3.71.35 \(\int \frac {e^{2 e^4} x^2-10 e^{3 e^4} x^2+25 e^{4 e^4} x^2-2 x \log (36)+10 e^{e^4} x \log (36)}{e^{4 e^4} x^2-10 e^{5 e^4} x^2+25 e^{6 e^4} x^2-2 e^{2 e^4} x \log (36)+10 e^{3 e^4} x \log (36)+\log ^2(36)} \, dx\)

Optimal. Leaf size=28 \[ \frac {x}{e^{2 e^4}-\frac {\log (36)}{x-5 e^{e^4} x}} \]

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Rubi [B]  time = 0.11, antiderivative size = 58, normalized size of antiderivative = 2.07, number of steps used = 10, number of rules used = 3, integrand size = 118, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {6, 27, 683} \begin {gather*} e^{-2 e^4} x+\frac {e^{-4 e^4} \log ^2(36)}{\left (1-5 e^{e^4}\right ) \left (e^{2 e^4} \left (1-5 e^{e^4}\right ) x-\log (36)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(2*E^4)*x^2 - 10*E^(3*E^4)*x^2 + 25*E^(4*E^4)*x^2 - 2*x*Log[36] + 10*E^E^4*x*Log[36])/(E^(4*E^4)*x^2 -
10*E^(5*E^4)*x^2 + 25*E^(6*E^4)*x^2 - 2*E^(2*E^4)*x*Log[36] + 10*E^(3*E^4)*x*Log[36] + Log[36]^2),x]

[Out]

x/E^(2*E^4) + Log[36]^2/(E^(4*E^4)*(1 - 5*E^E^4)*(E^(2*E^4)*(1 - 5*E^E^4)*x - Log[36]))

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 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 683

