3.11.74 \(\int \frac {e^{-4+x^2+4 x^5+4 x^3 \log (2)+x \log ^2(2)+(-4 x^4-2 x^2 \log (2)) \log (-25+\log (x^2))+x^3 \log ^2(-25+\log (x^2))} (-50 x-8 x^3-500 x^4+(-4 x-300 x^2) \log (2)-25 \log ^2(2)+(2 x+20 x^4+12 x^2 \log (2)+\log ^2(2)) \log (x^2)+(4 x^2+400 x^3+100 x \log (2)+(-16 x^3-4 x \log (2)) \log (x^2)) \log (-25+\log (x^2))+(-75 x^2+3 x^2 \log (x^2)) \log ^2(-25+\log (x^2)))}{-25+\log (x^2)} \, dx\)

Optimal. Leaf size=28 \[ e^{-4+x \left (x+\left (\log (2)+x \left (2 x-\log \left (-25+\log \left (x^2\right )\right )\right )\right )^2\right )} \]

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Rubi [F]  time = 27.36, 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 {\exp \left (-4+x^2+4 x^5+4 x^3 \log (2)+x \log ^2(2)+\left (-4 x^4-2 x^2 \log (2)\right ) \log \left (-25+\log \left (x^2\right )\right )+x^3 \log ^2\left (-25+\log \left (x^2\right )\right )\right ) \left (-50 x-8 x^3-500 x^4+\left (-4 x-300 x^2\right ) \log (2)-25 \log ^2(2)+\left (2 x+20 x^4+12 x^2 \log (2)+\log ^2(2)\right ) \log \left (x^2\right )+\left (4 x^2+400 x^3+100 x \log (2)+\left (-16 x^3-4 x \log (2)\right ) \log \left (x^2\right )\right ) \log \left (-25+\log \left (x^2\right )\right )+\left (-75 x^2+3 x^2 \log \left (x^2\right )\right ) \log ^2\left (-25+\log \left (x^2\right )\right )\right )}{-25+\log \left (x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(-4 + x^2 + 4*x^5 + 4*x^3*Log[2] + x*Log[2]^2 + (-4*x^4 - 2*x^2*Log[2])*Log[-25 + Log[x^2]] + x^3*Log[-
25 + Log[x^2]]^2)*(-50*x - 8*x^3 - 500*x^4 + (-4*x - 300*x^2)*Log[2] - 25*Log[2]^2 + (2*x + 20*x^4 + 12*x^2*Lo
g[2] + Log[2]^2)*Log[x^2] + (4*x^2 + 400*x^3 + 100*x*Log[2] + (-16*x^3 - 4*x*Log[2])*Log[x^2])*Log[-25 + Log[x
^2]] + (-75*x^2 + 3*x^2*Log[x^2])*Log[-25 + Log[x^2]]^2))/(-25 + Log[x^2]),x]

[Out]

