3.87.43 \(\int \frac {-2 x^2+2 x^3+4 x^4-2 x^5+e^3 (4 x^3-2 x^4)+e^6 (x^2-2 e^3 x^3-2 x^4)+(1+2 e^3 x+2 x^2) \log (2)}{2 x^3-e^6 x^3-x^4+x \log (2)} \, dx\)

Optimal. Leaf size=32 \[ \left (e^3+x\right )^2-\log \left (2 x-e^6 x-x^2+\frac {\log (2)}{x}\right ) \]

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Rubi [A]  time = 0.48, antiderivative size = 35, normalized size of antiderivative = 1.09, number of steps used = 5, number of rules used = 4, integrand size = 99, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {6, 1594, 6742, 1587} \begin {gather*} x^2-\log \left (x^3-\left (2-e^6\right ) x^2-\log (2)\right )+2 e^3 x+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2*x^2 + 2*x^3 + 4*x^4 - 2*x^5 + E^3*(4*x^3 - 2*x^4) + E^6*(x^2 - 2*E^3*x^3 - 2*x^4) + (1 + 2*E^3*x + 2*x
^2)*Log[2])/(2*x^3 - E^6*x^3 - x^4 + x*Log[2]),x]

[Out]

2*E^3*x + x^2 + Log[x] - Log[-((2 - E^6)*x^2) + x^3 - Log[2]]

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 1587

Int[(Pp_)/(Qq_), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*Log[RemoveConte
nt[Qq, x]])/(q*Coeff[Qq, x, q]), x] /; EqQ[p, q - 1] && EqQ[Pp, Simplify[(Coeff[Pp, x, p]*D[Qq, x])/(q*Coeff[Q
q, x, q])]]] /; PolyQ[Pp, x] && PolyQ[Qq, x]

Rule 1594

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

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2 x^2+2 x^3+4 x^4-2 x^5+e^3 \left (4 x^3-2 x^4\right )+e^6 \left (x^2-2 e^3 x^3-2 x^4\right )+\left (1+2 e^3 x+2 x^2\right ) \log (2)}{\left (2-e^6\right ) x^3-x^4+x \log (2)} \, dx\\ &=\int \frac {-2 x^2+2 x^3+4 x^4-2 x^5+e^3 \left (4 x^3-2 x^4\right )+e^6 \left (x^2-2 e^3 x^3-2 x^4\right )+\left (1+2 e^3 x+2 x^2\right ) \log (2)}{x \left (\left (2-e^6\right ) x^2-x^3+\log (2)\right )} \, dx\\ &=\int \left (2 e^3+\frac {1}{x}+2 x+\frac {x \left (-4+2 e^6+3 x\right )}{\left (2-e^6\right ) x^2-x^3+\log (2)}\right ) \, dx\\ &=2 e^3 x+x^2+\log (x)+\int \frac {x \left (-4+2 e^6+3 x\right )}{\left (2-e^6\right ) x^2-x^3+\log (2)} \, dx\\ &=2 e^3 x+x^2+\log (x)-\log \left (-\left (\left (2-e^6\right ) x^2\right )+x^3-\log (2)\right )\\ \end {aligned} \end {gather*}

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Mathematica [C]  time = 0.11, size = 48, normalized size = 1.50 \begin {gather*} x \left (2 e^3+x\right )+\log (x)-\text {RootSum}\left [\log (2)+2 \text {$\#$1}^2-e^6 \text {$\#$1}^2-\text {$\#$1}^3\&,\log (x-\text {$\#$1})\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*x^2 + 2*x^3 + 4*x^4 - 2*x^5 + E^3*(4*x^3 - 2*x^4) + E^6*(x^2 - 2*E^3*x^3 - 2*x^4) + (1 + 2*E^3*x
+ 2*x^2)*Log[2])/(2*x^3 - E^6*x^3 - x^4 + x*Log[2]),x]

[Out]

x*(2*E^3 + x) + Log[x] - RootSum[Log[2] + 2*#1^2 - E^6*#1^2 - #1^3 & , Log[x - #1] & ]

