3.41.59 \(\int \frac {e^{9 x+3 x^2+(3 x+x^2) \log (-1+e^{e^x}+x)} (-9+6 x+7 x^2+e^{e^x} (9+6 x+e^x (3 x+x^2))+(-3+x+2 x^2+e^{e^x} (3+2 x)) \log (-1+e^{e^x}+x))}{-1+e^{e^x}+x} \, dx\)

Optimal. Leaf size=18 \[ e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} \]

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Rubi [F]  time = 127.75, 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 (9 x+3 x^2+\left (3 x+x^2\right ) \log \left (-1+e^{e^x}+x\right )\right ) \left (-9+6 x+7 x^2+e^{e^x} \left (9+6 x+e^x \left (3 x+x^2\right )\right )+\left (-3+x+2 x^2+e^{e^x} (3+2 x)\right ) \log \left (-1+e^{e^x}+x\right )\right )}{-1+e^{e^x}+x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(9*x + 3*x^2 + (3*x + x^2)*Log[-1 + E^E^x + x])*(-9 + 6*x + 7*x^2 + E^E^x*(9 + 6*x + E^x*(3*x + x^2)) +
(-3 + x + 2*x^2 + E^E^x*(3 + 2*x))*Log[-1 + E^E^x + x]))/(-1 + E^E^x + x),x]

[Out]

