\(\int (-3-2 x-8 e^3 x^2-5 e^6 x^4+(-4-6 e^3 x^2) \log (x)-\log ^2(x)) \, dx\) [4090]

Optimal result

Integrand size = 40, antiderivative size = 23 \[ \int \left (-3-2 x-8 e^3 x^2-5 e^6 x^4+\left (-4-6 e^3 x^2\right ) \log (x)-\log ^2(x)\right ) \, dx=2-x^2-x \left (1+e^3 x^2+\log (x)\right )^2 \]

[Out]

2-x^2-(1+x^2*exp(3)+ln(x))^2*x

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(47\) vs. \(2(23)=46\).

Time = 0.02 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.04, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2350, 12, 2333, 2332} \[ \int \left (-3-2 x-8 e^3 x^2-5 e^6 x^4+\left (-4-6 e^3 x^2\right ) \log (x)-\log ^2(x)\right ) \, dx=-e^6 x^5-2 e^3 x^3-2 e^3 x^3 \log (x)-x^2-x-x \log ^2(x)-2 x \log (x) \]

[In]

Int[-3 - 2*x - 8*E^3*x^2 - 5*E^6*x^4 + (-4 - 6*E^3*x^2)*Log[x] - Log[x]^2,x]

[Out]

-x - x^2 - 2*E^3*x^3 - E^6*x^5 - 2*x*Log[x] - 2*E^3*x^3*Log[x] - x*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 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2333

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2350

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = IntHide[(d +
e*x^r)^q, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b
, c, d, e, n, r}, x] && IGtQ[q, 0]

Rubi steps \begin{align*} \text {integral}& = -3 x-x^2-\frac {8 e^3 x^3}{3}-e^6 x^5+\int \left (-4-6 e^3 x^2\right ) \log (x) \, dx-\int \log ^2(x) \, dx \\ & = -3 x-x^2-\frac {8 e^3 x^3}{3}-e^6 x^5-4 x \log (x)-2 e^3 x^3 \log (x)-x \log ^2(x)+2 \int \log (x) \, dx-\int 2 \left (-2-e^3 x^2\right ) \, dx \\ & = -5 x-x^2-\frac {8 e^3 x^3}{3}-e^6 x^5-2 x \log (x)-2 e^3 x^3 \log (x)-x \log ^2(x)-2 \int \left (-2-e^3 x^2\right ) \, dx \\ & = -x-x^2-2 e^3 x^3-e^6 x^5-2 x \log (x)-2 e^3 x^3 \log (x)-x \log ^2(x) \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(47\) vs. \(2(23)=46\).

Time = 0.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.04 \[ \int \left (-3-2 x-8 e^3 x^2-5 e^6 x^4+\left (-4-6 e^3 x^2\right ) \log (x)-\log ^2(x)\right ) \, dx=-x-x^2-2 e^3 x^3-e^6 x^5-2 x \log (x)-2 e^3 x^3 \log (x)-x \log ^2(x) \]

[In]

Integrate[-3 - 2*x - 8*E^3*x^2 - 5*E^6*x^4 + (-4 - 6*E^3*x^2)*Log[x] - Log[x]^2,x]

[Out]

-x - x^2 - 2*E^3*x^3 - E^6*x^5 - 2*x*Log[x] - 2*E^3*x^3*Log[x] - x*Log[x]^2

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.96

method	result	size
risch	\(-x -2 \,{\mathrm e}^{3} x^{3} \ln \left (x \right )-2 x^{3} {\mathrm e}^{3}-2 x \ln \left (x \right )-x^{2}-x \ln \left (x \right )^{2}-x^{5} {\mathrm e}^{6}\)	\(45\)
default	\(-x -2 \,{\mathrm e}^{3} x^{3} \ln \left (x \right )-2 x^{3} {\mathrm e}^{3}-2 x \ln \left (x \right )-x^{2}-x \ln \left (x \right )^{2}-x^{5} {\mathrm e}^{6}\)	\(47\)
norman	\(-x -2 \,{\mathrm e}^{3} x^{3} \ln \left (x \right )-2 x^{3} {\mathrm e}^{3}-2 x \ln \left (x \right )-x^{2}-x \ln \left (x \right )^{2}-x^{5} {\mathrm e}^{6}\)	\(47\)
parallelrisch	\(-x -2 \,{\mathrm e}^{3} x^{3} \ln \left (x \right )-2 x^{3} {\mathrm e}^{3}-2 x \ln \left (x \right )-x^{2}-x \ln \left (x \right )^{2}-x^{5} {\mathrm e}^{6}\)	\(47\)
parts	\(-x -2 \,{\mathrm e}^{3} x^{3} \ln \left (x \right )-2 x^{3} {\mathrm e}^{3}-2 x \ln \left (x \right )-x^{2}-x \ln \left (x \right )^{2}-x^{5} {\mathrm e}^{6}\)	\(47\)

