3.41.90 $\int (-3-2x-8e^3{x}^2-5e^6{x}^4+(-4-6e^3{x}^2)\log (x)-{\log }^2(x))dx$

Optimal. Leaf size=23 $$2-x^2-x{(1+e^3{x}^2+\log (x))}^2$$

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Rubi [B]  time = 0.03, antiderivative size = 52, normalized size of antiderivative = 2.26, number of steps used = 6, number of rules used = 4, integrand size = 40, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.100, Rules used = {2313, 12, 2296, 2295} $$\begin{array}{c}-e^6{x}^5-2e^3{x}^3-2(e^3{x}^3+2x)\log (x)-x^2-x-x{\log }^2(x)+2x\log (x)\end{array}$$

Antiderivative was successfully verified.

[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*(2*x + 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 2295

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

Rule 2296

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 2313

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = IntHide[(d +
e*x^r)^q, x]}, Simp[u*(a + b*Log[c*x^n]), 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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =-3x-x^2-\frac{8e^3{x}^3}{3}-e^6{x}^5+\int (-4-6e^3{x}^2)\log (x)dx-\int {\log }^2(x)dx\\ & =-3x-x^2-\frac{8e^3{x}^3}{3}-e^6{x}^5-2(2x+e^3{x}^3)\log (x)-x{\log }^2(x)+2\int \log (x)dx-\int 2(-2-e^3{x}^2)dx\\ & =-5x-x^2-\frac{8e^3{x}^3}{3}-e^6{x}^5+2x\log (x)-2(2x+e^3{x}^3)\log (x)-x{\log }^2(x)-2\int (-2-e^3{x}^2)dx\\ & =-x-x^2-2e^3{x}^3-e^6{x}^5+2x\log (x)-2(2x+e^3{x}^3)\log (x)-x{\log }^2(x)\end{array}\end{array}$$

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Mathematica [B]  time = 0.01, size = 47, normalized size = 2.04 $$\begin{array}{c}-x-x^2-2e^3{x}^3-e^6{x}^5-2x\log (x)-2e^3{x}^3\log (x)-x{\log }^2(x)\end{array}$$

Antiderivative was successfully verified.

[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

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fricas [A]  time = 0.70, size = 42, normalized size = 1.83 $$\begin{array}{c}-x^5{e}^6-2x^3{e}^3-x\log (x{)}^2-x^2-2(x^3{e}^3+x)\log (x)-x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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

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giac [A]  time = 0.22, size = 44, normalized size = 1.91 $$\begin{array}{c}-x^5{e}^6-2x^3{e}^3\log (x)-2x^3{e}^3-x\log (x{)}^2-x^2-2x\log (x)-x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [A]  time = 0.03, size = 45, normalized size = 1.96

method	result	size
risch	$-x-2{\mathrm{e}}^3\ln (x)x^3-2x^3{\mathrm{e}}^3-2x\ln (x)-x^2-x\ln (x{)}^2-x^5{\mathrm{e}}^6$	$45$
default	$-x-2{\mathrm{e}}^3\ln (x)x^3-2x^3{\mathrm{e}}^3-2x\ln (x)-x^2-x\ln (x{)}^2-x^5{\mathrm{e}}^6$	$47$
norman	$-x-2{\mathrm{e}}^3\ln (x)x^3-2x^3{\mathrm{e}}^3-2x\ln (x)-x^2-x\ln (x{)}^2-x^5{\mathrm{e}}^6$	$47$

Verification of antiderivative is not currently implemented for this CAS.

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

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maxima [B]  time = 0.38, size = 48, normalized size = 2.09 $$\begin{array}{c}-x^5{e}^6-2x^3{e}^3-(\log (x{)}^2-2\log (x)+2)x-x^2-2(x^3{e}^3+2x)\log (x)+x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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

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mupad [B]  time = 3.11, size = 36, normalized size = 1.57 $$\begin{array}{c}-x({\mathrm{e}}^6{x}^4+2{\mathrm{e}}^3{x}^2\ln (x)+2{\mathrm{e}}^3{x}^2+x+{\ln (x)}^2+2\ln (x)+1)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [B]  time = 0.13, size = 42, normalized size = 1.83 $$\begin{array}{c}-x^5{e}^6-2x^3{e}^3-x^2-x\log {(x)}^2-x+(-2x^3{e}^3-2x)\log (x)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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)

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