3.87.89 \(\int \frac {1-676 x-e^{2 e^2} x-104 x^2-315 x^3-16 x^4-20 x^5+e^{e^2} (-52 x-4 x^2-12 x^3)}{-676 x^2-e^{2 e^2} x^2-52 x^3-105 x^4-4 x^5-4 x^6+e^{e^2} (-52 x^2-2 x^3-4 x^4)+x \log (x)} \, dx\)

Optimal. Leaf size=22 \[ \log \left (-x \left (26+e^{e^2}+x+2 x^2\right )^2+\log (x)\right ) \]

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Rubi [A]  time = 0.59, antiderivative size = 23, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 3, integrand size = 122, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {6, 6688, 6684} \begin {gather*} \log \left (x \left (2 x^2+x+e^{e^2}+26\right )^2-\log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 676*x - E^(2*E^2)*x - 104*x^2 - 315*x^3 - 16*x^4 - 20*x^5 + E^E^2*(-52*x - 4*x^2 - 12*x^3))/(-676*x^2
- E^(2*E^2)*x^2 - 52*x^3 - 105*x^4 - 4*x^5 - 4*x^6 + E^E^2*(-52*x^2 - 2*x^3 - 4*x^4) + x*Log[x]),x]

[Out]

Log[x*(26 + E^E^2 + x + 2*x^2)^2 - Log[x]]

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 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

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 \frac {1+\left (-676-e^{2 e^2}\right ) x-104 x^2-315 x^3-16 x^4-20 x^5+e^{e^2} \left (-52 x-4 x^2-12 x^3\right )}{-676 x^2-e^{2 e^2} x^2-52 x^3-105 x^4-4 x^5-4 x^6+e^{e^2} \left (-52 x^2-2 x^3-4 x^4\right )+x \log (x)} \, dx\\ &=\int \frac {1+\left (-676-e^{2 e^2}\right ) x-104 x^2-315 x^3-16 x^4-20 x^5+e^{e^2} \left (-52 x-4 x^2-12 x^3\right )}{\left (-676-e^{2 e^2}\right ) x^2-52 x^3-105 x^4-4 x^5-4 x^6+e^{e^2} \left (-52 x^2-2 x^3-4 x^4\right )+x \log (x)} \, dx\\ &=\int \frac {-1+\left (26+e^{e^2}\right )^2 x+4 \left (26+e^{e^2}\right ) x^2+3 \left (105+4 e^{e^2}\right ) x^3+16 x^4+20 x^5}{x \left (x \left (26+e^{e^2}+x+2 x^2\right )^2-\log (x)\right )} \, dx\\ &=\log \left (x \left (26+e^{e^2}+x+2 x^2\right )^2-\log (x)\right )\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 2.67, size = 66, normalized size = 3.00 \begin {gather*} \log \left (676 x+52 e^{e^2} x+e^{2 e^2} x+52 x^2+2 e^{e^2} x^2+105 x^3+4 e^{e^2} x^3+4 x^4+4 x^5-\log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - 676*x - E^(2*E^2)*x - 104*x^2 - 315*x^3 - 16*x^4 - 20*x^5 + E^E^2*(-52*x - 4*x^2 - 12*x^3))/(-6
76*x^2 - E^(2*E^2)*x^2 - 52*x^3 - 105*x^4 - 4*x^5 - 4*x^6 + E^E^2*(-52*x^2 - 2*x^3 - 4*x^4) + x*Log[x]),x]

[Out]

Log[676*x + 52*E^E^2*x + E^(2*E^2)*x + 52*x^2 + 2*E^E^2*x^2 + 105*x^3 + 4*E^E^2*x^3 + 4*x^4 + 4*x^5 - Log[x]]

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fricas [B]  time = 0.71, size = 52, normalized size = 2.36 \begin {gather*} \log \left (-4 \, x^{5} - 4 \, x^{4} - 105 \, x^{3} - 52 \, x^{2} - x e^{\left (2 \, e^{2}\right )} - 2 \, {\left (2 \, x^{3} + x^{2} + 26 \, x\right )} e^{\left (e^{2}\right )} - 676 \, x + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x*exp(exp(2))^2+(-12*x^3-4*x^2-52*x)*exp(exp(2))-20*x^5-16*x^4-315*x^3-104*x^2-676*x+1)/(x*log(x)-
x^2*exp(exp(2))^2+(-4*x^4-2*x^3-52*x^2)*exp(exp(2))-4*x^6-4*x^5-105*x^4-52*x^3-676*x^2),x, algorithm="fricas")

[Out]

log(-4*x^5 - 4*x^4 - 105*x^3 - 52*x^2 - x*e^(2*e^2) - 2*(2*x^3 + x^2 + 26*x)*e^(e^2) - 676*x + log(x))

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giac [B]  time = 0.26, size = 57, normalized size = 2.59 \begin {gather*} \log \left (-4 \, x^{5} - 4 \, x^{4} - 4 \, x^{3} e^{\left (e^{2}\right )} - 105 \, x^{3} - 2 \, x^{2} e^{\left (e^{2}\right )} - 52 \, x^{2} - x e^{\left (2 \, e^{2}\right )} - 52 \, x e^{\left (e^{2}\right )} - 676 \, x + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x*exp(exp(2))^2+(-12*x^3-4*x^2-52*x)*exp(exp(2))-20*x^5-16*x^4-315*x^3-104*x^2-676*x+1)/(x*log(x)-
x^2*exp(exp(2))^2+(-4*x^4-2*x^3-52*x^2)*exp(exp(2))-4*x^6-4*x^5-105*x^4-52*x^3-676*x^2),x, algorithm="giac")

