3.6.15 \(\int \frac {4+8 e^{2 x}+e^x (12+4 x)-750 \log ^2(5)+(-2250 \log ^2(5)-3000 e^x \log ^2(5)) \log (x)+281250 \log ^4(5) \log ^2(x)}{4+8 e^x+4 e^{2 x}+(-1500 \log ^2(5)-1500 e^x \log ^2(5)) \log (x)+140625 \log ^4(5) \log ^2(x)} \, dx\)

Optimal. Leaf size=25 \[ 2 x+\frac {x}{-1-e^x+\frac {375}{2} \log ^2(5) \log (x)} \]

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Rubi [F]  time = 1.07, 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 {4+8 e^{2 x}+e^x (12+4 x)-750 \log ^2(5)+\left (-2250 \log ^2(5)-3000 e^x \log ^2(5)\right ) \log (x)+281250 \log ^4(5) \log ^2(x)}{4+8 e^x+4 e^{2 x}+\left (-1500 \log ^2(5)-1500 e^x \log ^2(5)\right ) \log (x)+140625 \log ^4(5) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4 + 8*E^(2*x) + E^x*(12 + 4*x) - 750*Log[5]^2 + (-2250*Log[5]^2 - 3000*E^x*Log[5]^2)*Log[x] + 281250*Log[
5]^4*Log[x]^2)/(4 + 8*E^x + 4*E^(2*x) + (-1500*Log[5]^2 - 1500*E^x*Log[5]^2)*Log[x] + 140625*Log[5]^4*Log[x]^2
),x]

[Out]

2*x - 4*Defer[Int][x/(2 + 2*E^x - 375*Log[5]^2*Log[x])^2, x] - 2*Defer[Int][(2 + 2*E^x - 375*Log[5]^2*Log[x])^
(-1), x] + 2*Defer[Int][x/(2 + 2*E^x - 375*Log[5]^2*Log[x]), x] - 750*Log[5]^2*Defer[Int][(-2 - 2*E^x + 375*Lo
g[5]^2*Log[x])^(-2), x] + 750*Log[5]^2*Defer[Int][(x*Log[x])/(-2 - 2*E^x + 375*Log[5]^2*Log[x])^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8 e^{2 x}+4 e^x (3+x)+4 \left (1-\frac {375 \log ^2(5)}{2}\right )-750 \left (3+4 e^x\right ) \log ^2(5) \log (x)+281250 \log ^4(5) \log ^2(x)}{\left (2+2 e^x-375 \log ^2(5) \log (x)\right )^2} \, dx\\ &=\int \left (2+\frac {2 (-1+x)}{2+2 e^x-375 \log ^2(5) \log (x)}+\frac {2 \left (-2 x-375 \log ^2(5)+375 x \log ^2(5) \log (x)\right )}{\left (2+2 e^x-375 \log ^2(5) \log (x)\right )^2}\right ) \, dx\\ &=2 x+2 \int \frac {-1+x}{2+2 e^x-375 \log ^2(5) \log (x)} \, dx+2 \int \frac {-2 x-375 \log ^2(5)+375 x \log ^2(5) \log (x)}{\left (2+2 e^x-375 \log ^2(5) \log (x)\right )^2} \, dx\\ &=2 x+2 \int \left (-\frac {1}{2+2 e^x-375 \log ^2(5) \log (x)}+\frac {x}{2+2 e^x-375 \log ^2(5) \log (x)}\right ) \, dx+2 \int \left (-\frac {2 x}{\left (2+2 e^x-375 \log ^2(5) \log (x)\right )^2}-\frac {375 \log ^2(5)}{\left (-2-2 e^x+375 \log ^2(5) \log (x)\right )^2}+\frac {375 x \log ^2(5) \log (x)}{\left (-2-2 e^x+375 \log ^2(5) \log (x)\right )^2}\right ) \, dx\\ &=2 x-2 \int \frac {1}{2+2 e^x-375 \log ^2(5) \log (x)} \, dx+2 \int \frac {x}{2+2 e^x-375 \log ^2(5) \log (x)} \, dx-4 \int \frac {x}{\left (2+2 e^x-375 \log ^2(5) \log (x)\right )^2} \, dx-\left (750 \log ^2(5)\right ) \int \frac {1}{\left (-2-2 e^x+375 \log ^2(5) \log (x)\right )^2} \, dx+\left (750 \log ^2(5)\right ) \int \frac {x \log (x)}{\left (-2-2 e^x+375 \log ^2(5) \log (x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.39, size = 24, normalized size = 0.96 \begin {gather*} 2 x+\frac {2 x}{-2-2 e^x+375 \log ^2(5) \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 + 8*E^(2*x) + E^x*(12 + 4*x) - 750*Log[5]^2 + (-2250*Log[5]^2 - 3000*E^x*Log[5]^2)*Log[x] + 28125
0*Log[5]^4*Log[x]^2)/(4 + 8*E^x + 4*E^(2*x) + (-1500*Log[5]^2 - 1500*E^x*Log[5]^2)*Log[x] + 140625*Log[5]^4*Lo
g[x]^2),x]

