3.101.18 \(\int \frac {(2+e^{15+6 x+x^2+(-6-2 x) \log (16 e^{-2 x})+\log ^2(16 e^{-2 x})} (36+12 x-12 \log (16 e^{-2 x}))) \log (e^{15+6 x+x^2+(-6-2 x) \log (16 e^{-2 x})+\log ^2(16 e^{-2 x})}+x)}{e^{15+6 x+x^2+(-6-2 x) \log (16 e^{-2 x})+\log ^2(16 e^{-2 x})}+x} \, dx\)

Optimal. Leaf size=26 \[ 5+\log ^2\left (e^{6+\left (3+x-\log \left (16 e^{-2 x}\right )\right )^2}+x\right ) \]

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Rubi [F]  time = 11.28, 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 {\left (2+\exp \left (15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )\right ) \left (36+12 x-12 \log \left (16 e^{-2 x}\right )\right )\right ) \log \left (\exp \left (15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )\right )+x\right )}{\exp \left (15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )\right )+x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((2 + E^(15 + 6*x + x^2 + (-6 - 2*x)*Log[16/E^(2*x)] + Log[16/E^(2*x)]^2)*(36 + 12*x - 12*Log[16/E^(2*x)])
)*Log[E^(15 + 6*x + x^2 + (-6 - 2*x)*Log[16/E^(2*x)] + Log[16/E^(2*x)]^2) + x])/(E^(15 + 6*x + x^2 + (-6 - 2*x
)*Log[16/E^(2*x)] + Log[16/E^(2*x)]^2) + x),x]

[Out]

