3.60.54 \(\int \frac {260+96 e^{2 x}+5 e^{4 x}+2 x-x^2}{256+32 e^{2 x}+e^{4 x}+4 x-x^2} \, dx\)

Optimal. Leaf size=27 \[ \log \left (e^x \left (x+\frac {1}{3} \left (\left (16+e^{2 x}\right )^2+x-x^2\right )\right )\right ) \]

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Rubi [F]  time = 0.54, 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 {260+96 e^{2 x}+5 e^{4 x}+2 x-x^2}{256+32 e^{2 x}+e^{4 x}+4 x-x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(260 + 96*E^(2*x) + 5*E^(4*x) + 2*x - x^2)/(256 + 32*E^(2*x) + E^(4*x) + 4*x - x^2),x]

[Out]

5*x - 1020*Defer[Int][(256 + 32*E^(2*x) + E^(4*x) + 4*x - x^2)^(-1), x] - 64*Defer[Int][E^(2*x)/(256 + 32*E^(2
*x) + E^(4*x) + 4*x - x^2), x] + 18*Defer[Int][x/(-256 - 32*E^(2*x) - E^(4*x) - 4*x + x^2), x] - 4*Defer[Int][
x^2/(-256 - 32*E^(2*x) - E^(4*x) - 4*x + x^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (5-\frac {2 \left (510+32 e^{2 x}+9 x-2 x^2\right )}{256+32 e^{2 x}+e^{4 x}+4 x-x^2}\right ) \, dx\\ &=5 x-2 \int \frac {510+32 e^{2 x}+9 x-2 x^2}{256+32 e^{2 x}+e^{4 x}+4 x-x^2} \, dx\\ &=5 x-2 \int \left (\frac {510}{256+32 e^{2 x}+e^{4 x}+4 x-x^2}+\frac {32 e^{2 x}}{256+32 e^{2 x}+e^{4 x}+4 x-x^2}-\frac {9 x}{-256-32 e^{2 x}-e^{4 x}-4 x+x^2}+\frac {2 x^2}{-256-32 e^{2 x}-e^{4 x}-4 x+x^2}\right ) \, dx\\ &=5 x-4 \int \frac {x^2}{-256-32 e^{2 x}-e^{4 x}-4 x+x^2} \, dx+18 \int \frac {x}{-256-32 e^{2 x}-e^{4 x}-4 x+x^2} \, dx-64 \int \frac {e^{2 x}}{256+32 e^{2 x}+e^{4 x}+4 x-x^2} \, dx-1020 \int \frac {1}{256+32 e^{2 x}+e^{4 x}+4 x-x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.19, size = 25, normalized size = 0.93 \begin {gather*} x+\log \left (256+32 e^{2 x}+e^{4 x}+4 x-x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(260 + 96*E^(2*x) + 5*E^(4*x) + 2*x - x^2)/(256 + 32*E^(2*x) + E^(4*x) + 4*x - x^2),x]

[Out]

x + Log[256 + 32*E^(2*x) + E^(4*x) + 4*x - x^2]

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fricas [A]  time = 0.77, size = 23, normalized size = 0.85 \begin {gather*} x + \log \left (-x^{2} + 4 \, x + e^{\left (4 \, x\right )} + 32 \, e^{\left (2 \, x\right )} + 256\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*exp(2*x)^2+96*exp(2*x)-x^2+2*x+260)/(exp(2*x)^2+32*exp(2*x)-x^2+4*x+256),x, algorithm="fricas")

[Out]

x + log(-x^2 + 4*x + e^(4*x) + 32*e^(2*x) + 256)

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giac [A]  time = 0.13, size = 23, normalized size = 0.85 \begin {gather*} x + \log \left (-x^{2} + 4 \, x + e^{\left (4 \, x\right )} + 32 \, e^{\left (2 \, x\right )} + 256\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*exp(2*x)^2+96*exp(2*x)-x^2+2*x+260)/(exp(2*x)^2+32*exp(2*x)-x^2+4*x+256),x, algorithm="giac")

[Out]

x + log(-x^2 + 4*x + e^(4*x) + 32*e^(2*x) + 256)

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maple [A]  time = 0.08, size = 24, normalized size = 0.89

method	result	size
risch	\(x +\ln \left ({\mathrm e}^{4 x}+32 \,{\mathrm e}^{2 x}-x^{2}+4 x +256\right )\)	\(24\)
norman	\(x +\ln \left (x^{2}-{\mathrm e}^{4 x}-4 x -32 \,{\mathrm e}^{2 x}-256\right )\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*exp(2*x)^2+96*exp(2*x)-x^2+2*x+260)/(exp(2*x)^2+32*exp(2*x)-x^2+4*x+256),x,method=_RETURNVERBOSE)

[Out]

x+ln(exp(4*x)+32*exp(2*x)-x^2+4*x+256)

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maxima [A]  time = 0.38, size = 23, normalized size = 0.85 \begin {gather*} x + \log \left (-x^{2} + 4 \, x + e^{\left (4 \, x\right )} + 32 \, e^{\left (2 \, x\right )} + 256\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*exp(2*x)^2+96*exp(2*x)-x^2+2*x+260)/(exp(2*x)^2+32*exp(2*x)-x^2+4*x+256),x, algorithm="maxima")

[Out]

x + log(-x^2 + 4*x + e^(4*x) + 32*e^(2*x) + 256)

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mupad [B]  time = 4.22, size = 23, normalized size = 0.85 \begin {gather*} x+\ln \left (x^2-32\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^{4\,x}-4\,x-256\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + 96*exp(2*x) + 5*exp(4*x) - x^2 + 260)/(4*x + 32*exp(2*x) + exp(4*x) - x^2 + 256),x)

[Out]

x + log(x^2 - 32*exp(2*x) - exp(4*x) - 4*x - 256)

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sympy [A]  time = 0.15, size = 22, normalized size = 0.81 \begin {gather*} x + \log {\left (- x^{2} + 4 x + e^{4 x} + 32 e^{2 x} + 256 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*exp(2*x)**2+96*exp(2*x)-x**2+2*x+260)/(exp(2*x)**2+32*exp(2*x)-x**2+4*x+256),x)

[Out]

x + log(-x**2 + 4*x + exp(4*x) + 32*exp(2*x) + 256)

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