[Out]

2*x + (2*x)/(-2 - 2*E^x + 375*Log[5]^2*Log[x])

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fricas [A]  time = 0.66, size = 36, normalized size = 1.44 \begin {gather*} \frac {2 \, {\left (375 \, x \log \relax (5)^{2} \log \relax (x) - 2 \, x e^{x} - x\right )}}{375 \, \log \relax (5)^{2} \log \relax (x) - 2 \, e^{x} - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((281250*log(5)^4*log(x)^2+(-3000*log(5)^2*exp(x)-2250*log(5)^2)*log(x)+8*exp(x)^2+(4*x+12)*exp(x)-75
0*log(5)^2+4)/(140625*log(5)^4*log(x)^2+(-1500*log(5)^2*exp(x)-1500*log(5)^2)*log(x)+4*exp(x)^2+8*exp(x)+4),x,
algorithm="fricas")

[Out]

2*(375*x*log(5)^2*log(x) - 2*x*e^x - x)/(375*log(5)^2*log(x) - 2*e^x - 2)

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giac [B]  time = 0.68, size = 49, normalized size = 1.96 \begin {gather*} \frac {2 \, {\left (375 \, x \log \relax (5)^{2} \log \relax (x) + 375 \, \log \relax (5)^{2} \log \relax (x) - 2 \, x e^{x} - x - 2 \, e^{x} - 2\right )}}{375 \, \log \relax (5)^{2} \log \relax (x) - 2 \, e^{x} - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((281250*log(5)^4*log(x)^2+(-3000*log(5)^2*exp(x)-2250*log(5)^2)*log(x)+8*exp(x)^2+(4*x+12)*exp(x)-75
0*log(5)^2+4)/(140625*log(5)^4*log(x)^2+(-1500*log(5)^2*exp(x)-1500*log(5)^2)*log(x)+4*exp(x)^2+8*exp(x)+4),x,
algorithm="giac")

[Out]

2*(375*x*log(5)^2*log(x) + 375*log(5)^2*log(x) - 2*x*e^x - x - 2*e^x - 2)/(375*log(5)^2*log(x) - 2*e^x - 2)

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maple [A]  time = 0.34, size = 24, normalized size = 0.96

method	result	size
risch	\(2 x +\frac {2 x}{375 \ln \relax (x ) \ln \relax (5)^{2}-2 \,{\mathrm e}^{x}-2}\)	\(24\)
norman	\(\frac {-2 x -4 \,{\mathrm e}^{x} x +750 x \ln \relax (5)^{2} \ln \relax (x )}{375 \ln \relax (x ) \ln \relax (5)^{2}-2 \,{\mathrm e}^{x}-2}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((281250*ln(5)^4*ln(x)^2+(-3000*ln(5)^2*exp(x)-2250*ln(5)^2)*ln(x)+8*exp(x)^2+(4*x+12)*exp(x)-750*ln(5)^2+4
)/(140625*ln(5)^4*ln(x)^2+(-1500*ln(5)^2*exp(x)-1500*ln(5)^2)*ln(x)+4*exp(x)^2+8*exp(x)+4),x,method=_RETURNVER
BOSE)

