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

Optimal. Leaf size=149 \[ \frac{\sqrt [3]{b} \log \left (a^{2/3} \log ^2\left (c x^n\right )-\sqrt [3]{a} \sqrt [3]{b} \log \left (c x^n\right )+b^{2/3}\right )}{6 a^{4/3} n}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a} \log \left (c x^n\right )+\sqrt [3]{b}\right )}{3 a^{4/3} n}+\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} \log \left (c x^n\right )}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} a^{4/3} n}+\frac{\log (x)}{a} \]

[Out]

(b^(1/3)*ArcTan[(b^(1/3) - 2*a^(1/3)*Log[c*x^n])/(Sqrt[3]*b^(1/3))])/(Sqrt[3]*a^(4/3)*n) + Log[x]/a - (b^(1/3)
*Log[b^(1/3) + a^(1/3)*Log[c*x^n]])/(3*a^(4/3)*n) + (b^(1/3)*Log[b^(2/3) - a^(1/3)*b^(1/3)*Log[c*x^n] + a^(2/3
)*Log[c*x^n]^2])/(6*a^(4/3)*n)

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Rubi [A]  time = 0.109425, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412, Rules used = {321, 200, 31, 634, 617, 204, 628} \[ \frac{\sqrt [3]{b} \log \left (a^{2/3} \log ^2\left (c x^n\right )-\sqrt [3]{a} \sqrt [3]{b} \log \left (c x^n\right )+b^{2/3}\right )}{6 a^{4/3} n}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a} \log \left (c x^n\right )+\sqrt [3]{b}\right )}{3 a^{4/3} n}+\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} \log \left (c x^n\right )}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} a^{4/3} n}+\frac{\log (x)}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b^(1/3)*ArcTan[(b^(1/3) - 2*a^(1/3)*Log[c*x^n])/(Sqrt[3]*b^(1/3))])/(Sqrt[3]*a^(4/3)*n) + Log[x]/a - (b^(1/3)
*Log[b^(1/3) + a^(1/3)*Log[c*x^n]])/(3*a^(4/3)*n) + (b^(1/3)*Log[b^(2/3) - a^(1/3)*b^(1/3)*Log[c*x^n] + a^(2/3
)*Log[c*x^n]^2])/(6*a^(4/3)*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 200

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 31

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

Rule 634

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]

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])

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]

Rubi steps

\begin{align*} \int \frac{1}{a x+\frac{b x}{\log ^3\left (c x^n\right )}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^3}{b+a x^3} \, 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^3} \, dx,x,\log \left (c x^n\right )\right )}{a n}\\ &=\frac{\log (x)}{a}-\frac{\sqrt [3]{b} \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx,x,\log \left (c x^n\right )\right )}{3 a n}-\frac{\sqrt [3]{b} \operatorname{Subst}\left (\int \frac{2 \sqrt [3]{b}-\sqrt [3]{a} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx,x,\log \left (c x^n\right )\right )}{3 a n}\\ &=\frac{\log (x)}{a}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{b}+\sqrt [3]{a} \log \left (c x^n\right )\right )}{3 a^{4/3} n}+\frac{\sqrt [3]{b} \operatorname{Subst}\left (\int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 a^{2/3} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx,x,\log \left (c x^n\right )\right )}{6 a^{4/3} n}-\frac{b^{2/3} \operatorname{Subst}\left (\int \frac{1}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx,x,\log \left (c x^n\right )\right )}{2 a n}\\ &=\frac{\log (x)}{a}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{b}+\sqrt [3]{a} \log \left (c x^n\right )\right )}{3 a^{4/3} n}+\frac{\sqrt [3]{b} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \log \left (c x^n\right )+a^{2/3} \log ^2\left (c x^n\right )\right )}{6 a^{4/3} n}-\frac{\sqrt [3]{b} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{a} \log \left (c x^n\right )}{\sqrt [3]{b}}\right )}{a^{4/3} n}\\ &=\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{a} \log \left (c x^n\right )}{\sqrt [3]{b}}}{\sqrt{3}}\right )}{\sqrt{3} a^{4/3} n}+\frac{\log (x)}{a}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{b}+\sqrt [3]{a} \log \left (c x^n\right )\right )}{3 a^{4/3} n}+\frac{\sqrt [3]{b} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \log \left (c x^n\right )+a^{2/3} \log ^2\left (c x^n\right )\right )}{6 a^{4/3} n}\\ \end{align*}

