3.10.5 \(\int \frac {36 x+12 x^2+3 x^5-3 x^6+e^{3 \log ^2(x)} (-3 x^2+3 x^3)+e^{2 \log ^2(x)} (9 x^3-9 x^4)+e^{\log ^2(x)} (-12-12 x-9 x^4+9 x^5-48 \log (x))}{8 x^2+2 e^{3 \log ^2(x)} x^3-6 e^{2 \log ^2(x)} x^4-2 x^6+e^{\log ^2(x)} (-8 x+6 x^5)} \, dx\)

Optimal. Leaf size=30 \[ \frac {3}{2} \left (x-\log \left (x \left (-1+\frac {4}{x^2 \left (-e^{\log ^2(x)}+x\right )^2}\right )\right )\right ) \]

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Rubi [F]  time = 4.26, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {36 x+12 x^2+3 x^5-3 x^6+e^{3 \log ^2(x)} \left (-3 x^2+3 x^3\right )+e^{2 \log ^2(x)} \left (9 x^3-9 x^4\right )+e^{\log ^2(x)} \left (-12-12 x-9 x^4+9 x^5-48 \log (x)\right )}{8 x^2+2 e^{3 \log ^2(x)} x^3-6 e^{2 \log ^2(x)} x^4-2 x^6+e^{\log ^2(x)} \left (-8 x+6 x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(36*x + 12*x^2 + 3*x^5 - 3*x^6 + E^(3*Log[x]^2)*(-3*x^2 + 3*x^3) + E^(2*Log[x]^2)*(9*x^3 - 9*x^4) + E^Log[
x]^2*(-12 - 12*x - 9*x^4 + 9*x^5 - 48*Log[x]))/(8*x^2 + 2*E^(3*Log[x]^2)*x^3 - 6*E^(2*Log[x]^2)*x^4 - 2*x^6 +
E^Log[x]^2*(-8*x + 6*x^5)),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-36 x-12 x^2-3 x^5+3 x^6-e^{3 \log ^2(x)} \left (-3 x^2+3 x^3\right )-e^{2 \log ^2(x)} \left (9 x^3-9 x^4\right )-e^{\log ^2(x)} \left (-12-12 x-9 x^4+9 x^5-48 \log (x)\right )}{2 x \left (4 e^{\log ^2(x)}-4 x-e^{3 \log ^2(x)} x^2+3 e^{2 \log ^2(x)} x^3-3 e^{\log ^2(x)} x^4+x^5\right )} \, dx\\ &=\frac {1}{2} \int \frac {-36 x-12 x^2-3 x^5+3 x^6-e^{3 \log ^2(x)} \left (-3 x^2+3 x^3\right )-e^{2 \log ^2(x)} \left (9 x^3-9 x^4\right )-e^{\log ^2(x)} \left (-12-12 x-9 x^4+9 x^5-48 \log (x)\right )}{x \left (4 e^{\log ^2(x)}-4 x-e^{3 \log ^2(x)} x^2+3 e^{2 \log ^2(x)} x^3-3 e^{\log ^2(x)} x^4+x^5\right )} \, dx\\ &=\frac {1}{2} \int \left (\frac {3 (-1+x)}{x}+\frac {6 (-1+2 \log (x))}{e^{\log ^2(x)}-x}+\frac {3 \left (-2-x^2-4 \log (x)+2 x^2 \log (x)\right )}{x \left (-2-e^{\log ^2(x)} x+x^2\right )}+\frac {3 \left (2-x^2+4 \log (x)+2 x^2 \log (x)\right )}{x \left (2-e^{\log ^2(x)} x+x^2\right )}\right ) \, dx\\ &=\frac {3}{2} \int \frac {-1+x}{x} \, dx+\frac {3}{2} \int \frac {-2-x^2-4 \log (x)+2 x^2 \log (x)}{x \left (-2-e^{\log ^2(x)} x+x^2\right )} \, dx+\frac {3}{2} \int \frac {2-x^2+4 \log (x)+2 x^2 \log (x)}{x \left (2-e^{\log ^2(x)} x+x^2\right )} \, dx+3 \int \frac {-1+2 \log (x)}{e^{\log ^2(x)}-x} \, dx\\ &=\frac {3}{2} \int \left (1-\frac {1}{x}\right ) \, dx+\frac {3}{2} \int \left (-\frac {2}{x \left (-2-e^{\log ^2(x)} x+x^2\right )}-\frac {x}{-2-e^{\log ^2(x)} x+x^2}-\frac {4 \log (x)}{x \left (-2-e^{\log ^2(x)} x+x^2\right )}+\frac {2 x \log (x)}{-2-e^{\log ^2(x)} x+x^2}\right ) \, dx+\frac {3}{2} \int \left (\frac {2}{x \left (2-e^{\log ^2(x)} x+x^2\right )}-\frac {x}{2-e^{\log ^2(x)} x+x^2}+\frac {4 \log (x)}{x \left (2-e^{\log ^2(x)} x+x^2\right )}+\frac {2 x \log (x)}{2-e^{\log ^2(x)} x+x^2}\right ) \, dx+3 \int \left (-\frac {1}{e^{\log ^2(x)}-x}+\frac {2 \log (x)}{e^{\log ^2(x)}-x}\right ) \, dx\\ &=\frac {3 x}{2}-\frac {3 \log (x)}{2}-\frac {3}{2} \int \frac {x}{-2-e^{\log ^2(x)} x+x^2} \, dx-\frac {3}{2} \int \frac {x}{2-e^{\log ^2(x)} x+x^2} \, dx-3 \int \frac {1}{e^{\log ^2(x)}-x} \, dx-3 \int \frac {1}{x \left (-2-e^{\log ^2(x)} x+x^2\right )} \, dx+3 \int \frac {1}{x \left (2-e^{\log ^2(x)} x+x^2\right )} \, dx+3 \int \frac {x \log (x)}{-2-e^{\log ^2(x)} x+x^2} \, dx+3 \int \frac {x \log (x)}{2-e^{\log ^2(x)} x+x^2} \, dx+6 \int \frac {\log (x)}{e^{\log ^2(x)}-x} \, dx-6 \int \frac {\log (x)}{x \left (-2-e^{\log ^2(x)} x+x^2\right )} \, dx+6 \int \frac {\log (x)}{x \left (2-e^{\log ^2(x)} x+x^2\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.17, size = 56, normalized size = 1.87 \begin {gather*} \frac {3}{2} \left (x+2 \log \left (e^{\log ^2(x)}-x\right )+\log (x)-\log \left (2+e^{\log ^2(x)} x-x^2\right )-\log \left (2-e^{\log ^2(x)} x+x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(36*x + 12*x^2 + 3*x^5 - 3*x^6 + E^(3*Log[x]^2)*(-3*x^2 + 3*x^3) + E^(2*Log[x]^2)*(9*x^3 - 9*x^4) +
E^Log[x]^2*(-12 - 12*x - 9*x^4 + 9*x^5 - 48*Log[x]))/(8*x^2 + 2*E^(3*Log[x]^2)*x^3 - 6*E^(2*Log[x]^2)*x^4 - 2*
x^6 + E^Log[x]^2*(-8*x + 6*x^5)),x]

