\(\int \frac {x^2}{\log ^3(c x)} \, dx\) [37]

Optimal result

Integrand size = 10, antiderivative size = 41 \[ \int \frac {x^2}{\log ^3(c x)} \, dx=\frac {9 \operatorname {ExpIntegralEi}(3 \log (c x))}{2 c^3}-\frac {x^3}{2 \log ^2(c x)}-\frac {3 x^3}{2 \log (c x)} \]

[Out]

9/2*Ei(3*ln(c*x))/c^3-1/2*x^3/ln(c*x)^2-3/2*x^3/ln(c*x)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 41, 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} \[ \int \frac {x^2}{\log ^3(c x)} \, dx=\frac {9 \operatorname {ExpIntegralEi}(3 \log (c x))}{2 c^3}-\frac {x^3}{2 \log ^2(c x)}-\frac {3 x^3}{2 \log (c x)} \]

[In]

Int[x^2/Log[c*x]^3,x]

[Out]

(9*ExpIntegralEi[3*Log[c*x]])/(2*c^3) - x^3/(2*Log[c*x]^2) - (3*x^3)/(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 {x^3}{2 \log ^2(c x)}+\frac {3}{2} \int \frac {x^2}{\log ^2(c x)} \, dx \\ & = -\frac {x^3}{2 \log ^2(c x)}-\frac {3 x^3}{2 \log (c x)}+\frac {9}{2} \int \frac {x^2}{\log (c x)} \, dx \\ & = -\frac {x^3}{2 \log ^2(c x)}-\frac {3 x^3}{2 \log (c x)}+\frac {9 \text {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (c x)\right )}{2 c^3} \\ & = \frac {9 \text {Ei}(3 \log (c x))}{2 c^3}-\frac {x^3}{2 \log ^2(c x)}-\frac {3 x^3}{2 \log (c x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{\log ^3(c x)} \, dx=\frac {9 \operatorname {ExpIntegralEi}(3 \log (c x))}{2 c^3}-\frac {x^3}{2 \log ^2(c x)}-\frac {3 x^3}{2 \log (c x)} \]

[In]

Integrate[x^2/Log[c*x]^3,x]

[Out]

(9*ExpIntegralEi[3*Log[c*x]])/(2*c^3) - x^3/(2*Log[c*x]^2) - (3*x^3)/(2*Log[c*x])

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.83

method	result	size
risch	\(-\frac {x^{3} \left (1+3 \ln \left (x c \right )\right )}{2 \ln \left (x c \right )^{2}}-\frac {9 \,\operatorname {Ei}_{1}\left (-3 \ln \left (x c \right )\right )}{2 c^{3}}\)	\(34\)
derivativedivides	\(\frac {-\frac {x^{3} c^{3}}{2 \ln \left (x c \right )^{2}}-\frac {3 x^{3} c^{3}}{2 \ln \left (x c \right )}-\frac {9 \,\operatorname {Ei}_{1}\left (-3 \ln \left (x c \right )\right )}{2}}{c^{3}}\)	\(44\)
default	\(\frac {-\frac {x^{3} c^{3}}{2 \ln \left (x c \right )^{2}}-\frac {3 x^{3} c^{3}}{2 \ln \left (x c \right )}-\frac {9 \,\operatorname {Ei}_{1}\left (-3 \ln \left (x c \right )\right )}{2}}{c^{3}}\)	\(44\)

[In]

int(x^2/ln(x*c)^3,x,method=_RETURNVERBOSE)

[Out]

-1/2*x^3*(1+3*ln(x*c))/ln(x*c)^2-9/2/c^3*Ei(1,-3*ln(x*c))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.15 \[ \int \frac {x^2}{\log ^3(c x)} \, dx=-\frac {3 \, c^{3} x^{3} \log \left (c x\right ) + c^{3} x^{3} - 9 \, \log \left (c x\right )^{2} \operatorname {log\_integral}\left (c^{3} x^{3}\right )}{2 \, c^{3} \log \left (c x\right )^{2}} \]

[In]

integrate(x^2/log(c*x)^3,x, algorithm="fricas")

[Out]

-1/2*(3*c^3*x^3*log(c*x) + c^3*x^3 - 9*log(c*x)^2*log_integral(c^3*x^3))/(c^3*log(c*x)^2)

Sympy [F]

\[ \int \frac {x^2}{\log ^3(c x)} \, dx=\frac {- 3 x^{3} \log {\left (c x \right )} - x^{3}}{2 \log {\left (c x \right )}^{2}} + \frac {9 \int \frac {x^{2}}{\log {\left (c x \right )}}\, dx}{2} \]

[In]

integrate(x**2/ln(c*x)**3,x)

[Out]

(-3*x**3*log(c*x) - x**3)/(2*log(c*x)**2) + 9*Integral(x**2/log(c*x), x)/2

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.32 \[ \int \frac {x^2}{\log ^3(c x)} \, dx=-\frac {9 \, \Gamma \left (-2, -3 \, \log \left (c x\right )\right )}{c^{3}} \]

[In]

integrate(x^2/log(c*x)^3,x, algorithm="maxima")

[Out]

-9*gamma(-2, -3*log(c*x))/c^3

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.85 \[ \int \frac {x^2}{\log ^3(c x)} \, dx=-\frac {3 \, x^{3}}{2 \, \log \left (c x\right )} - \frac {x^{3}}{2 \, \log \left (c x\right )^{2}} + \frac {9 \, {\rm Ei}\left (3 \, \log \left (c x\right )\right )}{2 \, c^{3}} \]

[In]

integrate(x^2/log(c*x)^3,x, algorithm="giac")

[Out]

-3/2*x^3/log(c*x) - 1/2*x^3/log(c*x)^2 + 9/2*Ei(3*log(c*x))/c^3

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{\log ^3(c x)} \, dx=\int \frac {x^2}{{\ln \left (c\,x\right )}^3} \,d x \]

[In]

int(x^2/log(c*x)^3,x)

[Out]

int(x^2/log(c*x)^3, x)