3.44.95 \(\int \frac {-480-84 x+21 x^2+3 x^3+e^{25+10 x+x^2} (-160-28 x+7 x^2+x^3) \log (5-x)+(60 x-12 x^2+e^{25+10 x+x^2} (32 x+12 x^2+x^3)+e^{25+10 x+x^2} (-1580 x-604 x^2+14 x^3+24 x^4+2 x^5) \log (5-x)) \log (x)}{(-480 x-84 x^2+21 x^3+3 x^4+e^{25+10 x+x^2} (-160 x-28 x^2+7 x^3+x^4) \log (5-x)) \log (x)} \, dx\)

Optimal. Leaf size=28 \[ \log \left (\frac {(8+x) \left (3+e^{(5+x)^2} \log (5-x)\right ) \log (x)}{4+x}\right ) \]

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Rubi [F]  time = 9.68, 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 {-480-84 x+21 x^2+3 x^3+e^{25+10 x+x^2} \left (-160-28 x+7 x^2+x^3\right ) \log (5-x)+\left (60 x-12 x^2+e^{25+10 x+x^2} \left (32 x+12 x^2+x^3\right )+e^{25+10 x+x^2} \left (-1580 x-604 x^2+14 x^3+24 x^4+2 x^5\right ) \log (5-x)\right ) \log (x)}{\left (-480 x-84 x^2+21 x^3+3 x^4+e^{25+10 x+x^2} \left (-160 x-28 x^2+7 x^3+x^4\right ) \log (5-x)\right ) \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-480 - 84*x + 21*x^2 + 3*x^3 + E^(25 + 10*x + x^2)*(-160 - 28*x + 7*x^2 + x^3)*Log[5 - x] + (60*x - 12*x^
2 + E^(25 + 10*x + x^2)*(32*x + 12*x^2 + x^3) + E^(25 + 10*x + x^2)*(-1580*x - 604*x^2 + 14*x^3 + 24*x^4 + 2*x
^5)*Log[5 - x])*Log[x])/((-480*x - 84*x^2 + 21*x^3 + 3*x^4 + E^(25 + 10*x + x^2)*(-160*x - 28*x^2 + 7*x^3 + x^
4)*Log[5 - x])*Log[x]),x]

[Out]