36*Defer[Int][Log[E^(15 + 6*x + x^2 - 2*(3 + x)*Log[16/E^(2*x)] + Log[16/E^(2*x)]^2) + x], x] + 12*Defer[Int][
x*Log[E^(15 + 6*x + x^2 - 2*(3 + x)*Log[16/E^(2*x)] + Log[16/E^(2*x)]^2) + x], x] + Defer[Int][(2^(25 + 8*x)*(
E^(-2*x))^(2*x)*Log[E^(15 + 6*x + x^2 - 2*(3 + x)*Log[16/E^(2*x)] + Log[16/E^(2*x)]^2) + x])/(E^(15 + 18*x + x
^2 + Log[16/E^(2*x)]^2) + 256^(3 + x)*(E^(-2*x))^(2*x)*x), x] - 9*Defer[Int][(2^(26 + 8*x)*(E^(-2*x))^(2*x)*x*
Log[E^(15 + 6*x + x^2 - 2*(3 + x)*Log[16/E^(2*x)] + Log[16/E^(2*x)]^2) + x])/(E^(15 + 18*x + x^2 + Log[16/E^(2
*x)]^2) + 256^(3 + x)*(E^(-2*x))^(2*x)*x), x] - 3*Defer[Int][(2^(26 + 8*x)*(E^(-2*x))^(2*x)*x^2*Log[E^(15 + 6*
x + x^2 - 2*(3 + x)*Log[16/E^(2*x)] + Log[16/E^(2*x)]^2) + x])/(E^(15 + 18*x + x^2 + Log[16/E^(2*x)]^2) + 256^
(3 + x)*(E^(-2*x))^(2*x)*x), x] - 12*Defer[Int][Log[16/E^(2*x)]*Log[E^(15 + 6*x + x^2 - 2*(3 + x)*Log[16/E^(2*
x)] + Log[16/E^(2*x)]^2) + x], x] + 3*Defer[Int][(2^(26 + 8*x)*(E^(-2*x))^(2*x)*x*Log[16/E^(2*x)]*Log[E^(15 +
6*x + x^2 - 2*(3 + x)*Log[16/E^(2*x)] + Log[16/E^(2*x)]^2) + x])/(E^(15 + 18*x + x^2 + Log[16/E^(2*x)]^2) + 25
6^(3 + x)*(E^(-2*x))^(2*x)*x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (12 \left (3+x-\log \left (16 e^{-2 x}\right )\right ) \log \left (\exp \left (15+6 x+x^2-2 (3+x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )\right )+x\right )-\frac {2^{25+8 x} \left (e^{-2 x}\right )^{2 x} \left (-1+18 x+6 x^2-6 x \log \left (16 e^{-2 x}\right )\right ) \log \left (\exp \left (15+6 x+x^2-2 (3+x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )\right )+x\right )}{e^{15+18 x+x^2+\log ^2\left (16 e^{-2 x}\right )}+2^{24+8 x} \left (e^{-2 x}\right )^{2 x} x}\right ) \, dx\\ &=12 \int \left (3+x-\log \left (16 e^{-2 x}\right )\right ) \log \left (\exp \left (15+6 x+x^2-2 (3+x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )\right )+x\right ) \, dx-\int \frac {2^{25+8 x} \left (e^{-2 x}\right )^{2 x} \left (-1+18 x+6 x^2-6 x \log \left (16 e^{-2 x}\right )\right ) \log \left (\exp \left (15+6 x+x^2-2 (3+x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )\right )+x\right )}{e^{15+18 x+x^2+\log ^2\left (16 e^{-2 x}\right )}+2^{24+8 x} \left (e^{-2 x}\right )^{2 x} x} \, dx\\ &=12 \int \left (3 \log \left (\exp \left (15+6 x+x^2-2 (3+x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )\right )+x\right )+x \log \left (\exp \left (15+6 x+x^2-2 (3+x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )\right )+x\right )-\log \left (16 e^{-2 x}\right ) \log \left (\exp \left (15+6 x+x^2-2 (3+x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )\right )+x\right )\right ) \, dx-\int \left (-\frac {2^{25+8 x} \left (e^{-2 x}\right )^{2 x} \log \left (\exp \left (15+6 x+x^2-2 (3+x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )\right )+x\right )}{e^{15+18 x+x^2+\log ^2\left (16 e^{-2 x}\right )}+256^{3+x} \left (e^{-2 x}\right )^{2 x} x}+\frac {9\ 2^{26+8 x} \left (e^{-2 x}\right )^{2 x} x \log \left (\exp \left (15+6 x+x^2-2 (3+x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )\right )+x\right )}{e^{15+18 x+x^2+\log ^2\left (16 e^{-2 x}\right )}+256^{3+x} \left (e^{-2 x}\right )^{2 x} x}+\frac {3\ 2^{26+8 x} \left (e^{-2 x}\right )^{2 x} x^2 \log \left (\exp \left (15+6 x+x^2-2 (3+x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )\right )+x\right )}{e^{15+18 x+x^2+\log ^2\left (16 e^{-2 x}\right )}+256^{3+x} \left (e^{-2 x}\right )^{2 x} x}-\frac {3\ 2^{26+8 x} \left (e^{-2 x}\right )^{2 x} x \log \left (16 e^{-2 x}\right ) \log \left (\exp \left (15+6 x+x^2-2 (3+x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )\right )+x\right )}{e^{15+18 x+x^2+\log ^2\left (16 e^{-2 x}\right )}+256^{3+x} \left (e^{-2 x}\right )^{2 x} x}\right ) \, dx\\ &=-\left (3 \int \frac {2^{26+8 x} \left (e^{-2 x}\right )^{2 x} x^2 \log \left (\exp \left (15+6 x+x^2-2 (3+x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )\right )+x\right )}{e^{15+18 x+x^2+\log ^2\left (16 e^{-2 x}\right )}+256^{3+x} \left (e^{-2 x}\right )^{2 x} x} \, dx\right )+3 \int \frac {2^{26+8 x} \left (e^{-2 x}\right )^{2 x} x \log \left (16 e^{-2 x}\right ) \log \left (\exp \left (15+6 x+x^2-2 (3+x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )\right )+x\right )}{e^{15+18 x+x^2+\log ^2\left (16 e^{-2 x}\right )}+256^{3+x} \left (e^{-2 x}\right )^{2 x} x} \, dx-9 \int \frac {2^{26+8 x} \left (e^{-2 x}\right )^{2 x} x \log \left (\exp \left (15+6 x+x^2-2 (3+x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )\right )+x\right )}{e^{15+18 x+x^2+\log ^2\left (16 e^{-2 x}\right )}+256^{3+x} \left (e^{-2 x}\right )^{2 x} x} \, dx+12 \int x \log \left (\exp \left (15+6 x+x^2-2 (3+x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )\right )+x\right ) \, dx-12 \int \log \left (16 e^{-2 x}\right ) \log \left (\exp \left (15+6 x+x^2-2 (3+x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )\right )+x\right ) \, dx+36 \int \log \left (\exp \left (15+6 x+x^2-2 (3+x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )\right )+x\right ) \, dx+\int \frac {2^{25+8 x} \left (e^{-2 x}\right )^{2 x} \log \left (\exp \left (15+6 x+x^2-2 (3+x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )\right )+x\right )}{e^{15+18 x+x^2+\log ^2\left (16 e^{-2 x}\right )}+256^{3+x} \left (e^{-2 x}\right )^{2 x} x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.37, size = 38, normalized size = 1.46 \begin {gather*} \log ^2\left (e^{15+6 x+x^2-2 (3+x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )}+x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((2 + E^(15 + 6*x + x^2 + (-6 - 2*x)*Log[16/E^(2*x)] + Log[16/E^(2*x)]^2)*(36 + 12*x - 12*Log[16/E^(
2*x)]))*Log[E^(15 + 6*x + x^2 + (-6 - 2*x)*Log[16/E^(2*x)] + Log[16/E^(2*x)]^2) + x])/(E^(15 + 6*x + x^2 + (-6
- 2*x)*Log[16/E^(2*x)] + Log[16/E^(2*x)]^2) + x),x]