[Out]

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

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fricas [A]  time = 0.71, size = 53, normalized size = 1.77 \begin {gather*} \frac {3}{2} \, x - \frac {3}{2} \, \log \relax (x) + 3 \, \log \left (-x + e^{\left (\log \relax (x)^{2}\right )}\right ) - \frac {3}{2} \, \log \left (\frac {x^{4} - 2 \, x^{3} e^{\left (\log \relax (x)^{2}\right )} + x^{2} e^{\left (2 \, \log \relax (x)^{2}\right )} - 4}{x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^3-3*x^2)*exp(log(x)^2)^3+(-9*x^4+9*x^3)*exp(log(x)^2)^2+(-48*log(x)+9*x^5-9*x^4-12*x-12)*exp(l
og(x)^2)-3*x^6+3*x^5+12*x^2+36*x)/(2*x^3*exp(log(x)^2)^3-6*x^4*exp(log(x)^2)^2+(6*x^5-8*x)*exp(log(x)^2)-2*x^6
+8*x^2),x, algorithm="fricas")

[Out]

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

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^3-3*x^2)*exp(log(x)^2)^3+(-9*x^4+9*x^3)*exp(log(x)^2)^2+(-48*log(x)+9*x^5-9*x^4-12*x-12)*exp(l
og(x)^2)-3*x^6+3*x^5+12*x^2+36*x)/(2*x^3*exp(log(x)^2)^3-6*x^4*exp(log(x)^2)^2+(6*x^5-8*x)*exp(log(x)^2)-2*x^6
+8*x^2),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.06, size = 49, normalized size = 1.63

