3.7.41 \(\int \frac {(4-12 x) \log (4)+x^{x^2} ((2-8 x^2) \log (4)-16 x^2 \log (4) \log (x))}{32+160 x+320 x^2+320 x^3+160 x^4+32 x^5+x^{5 x^2}+x^{4 x^2} (10+10 x)+x^{3 x^2} (40+80 x+40 x^2)+x^{2 x^2} (80+240 x+240 x^2+80 x^3)+x^{x^2} (80+320 x+480 x^2+320 x^3+80 x^4)} \, dx\)

Optimal. Leaf size=17 \[ \frac {2 x \log (4)}{\left (2+2 x+x^{x^2}\right )^4} \]

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Rubi [F]  time = 1.33, 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 {(4-12 x) \log (4)+x^{x^2} \left (\left (2-8 x^2\right ) \log (4)-16 x^2 \log (4) \log (x)\right )}{32+160 x+320 x^2+320 x^3+160 x^4+32 x^5+x^{5 x^2}+x^{4 x^2} (10+10 x)+x^{3 x^2} \left (40+80 x+40 x^2\right )+x^{2 x^2} \left (80+240 x+240 x^2+80 x^3\right )+x^{x^2} \left (80+320 x+480 x^2+320 x^3+80 x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((4 - 12*x)*Log[4] + x^x^2*((2 - 8*x^2)*Log[4] - 16*x^2*Log[4]*Log[x]))/(32 + 160*x + 320*x^2 + 320*x^3 +
160*x^4 + 32*x^5 + x^(5*x^2) + x^(4*x^2)*(10 + 10*x) + x^(3*x^2)*(40 + 80*x + 40*x^2) + x^(2*x^2)*(80 + 240*x
+ 240*x^2 + 80*x^3) + x^x^2*(80 + 320*x + 480*x^2 + 320*x^3 + 80*x^4)),x]

[Out]

