3.90.23 \(\int \frac {-3 e-3 e^6}{e x+e^6 x-4 x^4} \, dx\)

Optimal. Leaf size=15 \[ \log \left (1-\frac {e+e^6}{4 x^3}\right ) \]

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Rubi [A]  time = 0.03, antiderivative size = 19, normalized size of antiderivative = 1.27, number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {6, 12, 1593, 266, 36, 31, 29} \begin {gather*} \log \left (e \left (1+e^5\right )-4 x^3\right )-3 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3*E - 3*E^6)/(E*x + E^6*x - 4*x^4),x]

[Out]

-3*Log[x] + Log[E*(1 + E^5) - 4*x^3]

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 12

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

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

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 266

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

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 16, normalized size = 1.07 \begin {gather*} -3 \log (x)+\log \left (e+e^6-4 x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3*E - 3*E^6)/(E*x + E^6*x - 4*x^4),x]

[Out]

-3*Log[x] + Log[E + E^6 - 4*x^3]

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fricas [A]  time = 0.43, size = 20, normalized size = 1.33 \begin {gather*} \log \left (4 \, x^{3} - e^{6} - e\right ) - 3 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*exp(6)-3*exp(1))/(x*exp(6)+x*exp(1)-4*x^4),x, algorithm="fricas")

[Out]

log(4*x^3 - e^6 - e) - 3*log(x)

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giac [B]  time = 0.13, size = 43, normalized size = 2.87 \begin {gather*} {\left (\frac {\log \left ({\left | 4 \, x^{3} - e^{6} - e \right |}\right )}{e^{6} + e} - \frac {3 \, \log \left ({\left | x \right |}\right )}{e^{6} + e}\right )} {\left (e^{6} + e\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*exp(6)-3*exp(1))/(x*exp(6)+x*exp(1)-4*x^4),x, algorithm="giac")

[Out]

(log(abs(4*x^3 - e^6 - e))/(e^6 + e) - 3*log(abs(x))/(e^6 + e))*(e^6 + e)

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maple [A]  time = 0.49, size = 17, normalized size = 1.13

method	result	size
norman	\(-3 \ln \relax (x )+\ln \left (-4 x^{3}+{\mathrm e}+{\mathrm e}^{6}\right )\)	\(17\)
default	\(\left (-3 \,{\mathrm e}^{6}-3 \,{\mathrm e}\right ) \left (-\frac {\ln \left (4 x^{3}-{\mathrm e}-{\mathrm e}^{6}\right )}{3 \left ({\mathrm e}^{6}+{\mathrm e}\right )}+\frac {\ln \relax (x )}{{\mathrm e}^{6}+{\mathrm e}}\right )\)	\(46\)
risch	\(-\frac {3 \,{\mathrm e}^{-1} \ln \relax (x ) {\mathrm e}^{6}}{{\mathrm e}^{5}+1}-\frac {3 \,{\mathrm e}^{-1} \ln \relax (x ) {\mathrm e}}{{\mathrm e}^{5}+1}+\frac {{\mathrm e}^{-1} \ln \left (4 x^{3}-{\mathrm e}-{\mathrm e}^{6}\right ) {\mathrm e}^{6}}{{\mathrm e}^{5}+1}+\frac {{\mathrm e}^{-1} \ln \left (4 x^{3}-{\mathrm e}-{\mathrm e}^{6}\right ) {\mathrm e}}{{\mathrm e}^{5}+1}\)	\(82\)
meijerg	\(-\frac {{\mathrm e}^{6} \left (3 \ln \relax (x )+2 \ln \relax (2)-\ln \left ({\mathrm e}^{6}+{\mathrm e}\right )+i \pi -\ln \left (1-\frac {4 x^{3}}{{\mathrm e}^{6}+{\mathrm e}}\right )\right )}{{\mathrm e}^{6}+{\mathrm e}}-\frac {{\mathrm e} \left (3 \ln \relax (x )+2 \ln \relax (2)-\ln \left ({\mathrm e}^{6}+{\mathrm e}\right )+i \pi -\ln \left (1-\frac {4 x^{3}}{{\mathrm e}^{6}+{\mathrm e}}\right )\right )}{{\mathrm e}^{6}+{\mathrm e}}\)	\(100\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-3*exp(6)-3*exp(1))/(x*exp(6)+x*exp(1)-4*x^4),x,method=_RETURNVERBOSE)

[Out]

-3*ln(x)+ln(-4*x^3+exp(1)+exp(6))

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maxima [B]  time = 0.36, size = 41, normalized size = 2.73 \begin {gather*} {\left (\frac {\log \left (4 \, x^{3} - e^{6} - e\right )}{e^{6} + e} - \frac {3 \, \log \relax (x)}{e^{6} + e}\right )} {\left (e^{6} + e\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*exp(6)-3*exp(1))/(x*exp(6)+x*exp(1)-4*x^4),x, algorithm="maxima")

[Out]

(log(4*x^3 - e^6 - e)/(e^6 + e) - 3*log(x)/(e^6 + e))*(e^6 + e)

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mupad [B]  time = 0.16, size = 18, normalized size = 1.20 \begin {gather*} \ln \left (x^3-\frac {\mathrm {e}}{4}-\frac {{\mathrm {e}}^6}{4}\right )-3\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3*exp(1) + 3*exp(6))/(x*exp(1) + x*exp(6) - 4*x^4),x)

[Out]

log(x^3 - exp(6)/4 - exp(1)/4) - 3*log(x)

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sympy [B]  time = 0.79, size = 76, normalized size = 5.07 \begin {gather*} \left (3 e + 3 e^{6}\right ) \left (- \frac {\log {\relax (x )}}{e \left (1 + e\right ) \left (- e^{3} - e + 1 + e^{2} + e^{4}\right )} + \frac {\log {\left (x^{3} - \frac {e^{6}}{4} - \frac {e}{4} \right )}}{3 e \left (1 + e\right ) \left (- e^{3} - e + 1 + e^{2} + e^{4}\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*exp(6)-3*exp(1))/(x*exp(6)+x*exp(1)-4*x**4),x)

[Out]

(3*E + 3*exp(6))*(-exp(-1)*log(x)/((1 + E)*(-exp(3) - E + 1 + exp(2) + exp(4))) + exp(-1)*log(x**3 - exp(6)/4
- E/4)/(3*(1 + E)*(-exp(3) - E + 1 + exp(2) + exp(4))))

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