3.100.97 \(\int \frac {-6-12 e^{2 x}+x+e^{e^2+x} (-5+12 e^{2 x}+x)}{-5+12 e^{2 x}+x} \, dx\)

Optimal. Leaf size=22 \[ e^{e^2+x}+x-\log \left (-5+12 e^{2 x}+x\right ) \]

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Rubi [F]  time = 0.24, 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 {-6-12 e^{2 x}+x+e^{e^2+x} \left (-5+12 e^{2 x}+x\right )}{-5+12 e^{2 x}+x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-6 - 12*E^(2*x) + x + E^(E^2 + x)*(-5 + 12*E^(2*x) + x))/(-5 + 12*E^(2*x) + x),x]

[Out]

E^(E^2 + x) - x - 11*Defer[Int][(-5 + 12*E^(2*x) + x)^(-1), x] + 2*Defer[Int][x/(-5 + 12*E^(2*x) + x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-1+e^{e^2+x}+\frac {-11+2 x}{-5+12 e^{2 x}+x}\right ) \, dx\\ &=-x+\int e^{e^2+x} \, dx+\int \frac {-11+2 x}{-5+12 e^{2 x}+x} \, dx\\ &=e^{e^2+x}-x+\int \left (-\frac {11}{-5+12 e^{2 x}+x}+\frac {2 x}{-5+12 e^{2 x}+x}\right ) \, dx\\ &=e^{e^2+x}-x+2 \int \frac {x}{-5+12 e^{2 x}+x} \, dx-11 \int \frac {1}{-5+12 e^{2 x}+x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 24, normalized size = 1.09 \begin {gather*} e^{e^2+x}+x-\log \left (5-12 e^{2 x}-x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-6 - 12*E^(2*x) + x + E^(E^2 + x)*(-5 + 12*E^(2*x) + x))/(-5 + 12*E^(2*x) + x),x]

[Out]

E^(E^2 + x) + x - Log[5 - 12*E^(2*x) - x]

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fricas [A]  time = 0.61, size = 31, normalized size = 1.41 \begin {gather*} x + e^{\left (x + e^{2}\right )} - \log \left ({\left (x - 5\right )} e^{\left (2 \, e^{2}\right )} + 12 \, e^{\left (2 \, x + 2 \, e^{2}\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*exp(x)^2+x-5)*exp(x+exp(2))-12*exp(x)^2+x-6)/(12*exp(x)^2+x-5),x, algorithm="fricas")

[Out]

x + e^(x + e^2) - log((x - 5)*e^(2*e^2) + 12*e^(2*x + 2*e^2))

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giac [A]  time = 0.14, size = 21, normalized size = 0.95 \begin {gather*} x + e^{\left (x + e^{2}\right )} - \log \left (-x - 12 \, e^{\left (2 \, x\right )} + 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*exp(x)^2+x-5)*exp(x+exp(2))-12*exp(x)^2+x-6)/(12*exp(x)^2+x-5),x, algorithm="giac")

[Out]

x + e^(x + e^2) - log(-x - 12*e^(2*x) + 5)

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maple [A]  time = 0.05, size = 20, normalized size = 0.91

method	result	size
risch	\({\mathrm e}^{x +{\mathrm e}^{2}}+x -\ln \left ({\mathrm e}^{2 x}+\frac {x}{12}-\frac {5}{12}\right )\)	\(20\)
norman	\(x +{\mathrm e}^{x} {\mathrm e}^{{\mathrm e}^{2}}-\ln \left (12 \,{\mathrm e}^{2 x}+x -5\right )\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((12*exp(x)^2+x-5)*exp(x+exp(2))-12*exp(x)^2+x-6)/(12*exp(x)^2+x-5),x,method=_RETURNVERBOSE)

[Out]

exp(x+exp(2))+x-ln(exp(2*x)+1/12*x-5/12)

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maxima [A]  time = 0.38, size = 19, normalized size = 0.86 \begin {gather*} x + e^{\left (x + e^{2}\right )} - \log \left (\frac {1}{12} \, x + e^{\left (2 \, x\right )} - \frac {5}{12}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*exp(x)^2+x-5)*exp(x+exp(2))-12*exp(x)^2+x-6)/(12*exp(x)^2+x-5),x, algorithm="maxima")

[Out]

x + e^(x + e^2) - log(1/12*x + e^(2*x) - 5/12)

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mupad [B]  time = 0.16, size = 20, normalized size = 0.91 \begin {gather*} x-\ln \left (x+12\,{\mathrm {e}}^{2\,x}-5\right )+{\mathrm {e}}^{{\mathrm {e}}^2}\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x - 12*exp(2*x) + exp(x + exp(2))*(x + 12*exp(2*x) - 5) - 6)/(x + 12*exp(2*x) - 5),x)

[Out]

x - log(x + 12*exp(2*x) - 5) + exp(exp(2))*exp(x)

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sympy [A]  time = 0.17, size = 32, normalized size = 1.45 \begin {gather*} - \frac {5 x}{6} + \sqrt {e^{2 x}} e^{e^{2}} - \frac {\log {\left (\frac {x}{12} + e^{2 x} - \frac {5}{12} \right )}}{12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*exp(x)**2+x-5)*exp(x+exp(2))-12*exp(x)**2+x-6)/(12*exp(x)**2+x-5),x)

[Out]

-5*x/6 + sqrt(exp(2*x))*exp(exp(2)) - log(x/12 + exp(2*x) - 5/12)/12

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