Mathematica [A]  time = 0.0502176, size = 132, normalized size = 0.89 \[ \frac{\sqrt [3]{b} \left (\log \left (a^{2/3} \log ^2\left (c x^n\right )-\sqrt [3]{a} \sqrt [3]{b} \log \left (c x^n\right )+b^{2/3}\right )-2 \log \left (\sqrt [3]{a} \log \left (c x^n\right )+\sqrt [3]{b}\right )\right )+2 \sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{a} \log \left (c x^n\right )}{\sqrt [3]{b}}}{\sqrt{3}}\right )+6 \sqrt [3]{a} \log \left (c x^n\right )}{6 a^{4/3} n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[3]*b^(1/3)*ArcTan[(1 - (2*a^(1/3)*Log[c*x^n])/b^(1/3))/Sqrt[3]] + 6*a^(1/3)*Log[c*x^n] + b^(1/3)*(-2*L
og[b^(1/3) + a^(1/3)*Log[c*x^n]] + Log[b^(2/3) - a^(1/3)*b^(1/3)*Log[c*x^n] + a^(2/3)*Log[c*x^n]^2]))/(6*a^(4/
3)*n)

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Maple [A]  time = 0.006, size = 136, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( c{x}^{n} \right ) }{na}}-{\frac{b}{3\,n{a}^{2}}\ln \left ( \ln \left ( c{x}^{n} \right ) +\sqrt [3]{{\frac{b}{a}}} \right ) \left ({\frac{b}{a}} \right ) ^{-{\frac{2}{3}}}}+{\frac{b}{6\,n{a}^{2}}\ln \left ( \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{2}-\sqrt [3]{{\frac{b}{a}}}\ln \left ( c{x}^{n} \right ) + \left ({\frac{b}{a}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{b}{a}} \right ) ^{-{\frac{2}{3}}}}-{\frac{b\sqrt{3}}{3\,n{a}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\ln \left ( c{x}^{n} \right ){\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}-1 \right ) } \right ) \left ({\frac{b}{a}} \right ) ^{-{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Fricas [A]  time = 1.9596, size = 406, normalized size = 2.72 \begin{align*} \frac{6 \, n \log \left (x\right ) + 2 \, \sqrt{3} \left (-\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \,{\left (\sqrt{3} a n \log \left (x\right ) + \sqrt{3} a \log \left (c\right )\right )} \left (-\frac{b}{a}\right )^{\frac{2}{3}} - \sqrt{3} b}{3 \, b}\right ) - \left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left (n^{2} \log \left (x\right )^{2} + 2 \, n \log \left (c\right ) \log \left (x\right ) + \log \left (c\right )^{2} +{\left (n \log \left (x\right ) + \log \left (c\right )\right )} \left (-\frac{b}{a}\right )^{\frac{1}{3}} + \left (-\frac{b}{a}\right )^{\frac{2}{3}}\right ) + 2 \, \left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left (n \log \left (x\right ) - \left (-\frac{b}{a}\right )^{\frac{1}{3}} + \log \left (c\right )\right )}{6 \, a n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)**3),x)

[Out]