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {25 e^{4 e^4} x^2+\left (e^{2 e^4}-10 e^{3 e^4}\right ) x^2-2 x \log (36)+10 e^{e^4} x \log (36)}{e^{4 e^4} x^2-10 e^{5 e^4} x^2+25 e^{6 e^4} x^2-2 e^{2 e^4} x \log (36)+10 e^{3 e^4} x \log (36)+\log ^2(36)} \, dx\\ &=\int \frac {\left (e^{2 e^4}-10 e^{3 e^4}+25 e^{4 e^4}\right ) x^2-2 x \log (36)+10 e^{e^4} x \log (36)}{e^{4 e^4} x^2-10 e^{5 e^4} x^2+25 e^{6 e^4} x^2-2 e^{2 e^4} x \log (36)+10 e^{3 e^4} x \log (36)+\log ^2(36)} \, dx\\ &=\int \frac {\left (e^{2 e^4}-10 e^{3 e^4}+25 e^{4 e^4}\right ) x^2+\left (-2+10 e^{e^4}\right ) x \log (36)}{e^{4 e^4} x^2-10 e^{5 e^4} x^2+25 e^{6 e^4} x^2-2 e^{2 e^4} x \log (36)+10 e^{3 e^4} x \log (36)+\log ^2(36)} \, dx\\ &=\int \frac {\left (e^{2 e^4}-10 e^{3 e^4}+25 e^{4 e^4}\right ) x^2+\left (-2+10 e^{e^4}\right ) x \log (36)}{25 e^{6 e^4} x^2+\left (e^{4 e^4}-10 e^{5 e^4}\right ) x^2-2 e^{2 e^4} x \log (36)+10 e^{3 e^4} x \log (36)+\log ^2(36)} \, dx\\ &=\int \frac {\left (e^{2 e^4}-10 e^{3 e^4}+25 e^{4 e^4}\right ) x^2+\left (-2+10 e^{e^4}\right ) x \log (36)}{\left (e^{4 e^4}-10 e^{5 e^4}+25 e^{6 e^4}\right ) x^2-2 e^{2 e^4} x \log (36)+10 e^{3 e^4} x \log (36)+\log ^2(36)} \, dx\\ &=\int \frac {\left (e^{2 e^4}-10 e^{3 e^4}+25 e^{4 e^4}\right ) x^2+\left (-2+10 e^{e^4}\right ) x \log (36)}{\left (e^{4 e^4}-10 e^{5 e^4}+25 e^{6 e^4}\right ) x^2+\left (-2 e^{2 e^4}+10 e^{3 e^4}\right ) x \log (36)+\log ^2(36)} \, dx\\ &=\int \frac {\left (e^{2 e^4}-10 e^{3 e^4}+25 e^{4 e^4}\right ) x^2+\left (-2+10 e^{e^4}\right ) x \log (36)}{\left (-e^{2 e^4} x+5 e^{3 e^4} x+\log (36)\right )^2} \, dx\\ &=\int \frac {\left (e^{2 e^4}-10 e^{3 e^4}+25 e^{4 e^4}\right ) x^2+\left (-2+10 e^{e^4}\right ) x \log (36)}{\left (\left (-e^{2 e^4}+5 e^{3 e^4}\right ) x+\log (36)\right )^2} \, dx\\ &=\int \left (e^{-2 e^4}-\frac {e^{-2 e^4} \log ^2(36)}{\left (e^{2 e^4} \left (1-5 e^{e^4}\right ) x-\log (36)\right )^2}\right ) \, dx\\ &=e^{-2 e^4} x+\frac {e^{-4 e^4} \log ^2(36)}{\left (1-5 e^{e^4}\right ) \left (e^{2 e^4} \left (1-5 e^{e^4}\right ) x-\log (36)\right )}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.07, size = 118, normalized size = 4.21 \begin {gather*} \frac {e^{-4 e^4} \left (-1+5 e^{e^4}\right ) \left (e^{4 e^4} x^2-10 e^{5 e^4} x^2+25 e^{6 e^4} x^2-2 e^{2 e^4} x \log (36)+10 e^{3 e^4} x \log (36)+\log (36) \log (1296)\right )}{\left (1-5 e^{e^4}\right )^2 \left (-e^{2 e^4} x+5 e^{3 e^4} x+\log (36)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*E^4)*x^2 - 10*E^(3*E^4)*x^2 + 25*E^(4*E^4)*x^2 - 2*x*Log[36] + 10*E^E^4*x*Log[36])/(E^(4*E^4)*
x^2 - 10*E^(5*E^4)*x^2 + 25*E^(6*E^4)*x^2 - 2*E^(2*E^4)*x*Log[36] + 10*E^(3*E^4)*x*Log[36] + Log[36]^2),x]

[Out]

((-1 + 5*E^E^4)*(E^(4*E^4)*x^2 - 10*E^(5*E^4)*x^2 + 25*E^(6*E^4)*x^2 - 2*E^(2*E^4)*x*Log[36] + 10*E^(3*E^4)*x*
Log[36] + Log[36]*Log[1296]))/(E^(4*E^4)*(1 - 5*E^E^4)^2*(-(E^(2*E^4)*x) + 5*E^(3*E^4)*x + Log[36]))

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fricas [B]  time = 0.58, size = 101, normalized size = 3.61 \begin {gather*} \frac {25 \, x^{2} e^{\left (6 \, e^{4}\right )} - 10 \, x^{2} e^{\left (5 \, e^{4}\right )} + x^{2} e^{\left (4 \, e^{4}\right )} + 10 \, x e^{\left (3 \, e^{4}\right )} \log \relax (6) - 2 \, x e^{\left (2 \, e^{4}\right )} \log \relax (6) + 4 \, \log \relax (6)^{2}}{25 \, x e^{\left (8 \, e^{4}\right )} - 10 \, x e^{\left (7 \, e^{4}\right )} + x e^{\left (6 \, e^{4}\right )} + 10 \, e^{\left (5 \, e^{4}\right )} \log \relax (6) - 2 \, e^{\left (4 \, e^{4}\right )} \log \relax (6)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((25*x^2*exp(exp(4))^4-10*x^2*exp(exp(4))^3+x^2*exp(exp(4))^2+20*x*log(6)*exp(exp(4))-4*x*log(6))/(25
*x^2*exp(exp(4))^6-10*x^2*exp(exp(4))^5+x^2*exp(exp(4))^4+20*x*log(6)*exp(exp(4))^3-4*x*log(6)*exp(exp(4))^2+4
*log(6)^2),x, algorithm="fricas")