9*Defer[Int][E^(x*(3 + x)*(3 + Log[-1 + E^E^x + x])), x] + 6*Defer[Int][E^(x*(3 + x)*(3 + Log[-1 + E^E^x + x])
)*x, x] + 3*Defer[Int][(E^(x*(3 + x)*(3 + Log[-1 + E^E^x + x]))*x)/(-1 + E^E^x + x), x] + 3*Defer[Int][(E^(E^x
+ x + x*(3 + x)*(3 + Log[-1 + E^E^x + x]))*x)/(-1 + E^E^x + x), x] + Defer[Int][(E^(x*(3 + x)*(3 + Log[-1 + E
^E^x + x]))*x^2)/(-1 + E^E^x + x), x] + Defer[Int][(E^(E^x + x + x*(3 + x)*(3 + Log[-1 + E^E^x + x]))*x^2)/(-1
+ E^E^x + x), x] + 3*Defer[Int][E^(x*(3 + x)*(3 + Log[-1 + E^E^x + x]))*Log[-1 + E^E^x + x], x] + 2*Defer[Int
][E^(x*(3 + x)*(3 + Log[-1 + E^E^x + x]))*x*Log[-1 + E^E^x + x], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} \left (9-6 x-7 x^2-e^{e^x} \left (9+6 x+e^x \left (3 x+x^2\right )\right )-\left (-3+x+2 x^2+e^{e^x} (3+2 x)\right ) \log \left (-1+e^{e^x}+x\right )\right )}{1-e^{e^x}-x} \, dx\\ &=\int \left (\frac {e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x (3+x)}{-1+e^{e^x}+x}+\frac {e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} \left (-9+9 e^{e^x}+6 x+6 e^{e^x} x+7 x^2-3 \log \left (-1+e^{e^x}+x\right )+3 e^{e^x} \log \left (-1+e^{e^x}+x\right )+x \log \left (-1+e^{e^x}+x\right )+2 e^{e^x} x \log \left (-1+e^{e^x}+x\right )+2 x^2 \log \left (-1+e^{e^x}+x\right )\right )}{-1+e^{e^x}+x}\right ) \, dx\\ &=\int \frac {e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x (3+x)}{-1+e^{e^x}+x} \, dx+\int \frac {e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} \left (-9+9 e^{e^x}+6 x+6 e^{e^x} x+7 x^2-3 \log \left (-1+e^{e^x}+x\right )+3 e^{e^x} \log \left (-1+e^{e^x}+x\right )+x \log \left (-1+e^{e^x}+x\right )+2 e^{e^x} x \log \left (-1+e^{e^x}+x\right )+2 x^2 \log \left (-1+e^{e^x}+x\right )\right )}{-1+e^{e^x}+x} \, dx\\ &=\int \left (\frac {3 e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x}{-1+e^{e^x}+x}+\frac {e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x^2}{-1+e^{e^x}+x}\right ) \, dx+\int \frac {e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} \left (9-6 x-7 x^2-e^{e^x} (9+6 x)-\left (-1+e^{e^x}+x\right ) (3+2 x) \log \left (-1+e^{e^x}+x\right )\right )}{1-e^{e^x}-x} \, dx\\ &=3 \int \frac {e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x}{-1+e^{e^x}+x} \, dx+\int \frac {e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x^2}{-1+e^{e^x}+x} \, dx+\int \left (\frac {e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x (3+x)}{-1+e^{e^x}+x}+e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} (3+2 x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )\right ) \, dx\\ &=3 \int \frac {e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x}{-1+e^{e^x}+x} \, dx+\int \frac {e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x^2}{-1+e^{e^x}+x} \, dx+\int \frac {e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x (3+x)}{-1+e^{e^x}+x} \, dx+\int e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} (3+2 x) \left (3+\log \left (-1+e^{e^x}+x\right )\right ) \, dx\\ &=3 \int \frac {e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x}{-1+e^{e^x}+x} \, dx+\int \frac {e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x^2}{-1+e^{e^x}+x} \, dx+\int \left (\frac {3 e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x}{-1+e^{e^x}+x}+\frac {e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x^2}{-1+e^{e^x}+x}\right ) \, dx+\int \left (3 e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} (3+2 x)+e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} (3+2 x) \log \left (-1+e^{e^x}+x\right )\right ) \, dx\\ &=3 \int \frac {e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x}{-1+e^{e^x}+x} \, dx+3 \int \frac {e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x}{-1+e^{e^x}+x} \, dx+3 \int e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} (3+2 x) \, dx+\int \frac {e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x^2}{-1+e^{e^x}+x} \, dx+\int \frac {e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x^2}{-1+e^{e^x}+x} \, dx+\int e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} (3+2 x) \log \left (-1+e^{e^x}+x\right ) \, dx\\ &=3 \int \frac {e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x}{-1+e^{e^x}+x} \, dx+3 \int \frac {e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x}{-1+e^{e^x}+x} \, dx+3 \int \left (3 e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )}+2 e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x\right ) \, dx+\int \frac {e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x^2}{-1+e^{e^x}+x} \, dx+\int \frac {e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x^2}{-1+e^{e^x}+x} \, dx+\int \left (3 e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} \log \left (-1+e^{e^x}+x\right )+2 e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x \log \left (-1+e^{e^x}+x\right )\right ) \, dx\\ &=2 \int e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x \log \left (-1+e^{e^x}+x\right ) \, dx+3 \int \frac {e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x}{-1+e^{e^x}+x} \, dx+3 \int \frac {e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x}{-1+e^{e^x}+x} \, dx+3 \int e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} \log \left (-1+e^{e^x}+x\right ) \, dx+6 \int e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x \, dx+9 \int e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} \, dx+\int \frac {e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x^2}{-1+e^{e^x}+x} \, dx+\int \frac {e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x^2}{-1+e^{e^x}+x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.16, size = 23, normalized size = 1.28 \begin {gather*} e^{3 x (3+x)} \left (-1+e^{e^x}+x\right )^{x (3+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(9*x + 3*x^2 + (3*x + x^2)*Log[-1 + E^E^x + x])*(-9 + 6*x + 7*x^2 + E^E^x*(9 + 6*x + E^x*(3*x + x
^2)) + (-3 + x + 2*x^2 + E^E^x*(3 + 2*x))*Log[-1 + E^E^x + x]))/(-1 + E^E^x + x),x]

[Out]

E^(3*x*(3 + x))*(-1 + E^E^x + x)^(x*(3 + x))

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fricas [A]  time = 0.72, size = 25, normalized size = 1.39 \begin {gather*} e^{\left (3 \, x^{2} + {\left (x^{2} + 3 \, x\right )} \log \left (x + e^{\left (e^{x}\right )} - 1\right ) + 9 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x+3)*exp(exp(x))+2*x^2+x-3)*log(exp(exp(x))+x-1)+((x^2+3*x)*exp(x)+6*x+9)*exp(exp(x))+7*x^2+6*x
-9)*exp((x^2+3*x)*log(exp(exp(x))+x-1)+3*x^2+9*x)/(exp(exp(x))+x-1),x, algorithm="fricas")

