3.93.37 \(\int \frac {e^3 (-3 x^2-2 x^3)+e^3 (12+8 x) \log ^4(2)+(-6 x+e^3 (-6 x^2-2 x^3)) \log (3+e^3 (3 x+x^2))}{3+e^3 (3 x+x^2)} \, dx\)

Optimal. Leaf size=24 \[ \left (-x^2+4 \log ^4(2)\right ) \log \left (3+e^3 x (3+x)\right ) \]

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Rubi [B]  time = 0.46, antiderivative size = 90, normalized size of antiderivative = 3.75, number of steps used = 16, number of rules used = 9, integrand size = 79, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {6688, 1628, 634, 618, 206, 628, 2525, 12, 800} \begin {gather*} \frac {\left (6-e^3 \left (9-8 \log ^4(2)\right )\right ) \log \left (e^3 x^2+3 e^3 x+3\right )}{2 e^3}+x^2 \left (-\log \left (e^3 x^2+3 e^3 x+3\right )\right )-\frac {3 \left (2-3 e^3\right ) \log \left (e^3 x^2+3 e^3 x+3\right )}{2 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^3*(-3*x^2 - 2*x^3) + E^3*(12 + 8*x)*Log[2]^4 + (-6*x + E^3*(-6*x^2 - 2*x^3))*Log[3 + E^3*(3*x + x^2)])/
(3 + E^3*(3*x + x^2)),x]

[Out]

(-3*(2 - 3*E^3)*Log[3 + 3*E^3*x + E^3*x^2])/(2*E^3) - x^2*Log[3 + 3*E^3*x + E^3*x^2] + ((6 - E^3*(9 - 8*Log[2]
^4))*Log[3 + 3*E^3*x + E^3*x^2])/(2*E^3)

Rule 12

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 800

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + g*x))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rule 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 2525

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m
+ 1)*(a + b*Log[c*RFx^p])^n)/(e*(m + 1)), x] - Dist[(b*n*p)/(e*(m + 1)), Int[SimplifyIntegrand[((d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