[Out]

(25*x^2*e^(6*e^4) - 10*x^2*e^(5*e^4) + x^2*e^(4*e^4) + 10*x*e^(3*e^4)*log(6) - 2*x*e^(2*e^4)*log(6) + 4*log(6)
^2)/(25*x*e^(8*e^4) - 10*x*e^(7*e^4) + x*e^(6*e^4) + 10*e^(5*e^4)*log(6) - 2*e^(4*e^4)*log(6))

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((25*x^2*exp(exp(4))^4-10*x^2*exp(exp(4))^3+x^2*exp(exp(4))^2+20*x*log(6)*exp(exp(4))-4*x*log(6))/(25
*x^2*exp(exp(4))^6-10*x^2*exp(exp(4))^5+x^2*exp(exp(4))^4+20*x*log(6)*exp(exp(4))^3-4*x*log(6)*exp(exp(4))^2+4
*log(6)^2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: (250*exp(6*exp(4))*ln(6)*exp(exp(4))-50*
exp(6*exp(4))*ln(6)-100*exp(5*exp(4))*ln(6)*exp(exp(4))+20*exp(5*exp(4))*ln(6)-250*exp(4*exp(4))*exp(3*exp(4))
*ln(6)+50*exp(4*exp(4

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maple [A]  time = 0.28, size = 35, normalized size = 1.25