Log[2]^2*Defer[Int][E^(-4 + x^2 + 4*x^5 + 4*x^3*Log[2] + x*Log[2]^2 + (-4*x^4 - 2*x^2*Log[2])*Log[-25 + Log[x^
2]] + x^3*Log[-25 + Log[x^2]]^2), x] + 2*Defer[Int][E^(-4 + x^2 + 4*x^5 + 4*x^3*Log[2] + x*Log[2]^2 + (-4*x^4
- 2*x^2*Log[2])*Log[-25 + Log[x^2]] + x^3*Log[-25 + Log[x^2]]^2)*x, x] + 12*Log[2]*Defer[Int][E^(-4 + x^2 + 4*
x^5 + 4*x^3*Log[2] + x*Log[2]^2 + (-4*x^4 - 2*x^2*Log[2])*Log[-25 + Log[x^2]] + x^3*Log[-25 + Log[x^2]]^2)*x^2
, x] + 20*Defer[Int][E^(-4 + x^2 + 4*x^5 + 4*x^3*Log[2] + x*Log[2]^2 + (-4*x^4 - 2*x^2*Log[2])*Log[-25 + Log[x
^2]] + x^3*Log[-25 + Log[x^2]]^2)*x^4, x] - 4*Log[2]*Defer[Int][(E^(-4 + x^2 + 4*x^5 + 4*x^3*Log[2] + x*Log[2]
^2 + (-4*x^4 - 2*x^2*Log[2])*Log[-25 + Log[x^2]] + x^3*Log[-25 + Log[x^2]]^2)*x)/(-25 + Log[x^2]), x] - 8*Defe
r[Int][(E^(-4 + x^2 + 4*x^5 + 4*x^3*Log[2] + x*Log[2]^2 + (-4*x^4 - 2*x^2*Log[2])*Log[-25 + Log[x^2]] + x^3*Lo
g[-25 + Log[x^2]]^2)*x^3)/(-25 + Log[x^2]), x] + 100*Log[2]*Defer[Int][(E^(-4 + x^2 + 4*x^5 + 4*x^3*Log[2] + x
*Log[2]^2 + (-4*x^4 - 2*x^2*Log[2])*Log[-25 + Log[x^2]] + x^3*Log[-25 + Log[x^2]]^2)*x*Log[-25 + Log[x^2]])/(-
25 + Log[x^2]), x] + 4*Defer[Int][(E^(-4 + x^2 + 4*x^5 + 4*x^3*Log[2] + x*Log[2]^2 + (-4*x^4 - 2*x^2*Log[2])*L
og[-25 + Log[x^2]] + x^3*Log[-25 + Log[x^2]]^2)*x^2*Log[-25 + Log[x^2]])/(-25 + Log[x^2]), x] + 400*Defer[Int]
[(E^(-4 + x^2 + 4*x^5 + 4*x^3*Log[2] + x*Log[2]^2 + (-4*x^4 - 2*x^2*Log[2])*Log[-25 + Log[x^2]] + x^3*Log[-25
+ Log[x^2]]^2)*x^3*Log[-25 + Log[x^2]])/(-25 + Log[x^2]), x] - 4*Log[2]*Defer[Int][(E^(-4 + x^2 + 4*x^5 + 4*x^
3*Log[2] + x*Log[2]^2 + (-4*x^4 - 2*x^2*Log[2])*Log[-25 + Log[x^2]] + x^3*Log[-25 + Log[x^2]]^2)*x*Log[x^2]*Lo
g[-25 + Log[x^2]])/(-25 + Log[x^2]), x] - 16*Defer[Int][(E^(-4 + x^2 + 4*x^5 + 4*x^3*Log[2] + x*Log[2]^2 + (-4
*x^4 - 2*x^2*Log[2])*Log[-25 + Log[x^2]] + x^3*Log[-25 + Log[x^2]]^2)*x^3*Log[x^2]*Log[-25 + Log[x^2]])/(-25 +
Log[x^2]), x] + 3*Defer[Int][E^(-4 + x^2 + 4*x^5 + 4*x^3*Log[2] + x*Log[2]^2 + (-4*x^4 - 2*x^2*Log[2])*Log[-2
5 + Log[x^2]] + x^3*Log[-25 + Log[x^2]]^2)*x^2*Log[-25 + Log[x^2]]^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {\exp \left (-4+x^2+4 x^5+4 x^3 \log (2)+x \log ^2(2)+\left (-4 x^4-2 x^2 \log (2)\right ) \log \left (-25+\log \left (x^2\right )\right )+x^3 \log ^2\left (-25+\log \left (x^2\right )\right )\right ) \left (8 x^3+500 x^4+50 x \left (1+\frac {2 \log (2)}{25}\right )+300 x^2 \log (2)+25 \log ^2(2)-2 x \log \left (x^2\right )-20 x^4 \log \left (x^2\right )-12 x^2 \log (2) \log \left (x^2\right )-\log ^2(2) \log \left (x^2\right )\right )}{25-\log \left (x^2\right )}-\frac {4 \exp \left (-4+x^2+4 x^5+4 x^3 \log (2)+x \log ^2(2)+\left (-4 x^4-2 x^2 \log (2)\right ) \log \left (-25+\log \left (x^2\right )\right )+x^3 \log ^2\left (-25+\log \left (x^2\right )\right )\right ) x \left (-x-100 x^2-25 \log (2)+4 x^2 \log \left (x^2\right )+\log (2) \log \left (x^2\right )\right ) \log \left (-25+\log \left (x^2\right )\right )}{-25+\log \left (x^2\right )}+3 \exp \left (-4+x^2+4 x^5+4 x^3 \log (2)+x \log ^2(2)+\left (-4 x^4-2 x^2 \log (2)\right ) \log \left (-25+\log \left (x^2\right )\right )+x^3 \log ^2\left (-25+\log \left (x^2\right )\right )\right ) x^2 \log ^2\left (-25+\log \left (x^2\right )\right )\right ) \, dx\\ &=3 \int \exp \left (-4+x^2+4 x^5+4 x^3 \log (2)+x \log ^2(2)+\left (-4 x^4-2 x^2 \log (2)\right ) \log \left (-25+\log \left (x^2\right )\right )+x^3 \log ^2\left (-25+\log \left (x^2\right )\right )\right ) x^2 \log ^2\left (-25+\log \left (x^2\right )\right ) \, dx-4 \int \frac {\exp \left (-4+x^2+4 x^5+4 x^3 \log (2)+x \log ^2(2)+\left (-4 x^4-2 x^2 \log (2)\right ) \log \left (-25+\log \left (x^2\right )\right )+x^3 \log ^2\left (-25+\log \left (x^2\right )\right )\right ) x \left (-x-100 x^2-25 \log (2)+4 x^2 \log \left (x^2\right )+\log (2) \log \left (x^2\right )\right ) \log \left (-25+\log \left (x^2\right )\right )}{-25+\log \left (x^2\right )} \, dx+\int \frac {\exp \left (-4+x^2+4 x^5+4 x^3 \log (2)+x \log ^2(2)+\left (-4 x^4-2 x^2 \log (2)\right ) \log \left (-25+\log \left (x^2\right )\right )+x^3 \log ^2\left (-25+\log \left (x^2\right )\right )\right ) \left (8 x^3+500 x^4+50 x \left (1+\frac {2 \log (2)}{25}\right )+300 x^2 \log (2)+25 \log ^2(2)-2 x \log \left (x^2\right )-20 x^4 \log \left (x^2\right )-12 x^2 \log (2) \log \left (x^2\right )-\log ^2(2) \log \left (x^2\right )\right )}{25-\log \left (x^2\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [F]  time = 1.04, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{-4+x^2+4 x^5+4 x^3 \log (2)+x \log ^2(2)+\left (-4 x^4-2 x^2 \log (2)\right ) \log \left (-25+\log \left (x^2\right )\right )+x^3 \log ^2\left (-25+\log \left (x^2\right )\right )} \left (-50 x-8 x^3-500 x^4+\left (-4 x-300 x^2\right ) \log (2)-25 \log ^2(2)+\left (2 x+20 x^4+12 x^2 \log (2)+\log ^2(2)\right ) \log \left (x^2\right )+\left (4 x^2+400 x^3+100 x \log (2)+\left (-16 x^3-4 x \log (2)\right ) \log \left (x^2\right )\right ) \log \left (-25+\log \left (x^2\right )\right )+\left (-75 x^2+3 x^2 \log \left (x^2\right )\right ) \log ^2\left (-25+\log \left (x^2\right )\right )\right )}{-25+\log \left (x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(E^(-4 + x^2 + 4*x^5 + 4*x^3*Log[2] + x*Log[2]^2 + (-4*x^4 - 2*x^2*Log[2])*Log[-25 + Log[x^2]] + x^3
*Log[-25 + Log[x^2]]^2)*(-50*x - 8*x^3 - 500*x^4 + (-4*x - 300*x^2)*Log[2] - 25*Log[2]^2 + (2*x + 20*x^4 + 12*
x^2*Log[2] + Log[2]^2)*Log[x^2] + (4*x^2 + 400*x^3 + 100*x*Log[2] + (-16*x^3 - 4*x*Log[2])*Log[x^2])*Log[-25 +
Log[x^2]] + (-75*x^2 + 3*x^2*Log[x^2])*Log[-25 + Log[x^2]]^2))/(-25 + Log[x^2]),x]