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fricas [A]  time = 0.72, size = 33, normalized size = 1.03 \begin {gather*} x^{2} + 2 \, x e^{3} - \log \left (x^{3} + x^{2} e^{6} - 2 \, x^{2} - \log \relax (2)\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*exp(3)+2*x^2+1)*log(2)+(-2*x^3*exp(3)-2*x^4+x^2)*exp(6)+(-2*x^4+4*x^3)*exp(3)-2*x^5+4*x^4+2*x^
3-2*x^2)/(x*log(2)-x^3*exp(6)-x^4+2*x^3),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.17, size = 35, normalized size = 1.09 \begin {gather*} x^{2} + 2 \, x e^{3} - \log \left ({\left | x^{3} + x^{2} e^{6} - 2 \, x^{2} - \log \relax (2) \right |}\right ) + \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*exp(3)+2*x^2+1)*log(2)+(-2*x^3*exp(3)-2*x^4+x^2)*exp(6)+(-2*x^4+4*x^3)*exp(3)-2*x^5+4*x^4+2*x^
3-2*x^2)/(x*log(2)-x^3*exp(6)-x^4+2*x^3),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.09, size = 31, normalized size = 0.97

method	result	size
risch	\(2 x \,{\mathrm e}^{3}+x^{2}+\ln \relax (x )-\ln \left (x^{3}+\left ({\mathrm e}^{6}-2\right ) x^{2}-\ln \relax (2)\right )\)	\(31\)
default	\(x^{2}+2 x \,{\mathrm e}^{3}+\ln \relax (x )-\ln \left (x^{2} {\mathrm e}^{6}+x^{3}-2 x^{2}-\ln \relax (2)\right )\)	\(34\)
norman	\(x^{2}+2 x \,{\mathrm e}^{3}-\ln \left (-x^{2} {\mathrm e}^{6}-x^{3}+2 x^{2}+\ln \relax (2)\right )+\ln \relax (x )\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x*exp(3)+2*x^2+1)*ln(2)+(-2*x^3*exp(3)-2*x^4+x^2)*exp(6)+(-2*x^4+4*x^3)*exp(3)-2*x^5+4*x^4+2*x^3-2*x^2
)/(x*ln(2)-x^3*exp(6)-x^4+2*x^3),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.39, size = 30, normalized size = 0.94 \begin {gather*} x^{2} + 2 \, x e^{3} - \log \left (x^{3} + x^{2} {\left (e^{6} - 2\right )} - \log \relax (2)\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*exp(3)+2*x^2+1)*log(2)+(-2*x^3*exp(3)-2*x^4+x^2)*exp(6)+(-2*x^4+4*x^3)*exp(3)-2*x^5+4*x^4+2*x^
3-2*x^2)/(x*log(2)-x^3*exp(6)-x^4+2*x^3),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 5.41, size = 33, normalized size = 1.03 \begin {gather*} \ln \relax (x)-\ln \left (x^2\,{\mathrm {e}}^6-\ln \relax (2)-2\,x^2+x^3\right )+2\,x\,{\mathrm {e}}^3+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(3)*(4*x^3 - 2*x^4) - exp(6)*(2*x^3*exp(3) - x^2 + 2*x^4) + log(2)*(2*x*exp(3) + 2*x^2 + 1) - 2*x^2 +
2*x^3 + 4*x^4 - 2*x^5)/(x*log(2) - x^3*exp(6) + 2*x^3 - x^4),x)

[Out]

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

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sympy [A]  time = 1.15, size = 29, normalized size = 0.91 \begin {gather*} x^{2} + 2 x e^{3} + \log {\relax (x )} - \log {\left (x^{3} + x^{2} \left (-2 + e^{6}\right ) - \log {\relax (2 )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*exp(3)+2*x**2+1)*ln(2)+(-2*x**3*exp(3)-2*x**4+x**2)*exp(6)+(-2*x**4+4*x**3)*exp(3)-2*x**5+4*x*
*4+2*x**3-2*x**2)/(x*ln(2)-x**3*exp(6)-x**4+2*x**3),x)

[Out]

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

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