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

Optimal. Leaf size=144 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \log \left (c x^n\right )+b^{2/3} \log ^2\left (c x^n\right )\right )}{6 a^{2/3} \sqrt [3]{b} n}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} \log \left (c x^n\right )\right )}{3 a^{2/3} \sqrt [3]{b} n}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \log \left (c x^n\right )}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} \sqrt [3]{b} n} \]

[Out]

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

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Rubi [A]  time = 0.0944799, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353, Rules used = {200, 31, 634, 617, 204, 628} \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \log \left (c x^n\right )+b^{2/3} \log ^2\left (c x^n\right )\right )}{6 a^{2/3} \sqrt [3]{b} n}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} \log \left (c x^n\right )\right )}{3 a^{2/3} \sqrt [3]{b} n}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \log \left (c x^n\right )}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} \sqrt [3]{b} n} \]

Antiderivative was successfully verified.

[In]

Int[(a*x + b*x*Log[c*x^n]^3)^(-1),x]

[Out]

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

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

Mathematica [A]  time = 0.0494766, size = 112, normalized size = 0.78 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \log \left (c x^n\right )+b^{2/3} \log ^2\left (c x^n\right )\right )-2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \log \left (c x^n\right )\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \log \left (c x^n\right )}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{6 a^{2/3} \sqrt [3]{b} n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.006, size = 120, normalized size = 0.8 \begin{align*}{\frac{1}{3\,bn}\ln \left ( \ln \left ( c{x}^{n} \right ) +\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{1}{6\,bn}\ln \left ( \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{2}-\sqrt [3]{{\frac{a}{b}}}\ln \left ( c{x}^{n} \right ) + \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}}{3\,bn}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\ln \left ( c{x}^{n} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \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]

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

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

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

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Fricas [A]  time = 2.01421, size = 1335, normalized size = 9.27 \begin{align*} \text{result too large to display} \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*(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*log(c)*log(x)^2 + 3*b*n*log(c)^2*log(x) + b*log(c)^3 + a)) - (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), 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)]

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Sympy [A]  time = 79.709, size = 340, normalized size = 2.36 \begin{align*} \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{\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{1}{b \left (2 n^{3} \log{\left (x \right )}^{2} + 4 n^{2} \log{\left (c \right )} \log{\left (x \right )} + 2 n \log{\left (c \right )}^{2}\right )} & \text{for}\: a = 0 \\- \frac{\sqrt [3]{-1} \log{\left (- \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{\frac{1}{b}} + n \log{\left (x \right )} + \log{\left (c \right )} \right )}}{3 a^{\frac{2}{3}} b^{3} n \left (\frac{1}{b}\right )^{\frac{8}{3}}} + \frac{\sqrt [3]{-1} \log{\left (4 \left (-1\right )^{\frac{2}{3}} a^{\frac{2}{3}} \left (\frac{1}{b}\right )^{\frac{2}{3}} + 4 \sqrt [3]{-1} \sqrt [3]{a} n \sqrt [3]{\frac{1}{b}} \log{\left (x \right )} + 4 \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{\frac{1}{b}} \log{\left (c \right )} + 4 n^{2} \log{\left (x \right )}^{2} + 8 n \log{\left (c \right )} \log{\left (x \right )} + 4 \log{\left (c \right )}^{2} \right )}}{6 a^{\frac{2}{3}} b^{3} n \left (\frac{1}{b}\right )^{\frac{8}{3}}} - \frac{\sqrt [3]{-1} \sqrt{3} \operatorname{atan}{\left (- \frac{\sqrt{3}}{3} + \frac{2 \left (-1\right )^{\frac{2}{3}} \sqrt{3} n \log{\left (x \right )}}{3 \sqrt [3]{a} \sqrt [3]{\frac{1}{b}}} + \frac{2 \left (-1\right )^{\frac{2}{3}} \sqrt{3} \log{\left (c \right )}}{3 \sqrt [3]{a} \sqrt [3]{\frac{1}{b}}} \right )}}{3 a^{\frac{2}{3}} b^{3} n \left (\frac{1}{b}\right )^{\frac{8}{3}}} & \text{otherwise} \end{cases} \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]

Piecewise((zoo*log(x)/log(c)**3, Eq(a, 0) & Eq(b, 0) & Eq(n, 0)), (log(x)/a, Eq(b, 0)), (log(x)/(a + b*log(c)*
*3), Eq(n, 0)), (-1/(b*(2*n**3*log(x)**2 + 4*n**2*log(c)*log(x) + 2*n*log(c)**2)), Eq(a, 0)), (-(-1)**(1/3)*lo
g(-(-1)**(1/3)*a**(1/3)*(1/b)**(1/3) + n*log(x) + log(c))/(3*a**(2/3)*b**3*n*(1/b)**(8/3)) + (-1)**(1/3)*log(4
*(-1)**(2/3)*a**(2/3)*(1/b)**(2/3) + 4*(-1)**(1/3)*a**(1/3)*n*(1/b)**(1/3)*log(x) + 4*(-1)**(1/3)*a**(1/3)*(1/
b)**(1/3)*log(c) + 4*n**2*log(x)**2 + 8*n*log(c)*log(x) + 4*log(c)**2)/(6*a**(2/3)*b**3*n*(1/b)**(8/3)) - (-1)
**(1/3)*sqrt(3)*atan(-sqrt(3)/3 + 2*(-1)**(2/3)*sqrt(3)*n*log(x)/(3*a**(1/3)*(1/b)**(1/3)) + 2*(-1)**(2/3)*sqr
t(3)*log(c)/(3*a**(1/3)*(1/b)**(1/3)))/(3*a**(2/3)*b**3*n*(1/b)**(8/3)), True))

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Giac [B]  time = 1.30493, size = 328, normalized size = 2.28 \begin{align*} \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 (\frac{1}{4} \,{\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} + \frac{1}{4} \,{\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 ) \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)*(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(1/4*(sqrt(3)*pi*b*(sgn(c) - 1) - 2*b*n*log(x) - 2*b*log(a
bs(c)) - 2*(a*b^2)^(1/3))^2 + 1/4*(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)