-16*Log[4]*Defer[Int][x/(2 + 2*x + x^x^2)^5, x] + 16*Log[4]*Defer[Int][x^2/(2 + 2*x + x^x^2)^5, x] + 32*Log[4]
*Log[x]*Defer[Int][x^2/(2 + 2*x + x^x^2)^5, x] + 16*Log[4]*Defer[Int][x^3/(2 + 2*x + x^x^2)^5, x] + 32*Log[4]*
Log[x]*Defer[Int][x^3/(2 + 2*x + x^x^2)^5, x] + 2*Log[4]*Defer[Int][(2 + 2*x + x^x^2)^(-4), x] - 8*Log[4]*Defe
r[Int][x^2/(2 + 2*x + x^x^2)^4, x] - 16*Log[4]*Log[x]*Defer[Int][x^2/(2 + 2*x + x^x^2)^4, x] - 32*Log[4]*Defer
[Int][Defer[Int][x^2/(2 + 2*x + x^x^2)^5, x]/x, x] - 32*Log[4]*Defer[Int][Defer[Int][x^3/(2 + 2*x + x^x^2)^5,
x]/x, x] + 16*Log[4]*Defer[Int][Defer[Int][x^2/(2 + 2*x + x^x^2)^4, x]/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \log (4) \left (2-6 x+x^{x^2}-4 x^{2+x^2}-8 x^{2+x^2} \log (x)\right )}{\left (2+2 x+x^{x^2}\right )^5} \, dx\\ &=(2 \log (4)) \int \frac {2-6 x+x^{x^2}-4 x^{2+x^2}-8 x^{2+x^2} \log (x)}{\left (2+2 x+x^{x^2}\right )^5} \, dx\\ &=(2 \log (4)) \int \left (\frac {8 x \left (-1+x+x^2+2 x \log (x)+2 x^2 \log (x)\right )}{\left (2+2 x+x^{x^2}\right )^5}-\frac {-1+4 x^2+8 x^2 \log (x)}{\left (2+2 x+x^{x^2}\right )^4}\right ) \, dx\\ &=-\left ((2 \log (4)) \int \frac {-1+4 x^2+8 x^2 \log (x)}{\left (2+2 x+x^{x^2}\right )^4} \, dx\right )+(16 \log (4)) \int \frac {x \left (-1+x+x^2+2 x \log (x)+2 x^2 \log (x)\right )}{\left (2+2 x+x^{x^2}\right )^5} \, dx\\ &=-\left ((2 \log (4)) \int \left (-\frac {1}{\left (2+2 x+x^{x^2}\right )^4}+\frac {4 x^2}{\left (2+2 x+x^{x^2}\right )^4}+\frac {8 x^2 \log (x)}{\left (2+2 x+x^{x^2}\right )^4}\right ) \, dx\right )+(16 \log (4)) \int \left (-\frac {x}{\left (2+2 x+x^{x^2}\right )^5}+\frac {x^2}{\left (2+2 x+x^{x^2}\right )^5}+\frac {x^3}{\left (2+2 x+x^{x^2}\right )^5}+\frac {2 x^2 \log (x)}{\left (2+2 x+x^{x^2}\right )^5}+\frac {2 x^3 \log (x)}{\left (2+2 x+x^{x^2}\right )^5}\right ) \, dx\\ &=(2 \log (4)) \int \frac {1}{\left (2+2 x+x^{x^2}\right )^4} \, dx-(8 \log (4)) \int \frac {x^2}{\left (2+2 x+x^{x^2}\right )^4} \, dx-(16 \log (4)) \int \frac {x}{\left (2+2 x+x^{x^2}\right )^5} \, dx+(16 \log (4)) \int \frac {x^2}{\left (2+2 x+x^{x^2}\right )^5} \, dx+(16 \log (4)) \int \frac {x^3}{\left (2+2 x+x^{x^2}\right )^5} \, dx-(16 \log (4)) \int \frac {x^2 \log (x)}{\left (2+2 x+x^{x^2}\right )^4} \, dx+(32 \log (4)) \int \frac {x^2 \log (x)}{\left (2+2 x+x^{x^2}\right )^5} \, dx+(32 \log (4)) \int \frac {x^3 \log (x)}{\left (2+2 x+x^{x^2}\right )^5} \, dx\\ &=(2 \log (4)) \int \frac {1}{\left (2+2 x+x^{x^2}\right )^4} \, dx-(8 \log (4)) \int \frac {x^2}{\left (2+2 x+x^{x^2}\right )^4} \, dx-(16 \log (4)) \int \frac {x}{\left (2+2 x+x^{x^2}\right )^5} \, dx+(16 \log (4)) \int \frac {x^2}{\left (2+2 x+x^{x^2}\right )^5} \, dx+(16 \log (4)) \int \frac {x^3}{\left (2+2 x+x^{x^2}\right )^5} \, dx+(16 \log (4)) \int \frac {\int \frac {x^2}{\left (2+2 x+x^{x^2}\right )^4} \, dx}{x} \, dx-(32 \log (4)) \int \frac {\int \frac {x^2}{\left (2+2 x+x^{x^2}\right )^5} \, dx}{x} \, dx-(32 \log (4)) \int \frac {\int \frac {x^3}{\left (2+2 x+x^{x^2}\right )^5} \, dx}{x} \, dx-(16 \log (4) \log (x)) \int \frac {x^2}{\left (2+2 x+x^{x^2}\right )^4} \, dx+(32 \log (4) \log (x)) \int \frac {x^2}{\left (2+2 x+x^{x^2}\right )^5} \, dx+(32 \log (4) \log (x)) \int \frac {x^3}{\left (2+2 x+x^{x^2}\right )^5} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.24, size = 17, normalized size = 1.00 \begin {gather*} \frac {2 x \log (4)}{\left (2+2 x+x^{x^2}\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((4 - 12*x)*Log[4] + x^x^2*((2 - 8*x^2)*Log[4] - 16*x^2*Log[4]*Log[x]))/(32 + 160*x + 320*x^2 + 320*
x^3 + 160*x^4 + 32*x^5 + x^(5*x^2) + x^(4*x^2)*(10 + 10*x) + x^(3*x^2)*(40 + 80*x + 40*x^2) + x^(2*x^2)*(80 +
240*x + 240*x^2 + 80*x^3) + x^x^2*(80 + 320*x + 480*x^2 + 320*x^3 + 80*x^4)),x]

[Out]