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 \left (-\frac {e^3 (3+2 x) \left (x^2-4 \log ^4(2)\right )}{3+3 e^3 x+e^3 x^2}-2 x \log \left (3+3 e^3 x+e^3 x^2\right )\right ) \, dx\\ &=-\left (2 \int x \log \left (3+3 e^3 x+e^3 x^2\right ) \, dx\right )-e^3 \int \frac {(3+2 x) \left (x^2-4 \log ^4(2)\right )}{3+3 e^3 x+e^3 x^2} \, dx\\ &=-x^2 \log \left (3+3 e^3 x+e^3 x^2\right )-e^3 \int \left (-\frac {3}{e^3}+\frac {2 x}{e^3}-\frac {-3 \left (3-4 e^3 \log ^4(2)\right )+x \left (6-e^3 \left (9-8 \log ^4(2)\right )\right )}{e^3 \left (3+3 e^3 x+e^3 x^2\right )}\right ) \, dx+\int \frac {e^3 x^2 (3+2 x)}{3+3 e^3 x+e^3 x^2} \, dx\\ &=3 x-x^2-x^2 \log \left (3+3 e^3 x+e^3 x^2\right )+e^3 \int \frac {x^2 (3+2 x)}{3+3 e^3 x+e^3 x^2} \, dx+\int \frac {-3 \left (3-4 e^3 \log ^4(2)\right )+x \left (6-e^3 \left (9-8 \log ^4(2)\right )\right )}{3+3 e^3 x+e^3 x^2} \, dx\\ &=3 x-x^2-x^2 \log \left (3+3 e^3 x+e^3 x^2\right )+e^3 \int \left (-\frac {3}{e^3}+\frac {2 x}{e^3}+\frac {3 \left (3-\left (2-3 e^3\right ) x\right )}{e^3 \left (3+3 e^3 x+e^3 x^2\right )}\right ) \, dx-\frac {1}{2} \left (9 \left (4-3 e^3\right )\right ) \int \frac {1}{3+3 e^3 x+e^3 x^2} \, dx+\frac {\left (6-e^3 \left (9-8 \log ^4(2)\right )\right ) \int \frac {3 e^3+2 e^3 x}{3+3 e^3 x+e^3 x^2} \, dx}{2 e^3}\\ &=-x^2 \log \left (3+3 e^3 x+e^3 x^2\right )+\frac {\left (6-e^3 \left (9-8 \log ^4(2)\right )\right ) \log \left (3+3 e^3 x+e^3 x^2\right )}{2 e^3}+3 \int \frac {3-\left (2-3 e^3\right ) x}{3+3 e^3 x+e^3 x^2} \, dx+\left (9 \left (4-3 e^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-3 e^3 \left (4-3 e^3\right )-x^2} \, dx,x,3 e^3+2 e^3 x\right )\\ &=-\frac {3 \sqrt {3 \left (-4+3 e^3\right )} \tanh ^{-1}\left (\frac {e^{3/2} (3+2 x)}{\sqrt {3 \left (-4+3 e^3\right )}}\right )}{e^{3/2}}-x^2 \log \left (3+3 e^3 x+e^3 x^2\right )+\frac {\left (6-e^3 \left (9-8 \log ^4(2)\right )\right ) \log \left (3+3 e^3 x+e^3 x^2\right )}{2 e^3}+\frac {1}{2} \left (9 \left (4-3 e^3\right )\right ) \int \frac {1}{3+3 e^3 x+e^3 x^2} \, dx+\frac {\left (3 \left (-2+3 e^3\right )\right ) \int \frac {3 e^3+2 e^3 x}{3+3 e^3 x+e^3 x^2} \, dx}{2 e^3}\\ &=-\frac {3 \sqrt {3 \left (-4+3 e^3\right )} \tanh ^{-1}\left (\frac {e^{3/2} (3+2 x)}{\sqrt {3 \left (-4+3 e^3\right )}}\right )}{e^{3/2}}-\frac {3 \left (2-3 e^3\right ) \log \left (3+3 e^3 x+e^3 x^2\right )}{2 e^3}-x^2 \log \left (3+3 e^3 x+e^3 x^2\right )+\frac {\left (6-e^3 \left (9-8 \log ^4(2)\right )\right ) \log \left (3+3 e^3 x+e^3 x^2\right )}{2 e^3}-\left (9 \left (4-3 e^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-3 e^3 \left (4-3 e^3\right )-x^2} \, dx,x,3 e^3+2 e^3 x\right )\\ &=-\frac {3 \left (2-3 e^3\right ) \log \left (3+3 e^3 x+e^3 x^2\right )}{2 e^3}-x^2 \log \left (3+3 e^3 x+e^3 x^2\right )+\frac {\left (6-e^3 \left (9-8 \log ^4(2)\right )\right ) \log \left (3+3 e^3 x+e^3 x^2\right )}{2 e^3}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 23, normalized size = 0.96 \begin {gather*} -\left (\left (x^2-4 \log ^4(2)\right ) \log \left (3+e^3 x (3+x)\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^3*(-3*x^2 - 2*x^3) + E^3*(12 + 8*x)*Log[2]^4 + (-6*x + E^3*(-6*x^2 - 2*x^3))*Log[3 + E^3*(3*x + x
^2)])/(3 + E^3*(3*x + x^2)),x]

[Out]

-((x^2 - 4*Log[2]^4)*Log[3 + E^3*x*(3 + x)])

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fricas [A]  time = 0.92, size = 26, normalized size = 1.08 \begin {gather*} {\left (4 \, \log \relax (2)^{4} - x^{2}\right )} \log \left ({\left (x^{2} + 3 \, x\right )} e^{3} + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^3-6*x^2)*exp(3)-6*x)*log((x^2+3*x)*exp(3)+3)+(8*x+12)*exp(3)*log(2)^4+(-2*x^3-3*x^2)*exp(3))
/((x^2+3*x)*exp(3)+3),x, algorithm="fricas")

[Out]

(4*log(2)^4 - x^2)*log((x^2 + 3*x)*e^3 + 3)

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giac [A]  time = 0.28, size = 40, normalized size = 1.67 \begin {gather*} 4 \, \log \relax (2)^{4} \log \left (x^{2} e^{3} + 3 \, x e^{3} + 3\right ) - x^{2} \log \left (x^{2} e^{3} + 3 \, x e^{3} + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^3-6*x^2)*exp(3)-6*x)*log((x^2+3*x)*exp(3)+3)+(8*x+12)*exp(3)*log(2)^4+(-2*x^3-3*x^2)*exp(3))
/((x^2+3*x)*exp(3)+3),x, algorithm="giac")

[Out]

4*log(2)^4*log(x^2*e^3 + 3*x*e^3 + 3) - x^2*log(x^2*e^3 + 3*x*e^3 + 3)

