3.18.96 \(\int \frac {e^{\frac {1}{5} (x+x^2+5 e^8 x^8)} (1+2 x+40 e^8 x^7)}{2+e^{\frac {1}{5} (x+x^2+5 e^8 x^8)}} \, dx\)

Optimal. Leaf size=24 \[ 5 \log \left (2+e^{e^8 x^8+\frac {1}{5} \left (x+x^2\right )}\right ) \]

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Rubi [A]  time = 0.28, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 56, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {6684} \begin {gather*} 5 \log \left (e^{\frac {1}{5} \left (5 e^8 x^8+x^2+x\right )}+2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((x + x^2 + 5*E^8*x^8)/5)*(1 + 2*x + 40*E^8*x^7))/(2 + E^((x + x^2 + 5*E^8*x^8)/5)),x]

[Out]

5*Log[2 + E^((x + x^2 + 5*E^8*x^8)/5)]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=5 \log \left (2+e^{\frac {1}{5} \left (x+x^2+5 e^8 x^8\right )}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.30, size = 23, normalized size = 0.96 \begin {gather*} 5 \log \left (2+e^{\frac {1}{5} x \left (1+x+5 e^8 x^7\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((x + x^2 + 5*E^8*x^8)/5)*(1 + 2*x + 40*E^8*x^7))/(2 + E^((x + x^2 + 5*E^8*x^8)/5)),x]

[Out]

5*Log[2 + E^((x*(1 + x + 5*E^8*x^7))/5)]

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fricas [A]  time = 0.89, size = 21, normalized size = 0.88 \begin {gather*} 5 \, \log \left (e^{\left (x^{8} e^{8} + \frac {1}{5} \, x^{2} + \frac {1}{5} \, x\right )} + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((40*x^7*exp(4)^2+2*x+1)*exp(x^8*exp(4)^2+1/5*x^2+1/5*x)/(exp(x^8*exp(4)^2+1/5*x^2+1/5*x)+2),x, algor
ithm="fricas")

[Out]

5*log(e^(x^8*e^8 + 1/5*x^2 + 1/5*x) + 2)

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giac [A]  time = 0.33, size = 21, normalized size = 0.88 \begin {gather*} 5 \, \log \left (e^{\left (x^{8} e^{8} + \frac {1}{5} \, x^{2} + \frac {1}{5} \, x\right )} + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((40*x^7*exp(4)^2+2*x+1)*exp(x^8*exp(4)^2+1/5*x^2+1/5*x)/(exp(x^8*exp(4)^2+1/5*x^2+1/5*x)+2),x, algor
ithm="giac")

[Out]

5*log(e^(x^8*e^8 + 1/5*x^2 + 1/5*x) + 2)

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maple [A]  time = 0.10, size = 20, normalized size = 0.83

method	result	size
risch	\(5 \ln \left ({\mathrm e}^{\frac {x \left (5 x^{7} {\mathrm e}^{8}+x +1\right )}{5}}+2\right )\)	\(20\)
norman	\(5 \ln \left ({\mathrm e}^{x^{8} {\mathrm e}^{8}+\frac {x^{2}}{5}+\frac {x}{5}}+2\right )\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((40*x^7*exp(4)^2+2*x+1)*exp(x^8*exp(4)^2+1/5*x^2+1/5*x)/(exp(x^8*exp(4)^2+1/5*x^2+1/5*x)+2),x,method=_RETU
RNVERBOSE)

[Out]

5*ln(exp(1/5*x*(5*x^7*exp(8)+x+1))+2)

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maxima [A]  time = 0.76, size = 37, normalized size = 1.54 \begin {gather*} x^{2} + x + 5 \, \log \left ({\left (e^{\left (x^{8} e^{8} + \frac {1}{5} \, x^{2} + \frac {1}{5} \, x\right )} + 2\right )} e^{\left (-\frac {1}{5} \, x^{2} - \frac {1}{5} \, x\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((40*x^7*exp(4)^2+2*x+1)*exp(x^8*exp(4)^2+1/5*x^2+1/5*x)/(exp(x^8*exp(4)^2+1/5*x^2+1/5*x)+2),x, algor
ithm="maxima")

[Out]

x^2 + x + 5*log((e^(x^8*e^8 + 1/5*x^2 + 1/5*x) + 2)*e^(-1/5*x^2 - 1/5*x))

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mupad [B]  time = 0.15, size = 23, normalized size = 0.96 \begin {gather*} \ln \left ({\left ({\mathrm {e}}^{x^8\,{\mathrm {e}}^8}\,{\mathrm {e}}^{x/5}\,{\mathrm {e}}^{\frac {x^2}{5}}+2\right )}^5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x/5 + x^8*exp(8) + x^2/5)*(2*x + 40*x^7*exp(8) + 1))/(exp(x/5 + x^8*exp(8) + x^2/5) + 2),x)

[Out]

log((exp(x^8*exp(8))*exp(x/5)*exp(x^2/5) + 2)^5)

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sympy [A]  time = 0.16, size = 39, normalized size = 1.62 \begin {gather*} 4 x^{8} e^{8} + \frac {4 x^{2}}{5} + \frac {4 x}{5} + \log {\left (e^{x^{8} e^{8} + \frac {x^{2}}{5} + \frac {x}{5}} + 2 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((40*x**7*exp(4)**2+2*x+1)*exp(x**8*exp(4)**2+1/5*x**2+1/5*x)/(exp(x**8*exp(4)**2+1/5*x**2+1/5*x)+2),
x)

[Out]

4*x**8*exp(8) + 4*x**2/5 + 4*x/5 + log(exp(x**8*exp(8) + x**2/5 + x/5) + 2)

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