10*x + x^2 - Log[4 + x] + Log[8 + x] + Log[Log[5 - x]] + Log[Log[x]] - 30*Defer[Int][(3 + E^(5 + x)^2*Log[5 -
x])^(-1), x] - 6*Defer[Int][x/(3 + E^(5 + x)^2*Log[5 - x]), x] - 3*Defer[Int][1/((-5 + x)*Log[5 - x]*(3 + E^(5
+ x)^2*Log[5 - x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-3 \left (-160-28 x+7 x^2+x^3\right )-x \left (-12 (-5+x)+e^{(5+x)^2} \left (32+12 x+x^2\right )\right ) \log (x)-e^{(5+x)^2} (-5+x) \log (5-x) \left (32+12 x+x^2+2 x \left (158+92 x+17 x^2+x^3\right ) \log (x)\right )}{x \left (160+28 x-7 x^2-x^3\right ) \left (3+e^{(5+x)^2} \log (5-x)\right ) \log (x)} \, dx\\ &=\int \left (-\frac {3 \left (1-50 \log (5-x)+2 x^2 \log (5-x)\right )}{(-5+x) \log (5-x) \left (3+e^{(5+x)^2} \log (5-x)\right )}+\frac {-160 \log (5-x)-28 x \log (5-x)+7 x^2 \log (5-x)+x^3 \log (5-x)+32 x \log (x)+12 x^2 \log (x)+x^3 \log (x)-1580 x \log (5-x) \log (x)-604 x^2 \log (5-x) \log (x)+14 x^3 \log (5-x) \log (x)+24 x^4 \log (5-x) \log (x)+2 x^5 \log (5-x) \log (x)}{(-5+x) x (4+x) (8+x) \log (5-x) \log (x)}\right ) \, dx\\ &=-\left (3 \int \frac {1-50 \log (5-x)+2 x^2 \log (5-x)}{(-5+x) \log (5-x) \left (3+e^{(5+x)^2} \log (5-x)\right )} \, dx\right )+\int \frac {-160 \log (5-x)-28 x \log (5-x)+7 x^2 \log (5-x)+x^3 \log (5-x)+32 x \log (x)+12 x^2 \log (x)+x^3 \log (x)-1580 x \log (5-x) \log (x)-604 x^2 \log (5-x) \log (x)+14 x^3 \log (5-x) \log (x)+24 x^4 \log (5-x) \log (x)+2 x^5 \log (5-x) \log (x)}{(-5+x) x (4+x) (8+x) \log (5-x) \log (x)} \, dx\\ &=-\left (3 \int \left (-\frac {50}{(-5+x) \left (3+e^{(5+x)^2} \log (5-x)\right )}+\frac {2 x^2}{(-5+x) \left (3+e^{(5+x)^2} \log (5-x)\right )}+\frac {1}{(-5+x) \log (5-x) \left (3+e^{(5+x)^2} \log (5-x)\right )}\right ) \, dx\right )+\int \frac {\frac {32+12 x+x^2}{(-5+x) \log (5-x)}+\frac {32+12 x+x^2+2 x \left (158+92 x+17 x^2+x^3\right ) \log (x)}{x \log (x)}}{(4+x) (8+x)} \, dx\\ &=-\left (3 \int \frac {1}{(-5+x) \log (5-x) \left (3+e^{(5+x)^2} \log (5-x)\right )} \, dx\right )-6 \int \frac {x^2}{(-5+x) \left (3+e^{(5+x)^2} \log (5-x)\right )} \, dx+150 \int \frac {1}{(-5+x) \left (3+e^{(5+x)^2} \log (5-x)\right )} \, dx+\int \left (\frac {32+12 x+x^2-1580 \log (5-x)-604 x \log (5-x)+14 x^2 \log (5-x)+24 x^3 \log (5-x)+2 x^4 \log (5-x)}{(-5+x) (4+x) (8+x) \log (5-x)}+\frac {1}{x \log (x)}\right ) \, dx\\ &=-\left (3 \int \frac {1}{(-5+x) \log (5-x) \left (3+e^{(5+x)^2} \log (5-x)\right )} \, dx\right )-6 \int \left (\frac {5}{3+e^{(5+x)^2} \log (5-x)}+\frac {25}{(-5+x) \left (3+e^{(5+x)^2} \log (5-x)\right )}+\frac {x}{3+e^{(5+x)^2} \log (5-x)}\right ) \, dx+150 \int \frac {1}{(-5+x) \left (3+e^{(5+x)^2} \log (5-x)\right )} \, dx+\int \frac {32+12 x+x^2-1580 \log (5-x)-604 x \log (5-x)+14 x^2 \log (5-x)+24 x^3 \log (5-x)+2 x^4 \log (5-x)}{(-5+x) (4+x) (8+x) \log (5-x)} \, dx+\int \frac {1}{x \log (x)} \, dx\\ &=-\left (3 \int \frac {1}{(-5+x) \log (5-x) \left (3+e^{(5+x)^2} \log (5-x)\right )} \, dx\right )-6 \int \frac {x}{3+e^{(5+x)^2} \log (5-x)} \, dx-30 \int \frac {1}{3+e^{(5+x)^2} \log (5-x)} \, dx+\int \frac {-32-12 x-x^2-2 \left (-790-302 x+7 x^2+12 x^3+x^4\right ) \log (5-x)}{(5-x) (4+x) (8+x) \log (5-x)} \, dx+\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right )\\ &=\log (\log (x))-3 \int \frac {1}{(-5+x) \log (5-x) \left (3+e^{(5+x)^2} \log (5-x)\right )} \, dx-6 \int \frac {x}{3+e^{(5+x)^2} \log (5-x)} \, dx-30 \int \frac {1}{3+e^{(5+x)^2} \log (5-x)} \, dx+\int \left (\frac {2 \left (158+92 x+17 x^2+x^3\right )}{(4+x) (8+x)}+\frac {1}{(-5+x) \log (5-x)}\right ) \, dx\\ &=\log (\log (x))+2 \int \frac {158+92 x+17 x^2+x^3}{(4+x) (8+x)} \, dx-3 \int \frac {1}{(-5+x) \log (5-x) \left (3+e^{(5+x)^2} \log (5-x)\right )} \, dx-6 \int \frac {x}{3+e^{(5+x)^2} \log (5-x)} \, dx-30 \int \frac {1}{3+e^{(5+x)^2} \log (5-x)} \, dx+\int \frac {1}{(-5+x) \log (5-x)} \, dx\\ &=\log (\log (x))+2 \int \left (5+x-\frac {1}{2 (4+x)}+\frac {1}{2 (8+x)}\right ) \, dx-3 \int \frac {1}{(-5+x) \log (5-x) \left (3+e^{(5+x)^2} \log (5-x)\right )} \, dx-6 \int \frac {x}{3+e^{(5+x)^2} \log (5-x)} \, dx-30 \int \frac {1}{3+e^{(5+x)^2} \log (5-x)} \, dx+\operatorname {Subst}\left (\int \frac {1}{x \log (x)} \, dx,x,5-x\right )\\ &=10 x+x^2-\log (4+x)+\log (8+x)+\log (\log (x))-3 \int \frac {1}{(-5+x) \log (5-x) \left (3+e^{(5+x)^2} \log (5-x)\right )} \, dx-6 \int \frac {x}{3+e^{(5+x)^2} \log (5-x)} \, dx-30 \int \frac {1}{3+e^{(5+x)^2} \log (5-x)} \, dx+\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (5-x)\right )\\ &=10 x+x^2-\log (4+x)+\log (8+x)+\log (\log (5-x))+\log (\log (x))-3 \int \frac {1}{(-5+x) \log (5-x) \left (3+e^{(5+x)^2} \log (5-x)\right )} \, dx-6 \int \frac {x}{3+e^{(5+x)^2} \log (5-x)} \, dx-30 \int \frac {1}{3+e^{(5+x)^2} \log (5-x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.23, size = 32, normalized size = 1.14 \begin {gather*} -25-\log (4+x)+\log (8+x)+\log \left (3+e^{(5+x)^2} \log (5-x)\right )+\log (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-480 - 84*x + 21*x^2 + 3*x^3 + E^(25 + 10*x + x^2)*(-160 - 28*x + 7*x^2 + x^3)*Log[5 - x] + (60*x -
12*x^2 + E^(25 + 10*x + x^2)*(32*x + 12*x^2 + x^3) + E^(25 + 10*x + x^2)*(-1580*x - 604*x^2 + 14*x^3 + 24*x^4
+ 2*x^5)*Log[5 - x])*Log[x])/((-480*x - 84*x^2 + 21*x^3 + 3*x^4 + E^(25 + 10*x + x^2)*(-160*x - 28*x^2 + 7*x^
3 + x^4)*Log[5 - x])*Log[x]),x]

