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]

ln(x)/a-1/3*b^(1/3)*ln(b^(1/3)+a^(1/3)*ln(c*x^n))/a^(4/3)/n+1/6*b^(1/3)*ln(b^(2/3)-a^(1/3)*b^(1/3)*ln(c*x^n)+a
^(2/3)*ln(c*x^n)^2)/a^(4/3)/n+1/3*b^(1/3)*arctan(1/3*(b^(1/3)-2*a^(1/3)*ln(c*x^n))/b^(1/3)*3^(1/2))/a^(4/3)/n*
3^(1/2)

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Rubi [A]  time = 0.11, antiderivative size = 149, normalized size of antiderivative = 1.00, 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 31

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

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*}

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Mathematica [A]  time = 0.05, 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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fricas [A]  time = 0.47, size = 149, normalized size = 1.00 \[ \frac {6 \, n \log \relax (x) + 2 \, \sqrt {3} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, {\left (\sqrt {3} a n \log \relax (x) + \sqrt {3} a \log \relax (c)\right )} \left (-\frac {b}{a}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) - \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (n^{2} \log \relax (x)^{2} + 2 \, n \log \relax (c) \log \relax (x) + \log \relax (c)^{2} + {\left (n \log \relax (x) + \log \relax (c)\right )} \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right ) + 2 \, \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (n \log \relax (x) - \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \log \relax (c)\right )}{6 \, a n} \]

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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giac [B]  time = 0.22, size = 257, normalized size = 1.72 \[ \frac {\log \relax (x)}{a} + \frac {2 \, \sqrt {3} \left (-\frac {b n^{6}}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} \pi a {\left (\mathrm {sgn}\relax (c) - 1\right )} - 2 \, a n \log \relax (x) - 2 \, a \log \left ({\left | c \right |}\right ) + 2 \, \left (-a^{2} b\right )^{\frac {1}{3}}}{2 \, \sqrt {3} a n \log \relax (x) + \pi a {\left (\mathrm {sgn}\relax (c) - 1\right )} + 2 \, \sqrt {3} a \log \left ({\left | c \right |}\right ) + 2 \, \sqrt {3} \left (-a^{2} b\right )^{\frac {1}{3}}}\right ) + \left (-\frac {b n^{6}}{a}\right )^{\frac {1}{3}} \log \left (\frac {1}{4} \, {\left (\pi a n {\left (\mathrm {sgn}\relax (x) - 1\right )} + \pi a {\left (\mathrm {sgn}\relax (c) - 1\right )}\right )}^{2} + {\left (a n \log \left ({\left | x \right |}\right ) + a \log \left ({\left | c \right |}\right ) - \left (-a^{2} b\right )^{\frac {1}{3}}\right )}^{2}\right ) - \left (-\frac {b n^{6}}{a}\right )^{\frac {1}{3}} \log \left ({\left (\sqrt {3} \pi a {\left (\mathrm {sgn}\relax (c) - 1\right )} - 2 \, a n \log \relax (x) - 2 \, a \log \left ({\left | c \right |}\right ) + 2 \, \left (-a^{2} b\right )^{\frac {1}{3}}\right )}^{2} + {\left (2 \, \sqrt {3} a n \log \relax (x) + \pi a {\left (\mathrm {sgn}\relax (c) - 1\right )} + 2 \, \sqrt {3} a \log \left ({\left | c \right |}\right ) + 2 \, \sqrt {3} \left (-a^{2} b\right )^{\frac {1}{3}}\right )}^{2}\right )}{6 \, a n^{3}} \]

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]