[Out]

Integrate[(E^(-4 + x^2 + 4*x^5 + 4*x^3*Log[2] + x*Log[2]^2 + (-4*x^4 - 2*x^2*Log[2])*Log[-25 + Log[x^2]] + x^3
*Log[-25 + Log[x^2]]^2)*(-50*x - 8*x^3 - 500*x^4 + (-4*x - 300*x^2)*Log[2] - 25*Log[2]^2 + (2*x + 20*x^4 + 12*
x^2*Log[2] + Log[2]^2)*Log[x^2] + (4*x^2 + 400*x^3 + 100*x*Log[2] + (-16*x^3 - 4*x*Log[2])*Log[x^2])*Log[-25 +
Log[x^2]] + (-75*x^2 + 3*x^2*Log[x^2])*Log[-25 + Log[x^2]]^2))/(-25 + Log[x^2]), x]

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fricas [B]  time = 1.09, size = 58, normalized size = 2.07 \begin {gather*} e^{\left (4 \, x^{5} + x^{3} \log \left (\log \left (x^{2}\right ) - 25\right )^{2} + 4 \, x^{3} \log \relax (2) + x \log \relax (2)^{2} + x^{2} - 2 \, {\left (2 \, x^{4} + x^{2} \log \relax (2)\right )} \log \left (\log \left (x^{2}\right ) - 25\right ) - 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^2*log(x^2)-75*x^2)*log(log(x^2)-25)^2+((-4*x*log(2)-16*x^3)*log(x^2)+100*x*log(2)+400*x^3+4*x^
2)*log(log(x^2)-25)+(log(2)^2+12*x^2*log(2)+20*x^4+2*x)*log(x^2)-25*log(2)^2+(-300*x^2-4*x)*log(2)-500*x^4-8*x
^3-50*x)*exp(x^3*log(log(x^2)-25)^2+(-2*x^2*log(2)-4*x^4)*log(log(x^2)-25)+x*log(2)^2+4*x^3*log(2)+4*x^5+x^2-4
)/(log(x^2)-25),x, algorithm="fricas")

[Out]

e^(4*x^5 + x^3*log(log(x^2) - 25)^2 + 4*x^3*log(2) + x*log(2)^2 + x^2 - 2*(2*x^4 + x^2*log(2))*log(log(x^2) -
25) - 4)

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giac [B]  time = 9.39, size = 63, normalized size = 2.25 \begin {gather*} e^{\left (4 \, x^{5} - 4 \, x^{4} \log \left (\log \left (x^{2}\right ) - 25\right ) + x^{3} \log \left (\log \left (x^{2}\right ) - 25\right )^{2} + 4 \, x^{3} \log \relax (2) - 2 \, x^{2} \log \relax (2) \log \left (\log \left (x^{2}\right ) - 25\right ) + x \log \relax (2)^{2} + x^{2} - 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^2*log(x^2)-75*x^2)*log(log(x^2)-25)^2+((-4*x*log(2)-16*x^3)*log(x^2)+100*x*log(2)+400*x^3+4*x^
2)*log(log(x^2)-25)+(log(2)^2+12*x^2*log(2)+20*x^4+2*x)*log(x^2)-25*log(2)^2+(-300*x^2-4*x)*log(2)-500*x^4-8*x
^3-50*x)*exp(x^3*log(log(x^2)-25)^2+(-2*x^2*log(2)-4*x^4)*log(log(x^2)-25)+x*log(2)^2+4*x^3*log(2)+4*x^5+x^2-4
)/(log(x^2)-25),x, algorithm="giac")