[Out]

-25 - Log[4 + x] + Log[8 + x] + Log[3 + E^(5 + x)^2*Log[5 - x]] + Log[Log[x]]

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fricas [A]  time = 0.55, size = 51, normalized size = 1.82 \begin {gather*} x^{2} + 10 \, x + \log \left ({\left (e^{\left (x^{2} + 10 \, x + 25\right )} \log \left (-x + 5\right ) + 3\right )} e^{\left (-x^{2} - 10 \, x - 25\right )}\right ) + \log \left (x + 8\right ) - \log \left (x + 4\right ) + \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^5+24*x^4+14*x^3-604*x^2-1580*x)*exp(x^2+10*x+25)*log(5-x)+(x^3+12*x^2+32*x)*exp(x^2+10*x+25)-
12*x^2+60*x)*log(x)+(x^3+7*x^2-28*x-160)*exp(x^2+10*x+25)*log(5-x)+3*x^3+21*x^2-84*x-480)/((x^4+7*x^3-28*x^2-1
60*x)*exp(x^2+10*x+25)*log(5-x)+3*x^4+21*x^3-84*x^2-480*x)/log(x),x, algorithm="fricas")

[Out]

x^2 + 10*x + log((e^(x^2 + 10*x + 25)*log(-x + 5) + 3)*e^(-x^2 - 10*x - 25)) + log(x + 8) - log(x + 4) + log(l
og(x))

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giac [A]  time = 0.25, size = 33, normalized size = 1.18 \begin {gather*} \log \left (e^{\left (x^{2} + 10 \, x + 25\right )} \log \left (-x + 5\right ) + 3\right ) + \log \left (x + 8\right ) - \log \left (x + 4\right ) + \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^5+24*x^4+14*x^3-604*x^2-1580*x)*exp(x^2+10*x+25)*log(5-x)+(x^3+12*x^2+32*x)*exp(x^2+10*x+25)-
12*x^2+60*x)*log(x)+(x^3+7*x^2-28*x-160)*exp(x^2+10*x+25)*log(5-x)+3*x^3+21*x^2-84*x-480)/((x^4+7*x^3-28*x^2-1
60*x)*exp(x^2+10*x+25)*log(5-x)+3*x^4+21*x^3-84*x^2-480*x)/log(x),x, algorithm="giac")

[Out]

log(e^(x^2 + 10*x + 25)*log(-x + 5) + 3) + log(x + 8) - log(x + 4) + log(log(x))

