3.74.100 \(\int \frac {-6-2 \sqrt [5]{e}+\sqrt [5]{e} \log (-x)}{-864+432 \sqrt [5]{e} \log (-x)-72 e^{2/5} \log ^2(-x)+4 e^{3/5} \log ^3(-x)} \, dx\)

Optimal. Leaf size=22 \[ -1+\frac {x}{4 \left (6-\sqrt [5]{e} \log (-x)\right )^2} \]

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Rubi [A]  time = 0.25, antiderivative size = 20, normalized size of antiderivative = 0.91, number of steps used = 11, number of rules used = 6, integrand size = 61, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {6688, 12, 2360, 2297, 2299, 2178} \begin {gather*} \frac {x}{4 \left (6-\sqrt [5]{e} \log (-x)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-6 - 2*E^(1/5) + E^(1/5)*Log[-x])/(-864 + 432*E^(1/5)*Log[-x] - 72*E^(2/5)*Log[-x]^2 + 4*E^(3/5)*Log[-x]^
3),x]

[Out]

x/(4*(6 - E^(1/5)*Log[-x])^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2178

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

Rule 2297

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[(x*(a + b*Log[c*x^n])^(p + 1))/(b*n*(p + 1))
, x] - Dist[1/(b*n*(p + 1)), Int[(a + b*Log[c*x^n])^(p + 1), x], x] /; FreeQ[{a, b, c, n}, x] && LtQ[p, -1] &&
IntegerQ[2*p]

Rule 2299

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

Rule 2360

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(Log[(c_.)*(x_)^(n_.)]*(e_.) + (d_))^(q_.), x_Symbol] :> Int[E
xpandIntegrand[(a + b*Log[c*x^n])^p*(d + e*Log[c*x^n])^q, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IntegerQ[p
] && IntegerQ[q]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (3+\sqrt [5]{e}\right )-\sqrt [5]{e} \log (-x)}{4 \left (6-\sqrt [5]{e} \log (-x)\right )^3} \, dx\\ &=\frac {1}{4} \int \frac {2 \left (3+\sqrt [5]{e}\right )-\sqrt [5]{e} \log (-x)}{\left (6-\sqrt [5]{e} \log (-x)\right )^3} \, dx\\ &=\frac {1}{4} \int \left (\frac {-6+2 \left (3+\sqrt [5]{e}\right )}{\left (6-\sqrt [5]{e} \log (-x)\right )^3}+\frac {1}{\left (6-\sqrt [5]{e} \log (-x)\right )^2}\right ) \, dx\\ &=\frac {1}{4} \int \frac {1}{\left (6-\sqrt [5]{e} \log (-x)\right )^2} \, dx+\frac {1}{2} \sqrt [5]{e} \int \frac {1}{\left (6-\sqrt [5]{e} \log (-x)\right )^3} \, dx\\ &=\frac {x}{4 \left (6-\sqrt [5]{e} \log (-x)\right )^2}+\frac {x}{4 \sqrt [5]{e} \left (6-\sqrt [5]{e} \log (-x)\right )}-\frac {1}{4} \int \frac {1}{\left (6-\sqrt [5]{e} \log (-x)\right )^2} \, dx-\frac {\int \frac {1}{6-\sqrt [5]{e} \log (-x)} \, dx}{4 \sqrt [5]{e}}\\ &=\frac {x}{4 \left (6-\sqrt [5]{e} \log (-x)\right )^2}+\frac {\int \frac {1}{6-\sqrt [5]{e} \log (-x)} \, dx}{4 \sqrt [5]{e}}+\frac {\operatorname {Subst}\left (\int \frac {e^x}{6-\sqrt [5]{e} x} \, dx,x,\log (-x)\right )}{4 \sqrt [5]{e}}\\ &=-\frac {1}{4} e^{-\frac {2}{5}+\frac {6}{\sqrt [5]{e}}} \text {Ei}\left (-\frac {6-\sqrt [5]{e} \log (-x)}{\sqrt [5]{e}}\right )+\frac {x}{4 \left (6-\sqrt [5]{e} \log (-x)\right )^2}-\frac {\operatorname {Subst}\left (\int \frac {e^x}{6-\sqrt [5]{e} x} \, dx,x,\log (-x)\right )}{4 \sqrt [5]{e}}\\ &=\frac {x}{4 \left (6-\sqrt [5]{e} \log (-x)\right )^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 19, normalized size = 0.86 \begin {gather*} \frac {x}{4 \left (-6+\sqrt [5]{e} \log (-x)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-6 - 2*E^(1/5) + E^(1/5)*Log[-x])/(-864 + 432*E^(1/5)*Log[-x] - 72*E^(2/5)*Log[-x]^2 + 4*E^(3/5)*Lo
g[-x]^3),x]

