3.25.14 \(\int \frac {-6 e^{10+x} \log (5)-6 e^5 x \log (5)+(4 e^5 x+4 e^{10+x} x) \log (e^{5+x}+x)+(2 e^{10+x}+2 e^5 x) \log ^2(e^{5+x}+x)}{e^{5+x}+x} \, dx\)

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

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Rubi [F]  time = 1.41, 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 {-6 e^{10+x} \log (5)-6 e^5 x \log (5)+\left (4 e^5 x+4 e^{10+x} x\right ) \log \left (e^{5+x}+x\right )+\left (2 e^{10+x}+2 e^5 x\right ) \log ^2\left (e^{5+x}+x\right )}{e^{5+x}+x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-6*E^(10 + x)*Log[5] - 6*E^5*x*Log[5] + (4*E^5*x + 4*E^(10 + x)*x)*Log[E^(5 + x) + x] + (2*E^(10 + x) + 2
*E^5*x)*Log[E^(5 + x) + x]^2)/(E^(5 + x) + x),x]

[Out]

(-2*E^5*x^3)/3 - 6*E^5*x*Log[5] + 2*E^5*x^2*Log[E^(5 + x) + x] + 4*E^5*Log[E^(5 + x) + x]*Defer[Int][x/(E^(5 +
x) + x), x] - 2*E^5*Defer[Int][x^2/(E^(5 + x) + x), x] - 4*E^5*Log[E^(5 + x) + x]*Defer[Int][x^2/(E^(5 + x) +
x), x] + 2*E^5*Defer[Int][x^3/(E^(5 + x) + x), x] + 2*E^5*Defer[Int][Log[E^(5 + x) + x]^2, x] - 4*E^5*Defer[I
nt][Defer[Int][x/(E^(5 + x) + x), x], x] - 4*E^5*Defer[Int][Defer[Int][x/(E^(5 + x) + x), x]/(E^(5 + x) + x),
x] + 4*E^5*Defer[Int][(x*Defer[Int][x/(E^(5 + x) + x), x])/(E^(5 + x) + x), x] + 4*E^5*Defer[Int][Defer[Int][x
^2/(E^(5 + x) + x), x], x] + 4*E^5*Defer[Int][Defer[Int][x^2/(E^(5 + x) + x), x]/(E^(5 + x) + x), x] - 4*E^5*D
efer[Int][(x*Defer[Int][x^2/(E^(5 + x) + x), x])/(E^(5 + x) + x), x]

Rubi steps

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

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Mathematica [A]  time = 0.23, size = 23, normalized size = 0.88 \begin {gather*} 2 e^5 \left (-3 x \log (5)+x \log ^2\left (e^{5+x}+x\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-6*E^(10 + x)*Log[5] - 6*E^5*x*Log[5] + (4*E^5*x + 4*E^(10 + x)*x)*Log[E^(5 + x) + x] + (2*E^(10 +
x) + 2*E^5*x)*Log[E^(5 + x) + x]^2)/(E^(5 + x) + x),x]

[Out]

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

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fricas [A]  time = 0.68, size = 28, normalized size = 1.08 \begin {gather*} 2 \, x e^{5} \log \left ({\left (x e^{5} + e^{\left (x + 10\right )}\right )} e^{\left (-5\right )}\right )^{2} - 6 \, x e^{5} \log \relax (5) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*exp(5)*exp(5+x)+2*x*exp(5))*log(x+exp(5+x))^2+(4*x*exp(5)*exp(5+x)+4*x*exp(5))*log(x+exp(5+x))-6
*exp(5)*log(5)*exp(5+x)-6*x*exp(5)*log(5))/(x+exp(5+x)),x, algorithm="fricas")

[Out]

2*x*e^5*log((x*e^5 + e^(x + 10))*e^(-5))^2 - 6*x*e^5*log(5)

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giac [A]  time = 0.20, size = 22, normalized size = 0.85 \begin {gather*} 2 \, x e^{5} \log \left (x + e^{\left (x + 5\right )}\right )^{2} - 6 \, x e^{5} \log \relax (5) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*exp(5)*exp(5+x)+2*x*exp(5))*log(x+exp(5+x))^2+(4*x*exp(5)*exp(5+x)+4*x*exp(5))*log(x+exp(5+x))-6
*exp(5)*log(5)*exp(5+x)-6*x*exp(5)*log(5))/(x+exp(5+x)),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.10, size = 23, normalized size = 0.88

method	result	size
norman	\(-6 x \,{\mathrm e}^{5} \ln \relax (5)+2 x \,{\mathrm e}^{5} \ln \left (x +{\mathrm e}^{5+x}\right )^{2}\)	\(23\)
risch	\(-6 x \,{\mathrm e}^{5} \ln \relax (5)+2 x \,{\mathrm e}^{5} \ln \left (x +{\mathrm e}^{5+x}\right )^{2}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*exp(5)*exp(5+x)+2*x*exp(5))*ln(x+exp(5+x))^2+(4*x*exp(5)*exp(5+x)+4*x*exp(5))*ln(x+exp(5+x))-6*exp(5)*
ln(5)*exp(5+x)-6*x*exp(5)*ln(5))/(x+exp(5+x)),x,method=_RETURNVERBOSE)

[Out]

-6*x*exp(5)*ln(5)+2*x*exp(5)*ln(x+exp(5+x))^2

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maxima [A]  time = 0.62, size = 22, normalized size = 0.85 \begin {gather*} 2 \, x e^{5} \log \left (x + e^{\left (x + 5\right )}\right )^{2} - 6 \, x e^{5} \log \relax (5) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*exp(5)*exp(5+x)+2*x*exp(5))*log(x+exp(5+x))^2+(4*x*exp(5)*exp(5+x)+4*x*exp(5))*log(x+exp(5+x))-6
*exp(5)*log(5)*exp(5+x)-6*x*exp(5)*log(5))/(x+exp(5+x)),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.56, size = 20, normalized size = 0.77 \begin {gather*} -2\,x\,{\mathrm {e}}^5\,\left (\ln \left (125\right )-{\ln \left (x+{\mathrm {e}}^5\,{\mathrm {e}}^x\right )}^2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x + exp(x + 5))^2*(2*exp(x + 5)*exp(5) + 2*x*exp(5)) + log(x + exp(x + 5))*(4*x*exp(5) + 4*x*exp(x +
5)*exp(5)) - 6*exp(x + 5)*exp(5)*log(5) - 6*x*exp(5)*log(5))/(x + exp(x + 5)),x)

[Out]

-2*x*exp(5)*(log(125) - log(x + exp(5)*exp(x))^2)

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sympy [A]  time = 0.34, size = 26, normalized size = 1.00 \begin {gather*} 2 x e^{5} \log {\left (x + e^{x + 5} \right )}^{2} - 6 x e^{5} \log {\relax (5 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*exp(5)*exp(5+x)+2*x*exp(5))*ln(x+exp(5+x))**2+(4*x*exp(5)*exp(5+x)+4*x*exp(5))*ln(x+exp(5+x))-6*
exp(5)*ln(5)*exp(5+x)-6*x*exp(5)*ln(5))/(x+exp(5+x)),x)

[Out]

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

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