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} \operatorname{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} \operatorname{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} \operatorname{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} \operatorname{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} \operatorname{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} \operatorname{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} \operatorname{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]  time = 0.0664152, size = 211, normalized size = 0.91 \[ \frac{\sqrt{2} \sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt{a} \log ^2\left (c x^n\right )+\sqrt{b}\right )-\sqrt{2} \sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt{a} \log ^2\left (c x^n\right )+\sqrt{b}\right )+2 \sqrt{2} \sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a} \log \left (c x^n\right )}{\sqrt [4]{b}}\right )-2 \sqrt{2} \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{a} \log \left (c x^n\right )}{\sqrt [4]{b}}+1\right )+8 \sqrt [4]{a} \log \left (c x^n\right )}{8 a^{5/4} n} \]

Antiderivative was successfully verified.

[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)*Log[c*x^n] + 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]*Lo
g[c*x^n]^2])/(8*a^(5/4)*n)

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Maple [A]  time = 0.01, size = 181, normalized size = 0.8 \begin{align*}{\frac{\ln \left ( c{x}^{n} \right ) }{na}}-{\frac{\sqrt{2}}{8\,na}\sqrt [4]{{\frac{b}{a}}}\ln \left ({ \left ( \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{2}+\sqrt [4]{{\frac{b}{a}}}\ln \left ( c{x}^{n} \right ) \sqrt{2}+\sqrt{{\frac{b}{a}}} \right ) \left ( \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{2}-\sqrt [4]{{\frac{b}{a}}}\ln \left ( c{x}^{n} \right ) \sqrt{2}+\sqrt{{\frac{b}{a}}} \right ) ^{-1}} \right ) }-{\frac{\sqrt{2}}{4\,na}\sqrt [4]{{\frac{b}{a}}}\arctan \left ({\ln \left ( c{x}^{n} \right ) \sqrt{2}{\frac{1}{\sqrt [4]{{\frac{b}{a}}}}}}+1 \right ) }+{\frac{\sqrt{2}}{4\,na}\sqrt [4]{{\frac{b}{a}}}\arctan \left ( -{\ln \left ( c{x}^{n} \right ) \sqrt{2}{\frac{1}{\sqrt [4]{{\frac{b}{a}}}}}}+1 \right ) } \end{align*}

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -b \int \frac{1}{4 \, a^{2} x \log \left (c\right )^{3} \log \left (x^{n}\right ) + 6 \, a^{2} x \log \left (c\right )^{2} \log \left (x^{n}\right )^{2} + 4 \, a^{2} x \log \left (c\right ) \log \left (x^{n}\right )^{3} + a^{2} x \log \left (x^{n}\right )^{4} +{\left (a^{2} \log \left (c\right )^{4} + a b\right )} x}\,{d x} + \frac{\log \left (x\right )}{a} \end{align*}

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]

-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

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Fricas [A]  time = 1.97476, size = 505, normalized size = 2.17 \begin{align*} -\frac{4 \, a \left (-\frac{b}{a^{5} n^{4}}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{a^{2} n^{2} \sqrt{-\frac{b}{a^{5} n^{4}}} + n^{2} \log \left (x\right )^{2} + 2 \, n \log \left (c\right ) \log \left (x\right ) + \log \left (c\right )^{2}} a^{4} n^{3} \left (-\frac{b}{a^{5} n^{4}}\right )^{\frac{3}{4}} -{\left (a^{4} n^{4} \log \left (x\right ) + a^{4} n^{3} \log \left (c\right )\right )} \left (-\frac{b}{a^{5} n^{4}}\right )^{\frac{3}{4}}}{b}\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 ) - 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} \end{align*}

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]

-1/4*(4*a*(-b/(a^5*n^4))^(1/4)*arctan((sqrt(a^2*n^2*sqrt(-b/(a^5*n^4)) + n^2*log(x)^2 + 2*n*log(c)*log(x) + lo
g(c)^2)*a^4*n^3*(-b/(a^5*n^4))^(3/4) - (a^4*n^4*log(x) + a^4*n^3*log(c))*(-b/(a^5*n^4))^(3/4))/b) + a*(-b/(a^5
*n^4))^(1/4)*log(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

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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Giac [A]  time = 1.3349, size = 365, normalized size = 1.57 \begin{align*} \frac{1}{4} \, i \left (-\frac{b}{a^{5} n^{4}}\right )^{\frac{1}{4}} \log \left (a^{2} i n^{5} \log \left (x\right ) + a^{2} i n^{4} \log \left (c\right ) - \left (-a^{3} b\right )^{\frac{1}{4}} a n^{4}\right ) - \frac{1}{4} \, i \left (-\frac{b}{a^{5} n^{4}}\right )^{\frac{1}{4}} \log \left (-a^{2} i n^{5} \log \left (x\right ) - a^{2} i n^{4} \log \left (c\right ) - \left (-a^{3} b\right )^{\frac{1}{4}} a n^{4}\right ) + \frac{1}{8} \, \left (-\frac{b}{a^{5} n^{4}}\right )^{\frac{1}{4}} \log \left (\frac{1}{4} \,{\left (\pi a^{2} n^{5}{\left (\mathrm{sgn}\left (x\right ) - 1\right )} + \pi a^{2} n^{4}{\left (\mathrm{sgn}\left (c\right ) - 1\right )}\right )}^{2} +{\left (a^{2} n^{5} \log \left ({\left | x \right |}\right ) + a^{2} n^{4} \log \left ({\left | c \right |}\right ) + \left (-a^{3} b\right )^{\frac{1}{4}} a n^{4}\right )}^{2}\right ) - \frac{1}{8} \, \left (-\frac{b}{a^{5} n^{4}}\right )^{\frac{1}{4}} \log \left (\frac{1}{4} \,{\left (\pi a^{2} n^{5}{\left (\mathrm{sgn}\left (x\right ) - 1\right )} + \pi a^{2} n^{4}{\left (\mathrm{sgn}\left (c\right ) - 1\right )}\right )}^{2} +{\left (a^{2} n^{5} \log \left ({\left | x \right |}\right ) + a^{2} n^{4} \log \left ({\left | c \right |}\right ) - \left (-a^{3} b\right )^{\frac{1}{4}} a n^{4}\right )}^{2}\right ) + \frac{\log \left (x\right )}{a} \end{align*}

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]

1/4*i*(-b/(a^5*n^4))^(1/4)*log(a^2*i*n^5*log(x) + a^2*i*n^4*log(c) - (-a^3*b)^(1/4)*a*n^4) - 1/4*i*(-b/(a^5*n^
4))^(1/4)*log(-a^2*i*n^5*log(x) - a^2*i*n^4*log(c) - (-a^3*b)^(1/4)*a*n^4) + 1/8*(-b/(a^5*n^4))^(1/4)*log(1/4*
(pi*a^2*n^5*(sgn(x) - 1) + pi*a^2*n^4*(sgn(c) - 1))^2 + (a^2*n^5*log(abs(x)) + a^2*n^4*log(abs(c)) + (-a^3*b)^
(1/4)*a*n^4)^2) - 1/8*(-b/(a^5*n^4))^(1/4)*log(1/4*(pi*a^2*n^5*(sgn(x) - 1) + pi*a^2*n^4*(sgn(c) - 1))^2 + (a^
2*n^5*log(abs(x)) + a^2*n^4*log(abs(c)) - (-a^3*b)^(1/4)*a*n^4)^2) + log(x)/a