3.55.98 \(\int \frac {5 e^{2 x}+3 x \log (2)+e^{5+x} (3 e^x+3 \log (2))+e^x (3+5 x \log (2))+e^{5+x} (3 e^x+3 x \log (2)) \log (\frac {e^x+x \log (2)}{\log (2)})}{5 e^{2 x}+(6 x+3 x^2) \log (2)+e^x (6+3 x+5 x \log (2))+e^{5+x} (3 e^x+3 x \log (2)) \log (\frac {e^x+x \log (2)}{\log (2)})} \, dx\)

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

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Rubi [F]  time = 68.57, 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 {5 e^{2 x}+3 x \log (2)+e^{5+x} \left (3 e^x+3 \log (2)\right )+e^x (3+5 x \log (2))+e^{5+x} \left (3 e^x+3 x \log (2)\right ) \log \left (\frac {e^x+x \log (2)}{\log (2)}\right )}{5 e^{2 x}+\left (6 x+3 x^2\right ) \log (2)+e^x (6+3 x+5 x \log (2))+e^{5+x} \left (3 e^x+3 x \log (2)\right ) \log \left (\frac {e^x+x \log (2)}{\log (2)}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(5*E^(2*x) + 3*x*Log[2] + E^(5 + x)*(3*E^x + 3*Log[2]) + E^x*(3 + 5*x*Log[2]) + E^(5 + x)*(3*E^x + 3*x*Log
[2])*Log[(E^x + x*Log[2])/Log[2]])/(5*E^(2*x) + (6*x + 3*x^2)*Log[2] + E^x*(6 + 3*x + 5*x*Log[2]) + E^(5 + x)*
(3*E^x + 3*x*Log[2])*Log[(E^x + x*Log[2])/Log[2]]),x]

[Out]