method	result	size
gosper	\(\frac {x^{2} \left (5 \,{\mathrm e}^{{\mathrm e}^{4}}-1\right )}{5 x \,{\mathrm e}^{3 \,{\mathrm e}^{4}}-x \,{\mathrm e}^{2 \,{\mathrm e}^{4}}+2 \ln \relax (6)}\)	\(35\)
norman	\(\frac {x^{2} \left (5 \,{\mathrm e}^{{\mathrm e}^{4}}-1\right )}{5 x \,{\mathrm e}^{3 \,{\mathrm e}^{4}}-x \,{\mathrm e}^{2 \,{\mathrm e}^{4}}+2 \ln \relax (6)}\)	\(35\)
risch	\(x \,{\mathrm e}^{-2 \,{\mathrm e}^{4}}+\frac {2 \,{\mathrm e}^{-4 \,{\mathrm e}^{4}} \ln \relax (3)^{2}}{\left (5 \,{\mathrm e}^{{\mathrm e}^{4}}-1\right ) \left (\frac {5 x \,{\mathrm e}^{3 \,{\mathrm e}^{4}}}{2}-\frac {x \,{\mathrm e}^{2 \,{\mathrm e}^{4}}}{2}+\ln \relax (3)+\ln \relax (2)\right )}+\frac {4 \,{\mathrm e}^{-4 \,{\mathrm e}^{4}} \ln \relax (2) \ln \relax (3)}{\left (5 \,{\mathrm e}^{{\mathrm e}^{4}}-1\right ) \left (\frac {5 x \,{\mathrm e}^{3 \,{\mathrm e}^{4}}}{2}-\frac {x \,{\mathrm e}^{2 \,{\mathrm e}^{4}}}{2}+\ln \relax (3)+\ln \relax (2)\right )}+\frac {2 \,{\mathrm e}^{-4 \,{\mathrm e}^{4}} \ln \relax (2)^{2}}{\left (5 \,{\mathrm e}^{{\mathrm e}^{4}}-1\right ) \left (\frac {5 x \,{\mathrm e}^{3 \,{\mathrm e}^{4}}}{2}-\frac {x \,{\mathrm e}^{2 \,{\mathrm e}^{4}}}{2}+\ln \relax (3)+\ln \relax (2)\right )}\)	\(138\)
meijerg	\(\frac {2 \left (25 \,{\mathrm e}^{4 \,{\mathrm e}^{4}}-10 \,{\mathrm e}^{3 \,{\mathrm e}^{4}}+{\mathrm e}^{2 \,{\mathrm e}^{4}}\right ) {\mathrm e}^{-2 \,{\mathrm e}^{4}} \ln \relax (6) \left (\frac {x \,{\mathrm e}^{2 \,{\mathrm e}^{4}} \left (5 \,{\mathrm e}^{{\mathrm e}^{4}}-1\right ) \left (\frac {3 x \,{\mathrm e}^{2 \,{\mathrm e}^{4}} \left (5 \,{\mathrm e}^{{\mathrm e}^{4}}-1\right )}{2 \ln \relax (6)}+6\right )}{6 \ln \relax (6) \left (1+\frac {x \,{\mathrm e}^{2 \,{\mathrm e}^{4}} \left (5 \,{\mathrm e}^{{\mathrm e}^{4}}-1\right )}{2 \ln \relax (6)}\right )}-2 \ln \left (1+\frac {x \,{\mathrm e}^{2 \,{\mathrm e}^{4}} \left (5 \,{\mathrm e}^{{\mathrm e}^{4}}-1\right )}{2 \ln \relax (6)}\right )\right )}{\left (25 \,{\mathrm e}^{6 \,{\mathrm e}^{4}}-10 \,{\mathrm e}^{5 \,{\mathrm e}^{4}}+{\mathrm e}^{4 \,{\mathrm e}^{4}}\right ) \left (5 \,{\mathrm e}^{{\mathrm e}^{4}}-1\right )}+\frac {\left (20 \ln \relax (6) {\mathrm e}^{{\mathrm e}^{4}}-4 \ln \relax (6)\right ) \left (-\frac {x \,{\mathrm e}^{2 \,{\mathrm e}^{4}} \left (5 \,{\mathrm e}^{{\mathrm e}^{4}}-1\right )}{2 \ln \relax (6) \left (1+\frac {x \,{\mathrm e}^{2 \,{\mathrm e}^{4}} \left (5 \,{\mathrm e}^{{\mathrm e}^{4}}-1\right )}{2 \ln \relax (6)}\right )}+\ln \left (1+\frac {x \,{\mathrm e}^{2 \,{\mathrm e}^{4}} \left (5 \,{\mathrm e}^{{\mathrm e}^{4}}-1\right )}{2 \ln \relax (6)}\right )\right )}{25 \,{\mathrm e}^{6 \,{\mathrm e}^{4}}-10 \,{\mathrm e}^{5 \,{\mathrm e}^{4}}+{\mathrm e}^{4 \,{\mathrm e}^{4}}}\)	\(250\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((25*x^2*exp(exp(4))^4-10*x^2*exp(exp(4))^3+x^2*exp(exp(4))^2+20*x*ln(6)*exp(exp(4))-4*x*ln(6))/(25*x^2*exp
(exp(4))^6-10*x^2*exp(exp(4))^5+x^2*exp(exp(4))^4+20*x*ln(6)*exp(exp(4))^3-4*x*ln(6)*exp(exp(4))^2+4*ln(6)^2),
x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.37, size = 58, normalized size = 2.07 \begin {gather*} x e^{\left (-2 \, e^{4}\right )} + \frac {4 \, \log \relax (6)^{2}}{x {\left (25 \, e^{\left (8 \, e^{4}\right )} - 10 \, e^{\left (7 \, e^{4}\right )} + e^{\left (6 \, e^{4}\right )}\right )} + 2 \, {\left (5 \, e^{\left (5 \, e^{4}\right )} - e^{\left (4 \, e^{4}\right )}\right )} \log \relax (6)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((25*x^2*exp(exp(4))^4-10*x^2*exp(exp(4))^3+x^2*exp(exp(4))^2+20*x*log(6)*exp(exp(4))-4*x*log(6))/(25
*x^2*exp(exp(4))^6-10*x^2*exp(exp(4))^5+x^2*exp(exp(4))^4+20*x*log(6)*exp(exp(4))^3-4*x*log(6)*exp(exp(4))^2+4
*log(6)^2),x, algorithm="maxima")