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maple [A]  time = 0.06, size = 39, normalized size = 1.39

method	result	size
risch	\(x^{2}+10 x -\ln \left (4+x \right )+\ln \left (x +8\right )+\ln \left (\ln \relax (x )\right )+\ln \left (\ln \left (5-x \right )+3 \,{\mathrm e}^{-\left (5+x \right )^{2}}\right )\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*x^5+24*x^4+14*x^3-604*x^2-1580*x)*exp(x^2+10*x+25)*ln(5-x)+(x^3+12*x^2+32*x)*exp(x^2+10*x+25)-12*x^2+
60*x)*ln(x)+(x^3+7*x^2-28*x-160)*exp(x^2+10*x+25)*ln(5-x)+3*x^3+21*x^2-84*x-480)/((x^4+7*x^3-28*x^2-160*x)*exp
(x^2+10*x+25)*ln(5-x)+3*x^4+21*x^3-84*x^2-480*x)/ln(x),x,method=_RETURNVERBOSE)

[Out]

x^2+10*x-ln(4+x)+ln(x+8)+ln(ln(x))+ln(ln(5-x)+3*exp(-(5+x)^2))

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maxima [A]  time = 0.43, size = 51, normalized size = 1.82 \begin {gather*} x^{2} + 10 \, x + \log \left ({\left (e^{\left (x^{2} + 10 \, x + 25\right )} \log \left (-x + 5\right ) + 3\right )} e^{\left (-x^{2} - 10 \, x - 25\right )}\right ) + \log \left (x + 8\right ) - \log \left (x + 4\right ) + \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^5+24*x^4+14*x^3-604*x^2-1580*x)*exp(x^2+10*x+25)*log(5-x)+(x^3+12*x^2+32*x)*exp(x^2+10*x+25)-
12*x^2+60*x)*log(x)+(x^3+7*x^2-28*x-160)*exp(x^2+10*x+25)*log(5-x)+3*x^3+21*x^2-84*x-480)/((x^4+7*x^3-28*x^2-1
60*x)*exp(x^2+10*x+25)*log(5-x)+3*x^4+21*x^3-84*x^2-480*x)/log(x),x, algorithm="maxima")

[Out]

x^2 + 10*x + log((e^(x^2 + 10*x + 25)*log(-x + 5) + 3)*e^(-x^2 - 10*x - 25)) + log(x + 8) - log(x + 4) + log(l
og(x))

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mupad [B]  time = 4.72, size = 47, normalized size = 1.68 \begin {gather*} \ln \left (\ln \left (5-x\right )\right )+\ln \left (\ln \relax (x)\right )+\ln \left (\frac {{\mathrm {e}}^{{\left (x+5\right )}^2}\,\ln \left (5-x\right )+3}{\ln \left (5-x\right )}\right )-\mathrm {atan}\left (\frac {x\,1{}\mathrm {i}}{2}+3{}\mathrm {i}\right )\,2{}\mathrm {i} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((84*x - log(x)*(60*x + exp(10*x + x^2 + 25)*(32*x + 12*x^2 + x^3) - 12*x^2 + exp(10*x + x^2 + 25)*log(5 -
x)*(14*x^3 - 604*x^2 - 1580*x + 24*x^4 + 2*x^5)) - 21*x^2 - 3*x^3 + exp(10*x + x^2 + 25)*log(5 - x)*(28*x - 7*
x^2 - x^3 + 160) + 480)/(log(x)*(480*x + 84*x^2 - 21*x^3 - 3*x^4 + exp(10*x + x^2 + 25)*log(5 - x)*(160*x + 28
*x^2 - 7*x^3 - x^4))),x)

[Out]

log(log(5 - x)) + log(log(x)) - atan((x*1i)/2 + 3i)*2i + log((exp((x + 5)^2)*log(5 - x) + 3)/log(5 - x))

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sympy [A]  time = 0.80, size = 39, normalized size = 1.39 \begin {gather*} - \log {\left (x + 4 \right )} + \log {\left (x + 8 \right )} + \log {\left (e^{x^{2} + 10 x + 25} + \frac {3}{\log {\left (5 - x \right )}} \right )} + \log {\left (\log {\relax (x )} \right )} + \log {\left (\log {\left (5 - x \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x**5+24*x**4+14*x**3-604*x**2-1580*x)*exp(x**2+10*x+25)*ln(5-x)+(x**3+12*x**2+32*x)*exp(x**2+10
*x+25)-12*x**2+60*x)*ln(x)+(x**3+7*x**2-28*x-160)*exp(x**2+10*x+25)*ln(5-x)+3*x**3+21*x**2-84*x-480)/((x**4+7*
x**3-28*x**2-160*x)*exp(x**2+10*x+25)*ln(5-x)+3*x**4+21*x**3-84*x**2-480*x)/ln(x),x)

[Out]

-log(x + 4) + log(x + 8) + log(exp(x**2 + 10*x + 25) + 3/log(5 - x)) + log(log(x)) + log(log(5 - x))

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