x + 3*E^5*Defer[Int][(5 + 3*E^5*Log[x + E^x/Log[2]])^(-1), x] + 3*E^5*Log[2]^2*Defer[Int][x/((-E^x - x*Log[2])
*(6 + 3*x*(1 - Log[32]/3) - 3*E^5*x*Log[2]*Log[x + E^x/Log[2]])), x] + 3*E^5*Log[2]^2*Defer[Int][x^2/((E^x + x
*Log[2])*(6 + 3*x*(1 - Log[32]/3) - 3*E^5*x*Log[2]*Log[x + E^x/Log[2]])), x] + 18*(5 + E^5*(6 - Log[32]))*Defe
r[Int][1/((-5 - 3*E^5*Log[x + E^x/Log[2]])*(6 + 5*E^x + 3*x + 3*E^(5 + x)*Log[x + E^x/Log[2]])*(6 + 3*x*(1 - L
og[32]/3) - 3*E^5*x*Log[2]*Log[x + E^x/Log[2]])), x] + 9*(10 + E^5*(12 - 5*Log[2]) + (5*Log[2]*(3 - Log[32]))/
Log[8])*Defer[Int][x/((-5 - 3*E^5*Log[x + E^x/Log[2]])*(6 + 5*E^x + 3*x + 3*E^(5 + x)*Log[x + E^x/Log[2]])*(6
+ 3*x*(1 - Log[32]/3) - 3*E^5*x*Log[2]*Log[x + E^x/Log[2]])), x] + 3*(15 + 9*E^5 - 5*Log[32])*Defer[Int][x^2/(
(-5 - 3*E^5*Log[x + E^x/Log[2]])*(6 + 5*E^x + 3*x + 3*E^(5 + x)*Log[x + E^x/Log[2]])*(6 + 3*x*(1 - Log[32]/3)
- 3*E^5*x*Log[2]*Log[x + E^x/Log[2]])), x] + 54*E^5*(1 - E^5*Log[2])*Defer[Int][Log[x + E^x/Log[2]]/((-5 - 3*E
^5*Log[x + E^x/Log[2]])*(6 + 5*E^x + 3*x + 3*E^(5 + x)*Log[x + E^x/Log[2]])*(6 + 3*x*(1 - Log[32]/3) - 3*E^5*x
*Log[2]*Log[x + E^x/Log[2]])), x] - 9*E^5*(9 - E^5*Log[8] - Log[1024])*Defer[Int][(x*Log[x + E^x/Log[2]])/((5
+ 3*E^5*Log[x + E^x/Log[2]])*(6 + 5*E^x + 3*x + 3*E^(5 + x)*Log[x + E^x/Log[2]])*(6 + 3*x*(1 - Log[32]/3) - 3*
E^5*x*Log[2]*Log[x + E^x/Log[2]])), x] - 9*E^5*(3 - Log[1024])*Defer[Int][(x^2*Log[x + E^x/Log[2]])/((5 + 3*E^
5*Log[x + E^x/Log[2]])*(6 + 5*E^x + 3*x + 3*E^(5 + x)*Log[x + E^x/Log[2]])*(6 + 3*x*(1 - Log[32]/3) - 3*E^5*x*
Log[2]*Log[x + E^x/Log[2]])), x] + 27*E^10*Log[2]*Defer[Int][(x*Log[x + E^x/Log[2]]^2)/((5 + 3*E^5*Log[x + E^x
/Log[2]])*(6 + 5*E^x + 3*x + 3*E^(5 + x)*Log[x + E^x/Log[2]])*(6 + 3*x*(1 - Log[32]/3) - 3*E^5*x*Log[2]*Log[x
+ E^x/Log[2]])), x] + 27*E^10*Log[2]*Defer[Int][(x^2*Log[x + E^x/Log[2]]^2)/((5 + 3*E^5*Log[x + E^x/Log[2]])*(
6 + 5*E^x + 3*x + 3*E^(5 + x)*Log[x + E^x/Log[2]])*(6 + 3*x*(1 - Log[32]/3) - 3*E^5*x*Log[2]*Log[x + E^x/Log[2
]])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5 e^{2 x}+3 x \log (2)+e^{5+x} \left (3 e^x+3 \log (2)\right )+e^x (3+5 x \log (2))+e^{5+x} \left (3 e^x+3 x \log (2)\right ) \log \left (\frac {e^x+x \log (2)}{\log (2)}\right )}{\left (e^x+x \log (2)\right ) \left (6+5 e^x+3 x+3 e^{5+x} \log \left (x+\frac {e^x}{\log (2)}\right )\right )} \, dx\\ &=\int \left (\frac {5 \left (1+\frac {3 e^5}{5}\right )+3 e^5 \log \left (x+\frac {e^x}{\log (2)}\right )}{5+3 e^5 \log \left (x+\frac {e^x}{\log (2)}\right )}+\frac {3 e^5 (-1+x) x \log ^2(2)}{\left (e^x+x \log (2)\right ) \left (6+3 x \left (1-\frac {\log (32)}{3}\right )-3 e^5 x \log (2) \log \left (x+\frac {e^x}{\log (2)}\right )\right )}+\frac {3 \left (-15 x^2 \left (1+\frac {3 e^5}{5}-\frac {5 \log (2)}{3}\right )-30 \left (1-\frac {1}{5} e^5 (-6+\log (32))\right )-45 x \left (1+\frac {4}{5} e^5 \left (1-\frac {5}{12} \log (2) \left (1+\frac {\log (32)}{e^5 \log (8)}\right )\right )\right )-18 e^5 \left (1-e^5 \log (2)\right ) \log \left (x+\frac {e^x}{\log (2)}\right )-27 e^5 x \left (1+\frac {1}{9} \left (-e^5 \log (8)-\log (1024)\right )\right ) \log \left (x+\frac {e^x}{\log (2)}\right )-9 e^5 x^2 \left (1-\frac {\log (1024)}{3}\right ) \log \left (x+\frac {e^x}{\log (2)}\right )+9 e^{10} x \log (2) \log ^2\left (x+\frac {e^x}{\log (2)}\right )+9 e^{10} x^2 \log (2) \log ^2\left (x+\frac {e^x}{\log (2)}\right )\right )}{\left (5+3 e^5 \log \left (x+\frac {e^x}{\log (2)}\right )\right ) \left (6+5 e^x+3 x+3 e^{5+x} \log \left (x+\frac {e^x}{\log (2)}\right )\right ) \left (6+3 x \left (1-\frac {\log (32)}{3}\right )-3 e^5 x \log (2) \log \left (x+\frac {e^x}{\log (2)}\right )\right )}\right ) \, dx\\ &=3 \int \frac {-15 x^2 \left (1+\frac {3 e^5}{5}-\frac {5 \log (2)}{3}\right )-30 \left (1-\frac {1}{5} e^5 (-6+\log (32))\right )-45 x \left (1+\frac {4}{5} e^5 \left (1-\frac {5}{12} \log (2) \left (1+\frac {\log (32)}{e^5 \log (8)}\right )\right )\right )-18 e^5 \left (1-e^5 \log (2)\right ) \log \left (x+\frac {e^x}{\log (2)}\right )-27 e^5 x \left (1+\frac {1}{9} \left (-e^5 \log (8)-\log (1024)\right )\right ) \log \left (x+\frac {e^x}{\log (2)}\right )-9 e^5 x^2 \left (1-\frac {\log (1024)}{3}\right ) \log \left (x+\frac {e^x}{\log (2)}\right )+9 e^{10} x \log (2) \log ^2\left (x+\frac {e^x}{\log (2)}\right )+9 e^{10} x^2 \log (2) \log ^2\left (x+\frac {e^x}{\log (2)}\right )}{\left (5+3 e^5 \log \left (x+\frac {e^x}{\log (2)}\right )\right ) \left (6+5 e^x+3 x+3 e^{5+x} \log \left (x+\frac {e^x}{\log (2)}\right )\right ) \left (6+3 x \left (1-\frac {\log (32)}{3}\right )-3 e^5 x \log (2) \log \left (x+\frac {e^x}{\log (2)}\right )\right )} \, dx+\left (3 e^5 \log ^2(2)\right ) \int \frac {(-1+x) x}{\left (e^x+x \log (2)\right ) \left (6+3 x \left (1-\frac {\log (32)}{3}\right )-3 e^5 x \log (2) \log \left (x+\frac {e^x}{\log (2)}\right )\right )} \, dx+\int \frac {5 \left (1+\frac {3 e^5}{5}\right )+3 e^5 \log \left (x+\frac {e^x}{\log (2)}\right )}{5+3 e^5 \log \left (x+\frac {e^x}{\log (2)}\right )} \, dx\\ &=3 \int \frac {5 (1+x) (-6+x (-3+\log (32)))-3 e^5 \left (12+12 x+3 x^2-5 x \log (2)-2 \log (32)\right )+3 e^5 \left (-6+e^5 \log (64)+x^2 (-3+\log (1024))+x \left (-9+e^5 \log (8)+\log (1024)\right )\right ) \log \left (x+\frac {e^x}{\log (2)}\right )+9 e^{10} x (1+x) \log (2) \log ^2\left (x+\frac {e^x}{\log (2)}\right )}{\left (5+3 e^5 \log \left (x+\frac {e^x}{\log (2)}\right )\right ) \left (6+5 e^x+3 x+3 e^{5+x} \log \left (x+\frac {e^x}{\log (2)}\right )\right ) \left (6-x (-3+\log (32))-3 e^5 x \log (2) \log \left (x+\frac {e^x}{\log (2)}\right )\right )} \, dx+\left (3 e^5 \log ^2(2)\right ) \int \left (\frac {x}{\left (-e^x-x \log (2)\right ) \left (6+3 x \left (1-\frac {\log (32)}{3}\right )-3 e^5 x \log (2) \log \left (x+\frac {e^x}{\log (2)}\right )\right )}+\frac {x^2}{\left (e^x+x \log (2)\right ) \left (6+3 x \left (1-\frac {\log (32)}{3}\right )-3 e^5 x \log (2) \log \left (x+\frac {e^x}{\log (2)}\right )\right )}\right ) \, dx+\int \left (1+\frac {3 e^5}{5+3 e^5 \log \left (x+\frac {e^x}{\log (2)}\right )}\right ) \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [F]  time = 0.46, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {5 e^{2 x}+3 x \log (2)+e^{5+x} \left (3 e^x+3 \log (2)\right )+e^x (3+5 x \log (2))+e^{5+x} \left (3 e^x+3 x \log (2)\right ) \log \left (\frac {e^x+x \log (2)}{\log (2)}\right )}{5 e^{2 x}+\left (6 x+3 x^2\right ) \log (2)+e^x (6+3 x+5 x \log (2))+e^{5+x} \left (3 e^x+3 x \log (2)\right ) \log \left (\frac {e^x+x \log (2)}{\log (2)}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(5*E^(2*x) + 3*x*Log[2] + E^(5 + x)*(3*E^x + 3*Log[2]) + E^x*(3 + 5*x*Log[2]) + E^(5 + x)*(3*E^x + 3
*x*Log[2])*Log[(E^x + x*Log[2])/Log[2]])/(5*E^(2*x) + (6*x + 3*x^2)*Log[2] + E^x*(6 + 3*x + 5*x*Log[2]) + E^(5
+ x)*(3*E^x + 3*x*Log[2])*Log[(E^x + x*Log[2])/Log[2]]),x]