[Out]

x*e^(-2*e^4) + 4*log(6)^2/(x*(25*e^(8*e^4) - 10*e^(7*e^4) + e^(6*e^4)) + 2*(5*e^(5*e^4) - e^(4*e^4))*log(6))

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mupad [B]  time = 4.39, size = 95, normalized size = 3.39 \begin {gather*} x\,{\mathrm {e}}^{-2\,{\mathrm {e}}^4}-\frac {\mathrm {atan}\left (\frac {\ln \relax (6)\,2{}\mathrm {i}-x\,{\mathrm {e}}^{2\,{\mathrm {e}}^4}\,1{}\mathrm {i}+x\,{\mathrm {e}}^{3\,{\mathrm {e}}^4}\,5{}\mathrm {i}}{\sqrt {2\,\ln \relax (6)+\ln \left (36\right )}\,\sqrt {2\,\ln \relax (6)-\ln \left (36\right )}}\right )\,{\mathrm {e}}^{-4\,{\mathrm {e}}^4}\,{\ln \relax (6)}^2\,4{}\mathrm {i}}{\left (5\,{\mathrm {e}}^{{\mathrm {e}}^4}-1\right )\,\sqrt {2\,\ln \relax (6)+\ln \left (36\right )}\,\sqrt {2\,\ln \relax (6)-\ln \left (36\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*exp(2*exp(4)) - 4*x*log(6) - 10*x^2*exp(3*exp(4)) + 25*x^2*exp(4*exp(4)) + 20*x*exp(exp(4))*log(6))/(
x^2*exp(4*exp(4)) - 10*x^2*exp(5*exp(4)) + 25*x^2*exp(6*exp(4)) + 4*log(6)^2 - 4*x*exp(2*exp(4))*log(6) + 20*x
*exp(3*exp(4))*log(6)),x)

[Out]

x*exp(-2*exp(4)) - (atan((log(6)*2i - x*exp(2*exp(4))*1i + x*exp(3*exp(4))*5i)/((2*log(6) + log(36))^(1/2)*(2*
log(6) - log(36))^(1/2)))*exp(-4*exp(4))*log(6)^2*4i)/((5*exp(exp(4)) - 1)*(2*log(6) + log(36))^(1/2)*(2*log(6
) - log(36))^(1/2))

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sympy [B]  time = 0.48, size = 63, normalized size = 2.25 \begin {gather*} \frac {x}{e^{2 e^{4}}} + \frac {4 \log {\relax (6 )}^{2}}{x \left (- 10 e^{7 e^{4}} + e^{6 e^{4}} + 25 e^{8 e^{4}}\right ) - 2 e^{4 e^{4}} \log {\relax (6 )} + 10 e^{5 e^{4}} \log {\relax (6 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((25*x**2*exp(exp(4))**4-10*x**2*exp(exp(4))**3+x**2*exp(exp(4))**2+20*x*ln(6)*exp(exp(4))-4*x*ln(6))
/(25*x**2*exp(exp(4))**6-10*x**2*exp(exp(4))**5+x**2*exp(exp(4))**4+20*x*ln(6)*exp(exp(4))**3-4*x*ln(6)*exp(ex
p(4))**2+4*ln(6)**2),x)

[Out]

x*exp(-2*exp(4)) + 4*log(6)**2/(x*(-10*exp(7*exp(4)) + exp(6*exp(4)) + 25*exp(8*exp(4))) - 2*exp(4*exp(4))*log
(6) + 10*exp(5*exp(4))*log(6))

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