[Out]

Log[E^(15 + 6*x + x^2 - 2*(3 + x)*Log[16/E^(2*x)] + Log[16/E^(2*x)]^2) + x]^2

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fricas [A]  time = 1.76, size = 29, normalized size = 1.12 \begin {gather*} \log \left (x + e^{\left (9 \, x^{2} - 24 \, {\left (x + 1\right )} \log \relax (2) + 16 \, \log \relax (2)^{2} + 18 \, x + 15\right )}\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-12*log(16/exp(x)^2)+12*x+36)*exp(log(16/exp(x)^2)^2+(-2*x-6)*log(16/exp(x)^2)+x^2+6*x+15)+2)*log(
exp(log(16/exp(x)^2)^2+(-2*x-6)*log(16/exp(x)^2)+x^2+6*x+15)+x)/(exp(log(16/exp(x)^2)^2+(-2*x-6)*log(16/exp(x)
^2)+x^2+6*x+15)+x),x, algorithm="fricas")

[Out]

log(x + e^(9*x^2 - 24*(x + 1)*log(2) + 16*log(2)^2 + 18*x + 15))^2

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, {\left (6 \, {\left (x - \log \left (16 \, e^{\left (-2 \, x\right )}\right ) + 3\right )} e^{\left (x^{2} - 2 \, {\left (x + 3\right )} \log \left (16 \, e^{\left (-2 \, x\right )}\right ) + \log \left (16 \, e^{\left (-2 \, x\right )}\right )^{2} + 6 \, x + 15\right )} + 1\right )} \log \left (x + e^{\left (x^{2} - 2 \, {\left (x + 3\right )} \log \left (16 \, e^{\left (-2 \, x\right )}\right ) + \log \left (16 \, e^{\left (-2 \, x\right )}\right )^{2} + 6 \, x + 15\right )}\right )}{x + e^{\left (x^{2} - 2 \, {\left (x + 3\right )} \log \left (16 \, e^{\left (-2 \, x\right )}\right ) + \log \left (16 \, e^{\left (-2 \, x\right )}\right )^{2} + 6 \, x + 15\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-12*log(16/exp(x)^2)+12*x+36)*exp(log(16/exp(x)^2)^2+(-2*x-6)*log(16/exp(x)^2)+x^2+6*x+15)+2)*log(
exp(log(16/exp(x)^2)^2+(-2*x-6)*log(16/exp(x)^2)+x^2+6*x+15)+x)/(exp(log(16/exp(x)^2)^2+(-2*x-6)*log(16/exp(x)
^2)+x^2+6*x+15)+x),x, algorithm="giac")

[Out]

integrate(2*(6*(x - log(16*e^(-2*x)) + 3)*e^(x^2 - 2*(x + 3)*log(16*e^(-2*x)) + log(16*e^(-2*x))^2 + 6*x + 15)
+ 1)*log(x + e^(x^2 - 2*(x + 3)*log(16*e^(-2*x)) + log(16*e^(-2*x))^2 + 6*x + 15))/(x + e^(x^2 - 2*(x + 3)*lo
g(16*e^(-2*x)) + log(16*e^(-2*x))^2 + 6*x + 15)), x)

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maple [A]  time = 0.06, size = 37, normalized size = 1.42 \[\ln \left ({\mathrm e}^{\ln \left (16 \,{\mathrm e}^{-2 x}\right )^{2}+\left (-2 x -6\right ) \ln \left (16 \,{\mathrm e}^{-2 x}\right )+x^{2}+6 x +15}+x \right )^{2}\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-12*ln(16/exp(x)^2)+12*x+36)*exp(ln(16/exp(x)^2)^2+(-2*x-6)*ln(16/exp(x)^2)+x^2+6*x+15)+2)*ln(exp(ln(16/
exp(x)^2)^2+(-2*x-6)*ln(16/exp(x)^2)+x^2+6*x+15)+x)/(exp(ln(16/exp(x)^2)^2+(-2*x-6)*ln(16/exp(x)^2)+x^2+6*x+15
)+x),x)