[Out]

Integrate[(5*E^(2*x) + 3*x*Log[2] + E^(5 + x)*(3*E^x + 3*Log[2]) + E^x*(3 + 5*x*Log[2]) + E^(5 + x)*(3*E^x + 3
*x*Log[2])*Log[(E^x + x*Log[2])/Log[2]])/(5*E^(2*x) + (6*x + 3*x^2)*Log[2] + E^x*(6 + 3*x + 5*x*Log[2]) + E^(5
+ x)*(3*E^x + 3*x*Log[2])*Log[(E^x + x*Log[2])/Log[2]]), x]

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fricas [B]  time = 0.99, size = 49, normalized size = 1.75 \begin {gather*} x + \log \left ({\left (3 \, {\left (x + 2\right )} e^{5} + 3 \, e^{\left (x + 10\right )} \log \left (\frac {{\left (x e^{5} \log \relax (2) + e^{\left (x + 5\right )}\right )} e^{\left (-5\right )}}{\log \relax (2)}\right ) + 5 \, e^{\left (x + 5\right )}\right )} e^{\left (-x - 5\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*exp(x)+3*x*log(2))*exp(5+x)*log((exp(x)+x*log(2))/log(2))+(3*exp(x)+3*log(2))*exp(5+x)+5*exp(x)^
2+(5*x*log(2)+3)*exp(x)+3*x*log(2))/((3*exp(x)+3*x*log(2))*exp(5+x)*log((exp(x)+x*log(2))/log(2))+5*exp(x)^2+(
5*x*log(2)+6+3*x)*exp(x)+(3*x^2+6*x)*log(2)),x, algorithm="fricas")

[Out]

x + log((3*(x + 2)*e^5 + 3*e^(x + 10)*log((x*e^5*log(2) + e^(x + 5))*e^(-5)/log(2)) + 5*e^(x + 5))*e^(-x - 5))

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giac [A]  time = 0.21, size = 33, normalized size = 1.18 \begin {gather*} \log \left (3 \, e^{\left (x + 5\right )} \log \left (x \log \relax (2) + e^{x}\right ) - 3 \, e^{\left (x + 5\right )} \log \left (\log \relax (2)\right ) + 3 \, x + 5 \, e^{x} + 6\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*exp(x)+3*x*log(2))*exp(5+x)*log((exp(x)+x*log(2))/log(2))+(3*exp(x)+3*log(2))*exp(5+x)+5*exp(x)^
2+(5*x*log(2)+3)*exp(x)+3*x*log(2))/((3*exp(x)+3*x*log(2))*exp(5+x)*log((exp(x)+x*log(2))/log(2))+5*exp(x)^2+(
5*x*log(2)+6+3*x)*exp(x)+(3*x^2+6*x)*log(2)),x, algorithm="giac")

[Out]

log(3*e^(x + 5)*log(x*log(2) + e^x) - 3*e^(x + 5)*log(log(2)) + 3*x + 5*e^x + 6)

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maple [A]  time = 0.46, size = 30, normalized size = 1.07

method	result	size
norman	\(\ln \left (3 \,{\mathrm e}^{5} {\mathrm e}^{x} \ln \left (\frac {{\mathrm e}^{x}+x \ln \relax (2)}{\ln \relax (2)}\right )+3 x +5 \,{\mathrm e}^{x}+6\right )\)	\(30\)
risch	\(x +\ln \left (\ln \left (\frac {{\mathrm e}^{x}+x \ln \relax (2)}{\ln \relax (2)}\right )+\frac {\left (3 x +5 \,{\mathrm e}^{x}+6\right ) {\mathrm e}^{-x -5}}{3}\right )\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*exp(x)+3*x*ln(2))*exp(5+x)*ln((exp(x)+x*ln(2))/ln(2))+(3*exp(x)+3*ln(2))*exp(5+x)+5*exp(x)^2+(5*x*ln(2
)+3)*exp(x)+3*x*ln(2))/((3*exp(x)+3*x*ln(2))*exp(5+x)*ln((exp(x)+x*ln(2))/ln(2))+5*exp(x)^2+(5*x*ln(2)+6+3*x)*
exp(x)+(3*x^2+6*x)*ln(2)),x,method=_RETURNVERBOSE)

[Out]

ln(3*exp(5)*exp(x)*ln((exp(x)+x*ln(2))/ln(2))+3*x+5*exp(x)+6)