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

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maple [A]  time = 0.07, size = 136, normalized size = 0.91 \[ \frac {\ln \left (c \,x^{n}\right )}{a n}-\frac {\sqrt {3}\, b \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \ln \left (c \,x^{n}\right )}{\left (\frac {b}{a}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {b}{a}\right )^{\frac {2}{3}} a^{2} n}-\frac {b \ln \left (\ln \left (c \,x^{n}\right )+\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {b}{a}\right )^{\frac {2}{3}} a^{2} n}+\frac {b \ln \left (\ln \left (c \,x^{n}\right )^{2}-\left (\frac {b}{a}\right )^{\frac {1}{3}} \ln \left (c \,x^{n}\right )+\left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {b}{a}\right )^{\frac {2}{3}} a^{2} n} \]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -b \int \frac {1}{3 \, a^{2} x \log \relax (c)^{2} \log \left (x^{n}\right ) + 3 \, a^{2} x \log \relax (c) \log \left (x^{n}\right )^{2} + a^{2} x \log \left (x^{n}\right )^{3} + {\left (a^{2} \log \relax (c)^{3} + a b\right )} x}\,{d x} + \frac {\log \relax (x)}{a} \]

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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mupad [B]  time = 2.40, size = 174, normalized size = 1.17 \[ \frac {\ln \relax (x)}{a}+\frac {{\left (-b\right )}^{1/3}\,\ln \left (\frac {3\,{\left (-b\right )}^{4/3}\,n}{a^{7/3}\,x^2}+\frac {3\,b\,n\,\ln \left (c\,x^n\right )}{a^2\,x^2}\right )}{3\,a^{4/3}\,n}+\frac {{\left (-b\right )}^{1/3}\,\ln \left (\frac {3\,b\,n\,\ln \left (c\,x^n\right )}{a^2\,x^2}+\frac {3\,{\left (-b\right )}^{4/3}\,n\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{a^{7/3}\,x^2}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,a^{4/3}\,n}-\frac {{\left (-b\right )}^{1/3}\,\ln \left (\frac {3\,b\,n\,\ln \left (c\,x^n\right )}{a^2\,x^2}-\frac {3\,{\left (-b\right )}^{4/3}\,n\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{a^{7/3}\,x^2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,a^{4/3}\,n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x + (b*x)/log(c*x^n)^3),x)

[Out]

log(x)/a + ((-b)^(1/3)*log((3*(-b)^(4/3)*n)/(a^(7/3)*x^2) + (3*b*n*log(c*x^n))/(a^2*x^2)))/(3*a^(4/3)*n) + ((-
b)^(1/3)*log((3*b*n*log(c*x^n))/(a^2*x^2) + (3*(-b)^(4/3)*n*((3^(1/2)*1i)/2 - 1/2))/(a^(7/3)*x^2))*((3^(1/2)*1
i)/2 - 1/2))/(3*a^(4/3)*n) - ((-b)^(1/3)*log((3*b*n*log(c*x^n))/(a^2*x^2) - (3*(-b)^(4/3)*n*((3^(1/2)*1i)/2 +
1/2))/(a^(7/3)*x^2))*((3^(1/2)*1i)/2 + 1/2))/(3*a^(4/3)*n)