[Out]

e^(3*x^2 + (x^2 + 3*x)*log(x + e^(e^x) - 1) + 9*x)

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giac [B]  time = 2.39, size = 31, normalized size = 1.72 \begin {gather*} e^{\left (x^{2} \log \left (x + e^{\left (e^{x}\right )} - 1\right ) + 3 \, x^{2} + 3 \, x \log \left (x + e^{\left (e^{x}\right )} - 1\right ) + 9 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x+3)*exp(exp(x))+2*x^2+x-3)*log(exp(exp(x))+x-1)+((x^2+3*x)*exp(x)+6*x+9)*exp(exp(x))+7*x^2+6*x
-9)*exp((x^2+3*x)*log(exp(exp(x))+x-1)+3*x^2+9*x)/(exp(exp(x))+x-1),x, algorithm="giac")

[Out]

e^(x^2*log(x + e^(e^x) - 1) + 3*x^2 + 3*x*log(x + e^(e^x) - 1) + 9*x)

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maple [A]  time = 0.06, size = 21, normalized size = 1.17

method	result	size
risch	\(\left ({\mathrm e}^{{\mathrm e}^{x}}+x -1\right )^{\left (3+x \right ) x} {\mathrm e}^{3 \left (3+x \right ) x}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*x+3)*exp(exp(x))+2*x^2+x-3)*ln(exp(exp(x))+x-1)+((x^2+3*x)*exp(x)+6*x+9)*exp(exp(x))+7*x^2+6*x-9)*exp
((x^2+3*x)*ln(exp(exp(x))+x-1)+3*x^2+9*x)/(exp(exp(x))+x-1),x,method=_RETURNVERBOSE)

[Out]

(exp(exp(x))+x-1)^((3+x)*x)*exp(3*(3+x)*x)

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maxima [B]  time = 0.55, size = 31, normalized size = 1.72 \begin {gather*} e^{\left (x^{2} \log \left (x + e^{\left (e^{x}\right )} - 1\right ) + 3 \, x^{2} + 3 \, x \log \left (x + e^{\left (e^{x}\right )} - 1\right ) + 9 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x+3)*exp(exp(x))+2*x^2+x-3)*log(exp(exp(x))+x-1)+((x^2+3*x)*exp(x)+6*x+9)*exp(exp(x))+7*x^2+6*x
-9)*exp((x^2+3*x)*log(exp(exp(x))+x-1)+3*x^2+9*x)/(exp(exp(x))+x-1),x, algorithm="maxima")

[Out]

e^(x^2*log(x + e^(e^x) - 1) + 3*x^2 + 3*x*log(x + e^(e^x) - 1) + 9*x)

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mupad [B]  time = 0.22, size = 25, normalized size = 1.39 \begin {gather*} {\mathrm {e}}^{3\,x^2+9\,x}\,{\left (x+{\mathrm {e}}^{{\mathrm {e}}^x}-1\right )}^{x^2+3\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(9*x + 3*x^2 + log(x + exp(exp(x)) - 1)*(3*x + x^2))*(6*x + exp(exp(x))*(6*x + exp(x)*(3*x + x^2) + 9)
+ log(x + exp(exp(x)) - 1)*(x + 2*x^2 + exp(exp(x))*(2*x + 3) - 3) + 7*x^2 - 9))/(x + exp(exp(x)) - 1),x)

[Out]

exp(9*x + 3*x^2)*(x + exp(exp(x)) - 1)^(3*x + x^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((((2*x+3)*exp(exp(x))+2*x**2+x-3)*ln(exp(exp(x))+x-1)+((x**2+3*x)*exp(x)+6*x+9)*exp(exp(x))+7*x**2+6
*x-9)*exp((x**2+3*x)*ln(exp(exp(x))+x-1)+3*x**2+9*x)/(exp(exp(x))+x-1),x)

[Out]

Timed out

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