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maxima [A]  time = 0.55, size = 42, normalized size = 1.50 \begin {gather*} x + \log \left (-\frac {1}{3} \, {\left ({\left (3 \, e^{5} \log \left (\log \relax (2)\right ) - 5\right )} e^{x} - 3 \, e^{\left (x + 5\right )} \log \left (x \log \relax (2) + e^{x}\right ) - 3 \, x - 6\right )} e^{\left (-x - 5\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*exp(x)+3*x*log(2))*exp(5+x)*log((exp(x)+x*log(2))/log(2))+(3*exp(x)+3*log(2))*exp(5+x)+5*exp(x)^
2+(5*x*log(2)+3)*exp(x)+3*x*log(2))/((3*exp(x)+3*x*log(2))*exp(5+x)*log((exp(x)+x*log(2))/log(2))+5*exp(x)^2+(
5*x*log(2)+6+3*x)*exp(x)+(3*x^2+6*x)*log(2)),x, algorithm="maxima")

[Out]

x + log(-1/3*((3*e^5*log(log(2)) - 5)*e^x - 3*e^(x + 5)*log(x*log(2) + e^x) - 3*x - 6)*e^(-x - 5))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {5\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{x+5}\,\left (3\,\ln \relax (2)+3\,{\mathrm {e}}^x\right )+3\,x\,\ln \relax (2)+{\mathrm {e}}^x\,\left (5\,x\,\ln \relax (2)+3\right )+\ln \left (\frac {{\mathrm {e}}^x+x\,\ln \relax (2)}{\ln \relax (2)}\right )\,{\mathrm {e}}^{x+5}\,\left (3\,{\mathrm {e}}^x+3\,x\,\ln \relax (2)\right )}{5\,{\mathrm {e}}^{2\,x}+\ln \relax (2)\,\left (3\,x^2+6\,x\right )+{\mathrm {e}}^x\,\left (3\,x+5\,x\,\ln \relax (2)+6\right )+\ln \left (\frac {{\mathrm {e}}^x+x\,\ln \relax (2)}{\ln \relax (2)}\right )\,{\mathrm {e}}^{x+5}\,\left (3\,{\mathrm {e}}^x+3\,x\,\ln \relax (2)\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*exp(2*x) + exp(x + 5)*(3*log(2) + 3*exp(x)) + 3*x*log(2) + exp(x)*(5*x*log(2) + 3) + log((exp(x) + x*lo
g(2))/log(2))*exp(x + 5)*(3*exp(x) + 3*x*log(2)))/(5*exp(2*x) + log(2)*(6*x + 3*x^2) + exp(x)*(3*x + 5*x*log(2
) + 6) + log((exp(x) + x*log(2))/log(2))*exp(x + 5)*(3*exp(x) + 3*x*log(2))),x)

[Out]

int((5*exp(2*x) + exp(x + 5)*(3*log(2) + 3*exp(x)) + 3*x*log(2) + exp(x)*(5*x*log(2) + 3) + log((exp(x) + x*lo
g(2))/log(2))*exp(x + 5)*(3*exp(x) + 3*x*log(2)))/(5*exp(2*x) + log(2)*(6*x + 3*x^2) + exp(x)*(3*x + 5*x*log(2
) + 6) + log((exp(x) + x*log(2))/log(2))*exp(x + 5)*(3*exp(x) + 3*x*log(2))), x)

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sympy [A]  time = 0.54, size = 34, normalized size = 1.21 \begin {gather*} x + \log {\left (\frac {\left (3 x + 5 e^{x} + 6\right ) e^{- x}}{3 e^{5}} + \log {\left (\frac {x \log {\relax (2 )} + e^{x}}{\log {\relax (2 )}} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*exp(x)+3*x*ln(2))*exp(5+x)*ln((exp(x)+x*ln(2))/ln(2))+(3*exp(x)+3*ln(2))*exp(5+x)+5*exp(x)**2+(5
*x*ln(2)+3)*exp(x)+3*x*ln(2))/((3*exp(x)+3*x*ln(2))*exp(5+x)*ln((exp(x)+x*ln(2))/ln(2))+5*exp(x)**2+(5*x*ln(2)
+6+3*x)*exp(x)+(3*x**2+6*x)*ln(2)),x)

[Out]

x + log((3*x + 5*exp(x) + 6)*exp(-5)*exp(-x)/3 + log((x*log(2) + exp(x))/log(2)))

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