3.61.92 \(\int \frac {81+e^{e^x} (16+e^x (81-16 x))}{e^{2 e^x}+2 e^{e^x} x+x^2} \, dx\)

Optimal. Leaf size=17 \[ 2+\frac {-81+16 x}{e^{e^x}+x} \]

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Rubi [F]  time = 0.78, 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 {81+e^{e^x} \left (16+e^x (81-16 x)\right )}{e^{2 e^x}+2 e^{e^x} x+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(81 + E^E^x*(16 + E^x*(81 - 16*x)))/(E^(2*E^x) + 2*E^E^x*x + x^2),x]

[Out]

81*Defer[Int][(E^E^x + x)^(-2), x] + 81*Defer[Int][E^(E^x + x)/(E^E^x + x)^2, x] - 16*Defer[Int][x/(E^E^x + x)
^2, x] - 16*Defer[Int][(E^(E^x + x)*x)/(E^E^x + x)^2, x] + 16*Defer[Int][(E^E^x + x)^(-1), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {81+e^{e^x} \left (16+e^x (81-16 x)\right )}{\left (e^{e^x}+x\right )^2} \, dx\\ &=\int \left (\frac {81+16 e^{e^x}}{\left (e^{e^x}+x\right )^2}-\frac {e^{e^x+x} (-81+16 x)}{\left (e^{e^x}+x\right )^2}\right ) \, dx\\ &=\int \frac {81+16 e^{e^x}}{\left (e^{e^x}+x\right )^2} \, dx-\int \frac {e^{e^x+x} (-81+16 x)}{\left (e^{e^x}+x\right )^2} \, dx\\ &=-\int \left (-\frac {81 e^{e^x+x}}{\left (e^{e^x}+x\right )^2}+\frac {16 e^{e^x+x} x}{\left (e^{e^x}+x\right )^2}\right ) \, dx+\int \left (\frac {16}{e^{e^x}+x}-\frac {-81+16 x}{\left (e^{e^x}+x\right )^2}\right ) \, dx\\ &=-\left (16 \int \frac {e^{e^x+x} x}{\left (e^{e^x}+x\right )^2} \, dx\right )+16 \int \frac {1}{e^{e^x}+x} \, dx+81 \int \frac {e^{e^x+x}}{\left (e^{e^x}+x\right )^2} \, dx-\int \frac {-81+16 x}{\left (e^{e^x}+x\right )^2} \, dx\\ &=-\left (16 \int \frac {e^{e^x+x} x}{\left (e^{e^x}+x\right )^2} \, dx\right )+16 \int \frac {1}{e^{e^x}+x} \, dx+81 \int \frac {e^{e^x+x}}{\left (e^{e^x}+x\right )^2} \, dx-\int \left (-\frac {81}{\left (e^{e^x}+x\right )^2}+\frac {16 x}{\left (e^{e^x}+x\right )^2}\right ) \, dx\\ &=-\left (16 \int \frac {x}{\left (e^{e^x}+x\right )^2} \, dx\right )-16 \int \frac {e^{e^x+x} x}{\left (e^{e^x}+x\right )^2} \, dx+16 \int \frac {1}{e^{e^x}+x} \, dx+81 \int \frac {1}{\left (e^{e^x}+x\right )^2} \, dx+81 \int \frac {e^{e^x+x}}{\left (e^{e^x}+x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.25, size = 15, normalized size = 0.88 \begin {gather*} \frac {-81+16 x}{e^{e^x}+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(81 + E^E^x*(16 + E^x*(81 - 16*x)))/(E^(2*E^x) + 2*E^E^x*x + x^2),x]

[Out]

(-81 + 16*x)/(E^E^x + x)

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fricas [A]  time = 1.26, size = 13, normalized size = 0.76 \begin {gather*} \frac {16 \, x - 81}{x + e^{\left (e^{x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-16*x+81)*exp(x)+16)*exp(exp(x))+81)/(exp(exp(x))^2+2*x*exp(exp(x))+x^2),x, algorithm="fricas")

[Out]

(16*x - 81)/(x + e^(e^x))

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giac [A]  time = 0.13, size = 13, normalized size = 0.76 \begin {gather*} \frac {16 \, x - 81}{x + e^{\left (e^{x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-16*x+81)*exp(x)+16)*exp(exp(x))+81)/(exp(exp(x))^2+2*x*exp(exp(x))+x^2),x, algorithm="giac")

[Out]

(16*x - 81)/(x + e^(e^x))

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maple [A]  time = 0.06, size = 14, normalized size = 0.82

method	result	size
risch	\(\frac {16 x -81}{x +{\mathrm e}^{{\mathrm e}^{x}}}\)	\(14\)
norman	\(\frac {-81-16 \,{\mathrm e}^{{\mathrm e}^{x}}}{x +{\mathrm e}^{{\mathrm e}^{x}}}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-16*x+81)*exp(x)+16)*exp(exp(x))+81)/(exp(exp(x))^2+2*x*exp(exp(x))+x^2),x,method=_RETURNVERBOSE)

[Out]

(16*x-81)/(x+exp(exp(x)))

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maxima [A]  time = 0.37, size = 13, normalized size = 0.76 \begin {gather*} \frac {16 \, x - 81}{x + e^{\left (e^{x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-16*x+81)*exp(x)+16)*exp(exp(x))+81)/(exp(exp(x))^2+2*x*exp(exp(x))+x^2),x, algorithm="maxima")

[Out]

(16*x - 81)/(x + e^(e^x))

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mupad [B]  time = 4.41, size = 13, normalized size = 0.76 \begin {gather*} \frac {16\,x-81}{x+{\mathrm {e}}^{{\mathrm {e}}^x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(exp(x))*(exp(x)*(16*x - 81) - 16) - 81)/(exp(2*exp(x)) + 2*x*exp(exp(x)) + x^2),x)

[Out]

(16*x - 81)/(x + exp(exp(x)))

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sympy [A]  time = 0.11, size = 10, normalized size = 0.59 \begin {gather*} \frac {16 x - 81}{x + e^{e^{x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-16*x+81)*exp(x)+16)*exp(exp(x))+81)/(exp(exp(x))**2+2*x*exp(exp(x))+x**2),x)

[Out]

(16*x - 81)/(x + exp(exp(x)))

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