3.100.76 \(\int \frac {-24+8 e^6+e^{2 x}+2 x-x^2}{24-8 e^6+e^{2 x}+x^2} \, dx\)

Optimal. Leaf size=22 \[ \log \left (e^x+e^{-x} \left (-8 \left (-3+e^6\right )+x^2\right )\right ) \]

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Rubi [F]  time = 0.41, 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 {-24+8 e^6+e^{2 x}+2 x-x^2}{24-8 e^6+e^{2 x}+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-24 + 8*E^6 + E^(2*x) + 2*x - x^2)/(24 - 8*E^6 + E^(2*x) + x^2),x]

[Out]

x + 2*Defer[Int][x^2/(-E^(2*x) - 24*(1 - E^6/3) - x^2), x] - 16*(3 - E^6)*Defer[Int][(E^(2*x) + 24*(1 - E^6/3)
+ x^2)^(-1), x] + 2*Defer[Int][x/(E^(2*x) + 24*(1 - E^6/3) + x^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{2 x}-24 \left (1-\frac {e^6}{3}\right )+2 x-x^2}{e^{2 x}+24 \left (1-\frac {e^6}{3}\right )+x^2} \, dx\\ &=\int \left (1+\frac {2 \left (-8 \left (3-e^6\right )+x-x^2\right )}{e^{2 x}+24 \left (1-\frac {e^6}{3}\right )+x^2}\right ) \, dx\\ &=x+2 \int \frac {-8 \left (3-e^6\right )+x-x^2}{e^{2 x}+24 \left (1-\frac {e^6}{3}\right )+x^2} \, dx\\ &=x+2 \int \left (\frac {x^2}{-e^{2 x}-24 \left (1-\frac {e^6}{3}\right )-x^2}+\frac {8 \left (-3+e^6\right )}{e^{2 x}+24 \left (1-\frac {e^6}{3}\right )+x^2}+\frac {x}{e^{2 x}+24 \left (1-\frac {e^6}{3}\right )+x^2}\right ) \, dx\\ &=x+2 \int \frac {x^2}{-e^{2 x}-24 \left (1-\frac {e^6}{3}\right )-x^2} \, dx+2 \int \frac {x}{e^{2 x}+24 \left (1-\frac {e^6}{3}\right )+x^2} \, dx-\left (16 \left (3-e^6\right )\right ) \int \frac {1}{e^{2 x}+24 \left (1-\frac {e^6}{3}\right )+x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.15, size = 20, normalized size = 0.91 \begin {gather*} -x+\log \left (24-8 e^6+e^{2 x}+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-24 + 8*E^6 + E^(2*x) + 2*x - x^2)/(24 - 8*E^6 + E^(2*x) + x^2),x]

[Out]

-x + Log[24 - 8*E^6 + E^(2*x) + x^2]

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fricas [A]  time = 0.79, size = 18, normalized size = 0.82 \begin {gather*} -x + \log \left (x^{2} - 8 \, e^{6} + e^{\left (2 \, x\right )} + 24\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)^2+8*exp(3)^2-x^2+2*x-24)/(exp(x)^2-8*exp(3)^2+x^2+24),x, algorithm="fricas")

[Out]

-x + log(x^2 - 8*e^6 + e^(2*x) + 24)

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giac [A]  time = 0.19, size = 22, normalized size = 1.00 \begin {gather*} -x + \log \left (-x^{2} + 8 \, e^{6} - e^{\left (2 \, x\right )} - 24\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)^2+8*exp(3)^2-x^2+2*x-24)/(exp(x)^2-8*exp(3)^2+x^2+24),x, algorithm="giac")

[Out]

-x + log(-x^2 + 8*e^6 - e^(2*x) - 24)

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maple [A]  time = 0.06, size = 19, normalized size = 0.86

method	result	size
risch	\(-x +\ln \left ({\mathrm e}^{2 x}-8 \,{\mathrm e}^{6}+x^{2}+24\right )\)	\(19\)
norman	\(-x +\ln \left (8 \,{\mathrm e}^{6}-x^{2}-{\mathrm e}^{2 x}-24\right )\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)^2+8*exp(3)^2-x^2+2*x-24)/(exp(x)^2-8*exp(3)^2+x^2+24),x,method=_RETURNVERBOSE)

[Out]

-x+ln(exp(2*x)-8*exp(6)+x^2+24)

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maxima [A]  time = 0.38, size = 18, normalized size = 0.82 \begin {gather*} -x + \log \left (x^{2} - 8 \, e^{6} + e^{\left (2 \, x\right )} + 24\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)^2+8*exp(3)^2-x^2+2*x-24)/(exp(x)^2-8*exp(3)^2+x^2+24),x, algorithm="maxima")

[Out]

-x + log(x^2 - 8*e^6 + e^(2*x) + 24)

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mupad [B]  time = 0.13, size = 18, normalized size = 0.82 \begin {gather*} \ln \left ({\mathrm {e}}^{2\,x}-8\,{\mathrm {e}}^6+x^2+24\right )-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + exp(2*x) + 8*exp(6) - x^2 - 24)/(exp(2*x) - 8*exp(6) + x^2 + 24),x)

[Out]

log(exp(2*x) - 8*exp(6) + x^2 + 24) - x

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sympy [A]  time = 0.12, size = 17, normalized size = 0.77 \begin {gather*} - x + \log {\left (x^{2} + e^{2 x} - 8 e^{6} + 24 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)**2+8*exp(3)**2-x**2+2*x-24)/(exp(x)**2-8*exp(3)**2+x**2+24),x)

[Out]

-x + log(x**2 + exp(2*x) - 8*exp(6) + 24)

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