\(\int \frac {1}{x^3 \log ^3(c x)} \, dx\) [42]

Optimal result

Integrand size = 10, antiderivative size = 36 \[ \int \frac {1}{x^3 \log ^3(c x)} \, dx=2 c^2 \operatorname {ExpIntegralEi}(-2 \log (c x))-\frac {1}{2 x^2 \log ^2(c x)}+\frac {1}{x^2 \log (c x)} \]

[Out]

2*c^2*Ei(-2*ln(c*x))-1/2/x^2/ln(c*x)^2+1/x^2/ln(c*x)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2343, 2346, 2209} \[ \int \frac {1}{x^3 \log ^3(c x)} \, dx=2 c^2 \operatorname {ExpIntegralEi}(-2 \log (c x))-\frac {1}{2 x^2 \log ^2(c x)}+\frac {1}{x^2 \log (c x)} \]

[In]

Int[1/(x^3*Log[c*x]^3),x]

[Out]

2*c^2*ExpIntegralEi[-2*Log[c*x]] - 1/(2*x^2*Log[c*x]^2) + 1/(x^2*Log[c*x])

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2343

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

Rule 2346

Int[((a_.) + Log[(c_.)*(x_)]*(b_.))^(p_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[E^((m + 1)*x)*(a
+ b*x)^p, x], x, Log[c*x]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = -\frac {1}{2 x^2 \log ^2(c x)}-\int \frac {1}{x^3 \log ^2(c x)} \, dx \\ & = -\frac {1}{2 x^2 \log ^2(c x)}+\frac {1}{x^2 \log (c x)}+2 \int \frac {1}{x^3 \log (c x)} \, dx \\ & = -\frac {1}{2 x^2 \log ^2(c x)}+\frac {1}{x^2 \log (c x)}+\left (2 c^2\right ) \text {Subst}\left (\int \frac {e^{-2 x}}{x} \, dx,x,\log (c x)\right ) \\ & = 2 c^2 \text {Ei}(-2 \log (c x))-\frac {1}{2 x^2 \log ^2(c x)}+\frac {1}{x^2 \log (c x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^3 \log ^3(c x)} \, dx=2 c^2 \operatorname {ExpIntegralEi}(-2 \log (c x))-\frac {1}{2 x^2 \log ^2(c x)}+\frac {1}{x^2 \log (c x)} \]

[In]

Integrate[1/(x^3*Log[c*x]^3),x]

[Out]

2*c^2*ExpIntegralEi[-2*Log[c*x]] - 1/(2*x^2*Log[c*x]^2) + 1/(x^2*Log[c*x])

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.94

method	result	size
risch	\(\frac {-1+2 \ln \left (x c \right )}{2 x^{2} \ln \left (x c \right )^{2}}-2 c^{2} \operatorname {Ei}_{1}\left (2 \ln \left (x c \right )\right )\)	\(34\)
derivativedivides	\(c^{2} \left (-\frac {1}{2 x^{2} c^{2} \ln \left (x c \right )^{2}}+\frac {1}{x^{2} c^{2} \ln \left (x c \right )}-2 \,\operatorname {Ei}_{1}\left (2 \ln \left (x c \right )\right )\right )\)	\(43\)
default	\(c^{2} \left (-\frac {1}{2 x^{2} c^{2} \ln \left (x c \right )^{2}}+\frac {1}{x^{2} c^{2} \ln \left (x c \right )}-2 \,\operatorname {Ei}_{1}\left (2 \ln \left (x c \right )\right )\right )\)	\(43\)

[In]

int(1/x^3/ln(x*c)^3,x,method=_RETURNVERBOSE)

[Out]

1/2*(-1+2*ln(x*c))/x^2/ln(x*c)^2-2*c^2*Ei(1,2*ln(x*c))

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x^3 \log ^3(c x)} \, dx=\frac {4 \, c^{2} x^{2} \log \left (c x\right )^{2} \operatorname {log\_integral}\left (\frac {1}{c^{2} x^{2}}\right ) + 2 \, \log \left (c x\right ) - 1}{2 \, x^{2} \log \left (c x\right )^{2}} \]

[In]

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

[Out]

1/2*(4*c^2*x^2*log(c*x)^2*log_integral(1/(c^2*x^2)) + 2*log(c*x) - 1)/(x^2*log(c*x)^2)

Sympy [F]

\[ \int \frac {1}{x^3 \log ^3(c x)} \, dx=2 \int \frac {1}{x^{3} \log {\left (c x \right )}}\, dx + \frac {2 \log {\left (c x \right )} - 1}{2 x^{2} \log {\left (c x \right )}^{2}} \]

[In]

integrate(1/x**3/ln(c*x)**3,x)

[Out]

2*Integral(1/(x**3*log(c*x)), x) + (2*log(c*x) - 1)/(2*x**2*log(c*x)**2)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.36 \[ \int \frac {1}{x^3 \log ^3(c x)} \, dx=-4 \, c^{2} \Gamma \left (-2, 2 \, \log \left (c x\right )\right ) \]

[In]

integrate(1/x^3/log(c*x)^3,x, algorithm="maxima")

[Out]

-4*c^2*gamma(-2, 2*log(c*x))

Giac [F]

\[ \int \frac {1}{x^3 \log ^3(c x)} \, dx=\int { \frac {1}{x^{3} \log \left (c x\right )^{3}} \,d x } \]

[In]

integrate(1/x^3/log(c*x)^3,x, algorithm="giac")

[Out]

integrate(1/(x^3*log(c*x)^3), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^3 \log ^3(c x)} \, dx=\int \frac {1}{x^3\,{\ln \left (c\,x\right )}^3} \,d x \]

[In]

int(1/(x^3*log(c*x)^3),x)

[Out]

int(1/(x^3*log(c*x)^3), x)