3.7.61 \(\int \frac {9+e^x (96-144 x)}{512 e^{2 x}+e^x (16-144 x)-3 x+9 x^2} \, dx\) [661]

Optimal. Leaf size=26 \[ 3+3 \log \left (2-\frac {3+\frac {1}{x}}{3-\frac {16 e^x}{x}}\right ) \]

[Out]

3*ln(2-(1/x+3)/(3-16*exp(x)/x))+3

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Rubi [F]
time = 0.22, 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 {9+e^x (96-144 x)}{512 e^{2 x}+e^x (16-144 x)-3 x+9 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(9 + E^x*(96 - 144*x))/(512*E^(2*x) + E^x*(16 - 144*x) - 3*x + 9*x^2),x]

[Out]

9*Defer[Int][(16*E^x - 3*x)^(-1), x] - 12*Defer[Int][(1 + 32*E^x - 3*x)^(-1), x] + 9*Defer[Int][x/(1 + 32*E^x
- 3*x), x] + 9*Defer[Int][x/(-16*E^x + 3*x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {3 (-4+3 x)}{1+32 e^x-3 x}+\frac {9 (-1+x)}{-16 e^x+3 x}\right ) \, dx\\ &=3 \int \frac {-4+3 x}{1+32 e^x-3 x} \, dx+9 \int \frac {-1+x}{-16 e^x+3 x} \, dx\\ &=3 \int \left (-\frac {4}{1+32 e^x-3 x}+\frac {3 x}{1+32 e^x-3 x}\right ) \, dx+9 \int \left (\frac {1}{16 e^x-3 x}+\frac {x}{-16 e^x+3 x}\right ) \, dx\\ &=9 \int \frac {1}{16 e^x-3 x} \, dx+9 \int \frac {x}{1+32 e^x-3 x} \, dx+9 \int \frac {x}{-16 e^x+3 x} \, dx-12 \int \frac {1}{1+32 e^x-3 x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.26, size = 26, normalized size = 1.00 \begin {gather*} -3 \log \left (16 e^x-3 x\right )+3 \log \left (1+32 e^x-3 x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(9 + E^x*(96 - 144*x))/(512*E^(2*x) + E^x*(16 - 144*x) - 3*x + 9*x^2),x]

[Out]

-3*Log[16*E^x - 3*x] + 3*Log[1 + 32*E^x - 3*x]

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Maple [A]
time = 0.07, size = 21, normalized size = 0.81

method	result	size
risch	\(3 \ln \left (\frac {1}{32}-\frac {3 x}{32}+{\mathrm e}^{x}\right )-3 \ln \left (-\frac {3 x}{16}+{\mathrm e}^{x}\right )\)	\(21\)
norman	\(-3 \ln \left (3 x -16 \,{\mathrm e}^{x}\right )+3 \ln \left (3 x -32 \,{\mathrm e}^{x}-1\right )\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-144*x+96)*exp(x)+9)/(512*exp(x)^2+(-144*x+16)*exp(x)+9*x^2-3*x),x,method=_RETURNVERBOSE)

[Out]

3*ln(1/32-3/32*x+exp(x))-3*ln(-3/16*x+exp(x))

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Maxima [A]
time = 0.29, size = 20, normalized size = 0.77 \begin {gather*} 3 \, \log \left (-\frac {3}{32} \, x + e^{x} + \frac {1}{32}\right ) - 3 \, \log \left (-\frac {3}{16} \, x + e^{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-144*x+96)*exp(x)+9)/(512*exp(x)^2+(-144*x+16)*exp(x)+9*x^2-3*x),x, algorithm="maxima")

[Out]

3*log(-3/32*x + e^x + 1/32) - 3*log(-3/16*x + e^x)

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Fricas [A]
time = 0.37, size = 24, normalized size = 0.92 \begin {gather*} 3 \, \log \left (-3 \, x + 32 \, e^{x} + 1\right ) - 3 \, \log \left (-3 \, x + 16 \, e^{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-144*x+96)*exp(x)+9)/(512*exp(x)^2+(-144*x+16)*exp(x)+9*x^2-3*x),x, algorithm="fricas")

[Out]

3*log(-3*x + 32*e^x + 1) - 3*log(-3*x + 16*e^x)

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Sympy [A]
time = 0.12, size = 26, normalized size = 1.00 \begin {gather*} - 3 \log {\left (- \frac {3 x}{16} + e^{x} \right )} + 3 \log {\left (- \frac {3 x}{32} + e^{x} + \frac {1}{32} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-144*x+96)*exp(x)+9)/(512*exp(x)**2+(-144*x+16)*exp(x)+9*x**2-3*x),x)

[Out]

-3*log(-3*x/16 + exp(x)) + 3*log(-3*x/32 + exp(x) + 1/32)

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Giac [A]
time = 0.44, size = 24, normalized size = 0.92 \begin {gather*} 3 \, \log \left (3 \, x - 32 \, e^{x} - 1\right ) - 3 \, \log \left (-3 \, x + 16 \, e^{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-144*x+96)*exp(x)+9)/(512*exp(x)^2+(-144*x+16)*exp(x)+9*x^2-3*x),x, algorithm="giac")

[Out]

3*log(3*x - 32*e^x - 1) - 3*log(-3*x + 16*e^x)

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Mupad [B]
time = 0.66, size = 22, normalized size = 0.85 \begin {gather*} 3\,\ln \left (3\,x-32\,{\mathrm {e}}^x-1\right )-3\,\ln \left (x-\frac {16\,{\mathrm {e}}^x}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)*(144*x - 96) - 9)/(3*x - 512*exp(2*x) + exp(x)*(144*x - 16) - 9*x^2),x)

[Out]

3*log(3*x - 32*exp(x) - 1) - 3*log(x - (16*exp(x))/3)

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