[In]

int(-ln(x)^2+(-6*x^2*exp(3)-4)*ln(x)-5*x^4*exp(3)^2-8*x^2*exp(3)-2*x-3,x,method=_RETURNVERBOSE)

[Out]

-x-2*exp(3)*x^3*ln(x)-2*x^3*exp(3)-2*x*ln(x)-x^2-x*ln(x)^2-x^5*exp(6)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.83 \[ \int \left (-3-2 x-8 e^3 x^2-5 e^6 x^4+\left (-4-6 e^3 x^2\right ) \log (x)-\log ^2(x)\right ) \, dx=-x^{5} e^{6} - 2 \, x^{3} e^{3} - x \log \left (x\right )^{2} - x^{2} - 2 \, {\left (x^{3} e^{3} + x\right )} \log \left (x\right ) - x \]

[In]

integrate(-log(x)^2+(-6*x^2*exp(3)-4)*log(x)-5*x^4*exp(3)^2-8*x^2*exp(3)-2*x-3,x, algorithm="fricas")

[Out]

-x^5*e^6 - 2*x^3*e^3 - x*log(x)^2 - x^2 - 2*(x^3*e^3 + x)*log(x) - x

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (19) = 38\).

Time = 0.07 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.83 \[ \int \left (-3-2 x-8 e^3 x^2-5 e^6 x^4+\left (-4-6 e^3 x^2\right ) \log (x)-\log ^2(x)\right ) \, dx=- x^{5} e^{6} - 2 x^{3} e^{3} - x^{2} - x \log {\left (x \right )}^{2} - x + \left (- 2 x^{3} e^{3} - 2 x\right ) \log {\left (x \right )} \]

[In]

integrate(-ln(x)**2+(-6*x**2*exp(3)-4)*ln(x)-5*x**4*exp(3)**2-8*x**2*exp(3)-2*x-3,x)

[Out]

-x**5*exp(6) - 2*x**3*exp(3) - x**2 - x*log(x)**2 - x + (-2*x**3*exp(3) - 2*x)*log(x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (22) = 44\).

Time = 0.18 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.09 \[ \int \left (-3-2 x-8 e^3 x^2-5 e^6 x^4+\left (-4-6 e^3 x^2\right ) \log (x)-\log ^2(x)\right ) \, dx=-x^{5} e^{6} - 2 \, x^{3} e^{3} - {\left (\log \left (x\right )^{2} - 2 \, \log \left (x\right ) + 2\right )} x - x^{2} - 2 \, {\left (x^{3} e^{3} + 2 \, x\right )} \log \left (x\right ) + x \]

[In]

integrate(-log(x)^2+(-6*x^2*exp(3)-4)*log(x)-5*x^4*exp(3)^2-8*x^2*exp(3)-2*x-3,x, algorithm="maxima")

[Out]

-x^5*e^6 - 2*x^3*e^3 - (log(x)^2 - 2*log(x) + 2)*x - x^2 - 2*(x^3*e^3 + 2*x)*log(x) + x

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.91 \[ \int \left (-3-2 x-8 e^3 x^2-5 e^6 x^4+\left (-4-6 e^3 x^2\right ) \log (x)-\log ^2(x)\right ) \, dx=-x^{5} e^{6} - 2 \, x^{3} e^{3} \log \left (x\right ) - 2 \, x^{3} e^{3} - x \log \left (x\right )^{2} - x^{2} - 2 \, x \log \left (x\right ) - x \]

[In]

integrate(-log(x)^2+(-6*x^2*exp(3)-4)*log(x)-5*x^4*exp(3)^2-8*x^2*exp(3)-2*x-3,x, algorithm="giac")

[Out]

-x^5*e^6 - 2*x^3*e^3*log(x) - 2*x^3*e^3 - x*log(x)^2 - x^2 - 2*x*log(x) - x

Mupad [B] (verification not implemented)

Time = 9.88 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.57 \[ \int \left (-3-2 x-8 e^3 x^2-5 e^6 x^4+\left (-4-6 e^3 x^2\right ) \log (x)-\log ^2(x)\right ) \, dx=-x\,\left ({\mathrm {e}}^6\,x^4+2\,{\mathrm {e}}^3\,x^2\,\ln \left (x\right )+2\,{\mathrm {e}}^3\,x^2+x+{\ln \left (x\right )}^2+2\,\ln \left (x\right )+1\right ) \]

[In]

int(- 2*x - log(x)*(6*x^2*exp(3) + 4) - log(x)^2 - 8*x^2*exp(3) - 5*x^4*exp(6) - 3,x)

[Out]

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