[Out]

x/(4*(-6 + E^(1/5)*Log[-x])^2)

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fricas [A]  time = 0.57, size = 24, normalized size = 1.09 \begin {gather*} \frac {x}{4 \, {\left (e^{\frac {2}{5}} \log \left (-x\right )^{2} - 12 \, e^{\frac {1}{5}} \log \left (-x\right ) + 36\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(1/5)*log(-x)-2*exp(1/5)-6)/(4*exp(1/5)^3*log(-x)^3-72*exp(1/5)^2*log(-x)^2+432*exp(1/5)*log(-x)
-864),x, algorithm="fricas")

[Out]

1/4*x/(e^(2/5)*log(-x)^2 - 12*e^(1/5)*log(-x) + 36)

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giac [A]  time = 0.26, size = 24, normalized size = 1.09 \begin {gather*} \frac {x}{4 \, {\left (e^{\frac {2}{5}} \log \left (-x\right )^{2} - 12 \, e^{\frac {1}{5}} \log \left (-x\right ) + 36\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(1/5)*log(-x)-2*exp(1/5)-6)/(4*exp(1/5)^3*log(-x)^3-72*exp(1/5)^2*log(-x)^2+432*exp(1/5)*log(-x)
-864),x, algorithm="giac")

[Out]

1/4*x/(e^(2/5)*log(-x)^2 - 12*e^(1/5)*log(-x) + 36)

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maple [A]  time = 0.20, size = 15, normalized size = 0.68

method	result	size
norman	\(\frac {x}{4 \left ({\mathrm e}^{\frac {1}{5}} \ln \left (-x \right )-6\right )^{2}}\)	\(15\)
risch	\(\frac {x}{4 \left ({\mathrm e}^{\frac {1}{5}} \ln \left (-x \right )-6\right )^{2}}\)	\(15\)
derivativedivides	\(\frac {x}{4 \,{\mathrm e}^{\frac {2}{5}} \ln \left (-x \right )^{2}-48 \,{\mathrm e}^{\frac {1}{5}} \ln \left (-x \right )+144}\)	\(27\)
default	\(\frac {x}{4 \,{\mathrm e}^{\frac {2}{5}} \ln \left (-x \right )^{2}-48 \,{\mathrm e}^{\frac {1}{5}} \ln \left (-x \right )+144}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(1/5)*ln(-x)-2*exp(1/5)-6)/(4*exp(1/5)^3*ln(-x)^3-72*exp(1/5)^2*ln(-x)^2+432*exp(1/5)*ln(-x)-864),x,me
thod=_RETURNVERBOSE)

[Out]

1/4*x/(exp(1/5)*ln(-x)-6)^2

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(1/5)*log(-x)-2*exp(1/5)-6)/(4*exp(1/5)^3*log(-x)^3-72*exp(1/5)^2*log(-x)^2+432*exp(1/5)*log(-x)
-864),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

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mupad [B]  time = 5.03, size = 24, normalized size = 1.09 \begin {gather*} \frac {x}{4\,\left ({\mathrm {e}}^{2/5}\,{\ln \left (-x\right )}^2-12\,{\mathrm {e}}^{1/5}\,\ln \left (-x\right )+36\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*exp(1/5) - log(-x)*exp(1/5) + 6)/(432*log(-x)*exp(1/5) - 72*log(-x)^2*exp(2/5) + 4*log(-x)^3*exp(3/5)
- 864),x)

[Out]

x/(4*(log(-x)^2*exp(2/5) - 12*log(-x)*exp(1/5) + 36))

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sympy [A]  time = 0.16, size = 27, normalized size = 1.23 \begin {gather*} \frac {x}{4 e^{\frac {2}{5}} \log {\left (- x \right )}^{2} - 48 e^{\frac {1}{5}} \log {\left (- x \right )} + 144} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(1/5)*ln(-x)-2*exp(1/5)-6)/(4*exp(1/5)**3*ln(-x)**3-72*exp(1/5)**2*ln(-x)**2+432*exp(1/5)*ln(-x)
-864),x)

[Out]

x/(4*exp(2/5)*log(-x)**2 - 48*exp(1/5)*log(-x) + 144)

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