[Out]

e^(4*x^5 - 4*x^4*log(log(x^2) - 25) + x^3*log(log(x^2) - 25)^2 + 4*x^3*log(2) - 2*x^2*log(2)*log(log(x^2) - 25
) + x*log(2)^2 + x^2 - 4)

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maple [F]  time = 0.66, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (3 x^{2} \ln \left (x^{2}\right )-75 x^{2}\right ) \ln \left (\ln \left (x^{2}\right )-25\right )^{2}+\left (\left (-4 x \ln \relax (2)-16 x^{3}\right ) \ln \left (x^{2}\right )+100 x \ln \relax (2)+400 x^{3}+4 x^{2}\right ) \ln \left (\ln \left (x^{2}\right )-25\right )+\left (\ln \relax (2)^{2}+12 x^{2} \ln \relax (2)+20 x^{4}+2 x \right ) \ln \left (x^{2}\right )-25 \ln \relax (2)^{2}+\left (-300 x^{2}-4 x \right ) \ln \relax (2)-500 x^{4}-8 x^{3}-50 x \right ) {\mathrm e}^{x^{3} \ln \left (\ln \left (x^{2}\right )-25\right )^{2}+\left (-2 x^{2} \ln \relax (2)-4 x^{4}\right ) \ln \left (\ln \left (x^{2}\right )-25\right )+x \ln \relax (2)^{2}+4 x^{3} \ln \relax (2)+4 x^{5}+x^{2}-4}}{\ln \left (x^{2}\right )-25}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x^2*ln(x^2)-75*x^2)*ln(ln(x^2)-25)^2+((-4*x*ln(2)-16*x^3)*ln(x^2)+100*x*ln(2)+400*x^3+4*x^2)*ln(ln(x^2
)-25)+(ln(2)^2+12*x^2*ln(2)+20*x^4+2*x)*ln(x^2)-25*ln(2)^2+(-300*x^2-4*x)*ln(2)-500*x^4-8*x^3-50*x)*exp(x^3*ln
(ln(x^2)-25)^2+(-2*x^2*ln(2)-4*x^4)*ln(ln(x^2)-25)+x*ln(2)^2+4*x^3*ln(2)+4*x^5+x^2-4)/(ln(x^2)-25),x)

[Out]

int(((3*x^2*ln(x^2)-75*x^2)*ln(ln(x^2)-25)^2+((-4*x*ln(2)-16*x^3)*ln(x^2)+100*x*ln(2)+400*x^3+4*x^2)*ln(ln(x^2
)-25)+(ln(2)^2+12*x^2*ln(2)+20*x^4+2*x)*ln(x^2)-25*ln(2)^2+(-300*x^2-4*x)*ln(2)-500*x^4-8*x^3-50*x)*exp(x^3*ln
(ln(x^2)-25)^2+(-2*x^2*ln(2)-4*x^4)*ln(ln(x^2)-25)+x*ln(2)^2+4*x^3*ln(2)+4*x^5+x^2-4)/(ln(x^2)-25),x)

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maxima [B]  time = 1.19, size = 63, normalized size = 2.25 \begin {gather*} e^{\left (4 \, x^{5} - 4 \, x^{4} \log \left (2 \, \log \relax (x) - 25\right ) + x^{3} \log \left (2 \, \log \relax (x) - 25\right )^{2} + 4 \, x^{3} \log \relax (2) - 2 \, x^{2} \log \relax (2) \log \left (2 \, \log \relax (x) - 25\right ) + x \log \relax (2)^{2} + x^{2} - 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^2*log(x^2)-75*x^2)*log(log(x^2)-25)^2+((-4*x*log(2)-16*x^3)*log(x^2)+100*x*log(2)+400*x^3+4*x^
2)*log(log(x^2)-25)+(log(2)^2+12*x^2*log(2)+20*x^4+2*x)*log(x^2)-25*log(2)^2+(-300*x^2-4*x)*log(2)-500*x^4-8*x
^3-50*x)*exp(x^3*log(log(x^2)-25)^2+(-2*x^2*log(2)-4*x^4)*log(log(x^2)-25)+x*log(2)^2+4*x^3*log(2)+4*x^5+x^2-4
)/(log(x^2)-25),x, algorithm="maxima")

