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

Optimal. Leaf size=23 \[ 2-x^2-x \left (1+e^3 x^2+\log (x)\right )^2 \]

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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 {gather*} -e^6 x^5-2 e^3 x^3-2 \left (e^3 x^3+2 x\right ) \log (x)-x^2-x-x \log ^2(x)+2 x \log (x) \end {gather*}

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 {gather*} \begin {aligned} \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^5-2 \left (2 x+e^3 x^3\right ) \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 \left (2 x+e^3 x^3\right ) \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 \left (2 x+e^3 x^3\right ) \log (x)-x \log ^2(x)\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.01, size = 47, normalized size = 2.04 \begin {gather*} -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 {gather*}

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 {gather*} -x^{5} e^{6} - 2 \, x^{3} e^{3} - x \log \relax (x)^{2} - x^{2} - 2 \, {\left (x^{3} e^{3} + x\right )} \log \relax (x) - x \end {gather*}

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 {gather*} -x^{5} e^{6} - 2 \, x^{3} e^{3} \log \relax (x) - 2 \, x^{3} e^{3} - x \log \relax (x)^{2} - x^{2} - 2 \, x \log \relax (x) - x \end {gather*}

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	\(-x -2 \,{\mathrm e}^{3} \ln \relax (x ) x^{3}-2 x^{3} {\mathrm e}^{3}-2 x \ln \relax (x )-x^{2}-x \ln \relax (x )^{2}-x^{5} {\mathrm e}^{6}\)	\(45\)
default	\(-x -2 \,{\mathrm e}^{3} \ln \relax (x ) x^{3}-2 x^{3} {\mathrm e}^{3}-2 x \ln \relax (x )-x^{2}-x \ln \relax (x )^{2}-x^{5} {\mathrm e}^{6}\)	\(47\)
norman	\(-x -2 \,{\mathrm e}^{3} \ln \relax (x ) x^{3}-2 x^{3} {\mathrm e}^{3}-2 x \ln \relax (x )-x^{2}-x \ln \relax (x )^{2}-x^{5} {\mathrm e}^{6}\)	\(47\)

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 {gather*} -x^{5} e^{6} - 2 \, x^{3} e^{3} - {\left (\log \relax (x)^{2} - 2 \, \log \relax (x) + 2\right )} x - x^{2} - 2 \, {\left (x^{3} e^{3} + 2 \, x\right )} \log \relax (x) + x \end {gather*}

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 {gather*} -x\,\left ({\mathrm {e}}^6\,x^4+2\,{\mathrm {e}}^3\,x^2\,\ln \relax (x)+2\,{\mathrm {e}}^3\,x^2+x+{\ln \relax (x)}^2+2\,\ln \relax (x)+1\right ) \end {gather*}

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 {gather*} - x^{5} e^{6} - 2 x^{3} e^{3} - x^{2} - x \log {\relax (x )}^{2} - x + \left (- 2 x^{3} e^{3} - 2 x\right ) \log {\relax (x )} \end {gather*}

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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