3.98.99 \(\int \frac {(-36+12 e^4) \log (4 e^4-4 e^2 x+x^2)}{2 e^2-x} \, dx\)

Optimal. Leaf size=23 \[ 3 \left (3-e^4\right ) \log ^2\left (\left (2 e^2-x\right )^2\right ) \]

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Rubi [B]  time = 0.08, antiderivative size = 57, normalized size of antiderivative = 2.48, number of steps used = 6, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {12, 2524, 27, 21, 2390, 2301} \begin {gather*} 12 \left (3-e^4\right ) \log \left (2 e^2-x\right ) \log \left (x^2-4 e^2 x+4 e^4\right )-12 \left (3-e^4\right ) \log ^2\left (2 e^2-x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-36 + 12*E^4)*Log[4*E^4 - 4*E^2*x + x^2])/(2*E^2 - x),x]

[Out]

-12*(3 - E^4)*Log[2*E^2 - x]^2 + 12*(3 - E^4)*Log[2*E^2 - x]*Log[4*E^4 - 4*E^2*x + 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 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*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 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2524

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\left (\left (12 \left (3-e^4\right )\right ) \int \frac {\log \left (4 e^4-4 e^2 x+x^2\right )}{2 e^2-x} \, dx\right )\\ &=12 \left (3-e^4\right ) \log \left (2 e^2-x\right ) \log \left (4 e^4-4 e^2 x+x^2\right )-\left (12 \left (3-e^4\right )\right ) \int \frac {\left (-4 e^2+2 x\right ) \log \left (2 e^2-x\right )}{4 e^4-4 e^2 x+x^2} \, dx\\ &=12 \left (3-e^4\right ) \log \left (2 e^2-x\right ) \log \left (4 e^4-4 e^2 x+x^2\right )-\left (12 \left (3-e^4\right )\right ) \int \frac {\left (-4 e^2+2 x\right ) \log \left (2 e^2-x\right )}{\left (-2 e^2+x\right )^2} \, dx\\ &=12 \left (3-e^4\right ) \log \left (2 e^2-x\right ) \log \left (4 e^4-4 e^2 x+x^2\right )-\left (24 \left (3-e^4\right )\right ) \int \frac {\log \left (2 e^2-x\right )}{-2 e^2+x} \, dx\\ &=12 \left (3-e^4\right ) \log \left (2 e^2-x\right ) \log \left (4 e^4-4 e^2 x+x^2\right )-\left (24 \left (3-e^4\right )\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,2 e^2-x\right )\\ &=-12 \left (3-e^4\right ) \log ^2\left (2 e^2-x\right )+12 \left (3-e^4\right ) \log \left (2 e^2-x\right ) \log \left (4 e^4-4 e^2 x+x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 19, normalized size = 0.83 \begin {gather*} -3 \left (-3+e^4\right ) \log ^2\left (\left (-2 e^2+x\right )^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-36 + 12*E^4)*Log[4*E^4 - 4*E^2*x + x^2])/(2*E^2 - x),x]

[Out]

-3*(-3 + E^4)*Log[(-2*E^2 + x)^2]^2

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fricas [A]  time = 0.52, size = 22, normalized size = 0.96 \begin {gather*} -3 \, {\left (e^{4} - 3\right )} \log \left (x^{2} - 4 \, x e^{2} + 4 \, e^{4}\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*exp(4)-36)*log(4*exp(2)^2-4*exp(2)*x+x^2)/(2*exp(2)-x),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.18, size = 22, normalized size = 0.96 \begin {gather*} -3 \, {\left (e^{4} - 3\right )} \log \left (x^{2} - 4 \, x e^{2} + 4 \, e^{4}\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*exp(4)-36)*log(4*exp(2)^2-4*exp(2)*x+x^2)/(2*exp(2)-x),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.13, size = 26, normalized size = 1.13

method	result	size
norman	\(\left (-3 \,{\mathrm e}^{4}+9\right ) \ln \left (4 \,{\mathrm e}^{4}-4 \,{\mathrm e}^{2} x +x^{2}\right )^{2}\)	\(26\)
default	\(\left (12 \,{\mathrm e}^{4}-36\right ) \left (-\ln \left (-2 \,{\mathrm e}^{2}+x \right ) \ln \left (4 \,{\mathrm e}^{4}-4 \,{\mathrm e}^{2} x +x^{2}\right )+\ln \left (-2 \,{\mathrm e}^{2}+x \right )^{2}\right )\)	\(43\)
risch	\(-\left (12 \,{\mathrm e}^{4}-36\right ) \ln \left (-2 \,{\mathrm e}^{2}+x \right ) \ln \left (4 \,{\mathrm e}^{4}-4 \,{\mathrm e}^{2} x +x^{2}\right )+12 \ln \left (-2 \,{\mathrm e}^{2}+x \right )^{2} {\mathrm e}^{4}-36 \ln \left (-2 \,{\mathrm e}^{2}+x \right )^{2}\)	\(55\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((12*exp(4)-36)*ln(4*exp(2)^2-4*exp(2)*x+x^2)/(2*exp(2)-x),x,method=_RETURNVERBOSE)

[Out]

(-3*exp(4)+9)*ln(4*exp(2)^2-4*exp(2)*x+x^2)^2

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maxima [B]  time = 0.37, size = 68, normalized size = 2.96 \begin {gather*} -12 \, {\left (e^{4} - 3\right )} \log \left (x^{2} - 4 \, x e^{2} + 4 \, e^{4}\right ) \log \left (x - 2 \, e^{2}\right ) + 12 \, {\left (\log \left (x^{2} - 4 \, x e^{2} + 4 \, e^{4}\right ) \log \left (x - 2 \, e^{2}\right ) - \log \left (x - 2 \, e^{2}\right )^{2}\right )} {\left (e^{4} - 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*exp(4)-36)*log(4*exp(2)^2-4*exp(2)*x+x^2)/(2*exp(2)-x),x, algorithm="maxima")

[Out]

-12*(e^4 - 3)*log(x^2 - 4*x*e^2 + 4*e^4)*log(x - 2*e^2) + 12*(log(x^2 - 4*x*e^2 + 4*e^4)*log(x - 2*e^2) - log(
x - 2*e^2)^2)*(e^4 - 3)

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mupad [B]  time = 5.98, size = 24, normalized size = 1.04 \begin {gather*} -{\ln \left (x^2-4\,{\mathrm {e}}^2\,x+4\,{\mathrm {e}}^4\right )}^2\,\left (3\,{\mathrm {e}}^4-9\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(4*exp(4) - 4*x*exp(2) + x^2)*(12*exp(4) - 36))/(x - 2*exp(2)),x)

[Out]

-log(4*exp(4) - 4*x*exp(2) + x^2)^2*(3*exp(4) - 9)

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sympy [A]  time = 0.16, size = 24, normalized size = 1.04 \begin {gather*} \left (9 - 3 e^{4}\right ) \log {\left (x^{2} - 4 x e^{2} + 4 e^{4} \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*exp(4)-36)*ln(4*exp(2)**2-4*exp(2)*x+x**2)/(2*exp(2)-x),x)

[Out]

(9 - 3*exp(4))*log(x**2 - 4*x*exp(2) + 4*exp(4))**2

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