(2*x*Log[4])/(2 + 2*x + x^x^2)^4

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fricas [B]  time = 0.86, size = 83, normalized size = 4.88 \begin {gather*} \frac {4 \, x \log \relax (2)}{16 \, x^{4} + 64 \, x^{3} + 8 \, {\left (x + 1\right )} x^{3 \, x^{2}} + 24 \, {\left (x^{2} + 2 \, x + 1\right )} x^{2 \, x^{2}} + 32 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )} x^{\left (x^{2}\right )} + 96 \, x^{2} + 64 \, x + x^{4 \, x^{2}} + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-32*x^2*log(2)*log(x)+2*(-8*x^2+2)*log(2))*exp(x^2*log(x))+2*(-12*x+4)*log(2))/(exp(x^2*log(x))^5+
(10*x+10)*exp(x^2*log(x))^4+(40*x^2+80*x+40)*exp(x^2*log(x))^3+(80*x^3+240*x^2+240*x+80)*exp(x^2*log(x))^2+(80
*x^4+320*x^3+480*x^2+320*x+80)*exp(x^2*log(x))+32*x^5+160*x^4+320*x^3+320*x^2+160*x+32),x, algorithm="fricas")

[Out]

4*x*log(2)/(16*x^4 + 64*x^3 + 8*(x + 1)*x^(3*x^2) + 24*(x^2 + 2*x + 1)*x^(2*x^2) + 32*(x^3 + 3*x^2 + 3*x + 1)*
x^(x^2) + 96*x^2 + 64*x + x^(4*x^2) + 16)

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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(((-32*x^2*log(2)*log(x)+2*(-8*x^2+2)*log(2))*exp(x^2*log(x))+2*(-12*x+4)*log(2))/(exp(x^2*log(x))^5+
(10*x+10)*exp(x^2*log(x))^4+(40*x^2+80*x+40)*exp(x^2*log(x))^3+(80*x^3+240*x^2+240*x+80)*exp(x^2*log(x))^2+(80
*x^4+320*x^3+480*x^2+320*x+80)*exp(x^2*log(x))+32*x^5+160*x^4+320*x^3+320*x^2+160*x+32),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.06, size = 18, normalized size = 1.06

method	result	size
risch	\(\frac {4 x \ln \relax (2)}{\left (2 x +2+x^{x^{2}}\right )^{4}}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-32*x^2*ln(2)*ln(x)+2*(-8*x^2+2)*ln(2))*exp(x^2*ln(x))+2*(-12*x+4)*ln(2))/(exp(x^2*ln(x))^5+(10*x+10)*ex
p(x^2*ln(x))^4+(40*x^2+80*x+40)*exp(x^2*ln(x))^3+(80*x^3+240*x^2+240*x+80)*exp(x^2*ln(x))^2+(80*x^4+320*x^3+48
0*x^2+320*x+80)*exp(x^2*ln(x))+32*x^5+160*x^4+320*x^3+320*x^2+160*x+32),x,method=_RETURNVERBOSE)

[Out]

4*x*ln(2)/(2*x+2+x^(x^2))^4

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maxima [B]  time = 1.33, size = 83, normalized size = 4.88 \begin {gather*} \frac {4 \, x \log \relax (2)}{16 \, x^{4} + 64 \, x^{3} + 8 \, {\left (x + 1\right )} x^{3 \, x^{2}} + 24 \, {\left (x^{2} + 2 \, x + 1\right )} x^{2 \, x^{2}} + 32 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )} x^{\left (x^{2}\right )} + 96 \, x^{2} + 64 \, x + x^{4 \, x^{2}} + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-32*x^2*log(2)*log(x)+2*(-8*x^2+2)*log(2))*exp(x^2*log(x))+2*(-12*x+4)*log(2))/(exp(x^2*log(x))^5+
(10*x+10)*exp(x^2*log(x))^4+(40*x^2+80*x+40)*exp(x^2*log(x))^3+(80*x^3+240*x^2+240*x+80)*exp(x^2*log(x))^2+(80
*x^4+320*x^3+480*x^2+320*x+80)*exp(x^2*log(x))+32*x^5+160*x^4+320*x^3+320*x^2+160*x+32),x, algorithm="maxima")

[Out]

