3.2.79 \(\int \frac {2 x^2-4 x^5+(-2 x^2+4 x^4) \log (x)+(2-4 x^3+(-2+4 x^2) \log (x)) \log (\frac {-3 e^5 x+3 e^5 \log (x)}{2 x})}{-x^2+x \log (x)} \, dx\) [179]

Optimal. Leaf size=23 \[ \left (x^2+\log \left (\frac {3 e^5 (-x+\log (x))}{2 x}\right )\right )^2 \]

[Out]

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

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Rubi [A]
time = 0.25, antiderivative size = 21, normalized size of antiderivative = 0.91, number of steps used = 4, number of rules used = 4, integrand size = 78, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {2641, 6820, 12, 6818} \begin {gather*} \left (x^2+\log \left (-\frac {3}{2} \left (1-\frac {\log (x)}{x}\right )\right )+5\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(5 + x^2 + Log[(-3*(1 - Log[x]/x))/2])^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 2641

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

Rule 6818

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*(y^(m + 1)/(m + 1)), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6820

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

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Mathematica [A]
time = 0.02, size = 20, normalized size = 0.87 \begin {gather*} \left (5+x^2+\log \left (\frac {3}{2} \left (-1+\frac {\log (x)}{x}\right )\right )\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(5 + x^2 + Log[(3*(-1 + Log[x]/x))/2])^2

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 2.34, size = 1538, normalized size = 66.87

method	result	size
risch	\(\text {Expression too large to display}\)	\(1538\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25-Pi^2+2*ln(2)*ln(x)-2*ln(3)*ln(x)-2*x^2*ln(2)+2*ln(ln(x)-x)*ln(3)-2*ln(2)*ln(ln(x)-x)+10*ln(ln(x)-x)-2*ln(2)
*ln(3)+2*x^2*ln(3)+ln(3)^2-10*ln(2)+ln(2)^2-2*x^2*ln(x)+10*ln(3)-10*ln(x)+ln(x)^2+x^4+10*x^2-1/2*Pi^2*csgn(I/x
)*csgn(I*(ln(x)-x))^2*csgn(I*(ln(x)-x)/x)^3+2*I*Pi*ln(x)*csgn(I*(ln(x)-x)/x)^2+2*I*Pi*ln(2)*csgn(I*(ln(x)-x)/x
)^2-2*I*Pi*ln(ln(x)-x)*csgn(I*(ln(x)-x)/x)^2-2*I*Pi*x^2*csgn(I*(ln(x)-x)/x)^2-2*I*Pi*ln(3)*csgn(I*(ln(x)-x)/x)
^2-1/4*Pi^2*csgn(I/x)^2*csgn(I*(ln(x)-x))^2*csgn(I*(ln(x)-x)/x)^2+1/2*Pi^2*csgn(I/x)^2*csgn(I*(ln(x)-x))*csgn(
I*(ln(x)-x)/x)^3+Pi^2*csgn(I/x)*csgn(I*(ln(x)-x))*csgn(I*(ln(x)-x)/x)+I*Pi*x^2*csgn(I/x)*csgn(I*(ln(x)-x)/x)^2
-I*Pi*x^2*csgn(I*(ln(x)-x)/x)^3+I*Pi*ln(3)*csgn(I/x)*csgn(I*(ln(x)-x)/x)^2-I*Pi*ln(3)*csgn(I*(ln(x)-x)/x)^3-I*
Pi*ln(ln(x)-x)*csgn(I*(ln(x)-x)/x)^3+I*Pi*ln(ln(x)-x)*csgn(I/x)*csgn(I*(ln(x)-x)/x)^2-5*I*Pi*csgn(I*(ln(x)-x))
*csgn(I*(ln(x)-x)/x)^2+Pi^2*csgn(I/x)*csgn(I*(ln(x)-x)/x)^4-I*Pi*ln(2)*csgn(I/x)*csgn(I*(ln(x)-x)/x)^2+I*Pi*ln
(2)*csgn(I*(ln(x)-x)/x)^3+I*Pi*ln(x)*csgn(I*(ln(x)-x)/x)^3-I*Pi*ln(x)*csgn(I/x)*csgn(I*(ln(x)-x)/x)^2+I*Pi*ln(
2)*csgn(I/x)*csgn(I*(ln(x)-x))*csgn(I*(ln(x)-x)/x)+I*Pi*ln(x)*csgn(I/x)*csgn(I*(ln(x)-x))*csgn(I*(ln(x)-x)/x)-
I*Pi*ln(ln(x)-x)*csgn(I/x)*csgn(I*(ln(x)-x))*csgn(I*(ln(x)-x)/x)-I*Pi*ln(3)*csgn(I/x)*csgn(I*(ln(x)-x))*csgn(I
*(ln(x)-x)/x)-I*Pi*x^2*csgn(I/x)*csgn(I*(ln(x)-x))*csgn(I*(ln(x)-x)/x)-I*Pi*ln(ln(x)-x)*csgn(I*(ln(x)-x))*csgn
(I*(ln(x)-x)/x)^2-I*Pi*x^2*csgn(I*(ln(x)-x))*csgn(I*(ln(x)-x)/x)^2-I*Pi*ln(3)*csgn(I*(ln(x)-x))*csgn(I*(ln(x)-
x)/x)^2+I*Pi*ln(x)*csgn(I*(ln(x)-x))*csgn(I*(ln(x)-x)/x)^2+I*Pi*ln(2)*csgn(I*(ln(x)-x))*csgn(I*(ln(x)-x)/x)^2-
5*I*Pi*csgn(I/x)*csgn(I*(ln(x)-x))*csgn(I*(ln(x)-x)/x)-Pi^2*csgn(I/x)*csgn(I*(ln(x)-x))*csgn(I*(ln(x)-x)/x)^3-
1/4*Pi^2*csgn(I/x)^2*csgn(I*(ln(x)-x)/x)^4-1/4*Pi^2*csgn(I*(ln(x)-x)/x)^6-Pi^2*csgn(I/x)*csgn(I*(ln(x)-x)/x)^2
-Pi^2*csgn(I*(ln(x)-x)/x)^5+2*Pi^2*csgn(I*(ln(x)-x)/x)^2+Pi^2*csgn(I*(ln(x)-x)/x)^3+ln(x-ln(x))^2-10*I*Pi*csgn
(I*(ln(x)-x)/x)^2+1/2*Pi^2*csgn(I/x)*csgn(I*(ln(x)-x)/x)^5-Pi^2*csgn(I*(ln(x)-x)/x)^4+Pi^2*csgn(I*(ln(x)-x))*c
sgn(I*(ln(x)-x)/x)^2-Pi^2*csgn(I*(ln(x)-x))*csgn(I*(ln(x)-x)/x)^4+10*I*Pi-1/4*Pi^2*csgn(I*(ln(x)-x))^2*csgn(I*
(ln(x)-x)/x)^4-1/2*Pi^2*csgn(I*(ln(x)-x))*csgn(I*(ln(x)-x)/x)^5+5*I*Pi*csgn(I/x)*csgn(I*(ln(x)-x)/x)^2-5*I*Pi*
csgn(I*(ln(x)-x)/x)^3+(2*x^2-2*ln(x))*ln(x-ln(x))+2*I*Pi*ln(ln(x)-x)+2*I*Pi*x^2-2*I*Pi*ln(x)-2*I*Pi*ln(2)+2*I*
Pi*ln(3)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (20) = 40\).
time = 0.54, size = 69, normalized size = 3.00 \begin {gather*} x^{4} + 2 \, x^{2} {\left (\log \left (3\right ) - \log \left (2\right ) + 5\right )} - 2 \, {\left (x^{2} + \log \left (3\right ) - \log \left (2\right ) + 5\right )} \log \left (x\right ) + \log \left (x\right )^{2} + 2 \, {\left (x^{2} + \log \left (3\right ) - \log \left (2\right ) - \log \left (x\right ) + 5\right )} \log \left (-x + \log \left (x\right )\right ) + \log \left (-x + \log \left (x\right )\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^4 + 2*x^2*(log(3) - log(2) + 5) - 2*(x^2 + log(3) - log(2) + 5)*log(x) + log(x)^2 + 2*(x^2 + log(3) - log(2)
- log(x) + 5)*log(-x + log(x)) + log(-x + log(x))^2