Exception raised: TypeError

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Giac [B]  time = 1.39006, size = 431, normalized size = 2.89 \begin{align*} \frac{1}{3} \, \sqrt{3} \left (-\frac{b}{a^{4} n^{3}}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3} \pi a n^{3}{\left (\mathrm{sgn}\left (c\right ) - 1\right )} - 2 \, a n^{4} \log \left (x\right ) - 2 \, a n^{3} \log \left ({\left | c \right |}\right ) + 2 \, \left (-a^{2} b\right )^{\frac{1}{3}} n^{3}}{2 \, \sqrt{3} a n^{4} \log \left (x\right ) + \pi a n^{3}{\left (\mathrm{sgn}\left (c\right ) - 1\right )} + 2 \, \sqrt{3} a n^{3} \log \left ({\left | c \right |}\right ) + 2 \, \sqrt{3} \left (-a^{2} b\right )^{\frac{1}{3}} n^{3}}\right ) + \frac{1}{6} \, \left (-\frac{b}{a^{4} n^{3}}\right )^{\frac{1}{3}} \log \left (\frac{1}{4} \,{\left (\pi a^{2} n^{4}{\left (\mathrm{sgn}\left (x\right ) - 1\right )} + \pi a^{2} n^{3}{\left (\mathrm{sgn}\left (c\right ) - 1\right )}\right )}^{2} +{\left (a^{2} n^{4} \log \left ({\left | x \right |}\right ) + a^{2} n^{3} \log \left ({\left | c \right |}\right ) - \left (-a^{2} b\right )^{\frac{1}{3}} a n^{3}\right )}^{2}\right ) - \frac{1}{6} \, \left (-\frac{b}{a^{4} n^{3}}\right )^{\frac{1}{3}} \log \left (\frac{1}{4} \,{\left (\sqrt{3} \pi a n^{3}{\left (\mathrm{sgn}\left (c\right ) - 1\right )} - 2 \, a n^{4} \log \left (x\right ) - 2 \, a n^{3} \log \left ({\left | c \right |}\right ) + 2 \, \left (-a^{2} b\right )^{\frac{1}{3}} n^{3}\right )}^{2} + \frac{1}{4} \,{\left (2 \, \sqrt{3} a n^{4} \log \left (x\right ) + \pi a n^{3}{\left (\mathrm{sgn}\left (c\right ) - 1\right )} + 2 \, \sqrt{3} a n^{3} \log \left ({\left | c \right |}\right ) + 2 \, \sqrt{3} \left (-a^{2} b\right )^{\frac{1}{3}} n^{3}\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)^3),x, algorithm="giac")

[Out]

1/3*sqrt(3)*(-b/(a^4*n^3))^(1/3)*arctan((sqrt(3)*pi*a*n^3*(sgn(c) - 1) - 2*a*n^4*log(x) - 2*a*n^3*log(abs(c))
+ 2*(-a^2*b)^(1/3)*n^3)/(2*sqrt(3)*a*n^4*log(x) + pi*a*n^3*(sgn(c) - 1) + 2*sqrt(3)*a*n^3*log(abs(c)) + 2*sqrt
(3)*(-a^2*b)^(1/3)*n^3)) + 1/6*(-b/(a^4*n^3))^(1/3)*log(1/4*(pi*a^2*n^4*(sgn(x) - 1) + pi*a^2*n^3*(sgn(c) - 1)
)^2 + (a^2*n^4*log(abs(x)) + a^2*n^3*log(abs(c)) - (-a^2*b)^(1/3)*a*n^3)^2) - 1/6*(-b/(a^4*n^3))^(1/3)*log(1/4
*(sqrt(3)*pi*a*n^3*(sgn(c) - 1) - 2*a*n^4*log(x) - 2*a*n^3*log(abs(c)) + 2*(-a^2*b)^(1/3)*n^3)^2 + 1/4*(2*sqrt
(3)*a*n^4*log(x) + pi*a*n^3*(sgn(c) - 1) + 2*sqrt(3)*a*n^3*log(abs(c)) + 2*sqrt(3)*(-a^2*b)^(1/3)*n^3)^2) + lo
g(x)/a