\(\int \frac {x^3}{\log ^3(c x)} \, dx\) [36]

Optimal result

Integrand size = 10, antiderivative size = 37 \[ \int \frac {x^3}{\log ^3(c x)} \, dx=\frac {8 \operatorname {ExpIntegralEi}(4 \log (c x))}{c^4}-\frac {x^4}{2 \log ^2(c x)}-\frac {2 x^4}{\log (c x)} \]

[Out]

8*Ei(4*ln(c*x))/c^4-1/2*x^4/ln(c*x)^2-2*x^4/ln(c*x)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 37, 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} \[ \int \frac {x^3}{\log ^3(c x)} \, dx=\frac {8 \operatorname {ExpIntegralEi}(4 \log (c x))}{c^4}-\frac {x^4}{2 \log ^2(c x)}-\frac {2 x^4}{\log (c x)} \]

[In]

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

[Out]

(8*ExpIntegralEi[4*Log[c*x]])/c^4 - x^4/(2*Log[c*x]^2) - (2*x^4)/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^4}{2 \log ^2(c x)}+2 \int \frac {x^3}{\log ^2(c x)} \, dx \\ & = -\frac {x^4}{2 \log ^2(c x)}-\frac {2 x^4}{\log (c x)}+8 \int \frac {x^3}{\log (c x)} \, dx \\ & = -\frac {x^4}{2 \log ^2(c x)}-\frac {2 x^4}{\log (c x)}+\frac {8 \text {Subst}\left (\int \frac {e^{4 x}}{x} \, dx,x,\log (c x)\right )}{c^4} \\ & = \frac {8 \text {Ei}(4 \log (c x))}{c^4}-\frac {x^4}{2 \log ^2(c x)}-\frac {2 x^4}{\log (c x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00 \[ \int \frac {x^3}{\log ^3(c x)} \, dx=\frac {8 \operatorname {ExpIntegralEi}(4 \log (c x))}{c^4}-\frac {x^4}{2 \log ^2(c x)}-\frac {2 x^4}{\log (c x)} \]

[In]

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

[Out]

(8*ExpIntegralEi[4*Log[c*x]])/c^4 - x^4/(2*Log[c*x]^2) - (2*x^4)/Log[c*x]

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.92

method	result	size
risch	\(-\frac {x^{4} \left (1+4 \ln \left (x c \right )\right )}{2 \ln \left (x c \right )^{2}}-\frac {8 \,\operatorname {Ei}_{1}\left (-4 \ln \left (x c \right )\right )}{c^{4}}\)	\(34\)
derivativedivides	\(\frac {-\frac {x^{4} c^{4}}{2 \ln \left (x c \right )^{2}}-\frac {2 x^{4} c^{4}}{\ln \left (x c \right )}-8 \,\operatorname {Ei}_{1}\left (-4 \ln \left (x c \right )\right )}{c^{4}}\)	\(44\)
default	\(\frac {-\frac {x^{4} c^{4}}{2 \ln \left (x c \right )^{2}}-\frac {2 x^{4} c^{4}}{\ln \left (x c \right )}-8 \,\operatorname {Ei}_{1}\left (-4 \ln \left (x c \right )\right )}{c^{4}}\)	\(44\)

[In]

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

[Out]

-1/2*x^4*(1+4*ln(x*c))/ln(x*c)^2-8/c^4*Ei(1,-4*ln(x*c))

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.27 \[ \int \frac {x^3}{\log ^3(c x)} \, dx=-\frac {4 \, c^{4} x^{4} \log \left (c x\right ) + c^{4} x^{4} - 16 \, \log \left (c x\right )^{2} \operatorname {log\_integral}\left (c^{4} x^{4}\right )}{2 \, c^{4} \log \left (c x\right )^{2}} \]

[In]

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

[Out]

-1/2*(4*c^4*x^4*log(c*x) + c^4*x^4 - 16*log(c*x)^2*log_integral(c^4*x^4))/(c^4*log(c*x)^2)

Sympy [F]

\[ \int \frac {x^3}{\log ^3(c x)} \, dx=\frac {- 4 x^{4} \log {\left (c x \right )} - x^{4}}{2 \log {\left (c x \right )}^{2}} + 8 \int \frac {x^{3}}{\log {\left (c x \right )}}\, dx \]

[In]

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

[Out]

(-4*x**4*log(c*x) - x**4)/(2*log(c*x)**2) + 8*Integral(x**3/log(c*x), x)

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.35 \[ \int \frac {x^3}{\log ^3(c x)} \, dx=-\frac {16 \, \Gamma \left (-2, -4 \, \log \left (c x\right )\right )}{c^{4}} \]

[In]

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

[Out]

-16*gamma(-2, -4*log(c*x))/c^4

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.95 \[ \int \frac {x^3}{\log ^3(c x)} \, dx=-\frac {2 \, x^{4}}{\log \left (c x\right )} - \frac {x^{4}}{2 \, \log \left (c x\right )^{2}} + \frac {8 \, {\rm Ei}\left (4 \, \log \left (c x\right )\right )}{c^{4}} \]

[In]

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

[Out]

-2*x^4/log(c*x) - 1/2*x^4/log(c*x)^2 + 8*Ei(4*log(c*x))/c^4

Mupad [F(-1)]

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

[In]

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

[Out]

int(x^3/log(c*x)^3, x)