[Out]

2*x+2*x/(375*ln(x)*ln(5)^2-2*exp(x)-2)

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maxima [A]  time = 1.04, size = 36, normalized size = 1.44 \begin {gather*} \frac {2 \, {\left (375 \, x \log \relax (5)^{2} \log \relax (x) - 2 \, x e^{x} - x\right )}}{375 \, \log \relax (5)^{2} \log \relax (x) - 2 \, e^{x} - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((281250*log(5)^4*log(x)^2+(-3000*log(5)^2*exp(x)-2250*log(5)^2)*log(x)+8*exp(x)^2+(4*x+12)*exp(x)-75
0*log(5)^2+4)/(140625*log(5)^4*log(x)^2+(-1500*log(5)^2*exp(x)-1500*log(5)^2)*log(x)+4*exp(x)^2+8*exp(x)+4),x,
algorithm="maxima")

[Out]

2*(375*x*log(5)^2*log(x) - 2*x*e^x - x)/(375*log(5)^2*log(x) - 2*e^x - 2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {281250\,{\ln \relax (5)}^4\,{\ln \relax (x)}^2+\left (-3000\,{\mathrm {e}}^x\,{\ln \relax (5)}^2-2250\,{\ln \relax (5)}^2\right )\,\ln \relax (x)+8\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,\left (4\,x+12\right )-750\,{\ln \relax (5)}^2+4}{140625\,{\ln \relax (5)}^4\,{\ln \relax (x)}^2+\left (-1500\,{\mathrm {e}}^x\,{\ln \relax (5)}^2-1500\,{\ln \relax (5)}^2\right )\,\ln \relax (x)+4\,{\mathrm {e}}^{2\,x}+8\,{\mathrm {e}}^x+4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*exp(2*x) + 281250*log(5)^4*log(x)^2 + exp(x)*(4*x + 12) - 750*log(5)^2 - log(x)*(3000*exp(x)*log(5)^2 +
2250*log(5)^2) + 4)/(4*exp(2*x) + 8*exp(x) + 140625*log(5)^4*log(x)^2 - log(x)*(1500*exp(x)*log(5)^2 + 1500*l
og(5)^2) + 4),x)

[Out]

int((8*exp(2*x) + 281250*log(5)^4*log(x)^2 + exp(x)*(4*x + 12) - 750*log(5)^2 - log(x)*(3000*exp(x)*log(5)^2 +
2250*log(5)^2) + 4)/(4*exp(2*x) + 8*exp(x) + 140625*log(5)^4*log(x)^2 - log(x)*(1500*exp(x)*log(5)^2 + 1500*l
og(5)^2) + 4), x)

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sympy [A]  time = 0.36, size = 20, normalized size = 0.80 \begin {gather*} 2 x - \frac {x}{e^{x} - \frac {375 \log {\relax (5 )}^{2} \log {\relax (x )}}{2} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((281250*ln(5)**4*ln(x)**2+(-3000*ln(5)**2*exp(x)-2250*ln(5)**2)*ln(x)+8*exp(x)**2+(4*x+12)*exp(x)-75
0*ln(5)**2+4)/(140625*ln(5)**4*ln(x)**2+(-1500*ln(5)**2*exp(x)-1500*ln(5)**2)*ln(x)+4*exp(x)**2+8*exp(x)+4),x)

[Out]

2*x - x/(exp(x) - 375*log(5)**2*log(x)/2 + 1)

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