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sympy [A]  time = 54.11, size = 367, normalized size = 2.46 \[ \begin {cases} \tilde {\infty } \log {\relax (c )}^{3} \log {\relax (x )} & \text {for}\: a = 0 \wedge b = 0 \wedge n = 0 \\\frac {\log {\relax (c )}^{3} \log {\relax (x )}}{a \log {\relax (c )}^{3} + b} & \text {for}\: n = 0 \\\frac {\log {\relax (x )}}{a} & \text {for}\: b = 0 \\\frac {\begin {cases} \frac {\log {\left (c x^{n} \right )}^{4}}{4 n} & \text {for}\: \left |{c x^{n}}\right | < 1 \\\frac {\log {\left (\frac {x^{- n}}{c} \right )}^{4}}{4 n} & \text {for}\: \frac {1}{\left |{c x^{n}}\right |} < 1 \\\frac {6 {G_{5, 5}^{5, 0}\left (\begin {matrix} & 1, 1, 1, 1, 1 \\0, 0, 0, 0, 0 & \end {matrix} \middle | {c x^{n}} \right )}}{n} + \frac {6 {G_{5, 5}^{0, 5}\left (\begin {matrix} 1, 1, 1, 1, 1 & \\ & 0, 0, 0, 0, 0 \end {matrix} \middle | {c x^{n}} \right )}}{n} & \text {otherwise} \end {cases}}{b} & \text {for}\: a = 0 \\\frac {\sqrt [3]{-1} \sqrt [3]{b} \sqrt [3]{\frac {1}{a}} \log {\left (- \sqrt [3]{-1} \sqrt [3]{b} \sqrt [3]{\frac {1}{a}} + n \log {\relax (x )} + \log {\relax (c )} \right )}}{3 a n} - \frac {\sqrt [3]{-1} \sqrt [3]{b} \sqrt [3]{\frac {1}{a}} \log {\left (4 \left (-1\right )^{\frac {2}{3}} b^{\frac {2}{3}} \left (\frac {1}{a}\right )^{\frac {2}{3}} + 4 \sqrt [3]{-1} \sqrt [3]{b} n \sqrt [3]{\frac {1}{a}} \log {\relax (x )} + 4 \sqrt [3]{-1} \sqrt [3]{b} \sqrt [3]{\frac {1}{a}} \log {\relax (c )} + 4 n^{2} \log {\relax (x )}^{2} + 8 n \log {\relax (c )} \log {\relax (x )} + 4 \log {\relax (c )}^{2} \right )}}{6 a n} + \frac {\sqrt [3]{-1} \sqrt {3} \sqrt [3]{b} \sqrt [3]{\frac {1}{a}} \operatorname {atan}{\left (- \frac {\sqrt {3}}{3} + \frac {2 \left (-1\right )^{\frac {2}{3}} \sqrt {3} n \log {\relax (x )}}{3 \sqrt [3]{b} \sqrt [3]{\frac {1}{a}}} + \frac {2 \left (-1\right )^{\frac {2}{3}} \sqrt {3} \log {\relax (c )}}{3 \sqrt [3]{b} \sqrt [3]{\frac {1}{a}}} \right )}}{3 a n} + \frac {\log {\relax (x )}}{a} & \text {otherwise} \end {cases} \]

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(c)**3*log(x), Eq(a, 0) & Eq(b, 0) & Eq(n, 0)), (log(c)**3*log(x)/(a*log(c)**3 + b), Eq(n, 0
)), (log(x)/a, Eq(b, 0)), (Piecewise((log(c*x**n)**4/(4*n), Abs(c*x**n) < 1), (log(x**(-n)/c)**4/(4*n), 1/Abs(
c*x**n) < 1), (6*meijerg(((), (1, 1, 1, 1, 1)), ((0, 0, 0, 0, 0), ()), c*x**n)/n + 6*meijerg(((1, 1, 1, 1, 1),
()), ((), (0, 0, 0, 0, 0)), c*x**n)/n, True))/b, Eq(a, 0)), ((-1)**(1/3)*b**(1/3)*(1/a)**(1/3)*log(-(-1)**(1/
3)*b**(1/3)*(1/a)**(1/3) + n*log(x) + log(c))/(3*a*n) - (-1)**(1/3)*b**(1/3)*(1/a)**(1/3)*log(4*(-1)**(2/3)*b*
*(2/3)*(1/a)**(2/3) + 4*(-1)**(1/3)*b**(1/3)*n*(1/a)**(1/3)*log(x) + 4*(-1)**(1/3)*b**(1/3)*(1/a)**(1/3)*log(c
) + 4*n**2*log(x)**2 + 8*n*log(c)*log(x) + 4*log(c)**2)/(6*a*n) + (-1)**(1/3)*sqrt(3)*b**(1/3)*(1/a)**(1/3)*at
an(-sqrt(3)/3 + 2*(-1)**(2/3)*sqrt(3)*n*log(x)/(3*b**(1/3)*(1/a)**(1/3)) + 2*(-1)**(2/3)*sqrt(3)*log(c)/(3*b**
(1/3)*(1/a)**(1/3)))/(3*a*n) + log(x)/a, True))

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