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maple [A]  time = 0.62, size = 39, normalized size = 1.62

method	result	size
norman	\(4 \ln \relax (2)^{4} \ln \left (\left (x^{2}+3 x \right ) {\mathrm e}^{3}+3\right )-x^{2} \ln \left (\left (x^{2}+3 x \right ) {\mathrm e}^{3}+3\right )\)	\(39\)
risch	\(-x^{2} \ln \left (\left (x^{2}+3 x \right ) {\mathrm e}^{3}+3\right )+4 \ln \relax (2)^{4} \ln \left (x^{2} {\mathrm e}^{3}+3 x \,{\mathrm e}^{3}+3\right )\)	\(40\)
default	\(-x^{2} \ln \left (x^{2} {\mathrm e}^{3}+3 x \,{\mathrm e}^{3}+3\right )+4 \ln \relax (2)^{4} \ln \left (x^{2} {\mathrm e}^{3}+3 x \,{\mathrm e}^{3}+3\right )\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-2*x^3-6*x^2)*exp(3)-6*x)*ln((x^2+3*x)*exp(3)+3)+(8*x+12)*exp(3)*ln(2)^4+(-2*x^3-3*x^2)*exp(3))/((x^2+3
*x)*exp(3)+3),x,method=_RETURNVERBOSE)

[Out]

4*ln(2)^4*ln((x^2+3*x)*exp(3)+3)-x^2*ln((x^2+3*x)*exp(3)+3)

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maxima [B]  time = 0.52, size = 493, normalized size = 20.54 \begin {gather*} 4 \, {\left (e^{\left (-3\right )} \log \left (x^{2} e^{3} + 3 \, x e^{3} + 3\right ) - \frac {\sqrt {3} e^{\left (-\frac {3}{2}\right )} \log \left (\frac {2 \, x e^{3} - \sqrt {3} \sqrt {3 \, e^{3} - 4} e^{\frac {3}{2}} + 3 \, e^{3}}{2 \, x e^{3} + \sqrt {3} \sqrt {3 \, e^{3} - 4} e^{\frac {3}{2}} + 3 \, e^{3}}\right )}{\sqrt {3 \, e^{3} - 4}}\right )} e^{3} \log \relax (2)^{4} + \frac {4 \, \sqrt {3} e^{\frac {3}{2}} \log \relax (2)^{4} \log \left (\frac {2 \, x e^{3} - \sqrt {3} \sqrt {3 \, e^{3} - 4} e^{\frac {3}{2}} + 3 \, e^{3}}{2 \, x e^{3} + \sqrt {3} \sqrt {3 \, e^{3} - 4} e^{\frac {3}{2}} + 3 \, e^{3}}\right )}{\sqrt {3 \, e^{3} - 4}} - \frac {3}{2} \, \sqrt {3} \sqrt {3 \, e^{3} - 4} e^{\left (-\frac {3}{2}\right )} \log \left (\frac {2 \, x e^{3} - \sqrt {3} \sqrt {3 \, e^{3} - 4} e^{\frac {3}{2}} + 3 \, e^{3}}{2 \, x e^{3} + \sqrt {3} \sqrt {3 \, e^{3} - 4} e^{\frac {3}{2}} + 3 \, e^{3}}\right ) - {\left (3 \, {\left (3 \, e^{3} - 1\right )} e^{\left (-6\right )} \log \left (x^{2} e^{3} + 3 \, x e^{3} + 3\right ) - \frac {9 \, \sqrt {3} {\left (e^{3} - 1\right )} e^{\left (-\frac {9}{2}\right )} \log \left (\frac {2 \, x e^{3} - \sqrt {3} \sqrt {3 \, e^{3} - 4} e^{\frac {3}{2}} + 3 \, e^{3}}{2 \, x e^{3} + \sqrt {3} \sqrt {3 \, e^{3} - 4} e^{\frac {3}{2}} + 3 \, e^{3}}\right )}{\sqrt {3 \, e^{3} - 4}} + {\left (x^{2} - 6 \, x\right )} e^{\left (-3\right )}\right )} e^{3} - \frac {3}{2} \, {\left (\frac {\sqrt {3} {\left (3 \, e^{3} - 2\right )} e^{\left (-\frac {9}{2}\right )} \log \left (\frac {2 \, x e^{3} - \sqrt {3} \sqrt {3 \, e^{3} - 4} e^{\frac {3}{2}} + 3 \, e^{3}}{2 \, x e^{3} + \sqrt {3} \sqrt {3 \, e^{3} - 4} e^{\frac {3}{2}} + 3 \, e^{3}}\right )}{\sqrt {3 \, e^{3} - 4}} + 2 \, x e^{\left (-3\right )} - 3 \, e^{\left (-3\right )} \log \left (x^{2} e^{3} + 3 \, x e^{3} + 3\right )\right )} e^{3} + \frac {1}{2} \, {\left (2 \, x^{2} e^{3} - 6 \, x e^{3} - {\left (2 \, x^{2} e^{3} - 9 \, e^{3} + 6\right )} \log \left (x^{2} e^{3} + 3 \, x e^{3} + 3\right )\right )} e^{\left (-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^3-6*x^2)*exp(3)-6*x)*log((x^2+3*x)*exp(3)+3)+(8*x+12)*exp(3)*log(2)^4+(-2*x^3-3*x^2)*exp(3))
/((x^2+3*x)*exp(3)+3),x, algorithm="maxima")