4*x*log(2)/(16*x^4 + 64*x^3 + 8*(x + 1)*x^(3*x^2) + 24*(x^2 + 2*x + 1)*x^(2*x^2) + 32*(x^3 + 3*x^2 + 3*x + 1)*
x^(x^2) + 96*x^2 + 64*x + x^(4*x^2) + 16)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} \int -\frac {2\,\ln \relax (2)\,\left (12\,x-4\right )+{\mathrm {e}}^{x^2\,\ln \relax (x)}\,\left (2\,\ln \relax (2)\,\left (8\,x^2-2\right )+32\,x^2\,\ln \relax (2)\,\ln \relax (x)\right )}{160\,x+{\mathrm {e}}^{5\,x^2\,\ln \relax (x)}+{\mathrm {e}}^{4\,x^2\,\ln \relax (x)}\,\left (10\,x+10\right )+{\mathrm {e}}^{3\,x^2\,\ln \relax (x)}\,\left (40\,x^2+80\,x+40\right )+{\mathrm {e}}^{2\,x^2\,\ln \relax (x)}\,\left (80\,x^3+240\,x^2+240\,x+80\right )+{\mathrm {e}}^{x^2\,\ln \relax (x)}\,\left (80\,x^4+320\,x^3+480\,x^2+320\,x+80\right )+320\,x^2+320\,x^3+160\,x^4+32\,x^5+32} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*log(2)*(12*x - 4) + exp(x^2*log(x))*(2*log(2)*(8*x^2 - 2) + 32*x^2*log(2)*log(x)))/(160*x + exp(5*x^2*
log(x)) + exp(4*x^2*log(x))*(10*x + 10) + exp(3*x^2*log(x))*(80*x + 40*x^2 + 40) + exp(2*x^2*log(x))*(240*x +
240*x^2 + 80*x^3 + 80) + exp(x^2*log(x))*(320*x + 480*x^2 + 320*x^3 + 80*x^4 + 80) + 320*x^2 + 320*x^3 + 160*x
^4 + 32*x^5 + 32),x)

[Out]

int(-(2*log(2)*(12*x - 4) + exp(x^2*log(x))*(2*log(2)*(8*x^2 - 2) + 32*x^2*log(2)*log(x)))/(160*x + exp(5*x^2*
log(x)) + exp(4*x^2*log(x))*(10*x + 10) + exp(3*x^2*log(x))*(80*x + 40*x^2 + 40) + exp(2*x^2*log(x))*(240*x +
240*x^2 + 80*x^3 + 80) + exp(x^2*log(x))*(320*x + 480*x^2 + 320*x^3 + 80*x^4 + 80) + 320*x^2 + 320*x^3 + 160*x
^4 + 32*x^5 + 32), x)

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sympy [B]  time = 0.48, size = 95, normalized size = 5.59 \begin {gather*} \frac {4 x \log {\relax (2 )}}{16 x^{4} + 64 x^{3} + 96 x^{2} + 64 x + \left (8 x + 8\right ) e^{3 x^{2} \log {\relax (x )}} + \left (24 x^{2} + 48 x + 24\right ) e^{2 x^{2} \log {\relax (x )}} + \left (32 x^{3} + 96 x^{2} + 96 x + 32\right ) e^{x^{2} \log {\relax (x )}} + e^{4 x^{2} \log {\relax (x )}} + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-32*x**2*ln(2)*ln(x)+2*(-8*x**2+2)*ln(2))*exp(x**2*ln(x))+2*(-12*x+4)*ln(2))/(exp(x**2*ln(x))**5+(
10*x+10)*exp(x**2*ln(x))**4+(40*x**2+80*x+40)*exp(x**2*ln(x))**3+(80*x**3+240*x**2+240*x+80)*exp(x**2*ln(x))**
2+(80*x**4+320*x**3+480*x**2+320*x+80)*exp(x**2*ln(x))+32*x**5+160*x**4+320*x**3+320*x**2+160*x+32),x)

[Out]

4*x*log(2)/(16*x**4 + 64*x**3 + 96*x**2 + 64*x + (8*x + 8)*exp(3*x**2*log(x)) + (24*x**2 + 48*x + 24)*exp(2*x*
*2*log(x)) + (32*x**3 + 96*x**2 + 96*x + 32)*exp(x**2*log(x)) + exp(4*x**2*log(x)) + 16)

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