[Out]

e^(4*x^5 - 4*x^4*log(2*log(x) - 25) + x^3*log(2*log(x) - 25)^2 + 4*x^3*log(2) - 2*x^2*log(2)*log(2*log(x) - 25
) + x*log(2)^2 + x^2 - 4)

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mupad [B]  time = 1.53, size = 61, normalized size = 2.18 \begin {gather*} \frac {{16}^{x^3}\,{\mathrm {e}}^{x\,{\ln \relax (2)}^2}\,{\mathrm {e}}^{x^3\,{\ln \left (\ln \left (x^2\right )-25\right )}^2}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{-4}\,{\mathrm {e}}^{4\,x^5}}{{\left (\ln \left (x^2\right )-25\right )}^{4\,x^4+2\,\ln \relax (2)\,x^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x*log(2)^2 + 4*x^3*log(2) + x^3*log(log(x^2) - 25)^2 - log(log(x^2) - 25)*(2*x^2*log(2) + 4*x^4) + x
^2 + 4*x^5 - 4)*(50*x + log(2)*(4*x + 300*x^2) - log(x^2)*(2*x + 12*x^2*log(2) + log(2)^2 + 20*x^4) + 25*log(2
)^2 + 8*x^3 + 500*x^4 - log(log(x^2) - 25)^2*(3*x^2*log(x^2) - 75*x^2) - log(log(x^2) - 25)*(100*x*log(2) - lo
g(x^2)*(4*x*log(2) + 16*x^3) + 4*x^2 + 400*x^3)))/(log(x^2) - 25),x)

[Out]

(16^(x^3)*exp(x*log(2)^2)*exp(x^3*log(log(x^2) - 25)^2)*exp(x^2)*exp(-4)*exp(4*x^5))/(log(x^2) - 25)^(2*x^2*lo
g(2) + 4*x^4)

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sympy [B]  time = 1.35, size = 63, normalized size = 2.25 \begin {gather*} e^{4 x^{5} + x^{3} \log {\left (\log {\left (x^{2} \right )} - 25 \right )}^{2} + 4 x^{3} \log {\relax (2 )} + x^{2} + x \log {\relax (2 )}^{2} + \left (- 4 x^{4} - 2 x^{2} \log {\relax (2 )}\right ) \log {\left (\log {\left (x^{2} \right )} - 25 \right )} - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x**2*ln(x**2)-75*x**2)*ln(ln(x**2)-25)**2+((-4*x*ln(2)-16*x**3)*ln(x**2)+100*x*ln(2)+400*x**3+4*
x**2)*ln(ln(x**2)-25)+(ln(2)**2+12*x**2*ln(2)+20*x**4+2*x)*ln(x**2)-25*ln(2)**2+(-300*x**2-4*x)*ln(2)-500*x**4
-8*x**3-50*x)*exp(x**3*ln(ln(x**2)-25)**2+(-2*x**2*ln(2)-4*x**4)*ln(ln(x**2)-25)+x*ln(2)**2+4*x**3*ln(2)+4*x**
5+x**2-4)/(ln(x**2)-25),x)

[Out]

exp(4*x**5 + x**3*log(log(x**2) - 25)**2 + 4*x**3*log(2) + x**2 + x*log(2)**2 + (-4*x**4 - 2*x**2*log(2))*log(
log(x**2) - 25) - 4)

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