[Out]

4*(e^(-3)*log(x^2*e^3 + 3*x*e^3 + 3) - sqrt(3)*e^(-3/2)*log((2*x*e^3 - sqrt(3)*sqrt(3*e^3 - 4)*e^(3/2) + 3*e^3
)/(2*x*e^3 + sqrt(3)*sqrt(3*e^3 - 4)*e^(3/2) + 3*e^3))/sqrt(3*e^3 - 4))*e^3*log(2)^4 + 4*sqrt(3)*e^(3/2)*log(2
)^4*log((2*x*e^3 - sqrt(3)*sqrt(3*e^3 - 4)*e^(3/2) + 3*e^3)/(2*x*e^3 + sqrt(3)*sqrt(3*e^3 - 4)*e^(3/2) + 3*e^3
))/sqrt(3*e^3 - 4) - 3/2*sqrt(3)*sqrt(3*e^3 - 4)*e^(-3/2)*log((2*x*e^3 - sqrt(3)*sqrt(3*e^3 - 4)*e^(3/2) + 3*e
^3)/(2*x*e^3 + sqrt(3)*sqrt(3*e^3 - 4)*e^(3/2) + 3*e^3)) - (3*(3*e^3 - 1)*e^(-6)*log(x^2*e^3 + 3*x*e^3 + 3) -
9*sqrt(3)*(e^3 - 1)*e^(-9/2)*log((2*x*e^3 - sqrt(3)*sqrt(3*e^3 - 4)*e^(3/2) + 3*e^3)/(2*x*e^3 + sqrt(3)*sqrt(3
*e^3 - 4)*e^(3/2) + 3*e^3))/sqrt(3*e^3 - 4) + (x^2 - 6*x)*e^(-3))*e^3 - 3/2*(sqrt(3)*(3*e^3 - 2)*e^(-9/2)*log(
(2*x*e^3 - sqrt(3)*sqrt(3*e^3 - 4)*e^(3/2) + 3*e^3)/(2*x*e^3 + sqrt(3)*sqrt(3*e^3 - 4)*e^(3/2) + 3*e^3))/sqrt(
3*e^3 - 4) + 2*x*e^(-3) - 3*e^(-3)*log(x^2*e^3 + 3*x*e^3 + 3))*e^3 + 1/2*(2*x^2*e^3 - 6*x*e^3 - (2*x^2*e^3 - 9
*e^3 + 6)*log(x^2*e^3 + 3*x*e^3 + 3))*e^(-3)

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mupad [B]  time = 0.71, size = 27, normalized size = 1.12 \begin {gather*} \ln \left ({\mathrm {e}}^3\,x^2+3\,{\mathrm {e}}^3\,x+3\right )\,\left (4\,{\ln \relax (2)}^4-x^2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(3)*(3*x^2 + 2*x^3) + log(exp(3)*(3*x + x^2) + 3)*(6*x + exp(3)*(6*x^2 + 2*x^3)) - exp(3)*log(2)^4*(8
*x + 12))/(exp(3)*(3*x + x^2) + 3),x)

[Out]

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

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sympy [A]  time = 0.49, size = 39, normalized size = 1.62 \begin {gather*} - x^{2} \log {\left (\left (x^{2} + 3 x\right ) e^{3} + 3 \right )} + 4 \log {\relax (2 )}^{4} \log {\left (x^{2} e^{3} + 3 x e^{3} + 3 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x**3-6*x**2)*exp(3)-6*x)*ln((x**2+3*x)*exp(3)+3)+(8*x+12)*exp(3)*ln(2)**4+(-2*x**3-3*x**2)*exp
(3))/((x**2+3*x)*exp(3)+3),x)

[Out]

-x**2*log((x**2 + 3*x)*exp(3) + 3) + 4*log(2)**4*log(x**2*exp(3) + 3*x*exp(3) + 3)

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