3.20.98 \(\int \frac {e^{x \log (3) \log (x^2 \log (x^2))} ((-4 x-2 x^2) \log (3)+(4+3 x+(-4 x-2 x^2) \log (3)) \log (x^2)+(-2 x-x^2) \log (3) \log (x^2) \log (x^2 \log (x^2)))}{(4 x^3+4 x^4+x^5) \log (x^2)} \, dx\)

Optimal. Leaf size=25 \[ -\frac {e^{x \log (3) \log \left (x^2 \log \left (x^2\right )\right )}}{x^2 (2+x)} \]

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Rubi [F]  time = 6.37, 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 {e^{x \log (3) \log \left (x^2 \log \left (x^2\right )\right )} \left (\left (-4 x-2 x^2\right ) \log (3)+\left (4+3 x+\left (-4 x-2 x^2\right ) \log (3)\right ) \log \left (x^2\right )+\left (-2 x-x^2\right ) \log (3) \log \left (x^2\right ) \log \left (x^2 \log \left (x^2\right )\right )\right )}{\left (4 x^3+4 x^4+x^5\right ) \log \left (x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

Defer[Int][3^(x*Log[x^2*Log[x^2]])/x^3, x] - ((1 + Log[81])*Defer[Int][3^(x*Log[x^2*Log[x^2]])/x^2, x])/4 + (L
og[9]*Defer[Int][3^(x*Log[x^2*Log[x^2]])/x, x])/4 + Defer[Int][3^(x*Log[x^2*Log[x^2]])/(2 + x)^2, x]/4 - (Log[
9]*Defer[Int][3^(x*Log[x^2*Log[x^2]])/(2 + x), x])/4 - (Log[9]^2*Defer[Int][3^(x*Log[x^2*Log[x^2]])/(x^2*Log[x
^2]), x])/Log[81] + (Log[9]^3*Defer[Int][3^(x*Log[x^2*Log[x^2]])/(x*Log[x^2]), x])/Log[81]^2 - (Log[9]^4*Defer
[Int][3^(x*Log[x^2*Log[x^2]])/((x*Log[9] + Log[81])*Log[x^2]), x])/Log[81]^2 - (Log[3]*Defer[Int][(E^(x*Log[3]
*Log[x^2*Log[x^2]])*Log[x^2*Log[x^2]])/x^2, x])/2 + (Log[3]*Defer[Int][(E^(x*Log[3]*Log[x^2*Log[x^2]])*Log[x^2
*Log[x^2]])/x, x])/4 - (Log[3]*Defer[Int][(E^(x*Log[3]*Log[x^2*Log[x^2]])*Log[x^2*Log[x^2]])/(2 + x), x])/4

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{x \log (3) \log \left (x^2 \log \left (x^2\right )\right )} \left (\left (-4 x-2 x^2\right ) \log (3)+\left (4+3 x+\left (-4 x-2 x^2\right ) \log (3)\right ) \log \left (x^2\right )+\left (-2 x-x^2\right ) \log (3) \log \left (x^2\right ) \log \left (x^2 \log \left (x^2\right )\right )\right )}{x^3 \left (4+4 x+x^2\right ) \log \left (x^2\right )} \, dx\\ &=\int \frac {e^{x \log (3) \log \left (x^2 \log \left (x^2\right )\right )} \left (\left (-4 x-2 x^2\right ) \log (3)+\left (4+3 x+\left (-4 x-2 x^2\right ) \log (3)\right ) \log \left (x^2\right )+\left (-2 x-x^2\right ) \log (3) \log \left (x^2\right ) \log \left (x^2 \log \left (x^2\right )\right )\right )}{x^3 (2+x)^2 \log \left (x^2\right )} \, dx\\ &=\int \left (\frac {e^{x \log (3) \log \left (x^2 \log \left (x^2\right )\right )} \left (-4 x \log (3)-2 x^2 \log (3)+4 \log \left (x^2\right )+3 x \left (1-\frac {4 \log (3)}{3}\right ) \log \left (x^2\right )-x^2 \log (9) \log \left (x^2\right )\right )}{x^3 (2+x)^2 \log \left (x^2\right )}-\frac {e^{x \log (3) \log \left (x^2 \log \left (x^2\right )\right )} \log (3) \log \left (x^2 \log \left (x^2\right )\right )}{x^2 (2+x)}\right ) \, dx\\ &=-\left (\log (3) \int \frac {e^{x \log (3) \log \left (x^2 \log \left (x^2\right )\right )} \log \left (x^2 \log \left (x^2\right )\right )}{x^2 (2+x)} \, dx\right )+\int \frac {e^{x \log (3) \log \left (x^2 \log \left (x^2\right )\right )} \left (-4 x \log (3)-2 x^2 \log (3)+4 \log \left (x^2\right )+3 x \left (1-\frac {4 \log (3)}{3}\right ) \log \left (x^2\right )-x^2 \log (9) \log \left (x^2\right )\right )}{x^3 (2+x)^2 \log \left (x^2\right )} \, dx\\ &=-\left (\log (3) \int \left (\frac {e^{x \log (3) \log \left (x^2 \log \left (x^2\right )\right )} \log \left (x^2 \log \left (x^2\right )\right )}{2 x^2}-\frac {e^{x \log (3) \log \left (x^2 \log \left (x^2\right )\right )} \log \left (x^2 \log \left (x^2\right )\right )}{4 x}+\frac {e^{x \log (3) \log \left (x^2 \log \left (x^2\right )\right )} \log \left (x^2 \log \left (x^2\right )\right )}{4 (2+x)}\right ) \, dx\right )+\int \frac {3^{x \log \left (x^2 \log \left (x^2\right )\right )} \left (-x (x \log (9)+\log (81))-\left (-4+x^2 \log (9)+x (-3+\log (81))\right ) \log \left (x^2\right )\right )}{x^3 (2+x)^2 \log \left (x^2\right )} \, dx\\ &=\frac {1}{4} \log (3) \int \frac {e^{x \log (3) \log \left (x^2 \log \left (x^2\right )\right )} \log \left (x^2 \log \left (x^2\right )\right )}{x} \, dx-\frac {1}{4} \log (3) \int \frac {e^{x \log (3) \log \left (x^2 \log \left (x^2\right )\right )} \log \left (x^2 \log \left (x^2\right )\right )}{2+x} \, dx-\frac {1}{2} \log (3) \int \frac {e^{x \log (3) \log \left (x^2 \log \left (x^2\right )\right )} \log \left (x^2 \log \left (x^2\right )\right )}{x^2} \, dx+\int \left (\frac {3^{x \log \left (x^2 \log \left (x^2\right )\right )} \left (4-x^2 \log (9)+x (3-\log (81))\right )}{x^3 (2+x)^2}+\frac {3^{x \log \left (x^2 \log \left (x^2\right )\right )} \log ^2(9)}{x^2 (-x \log (9)-\log (81)) \log \left (x^2\right )}\right ) \, dx\\ &=\frac {1}{4} \log (3) \int \frac {e^{x \log (3) \log \left (x^2 \log \left (x^2\right )\right )} \log \left (x^2 \log \left (x^2\right )\right )}{x} \, dx-\frac {1}{4} \log (3) \int \frac {e^{x \log (3) \log \left (x^2 \log \left (x^2\right )\right )} \log \left (x^2 \log \left (x^2\right )\right )}{2+x} \, dx-\frac {1}{2} \log (3) \int \frac {e^{x \log (3) \log \left (x^2 \log \left (x^2\right )\right )} \log \left (x^2 \log \left (x^2\right )\right )}{x^2} \, dx+\log ^2(9) \int \frac {3^{x \log \left (x^2 \log \left (x^2\right )\right )}}{x^2 (-x \log (9)-\log (81)) \log \left (x^2\right )} \, dx+\int \frac {3^{x \log \left (x^2 \log \left (x^2\right )\right )} \left (4-x^2 \log (9)+x (3-\log (81))\right )}{x^3 (2+x)^2} \, dx\\ &=\frac {1}{4} \log (3) \int \frac {e^{x \log (3) \log \left (x^2 \log \left (x^2\right )\right )} \log \left (x^2 \log \left (x^2\right )\right )}{x} \, dx-\frac {1}{4} \log (3) \int \frac {e^{x \log (3) \log \left (x^2 \log \left (x^2\right )\right )} \log \left (x^2 \log \left (x^2\right )\right )}{2+x} \, dx-\frac {1}{2} \log (3) \int \frac {e^{x \log (3) \log \left (x^2 \log \left (x^2\right )\right )} \log \left (x^2 \log \left (x^2\right )\right )}{x^2} \, dx+\log ^2(9) \int \left (\frac {3^{x \log \left (x^2 \log \left (x^2\right )\right )} \log (9)}{x \log ^2(81) \log \left (x^2\right )}-\frac {3^{x \log \left (x^2 \log \left (x^2\right )\right )}}{x^2 \log (81) \log \left (x^2\right )}-\frac {3^{x \log \left (x^2 \log \left (x^2\right )\right )} \log ^2(9)}{\log ^2(81) (x \log (9)+\log (81)) \log \left (x^2\right )}\right ) \, dx+\int \left (\frac {3^{x \log \left (x^2 \log \left (x^2\right )\right )}}{x^3}+\frac {3^{x \log \left (x^2 \log \left (x^2\right )\right )}}{4 (2+x)^2}+\frac {3^{x \log \left (x^2 \log \left (x^2\right )\right )} \log (9)}{4 x}-\frac {3^{x \log \left (x^2 \log \left (x^2\right )\right )} \log (9)}{4 (2+x)}+\frac {3^{x \log \left (x^2 \log \left (x^2\right )\right )} (-1-\log (81))}{4 x^2}\right ) \, dx\\ &=\frac {1}{4} \int \frac {3^{x \log \left (x^2 \log \left (x^2\right )\right )}}{(2+x)^2} \, dx+\frac {1}{4} \log (3) \int \frac {e^{x \log (3) \log \left (x^2 \log \left (x^2\right )\right )} \log \left (x^2 \log \left (x^2\right )\right )}{x} \, dx-\frac {1}{4} \log (3) \int \frac {e^{x \log (3) \log \left (x^2 \log \left (x^2\right )\right )} \log \left (x^2 \log \left (x^2\right )\right )}{2+x} \, dx-\frac {1}{2} \log (3) \int \frac {e^{x \log (3) \log \left (x^2 \log \left (x^2\right )\right )} \log \left (x^2 \log \left (x^2\right )\right )}{x^2} \, dx+\frac {1}{4} \log (9) \int \frac {3^{x \log \left (x^2 \log \left (x^2\right )\right )}}{x} \, dx-\frac {1}{4} \log (9) \int \frac {3^{x \log \left (x^2 \log \left (x^2\right )\right )}}{2+x} \, dx+\frac {1}{4} (-1-\log (81)) \int \frac {3^{x \log \left (x^2 \log \left (x^2\right )\right )}}{x^2} \, dx+\frac {\log ^3(9) \int \frac {3^{x \log \left (x^2 \log \left (x^2\right )\right )}}{x \log \left (x^2\right )} \, dx}{\log ^2(81)}-\frac {\log ^4(9) \int \frac {3^{x \log \left (x^2 \log \left (x^2\right )\right )}}{(x \log (9)+\log (81)) \log \left (x^2\right )} \, dx}{\log ^2(81)}-\frac {\log ^2(9) \int \frac {3^{x \log \left (x^2 \log \left (x^2\right )\right )}}{x^2 \log \left (x^2\right )} \, dx}{\log (81)}+\int \frac {3^{x \log \left (x^2 \log \left (x^2\right )\right )}}{x^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 0.90, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{x \log (3) \log \left (x^2 \log \left (x^2\right )\right )} \left (\left (-4 x-2 x^2\right ) \log (3)+\left (4+3 x+\left (-4 x-2 x^2\right ) \log (3)\right ) \log \left (x^2\right )+\left (-2 x-x^2\right ) \log (3) \log \left (x^2\right ) \log \left (x^2 \log \left (x^2\right )\right )\right )}{\left (4 x^3+4 x^4+x^5\right ) \log \left (x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.59, size = 27, normalized size = 1.08 \begin {gather*} -\frac {e^{\left (x \log \relax (3) \log \left (x^{2} \log \left (x^{2}\right )\right )\right )}}{x^{3} + 2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left ({\left (x^{2} + 2 \, x\right )} \log \relax (3) \log \left (x^{2} \log \left (x^{2}\right )\right ) \log \left (x^{2}\right ) + 2 \, {\left (x^{2} + 2 \, x\right )} \log \relax (3) + {\left (2 \, {\left (x^{2} + 2 \, x\right )} \log \relax (3) - 3 \, x - 4\right )} \log \left (x^{2}\right )\right )} e^{\left (x \log \relax (3) \log \left (x^{2} \log \left (x^{2}\right )\right )\right )}}{{\left (x^{5} + 4 \, x^{4} + 4 \, x^{3}\right )} \log \left (x^{2}\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.12, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (-x^{2}-2 x \right ) \ln \relax (3) \ln \left (x^{2}\right ) \ln \left (x^{2} \ln \left (x^{2}\right )\right )+\left (\left (-2 x^{2}-4 x \right ) \ln \relax (3)+4+3 x \right ) \ln \left (x^{2}\right )+\left (-2 x^{2}-4 x \right ) \ln \relax (3)\right ) {\mathrm e}^{x \ln \relax (3) \ln \left (x^{2} \ln \left (x^{2}\right )\right )}}{\left (x^{5}+4 x^{4}+4 x^{3}\right ) \ln \left (x^{2}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.57, size = 35, normalized size = 1.40 \begin {gather*} -\frac {e^{\left (x \log \relax (3) \log \relax (2) + 2 \, x \log \relax (3) \log \relax (x) + x \log \relax (3) \log \left (\log \relax (x)\right )\right )}}{x^{3} + 2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {{\mathrm {e}}^{x\,\ln \relax (3)\,\ln \left (x^2\,\ln \left (x^2\right )\right )}\,\left (\ln \relax (3)\,\left (2\,x^2+4\,x\right )-\ln \left (x^2\right )\,\left (3\,x-\ln \relax (3)\,\left (2\,x^2+4\,x\right )+4\right )+\ln \left (x^2\right )\,\ln \relax (3)\,\ln \left (x^2\,\ln \left (x^2\right )\right )\,\left (x^2+2\,x\right )\right )}{\ln \left (x^2\right )\,\left (x^5+4\,x^4+4\,x^3\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.50, size = 26, normalized size = 1.04 \begin {gather*} - \frac {e^{x \log {\relax (3 )} \log {\left (x^{2} \log {\left (x^{2} \right )} \right )}}}{x^{3} + 2 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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