[Out]

ln(exp(ln(16/exp(x)^2)^2+(-2*x-6)*ln(16/exp(x)^2)+x^2+6*x+15)+x)^2

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 2 \, \int \frac {{\left (6 \, {\left (x - \log \left (16 \, e^{\left (-2 \, x\right )}\right ) + 3\right )} e^{\left (x^{2} - 2 \, {\left (x + 3\right )} \log \left (16 \, e^{\left (-2 \, x\right )}\right ) + \log \left (16 \, e^{\left (-2 \, x\right )}\right )^{2} + 6 \, x + 15\right )} + 1\right )} \log \left (x + e^{\left (x^{2} - 2 \, {\left (x + 3\right )} \log \left (16 \, e^{\left (-2 \, x\right )}\right ) + \log \left (16 \, e^{\left (-2 \, x\right )}\right )^{2} + 6 \, x + 15\right )}\right )}{x + e^{\left (x^{2} - 2 \, {\left (x + 3\right )} \log \left (16 \, e^{\left (-2 \, x\right )}\right ) + \log \left (16 \, e^{\left (-2 \, x\right )}\right )^{2} + 6 \, x + 15\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-12*log(16/exp(x)^2)+12*x+36)*exp(log(16/exp(x)^2)^2+(-2*x-6)*log(16/exp(x)^2)+x^2+6*x+15)+2)*log(
exp(log(16/exp(x)^2)^2+(-2*x-6)*log(16/exp(x)^2)+x^2+6*x+15)+x)/(exp(log(16/exp(x)^2)^2+(-2*x-6)*log(16/exp(x)
^2)+x^2+6*x+15)+x),x, algorithm="maxima")

[Out]

2*integrate((6*(x - log(16*e^(-2*x)) + 3)*e^(x^2 - 2*(x + 3)*log(16*e^(-2*x)) + log(16*e^(-2*x))^2 + 6*x + 15)
+ 1)*log(x + e^(x^2 - 2*(x + 3)*log(16*e^(-2*x)) + log(16*e^(-2*x))^2 + 6*x + 15))/(x + e^(x^2 - 2*(x + 3)*lo
g(16*e^(-2*x)) + log(16*e^(-2*x))^2 + 6*x + 15)), x)

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mupad [B]  time = 6.93, size = 118, normalized size = 4.54 \begin {gather*} 576\,{\ln \relax (2)}^2\,x^2-48\,\ln \relax (2)\,x\,\ln \left (16777216\,2^{24\,x}\,x+{\mathrm {e}}^{18\,x}\,{\mathrm {e}}^{15}\,{\mathrm {e}}^{16\,{\ln \relax (2)}^2}\,{\mathrm {e}}^{9\,x^2}\right )+1152\,{\ln \relax (2)}^2\,x+{\ln \left (16777216\,2^{24\,x}\,x+{\mathrm {e}}^{18\,x}\,{\mathrm {e}}^{15}\,{\mathrm {e}}^{16\,{\ln \relax (2)}^2}\,{\mathrm {e}}^{9\,x^2}\right )}^2-48\,\ln \relax (2)\,\ln \left (16777216\,2^{24\,x}\,x+{\mathrm {e}}^{18\,x}\,{\mathrm {e}}^{15}\,{\mathrm {e}}^{16\,{\ln \relax (2)}^2}\,{\mathrm {e}}^{9\,x^2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x + exp(6*x + log(16*exp(-2*x))^2 - log(16*exp(-2*x))*(2*x + 6) + x^2 + 15))*(exp(6*x + log(16*exp(-2
*x))^2 - log(16*exp(-2*x))*(2*x + 6) + x^2 + 15)*(12*x - 12*log(16*exp(-2*x)) + 36) + 2))/(x + exp(6*x + log(1
6*exp(-2*x))^2 - log(16*exp(-2*x))*(2*x + 6) + x^2 + 15)),x)

[Out]

576*x^2*log(2)^2 + log(16777216*2^(24*x)*x + exp(18*x)*exp(15)*exp(16*log(2)^2)*exp(9*x^2))^2 + 1152*x*log(2)^
2 - 48*log(16777216*2^(24*x)*x + exp(18*x)*exp(15)*exp(16*log(2)^2)*exp(9*x^2))*log(2) - 48*x*log(16777216*2^(
24*x)*x + exp(18*x)*exp(15)*exp(16*log(2)^2)*exp(9*x^2))*log(2)

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sympy [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(((-12*ln(16/exp(x)**2)+12*x+36)*exp(ln(16/exp(x)**2)**2+(-2*x-6)*ln(16/exp(x)**2)+x**2+6*x+15)+2)*ln
(exp(ln(16/exp(x)**2)**2+(-2*x-6)*ln(16/exp(x)**2)+x**2+6*x+15)+x)/(exp(ln(16/exp(x)**2)**2+(-2*x-6)*ln(16/exp
(x)**2)+x**2+6*x+15)+x),x)

[Out]

Timed out

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