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (20) = 40\).
time = 0.31, size = 45, normalized size = 1.96 \begin {gather*} x^{4} + 2 \, x^{2} \log \left (-\frac {3 \, {\left (x e^{5} - e^{5} \log \left (x\right )\right )}}{2 \, x}\right ) + \log \left (-\frac {3 \, {\left (x e^{5} - e^{5} \log \left (x\right )\right )}}{2 \, x}\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^4 + 2*x^2*log(-3/2*(x*e^5 - e^5*log(x))/x) + log(-3/2*(x*e^5 - e^5*log(x))/x)^2

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (19) = 38\).
time = 0.15, size = 53, normalized size = 2.30 \begin {gather*} x^{4} + 2 x^{2} \log {\left (\frac {- \frac {3 x e^{5}}{2} + \frac {3 e^{5} \log {\left (x \right )}}{2}}{x} \right )} + \log {\left (\frac {- \frac {3 x e^{5}}{2} + \frac {3 e^{5} \log {\left (x \right )}}{2}}{x} \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**4 + 2*x**2*log((-3*x*exp(5)/2 + 3*exp(5)*log(x)/2)/x) + log((-3*x*exp(5)/2 + 3*exp(5)*log(x)/2)/x)**2

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (20) = 40\).
time = 0.42, size = 75, normalized size = 3.26 \begin {gather*} x^{4} - 2 \, x^{2} {\left (\log \left (2\right ) - 5\right )} - 2 \, x^{2} \log \left (x\right ) + 2 \, {\left (\log \left (2\right ) - 5\right )} \log \left (x\right ) + \log \left (x\right )^{2} - 2 \, {\left (\log \left (2\right ) - 5\right )} \log \left (-x + \log \left (x\right )\right ) + 2 \, {\left (x^{2} - \log \left (x\right )\right )} \log \left (-3 \, x + 3 \, \log \left (x\right )\right ) + \log \left (-3 \, x + 3 \, \log \left (x\right )\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^4 - 2*x^2*(log(2) - 5) - 2*x^2*log(x) + 2*(log(2) - 5)*log(x) + log(x)^2 - 2*(log(2) - 5)*log(-x + log(x)) +
2*(x^2 - log(x))*log(-3*x + 3*log(x)) + log(-3*x + 3*log(x))^2

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Mupad [B]
time = 0.50, size = 24, normalized size = 1.04 \begin {gather*} {\left (\ln \left (-\frac {3\,x\,{\mathrm {e}}^5-3\,{\mathrm {e}}^5\,\ln \left (x\right )}{2\,x}\right )+x^2\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(-(3*x*exp(5) - 3*exp(5)*log(x))/(2*x)) + x^2)^2

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