method	result	size
risch	\(\frac {3 x}{2}-\frac {3 \ln \relax (x )}{2}+3 \ln \left (-x +{\mathrm e}^{\ln \relax (x )^{2}}\right )-\frac {3 \ln \left ({\mathrm e}^{2 \ln \relax (x )^{2}}-2 \,{\mathrm e}^{\ln \relax (x )^{2}} x +\frac {x^{4}-4}{x^{2}}\right )}{2}\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x^3-3*x^2)*exp(ln(x)^2)^3+(-9*x^4+9*x^3)*exp(ln(x)^2)^2+(-48*ln(x)+9*x^5-9*x^4-12*x-12)*exp(ln(x)^2)-3
*x^6+3*x^5+12*x^2+36*x)/(2*x^3*exp(ln(x)^2)^3-6*x^4*exp(ln(x)^2)^2+(6*x^5-8*x)*exp(ln(x)^2)-2*x^6+8*x^2),x,met
hod=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.66, size = 62, normalized size = 2.07 \begin {gather*} \frac {3}{2} \, x - \frac {3}{2} \, \log \relax (x) + 3 \, \log \left (-x + e^{\left (\log \relax (x)^{2}\right )}\right ) - \frac {3}{2} \, \log \left (-\frac {x^{2} - x e^{\left (\log \relax (x)^{2}\right )} + 2}{x}\right ) - \frac {3}{2} \, \log \left (-\frac {x^{2} - x e^{\left (\log \relax (x)^{2}\right )} - 2}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^3-3*x^2)*exp(log(x)^2)^3+(-9*x^4+9*x^3)*exp(log(x)^2)^2+(-48*log(x)+9*x^5-9*x^4-12*x-12)*exp(l
og(x)^2)-3*x^6+3*x^5+12*x^2+36*x)/(2*x^3*exp(log(x)^2)^3-6*x^4*exp(log(x)^2)^2+(6*x^5-8*x)*exp(log(x)^2)-2*x^6
+8*x^2),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {36\,x-{\mathrm {e}}^{3\,{\ln \relax (x)}^2}\,\left (3\,x^2-3\,x^3\right )+{\mathrm {e}}^{2\,{\ln \relax (x)}^2}\,\left (9\,x^3-9\,x^4\right )+12\,x^2+3\,x^5-3\,x^6-{\mathrm {e}}^{{\ln \relax (x)}^2}\,\left (12\,x+48\,\ln \relax (x)+9\,x^4-9\,x^5+12\right )}{6\,x^4\,{\mathrm {e}}^{2\,{\ln \relax (x)}^2}-2\,x^3\,{\mathrm {e}}^{3\,{\ln \relax (x)}^2}+{\mathrm {e}}^{{\ln \relax (x)}^2}\,\left (8\,x-6\,x^5\right )-8\,x^2+2\,x^6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(36*x - exp(3*log(x)^2)*(3*x^2 - 3*x^3) + exp(2*log(x)^2)*(9*x^3 - 9*x^4) + 12*x^2 + 3*x^5 - 3*x^6 - exp(
log(x)^2)*(12*x + 48*log(x) + 9*x^4 - 9*x^5 + 12))/(6*x^4*exp(2*log(x)^2) - 2*x^3*exp(3*log(x)^2) + exp(log(x)
^2)*(8*x - 6*x^5) - 8*x^2 + 2*x^6),x)

[Out]

int(-(36*x - exp(3*log(x)^2)*(3*x^2 - 3*x^3) + exp(2*log(x)^2)*(9*x^3 - 9*x^4) + 12*x^2 + 3*x^5 - 3*x^6 - exp(
log(x)^2)*(12*x + 48*log(x) + 9*x^4 - 9*x^5 + 12))/(6*x^4*exp(2*log(x)^2) - 2*x^3*exp(3*log(x)^2) + exp(log(x)
^2)*(8*x - 6*x^5) - 8*x^2 + 2*x^6), x)

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sympy [A]  time = 0.74, size = 54, normalized size = 1.80 \begin {gather*} \frac {3 x}{2} - \frac {3 \log {\relax (x )}}{2} + 3 \log {\left (- x + e^{\log {\relax (x )}^{2}} \right )} - \frac {3 \log {\left (- 2 x e^{\log {\relax (x )}^{2}} + e^{2 \log {\relax (x )}^{2}} + \frac {x^{4} - 4}{x^{2}} \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x**3-3*x**2)*exp(ln(x)**2)**3+(-9*x**4+9*x**3)*exp(ln(x)**2)**2+(-48*ln(x)+9*x**5-9*x**4-12*x-12
)*exp(ln(x)**2)-3*x**6+3*x**5+12*x**2+36*x)/(2*x**3*exp(ln(x)**2)**3-6*x**4*exp(ln(x)**2)**2+(6*x**5-8*x)*exp(
ln(x)**2)-2*x**6+8*x**2),x)

[Out]

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

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