[Out]

log(-4*x^5 - 4*x^4 - 4*x^3*e^(e^2) - 105*x^3 - 2*x^2*e^(e^2) - 52*x^2 - x*e^(2*e^2) - 52*x*e^(e^2) - 676*x + l
og(x))

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maple [B]  time = 0.10, size = 58, normalized size = 2.64

method	result	size
risch	\(\ln \left (-4 x^{5}-4 x^{3} {\mathrm e}^{{\mathrm e}^{2}}-4 x^{4}-x \,{\mathrm e}^{2 \,{\mathrm e}^{2}}-2 x^{2} {\mathrm e}^{{\mathrm e}^{2}}-105 x^{3}-52 x \,{\mathrm e}^{{\mathrm e}^{2}}-52 x^{2}+\ln \relax (x )-676 x \right )\)	\(58\)
norman	\(\ln \left (4 x^{5}+4 x^{3} {\mathrm e}^{{\mathrm e}^{2}}+4 x^{4}+x \,{\mathrm e}^{2 \,{\mathrm e}^{2}}+2 x^{2} {\mathrm e}^{{\mathrm e}^{2}}+105 x^{3}+52 x \,{\mathrm e}^{{\mathrm e}^{2}}+52 x^{2}+676 x -\ln \relax (x )\right )\)	\(59\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x*exp(exp(2))^2+(-12*x^3-4*x^2-52*x)*exp(exp(2))-20*x^5-16*x^4-315*x^3-104*x^2-676*x+1)/(x*ln(x)-x^2*exp
(exp(2))^2+(-4*x^4-2*x^3-52*x^2)*exp(exp(2))-4*x^6-4*x^5-105*x^4-52*x^3-676*x^2),x,method=_RETURNVERBOSE)

[Out]

ln(-4*x^5-4*x^3*exp(exp(2))-4*x^4-x*exp(2*exp(2))-2*x^2*exp(exp(2))-105*x^3-52*x*exp(exp(2))-52*x^2+ln(x)-676*
x)

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maxima [B]  time = 0.39, size = 51, normalized size = 2.32 \begin {gather*} \log \left (-4 \, x^{5} - 4 \, x^{4} - x^{3} {\left (4 \, e^{\left (e^{2}\right )} + 105\right )} - 2 \, x^{2} {\left (e^{\left (e^{2}\right )} + 26\right )} - x {\left (e^{\left (2 \, e^{2}\right )} + 52 \, e^{\left (e^{2}\right )} + 676\right )} + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x*exp(exp(2))^2+(-12*x^3-4*x^2-52*x)*exp(exp(2))-20*x^5-16*x^4-315*x^3-104*x^2-676*x+1)/(x*log(x)-
x^2*exp(exp(2))^2+(-4*x^4-2*x^3-52*x^2)*exp(exp(2))-4*x^6-4*x^5-105*x^4-52*x^3-676*x^2),x, algorithm="maxima")

[Out]

log(-4*x^5 - 4*x^4 - x^3*(4*e^(e^2) + 105) - 2*x^2*(e^(e^2) + 26) - x*(e^(2*e^2) + 52*e^(e^2) + 676) + log(x))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {676\,x+x\,{\mathrm {e}}^{2\,{\mathrm {e}}^2}+{\mathrm {e}}^{{\mathrm {e}}^2}\,\left (12\,x^3+4\,x^2+52\,x\right )+104\,x^2+315\,x^3+16\,x^4+20\,x^5-1}{{\mathrm {e}}^{{\mathrm {e}}^2}\,\left (4\,x^4+2\,x^3+52\,x^2\right )+x^2\,{\mathrm {e}}^{2\,{\mathrm {e}}^2}-x\,\ln \relax (x)+676\,x^2+52\,x^3+105\,x^4+4\,x^5+4\,x^6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((676*x + x*exp(2*exp(2)) + exp(exp(2))*(52*x + 4*x^2 + 12*x^3) + 104*x^2 + 315*x^3 + 16*x^4 + 20*x^5 - 1)/
(exp(exp(2))*(52*x^2 + 2*x^3 + 4*x^4) + x^2*exp(2*exp(2)) - x*log(x) + 676*x^2 + 52*x^3 + 105*x^4 + 4*x^5 + 4*
x^6),x)

[Out]

int((676*x + x*exp(2*exp(2)) + exp(exp(2))*(52*x + 4*x^2 + 12*x^3) + 104*x^2 + 315*x^3 + 16*x^4 + 20*x^5 - 1)/
(exp(exp(2))*(52*x^2 + 2*x^3 + 4*x^4) + x^2*exp(2*exp(2)) - x*log(x) + 676*x^2 + 52*x^3 + 105*x^4 + 4*x^5 + 4*
x^6), x)

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sympy [B]  time = 0.28, size = 65, normalized size = 2.95 \begin {gather*} \log {\left (- 4 x^{5} - 4 x^{4} - 4 x^{3} e^{e^{2}} - 105 x^{3} - 2 x^{2} e^{e^{2}} - 52 x^{2} - x e^{2 e^{2}} - 52 x e^{e^{2}} - 676 x + \log {\relax (x )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x*exp(exp(2))**2+(-12*x**3-4*x**2-52*x)*exp(exp(2))-20*x**5-16*x**4-315*x**3-104*x**2-676*x+1)/(x*
ln(x)-x**2*exp(exp(2))**2+(-4*x**4-2*x**3-52*x**2)*exp(exp(2))-4*x**6-4*x**5-105*x**4-52*x**3-676*x**2),x)

[Out]

log(-4*x**5 - 4*x**4 - 4*x**3*exp(exp(2)) - 105*x**3 - 2*x**2*exp(exp(2)) - 52*x**2 - x*exp(2*exp(2)) - 52*x*e
xp(exp(2)) - 676*x + log(x))

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